Arch Computat Methods Eng (2018) 25:591–645 DOI 10.1007/s11831-016-9207-y ORIGINAL PAPER Multiscale Molecular Simulations of Polymer‑Matrix Nanocomposites or What Molecular Simulations Have Taught us About the Fascinating Nanoworld 1,2 1 Georgios G. Vogiatzis · Doros N. Theodorou Received: 8 December 2016 / Accepted: 20 December 2016 / Published online: 22 February 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract Following the substantial progress in molecular the structure, dynamics, thermodynamics, rheology and simulations of polymer-matrix nanocomposites, now is the mechanical properties of polymer-nanoparticle (NP) time to reconsider this topic from a critical point of view. A mixtures. comprehensive survey is reported herein providing an over- There are numerous excellent reviews of the field avail- view of classical molecular simulations, reviewing their able [1–19]. The present overview, organized according to major achievements in modeling polymer matrix nanocom- the answers to specific questions posed and not according posites, and identifying several open challenges. Molecu- to the simulation methods employed, aims at illustrating lar simulations at multiple length and time scales, working how molecular simulations have enhanced our understand- hand-in-hand with sensitive experiments, have enhanced ing of the complex and fascinating field of PNCs. our understanding of how nanofillers alter the structure, dynamics, thermodynamics, rheology and mechanical 1.1 Polymer‑Matrix Nanocomposites properties of the surrounding polymer matrices. In the simplest sense, a composite is an object made up of two or more distinct parts. Within materials science and 1 Introduction engineering, composite materials are put together from two or more components that remain distinct or separate within Polymer-matrix nanocomposites (PNCs) have drawn the final product. Composites can be found anywhere, intense research interest over the last decade owing to being as simple as a matrix material that envelops a rein- both the rich fundamental physics associated with mixing forcing material, such as concrete surrounding steel bars, macromolecules and particles and their unique mechani- the latter preventing failure under tension. The real chal- cal, optical, magnetic and other material properties [1]. lenge is that the options in making a composite material Driven by the need to develop functionally superior mate- are almost limitless, but only a few sets of materials will rials, significant effort has been invested in understanding combine synergistically, and the design criteria may not be obvious. The observation that, other things being equal, the effectiveness of the filler increases with an increase in sur - face to volume ratio has provided large impetus to the shift * Georgios G. Vogiatzis gvog@chemeng.ntua.gr from micron- to nanosized particles. With the appearance of synthetic methods that can produce nanometer sized fill- Doros N. Theodorou doros@central.ntua.gr ers, resulting in an enormous increase of surface area, a new class of materials emerged, known as PNCs, i.e., poly- School of Chemical Engineering, National Technical mer hosts filled with nanoparticles, which possess proper - University of Athens, 9 Heroon Polytechniou Street, ties that typically differ significantly from those of the pure Zografou Campus, 15780 Athens, Greece polymer, even at low nanoparticle concentrations [1, 15]. Present Address: Department of Mechanical Engineering, Nanocomposite materials contain particles of size Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands ∼ 10 nm dispersed at a volume fraction, , often lower Vol.:(0123456789) 1 3 592 G. G. Vogiatzis, D. N. Theodorou −3 commercialization of nanocomposites is the dearth of than 10 within a polymer matrix. They are thus character- cost-effective methods for controlling the dispersion of the ized by particle number densities 3 20 −3 nanoparticles in polymeric hosts. The nanoscale particles = 3∕ 4 ≈ 10 m , interfacial areas per unit typically aggregate, which negates any benefits associated 6 −1 volume 3∕ ≈ 10 m , and interparticle spacings, p with the nanoscopic dimension. PNCs generally possess −1∕3 − 2 ≈ 100 nm that are commensurate with the par- n p nonequilibrium morphologies due to the complex interplay ticle dimensions, and the radii of gyration of matrix p of enthalpic and entropic interactions leading to particle chains, R ≈ 10 nm. aggregation, particle bridging interactions, and phase sepa- The practice of adding nanoscale filler particles to rein- ration at various length scales [32, 33]. The second chal- force polymeric materials can be traced back to the early lenge is associated with understanding and predicting prop- years of the composite industry, in the second half of the erty enhancements in these materials, which are intimately 19th century. Charles Goodyear, inventor of vulcanized connected to their morphology. rubber, attempted to prepare nanoparticle-toughened auto- Nanocomposite research has recently expanded to con- mobile tires by blending carbon black, zinc oxide, and/or sider more complicated systems involving polymer blends magnesium sulfate particles with vulcanized rubber [20]. and block copolymers, where novel electrical, magnetic Another example was the clay-reinforced resin known as and optical properties arise [15, 34, 35]. Bakelite that was introduced in the early 1900s as one of the first mass-produced polymer–nanoparticle compos- 1.2 Multiscale Modeling ites and fundamentally transformed the nature of practical household materials [21–24]. Then, a long period of time Understanding the fascinating and complex structure and passed till the early 1990s when it was first demonstrated dynamics of polymeric materials has been an ongoing that the thermal and mechanical properties of Nylon-6 were challenge for many decades. From the point of view of improved by the addition of a few percent (2–4 % w/w) molecular simulations, the spectrum of length and time mica-type layered silicates to the extent that it could be scales associated with polymer melts of long chains poses used in an automotive engine compartment [25, 26]. a formidable challenge to studying their long-time dynam- Even though some property improvements have been ics [36, 37]. The topological constraints arising from chain achieved in nanocomposites, nanoparticle dispersion is dif- connectivity and uncrossability (entanglements) domi- ficult to control, with both thermodynamic and kinetic pro- nate intermediate and long-time relaxation [38] and trans- cesses playing significant roles. It has been demonstrated port phenomena when polymers become sufficiently long. that dispersed spherical nanoparticles can yield a range of Atomistic molecular simulations of dense phases of soft multifunctional behavior, including a viscosity decrease, matter prove to be difficult for many systems across length reduction of thermal degradation, increased mechanical and time scales of practical interest. Even coarse-grained damping, enriched electrical and/or magnetic performance, particle-based simulation methods may not be applicable and control of thermomechanical properties [27–31]. The due to the lack of faithful descriptions of polymer–polymer tailor-made properties of these systems are very impor- and polymer–surface interactions. Since complex interac- tant to the manufacturing procedure, as they fully over- tions between constituent phases at the atomic level ulti- come many of the existing operational limitations. As a mately manifest themselves in macroscopic properties, a final product, a polymeric matrix enriched with dispersed broad range of length and time scales must be addressed particles may have better properties than the neat poly- and a combination of modeling techniques is therefore meric material and can be used in more demanding and required to simulate meaningfully the bulk-level behavior novel applications. Therefore, an understanding and quan- of nanocomposites [9]. titative description of the physicochemical properties of Soft condensed matter is a relatively new term describ- these materials is of major importance for their successful ing a huge class of rather different materials such as col- production. loids, polymers, membranes, complex molecular assem- As part of this renewed interest in nanocomposites, blies, complex fluids etc. Though these materials are rather researchers also began seeking design rules that would different in their structures, there is one unifying aspect, allow them to engineer materials that combine the desir- which makes it very reasonable to treat such systems from able properties of nanoparticles and polymers. In light of a common point of view. Compared to “hard matter” the the diversity of polymers and nanoparticles, the potential characteristic energy density is much smaller. While the for use of PNCs is nearly limitless. The ensuing research typical energy of a chemical bond (C–C bond) is about revealed a number of key challenges in producing nano- −18 10 J ≈ 250k T at room temperature of 300 K, the non- composites that exhibit a desired behavior. The great- bonded interactions are of the order of k T and allow for est stumbling block to the large-scale production and strong density fluctuations even though the molecular 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 593 connectivity is never affected ( k is the Boltzmann’s con- up to the scale of seconds for long-chain polymer melts), stant). It is instructive to compare the cohesive energy den- would consist of billions of atoms and would require bil- sity, which gives a first estimate of the elastic constants, lions of time steps to run, which is obviously beyond the between a typical “hard matter” crystal to soft matter. The capability of the technique, even with the most sophisti- ratio between the two shows that polymeric systems are cated supercomputers available today. typically 100–10,000 times softer than classical crystals. As a consequence the average thermal energyk T is not negligible for these systems any more, but rather defines 1.2.2 Monte Carlo (MC) the essential energy scale. This means that entropy, which typically contributes to the free energy a term of the order A robust sampling of the configuration space of polymeric of k T per degree of freedom, plays a crucial role. Espe- substances is a prerequisite for the reliable prediction of cially in the case of macromolecules, this is mainly intra- their physical properties. The constraints posed by atomis- molecular entropy, which for a linear polymer of length tic MD simulations can be overcome by resorting to MC N contributes to the free energy a term of order Nk T, simulations, which enable us to use the complete arsenal of representing about 90% of the free energy of polymeric equilibrium statistical mechanics, e.g. perform sampling in materials [39]. As an immediate consequence it is clear all sorts of ensembles [36, 37, 40–42]. Through the design that typical quantum chemical electronic structure calcula- of efficient unphysical moves, configurational sampling can tions (Hartree-Fock or DFT) which focus on obtaining the be dramatically enhanced. MC moves such as concerted energy as a function of nuclear coordinates cannot be suf- rotation [43], configurational bias [44, 45], and internal ficient to characterize soft condensed matter and will even configurational bias [46] have thus successfully addressed be less sufficient to properly predict/interpret macroscopic the problem of equilibrating polymer systems of moderate properties. Molecular theoretical and simulation methods chain lengths. which incorporate entropic effects are required for this. Even these moves prove incapable of providing equi- The length and time scales governing polymer physics libration when applied to long-chain polymer melts, range from Å and femtoseconds for the vibrations of atomic however. A solution to this problem was given by the bonds to millimeters and seconds for crack propagation in development and efficient implementation of a chain con- polymer composites. The entities used as basic degrees of nectivity-altering MC move, end-bridging [47, 48]. Using freedom are: electrons (quantum chemistry), atoms (classi- end-bridging, atomistic systems consisting of a large num- cal forcefields), monomers or groups of monomers (coarse- ber of long chains, up to C , have been simulated in full grained or mesoscopic models) and entire polymer chains atomistic detail [48, 49]. Despite its efficiency in equilibrat- (soft fluids). All these methods and many others have been ing long-chain polymer melts, end-bridging cannot equili- applied side by side to polymers. Until recently, however, brate monodisperse polymer melts; a finite degree of poly - multiscale methods with rigorous bridging between the dif- dispersity is necessary for the move to operate. While this ferent scales have been few. is not a drawback in modeling industrial polymers, which are typically polydisperse, an ability to equilibrate strictly 1.2.1 Atomistic Molecular Dynamics (MD) monodisperse polymers is highly desirable for comparing against theory or model experimental systems. Morover, The stepping stone of classical molecular simulations is end-bridginig relies on the existence of chain ends, ren- atomistic Molecular Dynamics (MD). As accurate MD dering itself inappropriate for dense phases of chains with potentials are developed for a broad range of materials nonlinear architectures. These limitations have been over- based on quantum chemistry calculations and with the come by the introduction of Double Bridging (DB) and increase of supercomputer performance, atomistic MD Intramolecular Double Rebridging (IDR) [50, 51]. The simulations have become a very powerful tool for analyz- key innovation of those moves is the construction of two ing complex physical phenomena in polymeric materials, bridging trimers between two different chains, as far as the including dynamics, viscosity and shear thinning. However, former is concerned, or along the same chain, as far as the as discussed above, entangled polymer systems are charac- latter move is concerned, thus preserving the initial chain terized by a wide range of spatial and temporal scales. It is lengths. still not feasible to equilibrate atomistic MD simulations of MC simulations using atomistic forcefields have inher - highly entangled polymer chain systems, due to their long ent limitations, as Doxastakis et al. have shown [52]. The relaxation times, long-range electrostatic interactions and hard interactions between atoms reduce the acceptance rate tremendous number of atoms. The atomistic MD model for of the moves. Thus, it is essential to resort to parallel tem- such a system, with a typical size of about a micrometer pering techniques in order to allow motion of the system in and a relaxation time on the scale of microseconds (or even its phase space [53]. 1 3 594 G. G. Vogiatzis, D. N. Theodorou 1.2.3 Coarse Graining (CG) well-founded tool [54]. This method adopts a field-theoretic description of the polymeric fluids and makes a saddle- Polymers show a hierarchy of length and time scales. How- point (mean-field) approximation. An alternative to invok - ever, the connectivity in a polymer molecule enforces an ing the saddle-point approximation is performing a normal interdependence between features on different scales. As Metropolis Monte Carlo (MC) simulation, with the effec- a consequence, the choice of where one building block tive potential energy of the system given by field-theoretic ends and where the next one begins is not unique, and it is functionals. One of the first attempts has been made by not obvious how to abstract from a fundamental degree of Laradji et al. [55] for polymer brushes and then by Daoulas freedom and use it in an implicit way in a coarser model. and Müller [56] and Detcheverry et al. [57, 58] for poly- Thus, we will use the generic term “coarse-grained” for meric melts. The coordinates of all particles in the system any model employing the idea of soft interacting particles are explicitly retained as degrees of freedom and evolve (blobs) equal to or larger than the monomers constituting through MC moves. Tracking the motion of mesoscopic the polymeric chains. particles requires the use of stochastic dynamics [59]. The degree of coarse-graining is application-driven and describes the number of atoms/molecules in a typical blob considered by the coarse-grained model. It is closely 2 Selected Unresolved Issues in PNCs related to the minimal features of the atomistic model that should be retained in order to reproduce the desired proper- PNCs have been an area of intense industrial and academic ties from the coarse-grained model. Mapping an atomistic research for the past twenty years. Irrespectively of the model to a coarse-grained one is very important in defin- measure employed - articles, patents, or funding-efforts in ing the positions of coarse-grained particles and directly PNCs have been exponentially growing worldwide over influences the parameterization of the coarse-grained force the last 10 years. PNCs represent a radical alternative to field. conventional filled polymers or polymer blends-a staple of A general procedure in coarse-graining usually involves: the modern plastics industry. Considering the multitude of defining the observable of interest and determining the potential nanoparticles, polymeric resins, and applications, degree of coarse-graining; deciding an appropriate map- the field of PNCs is immense [60]. The restricted class ping of the atomistic model to the coarse-grained one; of polymer nanocomposites defined above still presents a deriving interactions between the coarse-grained parti- complex and fascinating problem in statistical mechan- cles; reproducing target functions with the coarse-grained ics due to the richness of physical phenomena in mixtures model; optimizing parameters/functions in the coarse- of flexible polymer coils and hard impenetrable objects. grained model and validating its range of applicability; Despite the unprecedented efforts placed on PNCs research conducting coarse-grained simulations. there are still open questions which have not been definitely addressed yet. In the following we will summarize a few of 1.2.4 Mesoscopic Simulations them; later we will analyze the perspective simulations and theoretical calculations have provided us with. A major challenge in simulating realistic PNCs is that Fundamental issues and questions include, but are not neither the length nor the time scales can be adequately limited to: the packing and structure of dense mixtures of addressed by atomistic simulations alone, because of the long polymer chains and hard impenetrable fillers, in the extensive computational load. Until relatively recently, a presence of attractive, neutral or repulsive interactions; per- somewhat neglected level of description in materials mode- turbation of polymer packing and the possible nonexistence ling has been the mesoscopic regime, lying between atomic of a bulk region of the polymer matrix, especially in the (or super-atomic like) particles and finite element-based case of PNC films; non-universal filler-induced polymer representations of a continuum, and covering characteristic conformational changes triggered by interfacial effects and/ −8 −5 length scales of 10 –10 m. At this scale, the system is or modification of the excluded volume screening mecha- still too small to be regarded as a continuum, yet too large nism of a pure polymer melt; the way in which geometric to be simulated efficiently using atomic models. In a more and chemical factors determine, in a nonadditive manner, precise way, a mesoscale can be defined as an intermedi- the competing entropic and enthalpic contributions to the ate length scale at which the phenomena at the next level mixture free energy, miscibility and the physical nature of below (e.g. particle motions) can be regarded as having phase separated states. In all cases, the large particle sur- been equilibrated, and at which new phenomena emerge face-to-volume ratio leads to an amplification of a number with their own characteristic time scales. of rather distinct molecular processes, implying pervasive Among the several mesoscopic methods applied to the interference between layers of polymers around nanoparti- study of polymers, Self Consistent Field theory has been a cle surfaces. 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 595 2.1 Segmental Dynamics and the Glass Transition 2.2 Enhancing Nanoparticle Dispersion by Surface Temperature Grafting When cooling a glass forming liquid, instead of freez- One of the biggest challenges is the rational control of filler ing at a well defined temperature, one observes a huge clustering or aggregation, which often adversely affects increase of the viscosity which takes place continuously. material properties. The idea of achieving a good, uniform Such glass formers can be either simple liquids or poly- nanoparticle dispersion state has been the focus of consid- mer liquids, and many features are similar in both regard- erable research, especially because of its favorable impact ing the glass transition. One defines the glass transition on optical and some mechanical properties of the resulting temperature, T , as the temperature at which the dominant composites [70, 71]. In the past few years, several research relaxation time on the molecular scale (or monomeric groups have modified the surface of nanoparticle fillers in scale in the case of polymers) reaches about ~100 s, an effort to improve their dispersion in a polymer matrix. A which corresponds typically to a viscosity of 10 Pa s in promising strategy for controlling the dispersion and mor- the case of simple liquids. Typically for such glass form- phology of PNCs is to graft polymer chains onto the nano- ing liquids, the viscosity increases by twelve orders of particles to form a brush layer [33]. The free chain/brush magnitude over a change of temperature of about 100 K interfacial interactions may be “tuned” by controlling graft- down to T . The underlying mechanisms involved in this ing density, , the degree of polymerization of the grafted g g dramatic increase are still poorly understood [61, 62]. chains, N , and of the polymer host, N , the nanoparticle g f Experimental results on polymer dynamics and the size, , and its shape. For example, if nanoparticles are glass transition in PNCs are not conclusive concern- grafted with chains compatible with the matrix polymer, ing the mechanism and the details of this modification. filler dispersion is favored [72–76]. Motivated by this con- Increases or decreases in T by as much as 30 K [63] have cept, experimentalists have synthesized nanometer sized been reported depending on polymer–nanoparticle inter- particles with high surface grafting density [74, 77, 78]. actions. Reduction of T has been reported in the case At fixed polymer chemistry, when the molecular weight of of weak interactions between filler and polymer [64]. In matrix polymer is lower than that of grafted polymer, nano- other cases the addition of nanoparticles causes no sig- particles disperse. On the contrary, if the molecular weight nificant change to the glass transition of the polymer, pre- of the matrix polymer is higher than that of the grafted sumably because effects causing increase and decrease of polymer, nanoparticles are thought to aggregate [74]. polymer mobility are present simultaneously, effectively Since both the matrix and the brush have the same chemi- canceling out each other [65]. Moreover, strong interac- cal structure, the immiscibility for longer matrix chains is tions between the filler particles and the polymer sup- entropic in origin and attributable to the concept of “brush press crystallinity, yielding new segmental relaxation autophobicity” [72, 79–83]. mechanisms in semicrystalline polymers, originating from polymer chains restricted between condensed crys- 2.3 Mechanical and Rheological Properties tal regions and the semi-bound polymer in an interfacial layer with strongly reduced mobility [66]. The dispersion of micro- or nano-scale rigid particles Concerning the spatial extent of the T -shift, several within a polymer matrix often—but by no means always— studies [67, 68] on PNCs show an increase of the glass produces an enhancement in the mechanical properties of transition temperature, suggesting that the mobility of the these materials. As mentioned earlier, the most important entire volume of the polymer is restricted by the presence application of this sort involves rigid inorganic particles of the nanoparticles. However, there are many experi- (originally carbon black, later also silica) in a cross-linked mental results suggesting that the restriction of chain elastomer matrix, where an improvement of mechanical mobility caused by the nanoparticles does not extend properties is sought. This so-called rubber reinforcement is throughout the material but affects only the chains within a complex phenomenon, which may involve an enhanced a few nanometers of the filler surface [69]. The existence grip of tires on wet roads, an improved resistance to wear of such an interfacial layer seems relatively well-estab- and abrasion, low rolling resistance, and an increase of lished in the case of silica-filled elastomers, however its tires’ ultimate mechanical strength (toughness, tearing exact nature is not well understood: experimental results resistance). have been described in terms of one or two distinct inter- There is a variety of phenomena seeking an explana- facial layers or a gradual change in dynamics with chang- tion. For the sake of readability, we will focus on a subset ing distance from the particle. of them. Under very small cyclic deformations, there is a linear viscoelastic regime characterized by a very signifi- cant increase (sometimes even by two orders of magnitude 1 3 596 G. G. Vogiatzis, D. N. Theodorou compared to the reference unfilled network) of the in-phase as configuration space, and to the 3N -dimensional set from storage modulus, both under elongation and under shear which the generalized momenta {} ≡ , , … , take 1 2 N [84]. At medium-to-large strains, filled elastomers exhibit on values as momentum space. Any instantaneous micro- a markedly non-linear response which is absent in unfilled scopic state of the system can be written as a point: elastomers (“Payne effect”) [85]. The degree of non-line- Γ= , (1) i i arity increases strongly with particle loading. An order-of- in the phase space of the system. The set of values of the magnitude drop in the modulus is often observed on going macroscopic observables, such as temperature, pressure, to 5–10 % deformation (under shear), bringing the asymp- etc., describes the system’s macroscopic state. One macro- totic modulus of the filled systems much closer to that of scopic state corresponds to all the microscopic states that the reference unfilled network. provide the same values of the macroscopic observables, Other related effects are commonly observed in filled defined by the macroscopic state. elastomers. One is deformation hysteresis (“Mullins If we know the Hamiltonian, ℋ , , t , for the effect”): under cyclic deformation, the elastic modulus i i system, then the time evolution of the quantities and in the first cycle is higher than that in the following ones i i (i = 1,… , N) is given by Hamilton’s equations of motion [86]. This points to some kind of “damage” of the material, which, however, is often reversible. The original properties 𝜕 ℋ , , t can be recovered within a few hours, by high-temperature i i ̇ ≡ =− (2) annealing of the sample. Secondly, fillers affect also the 𝜕 t 𝜕 dissipative, out-of-phase components of the modulus. This and is expected, since, probably, friction of the polymer chains against the filler surfaces, or of two particles against each 𝜕 ℋ , , t i i ̇ ≡ = (3) other produces new energy dissipation mechanisms, which 𝜕 t 𝜕 are absent in unfilled elastomers. Elastic and dissipative where i = 1, 2,… , N and ∕ ≡ ∇ symbolizes the gra- effects likely share a common origin. Finally, reinforcement dient operator with respect to the vectorial quantity . As effects have a remarkable temperature dependence. The the system evolves in time and its state changes, the system small-strain (linear) modulus of filled rubbers decreases point traces out a trajectory in Γ-space. Since the subse- with temperature, pointing to important enthalpic effects. quent motion of a classical system is uniquely determined The situation is completely reversed compared to unfilled by the initial conditions, it follows that no two trajecto- elastomers, where the modulus increases linearly with ries in phase space can cross. If the Hamiltonian ℋ does absolute temperature due to the entropic nature of rubber not depend explicitly on time, then ℋ is a constant of the elasticity. motion. Such is the case for conservative systems. 3.1.1 Time Average 3 From Statistical Mechanics to Computer Simulations Any property of the system, , is then a function of the points traversed by the system in phase space. The instantane- Our discussion starts by introducing the formalism of sta- ous property at a time t is (Γ(t)) and the macroscopically tistical mechanics and briefly describing the basic methods meaningful observable property is the time average of obs used in computer simulations. We limit ourselves to the this, absolute minimum of definitions and methods to be pre- sented, trying not to sacrifice consistency and rigor. obs = ⟨(Γ(t)) ⟩ = lim (Γ(t))dt (4) obs t t →∞ obs obs 3.1 Motion in Phase Space In experiments, the time average comes about quite natu- rally, since almost all experimental methods measure over Statistical physics describes a system of N particles at a much longer time scales than the longest relaxation time of given state as one point in 6N/dimensional phase space, the system. A straightforward approach, in order to get containing the atom positions and momenta (and neglecting from molecular simulations, is to determine a time average, the internal degrees of freedom of atoms) [87]. In classical taking a discrete sum over M time steps of length Δt: mechanics, the state of the system is completely specified in terms of a set of generalized coordinates{ } and gener- M alized momenta{ }, where i = 1,… , N [88]. We will refer ≃ lim (Γ(jΔt))Δt (5) i obs M→∞ MΔt j=1 to the 3N-dimensional set from which the generalized coor- dinates of the system {} ≡ , , … , take on values 1 2 N 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 597 This is the approach undertaken in Molecular Dynamics in phase space, the density distribution of points in phase (MD) simulations, where the atoms’ trajectory is followed space traversed by the trajectory converging to a stationary as a function of time, so it is straightforward to obtain the distribution. average. According to the ergodic hypothesis we can calculate the observables of a system in equilibrium as averages over phase space with respect to the probability density of an equilib- 3.1.2 Phase Space Probability Density rium ensemble, (Γ). If (Γ) obeys the normalization ens ens condition, Eq. (6), on the entire phase space Γ and also is zero When we deal with real systems, we can never specify for all points outside the hypersurface ℋ(Γ) = E, the ensem- exactly the state of the system, despite the deterministic ble average can be defined as: character of its motion in phase space. There will always be some uncertainty in the initial conditions. Therefore, = ⟨⟩ = (Γ) (Γ)dΓ (8) it is useful to consider Γ as a stochastic variable and to obs ens ens introduce a probability density (Γ, t) on the phase space. In Monte Carlo (MC) simulations, the desired thermody- In doing so, we envision the phase space filled with a con- namic quantities are determined as ensemble averages: tinuum (or fluid) of state points. If the fluid were composed of individual discrete points, then each point would be (Γ) (Γ) ens equipped with a probability in accordance with our initial ⟨⟩ = ∑ (9) ens knowledge of the system and would carry this probability (Γ) ens for all time, since probability is conserved. Because state If we wish to obtain an average over points in phase space, points must always lie somewhere in the phase space, we there is no need to simulate any real time dependence of have the normalization condition the system; one need only construct a sequence of states in 3N 3N phase space in the correct ensemble. In the context of equi- (Γ, t)dΓ ≡ { }, { }, t d p d q = 1 (6) i i � � Γ Γ librium simulations, it is always important to make sure where the integration takes place over the entire phase that the algorithm used in the simulation is ergodic. This space. Similarly, the probability of finding the system in a means that no particular region in phase space should be small finite region D of Γ-space at time t is found by inte- excluded from sampling by the algorithm. Such an exclu- grating the probability density over that region: sion would render the simulation wrong, even if the simu- lated object itself is ergodic. From a practical point of view, 3N 3N P(D, t) = { }, { }, t d p d q (7) the ergodicity of the system can and should be checked i i through reproducibility of the calculated thermodynamic The probability density for finding a system in the vicinity properties (pressure, temperature, etc.) in runs with differ - of Γ depends on the macroscopic state of the system, i.e. ent initial conditions. on the macroscopic constraints defining the system’s size, spatial extent, and interactions with its environment. A set 3.2 Statistical Ensembles of microscopic states distributed in phase space accord- ing to a certain probability density is called an ensemble. 3.2.1 Microcanonical (NVE) Ensemble A very important measure of the probability distribution of an equilibrium ensemble is the partition functionQ. This In the microcanonical (NVE) ensemble the number of par- appears as a normalizing factor in the probability distribu- ticles, N, the volume of the system, V and the total energy, tion defined by the ensemble. E, are conserved. This corresponds to a completely closed system which does not interact in any way with the environ- 3.1.3 Ensemble Average ment and lies in a container of fixed volume, V . For simplic- ity, we neglect the intramolecular degrees of freedom. Then, The ergodic hypothesis, originally due to L. Boltzmann [89], the system energy will be a sum of kinetic, , and potential, states that, over long periods of time, the time spent by a sys- energies. Since the total energy E must be conserved, the tem in some region of the phase space of microstates with the criterion for adding states in the ensemble would be same energy is proportional to the volume of that region, i.e., that all accessible microstates are equiprobable over a long ℋ { }, { } =𝒦 { }, { } +𝒱 { }, { } i i i i i i (10) period of time. Ergodicity is based on the assumption (prov- = constant = E able for some Hamiltonians) that any dynamical trajectory, given sufficient time, will visit all “representative” regions 1 3 598 G. G. Vogiatzis, D. N. Theodorou which means that not all, but only those states in phase with k being the Boltzmann’s constant and Q the parti- B NVT space Γ that have total energy E are allowed. This can also tion function in the NVT ensemble: be stated so that the probability density of the ensemble is ℋ { }, { } 1 i i 3 3 1 Q = d q d p exp − . NVT i i 3N { }, { } = ℋ { }, { } − E N!h k T (11) NVE i i i i 0 B i=1 NVE (16) where is the Kronecker delta for a discrete system, and The thermodynamic function of the system is the Helm- the Dirac delta function for a continuous one. The partition holtz energy: function in the microcanonical ensemble, Q , is: NVE A =−k T ln Q (17) B NVT Q = ℋ(Γ) − E NVE 0 (12) Eq. (17) defines a fundamental equation in the Helmholtz energy representation by expressing A as a function of N, The summation over states, , is used if microscopic V, T. states are discrete and (Γ) has the meaning of a probabil- ity. For one-component classical systems, the sum can be replaced by an integral, yielding 3.2.3 Isothermal - Isobaric Ensemble (NpT) 1 1 Q = dΓ ℋ(Γ) − E The isothermal-isobaric ensemble describes the equilib- NVE 0 3N N! h rium distribution in phase space of a system under constant 1 1 number of particles, temperature, and pressure. The volume 3 3 = d r d p ℋ { }, { } − E i i i i 0 3N N! h of the system is allowed to fluctuate. Thus, a point in phase i=1 space is defined by specifying V, and , where the (13) i i domain from which the s take on values depend on the where N! takes care of the indistinguishability of particles 3N value of V. of the same species and h is the ultimate resolution for The probability density of the NpT ensemble can be counting states allowed by the uncertainty principle. derived from that of the microcanonical ensemble, by con- The proper thermodynamic potential for the microca- sidering a bath around the system which acts as both a heat nonical ensemble is the entropy: and a work reservoir for the system under study. The prob- S = k ln Q (14) B NVE ability density, in a classical statistical mechanical formula- where k is the Boltzmann constant. We therefore have a tion, is: statistical thermodynamic definition of entropy as a quan- tity proportional to the logarithm of the number of micro- { }, { };V = NpT i i NpT scopic states under given N, V, E. Eq. (14) establishes (18) ℋ { }, { } + pV a fundamental thermodynamic equation in the entropy i i × exp − representation. k T where Q is the isothermal-isobaric partition function: NpT 3.2.2 Canonical (NVT) Ensemble 1 1 In the canonical (NVT) ensemble the number of parti- 3 3 Q = dV d q d p NpT i i 3N N!h V cles, N, the volume of the system, V, and temperature, T i=1 (19) are conserved. This corresponds to a closed system, which, ℋ { }, { } + pV i i however, can exchange heat with a large surrounding bath. × exp − k T The energy is fluctuating, but the temperature is constant, describing the probability distribution of energy fluctua- where V denotes some basic unit of volume introduced to tions. The total energy of the system is given by its Ham- make the partition function dimensionless (the exact mag- iltonian, ℋ { }, { } . The probability density of the i i nitude of V is immaterial). ensemble is: The connection between the formalism of the isother- mal-isobaric ensemble and macroscopic thermodynamic ℋ { }, { } 1 1 i i properties is established via the Gibbs energy: { }, { } = exp − NVT i i 3N Q N!h k T NVT B G(N, p, T) =−k T ln Q (N, p, T) (20) (15) B NpT 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 599 3.2.4 Configurational Integral independent of the interactions in the system. Indeed, com- puting the average of As long as the Born-Oppenheimer approximation [90] is N 2 valid (as it practically always is in equilibrium thermody- = (27) namics) the potential energy of the system, (Γ), depends 2m i=1 only on the generalized coordinates, . Similarly, the with respect to the probability distribution of Eq. (15) and kinetic energy, (Γ), depends only on the momenta . using the factorization of Eq. (23) we obtain that [87]: Hence we can rewrite the expression for the system Ham- iltonian as: ⟨⟩ = Nk T (28) ℋ(Γ) = 𝒦 +𝒱 (21) i i or, more generally ⟨⟩ = 1∕2N k T for a system of N dof B dof It can be now seen that, in a classical (as opposed to quan- degrees of freedom. tum mechanical) treatment, the partition function, e.g. of the NVT ensemble, factorizes into a product of kinetic (ideal gas) and potential (excess) parts: 4 Simulation Methods 1 1 i 4.1 Molecular Dynamics Q = d p exp − NVT i 3N N! h k T i=1 (22) N In Cartesian coordinates, and under the assumption that the potential energy is independent of velocities and time, × d q exp − k T i=1 Hamilton’s equations of motion read: This can be written as a product of the ideal gas contribu- ≡ = tion, and the excess contribution as: i i (29) id −N Q = Q V (23) NVT NVT NVT ̇ ≡ − = i i (30) where: N hence = d q exp − (24) NVT i m ̈ = k T (31) i i i i=1 where is the force acting on atom i: is the so called configurational integral. The partition func- tion of the ideal gas is: = −∇ i (32) with the gradient being taken keeping all positions other Q = (25) NVT 3N than constant. Solving the equations of motion then N!Λ i involves the integration of 3N second-order differential with Λ being the de Broglie or thermal wavelength: equations Eq. (31) which are Newton’s equations of motion. 1∕2 The classical equations of motion possess some inter- Λ= (26) esting properties, the most important one being the con- 2mk T servation law. If we assume that and do not depend From the perspective of a particle-based model, the fun- explicitly on time, then it is straightforward to verify that damental problem of equilibrium statistical mechanics, ℋ = dℋ∕dt is zero, i.e., the Hamiltonian is a constant of according to Chandler [91], is to evaluate a configurational the motion. In actual calculations this conservation law partition function of the form of Eq. (24). is satisfied if there exist no explicitly time- or velocity- Two important consequences arise from Eq. (23). First, dependent forces acting on the system. A second important all the thermodynamic properties can be expressed as a property is that Hamilton’s equations of motion are revers- sum of an ideal gas part and an excess part. The chemical ible in time. This means that, if we change the signs of details which govern the interactions between the atoms of all velocities, we will cause the molecules to retrace their the system are included in the latter. In fact, in MC simu- lations the momentum part of the phase space is usually omitted, and all calculations are performed in configuration If the kinetic energy can be separated into a sum of terms, each of space. The second important consequence of Eq. (23) is them being quadratic in only one momentum component, the average that the total average kinetic energy is a universal quantity, kinetic energy per degree of freedom is 1∕2k T, which is a special case of the equipartition theorem [92]. 1 3 600 G. G. Vogiatzis, D. N. Theodorou trajectories backwards. The computer-generated trajecto- our MD scheme. It should be noted that the Verlet algo- ries should also possess this property. rithm does not use the velocities to compute the new posi- Concerning the solution of equations of motion, in the tions. One can, however, derive the velocities from knowl- limit of very long times, it is clear that no algorithm pro- edge of the trajectory, using vides an essentially exact solution. However, this turns out (t +Δt) − (t −Δt) i i to be not a serious problem, because the main objective of (t) = + Δt (36) 2Δt an MD simulation is not to trace the exact configuration of which is only accurate to order Δt . a system after a long time, but rather to predict thermody- namic properties as time averages and calculate time cor- 4.1.2 Velocity-Verlet Algorithm relation functions representative of the dynamics. In the following we briefly describe the most popular The problem of defining the positions and velocities at family of algorithms used in MD simulations for the solu- the same time can be overcome by casting the Verlet tion of classical equations of motion: the Verlet algorithms. algorithm in a different way. This is the Velocity-Verlet Another family of algorithms comprises higher-order meth- algorithm [95], according to which positions are obtained ods, whose basic idea is to use information about positions through the usual Taylor expansion and their first, second, and higher order time derivatives at time t in order to estimate the positions and their deriva- Δt (t +Δt) = (t) + (t)Δt + ̈ (t) (37) tives at time t +Δt [93]. i i i i In general, higher-order methods are characterized by whereas velocities are calculated through a much better accuracy than the Verlet algorithms, par- ticularly at small times. However, their main drawback is Δt ̈ ̈ (t +Δt) = (t) + (t) + (t +Δt) (38) i i i i that they are not reversible in time, which results in insuf- 2 ficient energy conservation, especially in very long-time with all accelerations computed from the forces at the con- MD simulations. On the contrary, the Verlet methods are figuration corresponding to the considered time. not essentially exact for small times but their inherent time reversibility guarantees that the energy conservation law is satisfied even for very long times. This feature renders the 4.2 Langevin Dynamics Verlet methods, and particularly the velocity-Verlet algo- rithm, the most appropriate ones to use in long atomistic When a large system is simulated, it is generally desired MD simulations. to keep the number of degrees of freedom as low as pos- sible. If a certain subset of particles can be distinguished, 4.1.1 Verlet Algorithm of which details of the motion are not relevant, these particles can be omitted from a detailed MD simulation. The initial Verlet algorithm [94] ends up calculating the However, the forces they exert on the remaining particles positions at time t +Δt by using two Taylor expansions must be represented as faithfully as possible. This means around times t −Δt and t +Δt, respectively: that correlations of such forces with positions and veloci- ties of particle i must be incorporated in the equations (t) i 2 (t −Δt) = (t) − (t)Δt + Δt of motion of particle i, while uncorrelated contributions i i i 2m (33) can be represented by random forces. This brings us to Δt the field of Langevin Dynamics (LD) [96, 97]. In LD a − ⃛ (t) + Δt 3! frictional force, proportional to the velocity, is added to the conservative force, in order to mimic an implicitly (t) (t +Δt) = (t) + (t)Δt + Δt i i i treated background (e.g. solvent). The friction removes 2m (34) kinetic energy from the system. In order to compensate Δt + ⃛ (t) + Δt for the friction, a random force adds kinetic energy to the 3! system. Summing these two equations we obtain: In the simplest case of LD, the random force is taken to (t) have white-noise character, and no correlations between (t +Δt) ≈ 2 (t) − (t −Δt) + Δt (35) i i i the various degrees of freedom are assumed to exist. Under these conditions, the velocity dependent frictional The estimate of the new positions contains an error that is forces become proportional to the instantaneous velocity in the order of Δt , where Δt is the time step employed in 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 601 of the particle considered. Thus, the equation of motion According to Kubo [103], two different kinds of the of a particle i is transformed into the stochastic equation: fluctuation-dissipation theorem can be distinguished. The fluctuation-dissipation theorem of the first kind relates the m ̇ t = t − 𝜁 t + t ( ) ( ) ( ) ( ) (39) i i i i i i i linear response of a system to an externally applied pertur- where the friction coefficient of a particle is denoted by bation and a two-time correlation function of the system and the random force by . The systematic force is the i i in the absence of external forces. The latter form is closely explicit mutual force between the N particles of the system, related to the famous Green-Kubo expressions for transport which is to be derived from the potential (or free) energy of coefficients. The fluctuation-dissipation theorem of the sec- the system, which depends on the positions of all particles, ond kind constitutes a relationship between the frictional denoted by . i and random forces in the system, relying on the assump- The stochastic force, (t), is assumed to be a station- tion that the response of a system in thermodynamic equi- ary Gaussian random variable with zero mean and to librium to a small applied perturbation is the same as its have no correlations with prior velocities or with the sys- response to a spontaneous fluctuation [59]. tematic force: 4.2.2 Mori-Zwanzig Projection Operator Formalism 0 t = 2 k T t ( ) ( ) ( ) (40) i, j, i B ref ij ens A formal way of deriving LD is the projection operator ⎛ ⎞ formalism of Zwanzig [104, 105] and Mori [106, 107]. � � � � −1∕2 � � ⎜ ⎟ i, The basis of the formulation is the assumption that we W = 2 exp − � � (41) ⎜ ⎟ i, i, ens 2 ⎜ ⎟ have partial knowledge of the evolving system, for exam- i, ⎝ ⎠ ens ple we can only track certain variables, while the effect of the other variables is modeled or approximated in a rigor- ⟨ ⟩ = 0 (42) i ens ous way. In this approach the phase space is divided into two parts, which we are called interesting and uninterest- v (0) (t) = 0, t ≥ 0 (43) i, j, ens ing degrees of freedom. For the approach to be useful, the uninteresting degrees of freedom should be rapidly varying F (0) (t) = 0, t ≥ 0 in comparison to the interesting ones. Mori introduced two (44) i, j, ens projection operators, which project the whole phase space where ⟨… ⟩ denotes averaging over an equilibrium ens onto the sets of interesting and uninteresting degrees of ensemble, indices , run over the Cartesian components freedom. The full equations of motion are projected only (x, y and z), k is Boltzmann’s constant, T is the reference B ref onto the set of interesting degrees of freedom. The result is temperature of the LD simulation and W() is the (Gauss- a differential equation with three force terms: a mean force ian) probability distribution of the stochastic force. Accord- between the interesting degrees of freedom, a dissipative or ing to van Gunsteren et al. [98], the solution of the linear, frictional force exerted by the uninteresting degrees of free- inhomogeneous, first order differential equation, Eq. (39), dom onto the interesting ones and a third term containing is: forces not correlated with the interesting degrees. When the uncorrelated force is approximated by a random force the (t) = (0) exp − t interesting degrees of freedom are considered independent (45) of the uninteresting degrees of freedom. � � � � + exp − t − t t + t dt i i m m i 0 i 4.3 Brownian Dynamics 4.2.1 Fluctuation-Dissipation Theorem If the friction exerted by the background to the particles To generate a canonical ensemble, the friction and random under consideration is high, correlations in the velocity force have to obey the fluctuation - dissipation theorem will decay in a time period over which changes in the sys- [99]. Einstein was the first to extract the diffusion coeffi- tematic force are negligible. Such a system can be called cient and mobility in a special case of Brownian motion overdamped. In this case, the left-hand side of Eq. (39) can [100], and made allusions to the existence of a balance be neglected, after averaging over short times. The result is between random forces and friction. Then, Nyquist [101] Brownian Dynamics (BD), which is described by the posi- formulated a limited version of the theorem, in his study of tion Langevin equation: noise in resistors. Later, Callen and Welton [102] proved 1 1 (t) = (t) + (t) the theorem in a generalized form. i i i i (46) i i 1 3 602 G. G. Vogiatzis, D. N. Theodorou The time scale separation makes possible the exchange of simulations. This approach is ultimately based on the Lan- the second order stochastic differential equation Eq. (39) gevin equation, the stochastic differential equation describing for a first order stochastic differential equation, Eq. (46), Brownian motion accounting for the omitted degrees of free- without affecting the dynamics on time scales longer than dom by a viscous force and a noise term. m ∕ . The original DPD model tracks the equation of motion i i Van Gunsteren and Berendsen [108, 109] have proposed of the particles: several algorithms for integrating Eq. (46). We will pay a closer look to the one which reduces to the Verlet algorithm m = , (49) i i for zero friction. If we assume a timestep of Δt, for large val- where m , and = are the mass, position and velocity i i i ues of ∕m Δt in the diffusive regime, when the friction is so i i of particle i, respectively. The total force, , acting on each strong that the velocities relax within Δt, the BD algorithm particle consists of three parts: reduces to: C D S = + + , r t +Δt =r t ij ij ij (50) i,𝛼 n i,𝛼 n j≠i 1 1 ̇ C D S + F t Δt + F t (Δt) where , and represent the conservative, dissipative i,𝛼 n i,𝛼 n (47) ij ij ij 𝜁 2 and stochastic forces between particles i and j, respectively. +ℛ (Δt) i,𝛼 The conservative force depends on the distance between with i enumerating the particles, 1 ≤ i ≤ N, and marking particles i and j, r and is directed along the unit vector of ij a Cartesian component of the vectors. The components of their separation, ̂ : ij the random displacement ℛ(Δt) are sampled from a Gauss- ian distribution with zero mean and width: C C = f r ̂ (51) ij ij ij 2k T B ref where f r is a non-negative (i.e. neutral or repulsive) ℛ (Δt) = Δt ij (48) i, scalar function determining the form of the conservative The reader is reminded that the integration timestep Δt interactions, depending on the particular system of interest. should be small enough, such that systematic forces do In literature it is frequently implemented as a soft repulsion not change significantly over its duration. The integration of the form: scheme for BD Eq. (47) resembles a MC algorithm, except ij that there is no acceptance criterion. Rossky et al. [110] 𝛼 1 − r ≤ r C ij ij c f r = c (52) ij have derived the correct acceptance probability and intro- 0 r > r ij c duced their method under the name “Smart Monte Carlo”. where is a parameter determining the maximum repul- ij sion between the particles and r is a cut-off distance. 4.4 Dissipative Particle Dynamics The dissipative force, , represents the effect of vis- ij Molecular Dynamics (MD) is a powerful simulation tech- cosity and depends on the relative positions and veloci- nique capable of producing realistic results in a wide spec- ties of the particles. The form usually used for this inter- trum of applications. However, the computational cost of a action in DPD simulations is [113]: detailed atomistic interaction model in that paradigm severely D D limits its applicability beyond extremely small spatial and =−𝛾 w r ̂ ⋅ ̂ ij ij ij ij (53) ij short temporal scales. Within the family of simulation tech- where is a friction coefficient, = − and w r is ij i j ij niques designed to overcome the limitations of MD, we turn a distance-dependent weighting function. The fluctuating our attention to Dissipative Particle Dynamics (DPD), which random force depends on the relative positions of the parti- allows the study of complex hydrodynamic phenomena in cles, and is defined as: extensive scales. The DPD method was introduced in 1990s as a novel scheme for mesoscopic simulations of complex S S = 𝜎 w r 𝜉 ̂ , (54) ij i ij ij fluids [111, 112]. In DPD simulations, the particles represent with being a coefficient, w r is a distance-dependent clusters of molecules that interact via conservative (non-dissi- ij weighting function and is a random variable sampled pative), dissipative and fluctuating stochastic forces. Because i from a Gaussian distribution with zero mean and unit vari- the effective interactions between clusters of molecules are ance. It should be noted that both the dissipative and the much softer than the interactions between individual mol- random force act along the particle separation vector and ecules, much longer time steps can be taken relative to MD therefore conserve linear and angular momentum. Also, the 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 603 resulting model fluids are Galilean invariant because the by Pagonabarraga and Frenkel [117] assumes that the con- particle–particle interactions depend only on relative posi- servative force depends on the instantaneous local particle tions and velocities. The fluctuating stochastic force, , density, which in turn depends on the positions of many ij neighboring particles. As far as the integration of the DPD heats up the system, whereas the dissipative force, , ij equations of motion is concerned, Pagonabarraga et al. reduces the relative velocity of the particles, thus removing [118] proposed a leap-frog scheme which was self-consist- kinetic energy and cooling down the system. Therefore, the ent and satisfied a form of microscopic reversibility. Thus, stochastic and dissipative forces act together to maintain an the correct equilibrium properties could be recovered from essentially constant temperature which fluctuates around trajectories generated with that algorithm. the nominal temperature of the simulation, T. Dissipative particle dynamics simulations can be thought of as thermo- 4.5 Monte Carlo statted molecular dynamics simulations with soft parti- cle–particle interactions. The Monte Carlo (MC) method is a statistical approach for Despite qualitative observations, there was no theoreti- finding approximate solutions to problems by means of ran- cal justification that DPD produces the correct hydrody - dom sampling. In addition to molecular simulations and namic behavior until Español and Warren [114] formulated physics, it is widely applied in other natural sciences, math- the Fokker-Planck equation for studying the equilibrium ematics, engineering and social sciences [119]. The earliest properties of the stochastic differential equation describing treatments in the subject, such as this by Babier [120], were DPD. Later, Español [115] derived the macroscopic hydro- made in connection with the “Buffon’s needle problem”. dynamic variables starting from the microscopic descrip- According to Metropolis [121], the invention of the modern tion. In order to recover the proper thermodynamic equi- class of MC algorithms is due to Enrico Fermi, when he librium for a DPD fluid at a temperature T, the coefficients was studying the properties of the then newly-discovered and the weighting functions for the dissipative and random neutron in 1930. It was further developed during the 1940s forces should be related by: by physicists working in the nuclear weapons program of the United States, at the Los Alamos National Laboratory. D S w r = w r (55) The technique was given its name by Nicholas Metropolis, ij ij in reference to the famous casino in Monaco, considering and the use of randomness and the repetitive nature of the sam- pling process. (56) 2k T B In their simplest version, MC simulations of simple flu- ids are carried out by sampling trial moves for the mole- as required by the fluctuation-dissipation theorem. All interaction energies are expressed in units of k T , which is cules from a uniform distribution. For example, in a canon- ical (NVT) ensemble simulation, a molecule is chosen at usually assigned a value of unity. One straightforward and commonly used choice is: random, and then displaced, also randomly to a new posi- tion. The trial move is accepted or rejected according to an ij importance sampling scheme [93, 122, 123]. A frequently 2 1 − r < r D S w r = w r = c (57) ij ij used importance sampling algorithm is the Metropolis 0 r ≥ r algorithm, originally derived for the specific case of the where r is the cut-off distance of the the dissipative and Boltzmann distribution [122] and later generalized to other the random force. In conventional DPD formulation, it distributions [124] which need not to be analytical (e.g. the usually takes the same value as the cut-off distance of the force-bias method of Pangali et al. [123] provides a classi- conservative force but it can vary in order to modify the cal example of such algorithms). dynamic properties in DPD simulations. For conventional The probability of accepting a move, P , of the form: accept DPD simulations, the exponent of the weighting function, P()P() s, is set equal to 2 with w and its gradient being continu- P = min 1, (58) accept P()P() ous at r ∕r = 1. ij c Summarizing, Español and Warren [114] established a sound theoretical basis for DPD and Groot and War- ren [116] obtained parameter ranges to achieve a sat- isfactory compromise between speed, stability, rate of 2 The following problem was posed by Georges-Louis Leclerc, temperature equilibration, and compressibility. Unlike tra- Comte de Buffon: given a needle of length l dropped on a floor striped with parallel lines t units apart, to find the probability that ditional DPD methods using conservative pairwise forces the needle will land such that it crosses a line. (The answer being between particles, the multi-body DPD model presented (2l)∕(t).) 1 3 604 G. G. Vogiatzis, D. N. Theodorou Table 1 Conversion to reduced units for some commonly used quan- are physically easier to interpret, and the results obtained tities with , and m as the basis units for energy, length and mass, become applicable to all materials modeled by the same respectively potential. Reduced units are obtained by expressing all Quantity In reduced units quantities of the simulation in terms of selected base units which are characterizing the system, in order to make Energy E = E∕ equations dimensionless. Table 1 presents some reduced Length L = L∕ quantities. For example, in the case of the Lennard-Jones Mass M = M∕m potential, the particle diameter, , the depth of the potential ∗ 3 Density well, , together with the mass of the simulated particles, Temperature T =(k T)∕ m, provide a meaningful and complete set of base units for Force F =(F)∕ simulations. ∗ 3 Pressure p =( p)∕ Time 3 t = t ∕(m ) 5 Structural Predictions will asymptotically sample the configuration space accord- 5.1 Chain Dimensions in the Bulk ing to a probability P. In Eq. (58), P is the probability accept with which trial moves are accepted or rejected, P() is One of the most important and probably the most funda- the transition probability of making a trial move from state mental question in the area of PNCs is how the size of to state , and P() is the probability of being at state the polymer chains is affected by the dispersion of nano- . This means that, at equilibrium, the average number of particles. There has been considerable controversy in the accepted trial moves that result in the system leaving state experimental literature over whether nanoparticles cause must be exactly equal to the number of accepted trial chain expansion (swelling) [125, 126], contraction, [127] moves from all other states to the state . This is a looser or neither [128–133]. The sign (attractive or repulsive) and statement of the detailed balance condition, reflected in Eq. strength of the nanoparticle/polymer interactions, the rela- (58), that at equilibrium the average number of accepted tive dimensions of the chains with respect to the size of the moves from to any other state should be exactly nanoparticle, R ∕R , and the exact state of dispersion, have g n canceled by the number of reverse moves. been identified as the key factors that can account for the In the original Metropolis scheme [122], the prob- aforementioned differences in the structure of the matrix abilities P() form a symmetric matrix, constructing chains. a Markov chain that has the Boltzmann distribution as its equilibrium distribution. In this case, there is no bias 5.1.1 Experimental Findings involved in making the move and Eq. (58) reduces to the standard Metropolis acceptance criterion: Chain conformations in PNCs have been primarily meas- ured by small angle neutron scattering (SANS). These () − () P = min 1, exp − measurements are greatly facilitated by combining deuter- (59) accept k T ated and hydrogenated chains such that the average scat- The advanced MC methods are based on judicious choices tering length density of the polymer blend matches that of P() [93]. It should be noted that the simulation of the filler. This zero average contrast condition [134], steps in the MC technique are steps in configuration space which is hard to achieve, minimizes the scattering due to and there is no notion of “time” in MC simulations. This is the nanoparticles. To date, studies which report increases contrast to MD, where the simulation steps are explicit time in polymer dimensions, in the case of spherical nanopar- steps. Moreover, a computational advantage of MC over ticles, invoke the presence of attractive nanoparticle/poly- MD is that only the energy needs to be calculated, not the mer interactions, combined with R > R , and good nano- g n forces, rendering the Central Processing Unit (CPU) time particle dispersion [135], to conclude that the nanoparticles needed per step smaller than that of an MD simulation. behave as a good solvent for the polymer chains. However, even though the existence of a shell containing polymer of 4.6 Reduced Units reduced mobility is acceptable in nanocomposites com- posed of strongly interacting particles and polymer, e.g. Molecular simulations can conveniently be performed in composed by silica and PMMA, the size of the chains, e.g. non-dimensional or reduced units, based on the character- in terms of their radius of gyration, R , is intrinsically inde- istic physical dimensions of the system under study. Work- pendent of the the volume fraction, , (up to ≃ 0.1) and ing with reduced units is preferred mainly because they the polymer-to-particle size ratio [132]. All other studies 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 605 on spherical nanoparticles showed little if any changes in Sen et al. [128] employed polymer reference interac- polymer R , that is where the nanoparticle–polymer intrac- tion site model (PRISM) [144, 145] calculations in order tions are believed to be athermal, or significant nanoparti- to interpret small angle neutron scattering findings on poly - cle aggregation was present, due to unfavorable nanoparti- styrene loaded with spherical silica nanoparticles under cle/polymer interactions [136]. contrast-matched conditions. They considered blends with In an early study of a poly(dimethylosiloxane) / poly- 66 wt% hPS and 34% dPS, which almost contrast match silicate (R = 1 nm) nanocomposite [125], a significant the silica. Nanocomposites with 0, 2.9, 6.1 and 9.1% vol increase of the polymer chain dimensions (reaching 60% silica were prepared for each molecular weight and 15.9 expansion at nanoparticle volume fraction, = 40%) was and 27.4 vol% for the higher molecular weights considered. observed for R > R and a decrease in polymer dimen- However, in their experiments, as in earlier studies [146, g n sions for R ≤ R . Neutron scattering studies of an ather- 147], the particles were imperfectly mixed with the poly- g n mal polystyrene (PS) PNC, indicated swelling of the matrix meric matrix, with particles being surrounded by “voids”, chains, induced by dispersed tightly cross-linked PS nano- especially at large filler contents. In parallel, the PRISM particles [126]. PS chains around crosslinked PS particles theory was applied, by modeling the fillers as hard spheres (R = 2 − 3.6 nm) were found to be expanded by up to 20 and polymers as freely jointed chains with a realistic per- % relative to their unperturbed size, when their unperturbed sistence length. Polymer–polymer and particle–particle radius of gyration was comparable to or larger than the interactions were taken to be hard-core, while monomers radius of the dispersed particles. More recent studies of PS/ and filler interact via an exponentially decreasing attrac- silica nanocomposite [128, 129, 131] for R ∕R = 2 − 4, tion over a predefined spatial range. From the experimen- g n [131] and poly(ethylene-propylene)/silica nanocompos- tal point of view (Fig. 1), the low-q intensity increases ites, [127] found no perturbation of the matrix chain dramatically with increasing silica content, especially for dimensions. loadings ≤10 vol%, implying that the matrix is not totally We may attribute the qualitatively different trends contrast matched to the filler (unsurprising in light of voids deduced by different experimental studies to several fac- surrounding particles [146]). However, the scaling and tors, including, but not limited to the following: (a) most dimensions of the polymer chains can be obtained from of the polymers used exhibit significant polydispersity, (b) the high-q intensity which is expected to be independent of particle dispersion/agglomeration cannot be adequately the filler structure [146]. Their results (Fig. 1) showed that quantified, (c) the molecular weight of the isotopic poly - chain conformations follow unperturbed Gaussian statistics mers blended with the filler was quite different in at least independent of chain molecular weight and filler composi- one case. The compound effect of these factors can result in tion. Liquid state theory calculations were consistent with significant uncertainty in the chain dimensions measured. this conclusion and also predicted filler-induced modifica- Molecular simulations can shed some light on the role of tion of interchain polymer correlations which had a dis- nanoparticles on chain dimensions, especially in regimes tinctive scattering signature that was in nearly quantitative where it is hard to conduct experiments. agreement with the experimental observations. The chain R varied from ~8 nm (90 kg/mol) to 22 nm (620 kg/mol), 5.1.2 Insight Obtained from Simulations bracketing the nanoparticle diameter (~14 nm), suggesting that the ratio of the particle size to R was not an important From the point of view of molecular simulations, there variable in that context. also exists controversy as to whether the incorporation of The structure of a polystyrene matrix filled with tightly nanoparticles in a polymer melt causes polymer chains to cross-linked polystyrene nanoparticles, forming an ather- expand, remain unaltered or reduce [137–139] their dimen- mal nanocomposite system, has been investigated by sions compared to their size in the bulk material. To date, means of a Monte Carlo sampling formalism by Vogiatzis all studies have indicated that, irrespectively of the absolute et al. [148]. Although the level of description is coarse- dimensions of the chains in the interparticle region, these grained (e.g., employing freely jointed chains to represent retain their unperturbed Gaussian scaling. This is a striking the matrix), the approach developed aims at predicting the feature, resembling the scaling behavior of chains in thin behavior of a nanocomposite with specific chemistry quan- films [140, 141], where chain conformations parallel to the titatively, in contrast to previous coarse-grained simula- surface assume their unperturbed values even for film thick - tions. A main characteristic of the method was that it treats nesses < R . Most of the simulation works have addressed polymer–polymer and polymer–particle interactions in a polymer dimensions in nanocomposites below the percola- different manner: the former are accounted for through a tion threshold ( = 31% [142]), except the early works of suitable functional of the local polymer density, while the Vacatello [137–139, 143] that were implemented at con- latter are described directly by an explicit interaction poten- stant density and spatially frozen nanoparticles. tial. The simulation methodology was parameterized in a 1 3 606 G. G. Vogiatzis, D. N. Theodorou Fig. 2 Radius of gyration of polystyrene chains in melts filled with tightly cross-linked PS nanoparticles of radius R = 3.6 nm, normal- ized by its value in the bulk, R , as a function of the particle volume g,0 fraction. The corresponding molecular weights are 23 (a), 47 (b), 93 (c) and 187 (d) kg/mol, respectively. (Reprinted from [148] with per- mission from Elsevier.) where the unperturbed radius of gyration R ≃ 42 Å is g,0 comparable to the radius of the nanoparticle, R = 36 Å. It seems that there is a tendency of chains to swell when their dimension is equal to or approaches the dimension of the Fig. 1 a Transmission electron microscopy (TEM) micrographs of nanoparticle. This observation is in very good quantitative nanocomposites formed from PS of 250 kg/mol molecular weight agreement with experimental data reported for the same and indicating % vol loading of silica in each sample. b Ratio of the system [126]. In all other cases, the swelling due to the radius of gyration of the PS chains in the presence of particles to their corresponding value in the pure blend for 90 kg/mol PS (green presence of the nanoparticles could hardly be discerned. squares), 250 kg/mol PS (blue circles) and 620 kg/mol PS (red tri- Karatrantos et al. [135] have investigated the effect of angles) as functions of the silica volume fraction. In the inset to the various spherical nanoparticles on chain dimensions in figure, a plot of small angle neutron scattering intensities in abso- polymer melts for high nanoparticle loading which was lute units as a function of q for the 250 kg/mol PS nanocomposites. The interested reader can refer to [128] for more details. (Color fig- larger than the percolation threshold, using coarse-grained ure online) (Reprinted figure with permission from [128]. Copyright molecular dynamics simulations of the Kremer-Grest 2007 by the American Physical Society) model [149]. Their results, presented in Fig. 3, revealed different behavior of the polymer chains in the presence of bottom-up fashion in order to mimic the experimental stud- repulsive or attractive particles. In nanocomposites con- ies. Many particle systems, with volume fractions up to 15 taining repulsive nanoparticles (black symbols), the poly- vol%, were simulated. The positions of the nanoparticles mer dimensions were not altered by the particle loading. were held constant in the course of the simulation, while These authors reported that the polymers were phase sep- polymeric chains were allowed to equilibrate via a combi- arated from the repulsive nanoparticles (of R = 2) in the nation of MC moves. The generation of many independent nanocomposites, thus, there were no changes in the radius initial configurations compensated for the immobility of of gyration values. On the contrary, in the nanocomposites the particles along the simulation. The values of the radius containing attractive nanoparticles, the overall polymer of gyration R , relative to the value for the pure polymer dimensions increased dramatically at high particle load- melt R , are shown in Fig. 2 as a function of the nanopar- g0 ing. In particular, the magnitude of expansion of polymer ticle volume fraction for the four different chain molecular dimensions was larger for polymers with N = 200 follow- weights used in that work (23, 47, 93 and 187 kg/mol). In ing qualitatively the experimental data [125, 150]. The −1∕3 general, an expansion of polymeric chains with increas- relation R ∕R = (1 − ) , included in Fig. 3 was pro- g g,0 ing nanoparticle volume fraction can be observed for all posed by Frischknecht et al. [151] for predicting the poly- chain lengths. This expansion is maximal for 23 kg/mol, mer expansion due to the excluded volume introduced by 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 607 Fig. 3 Radius of gyration of polymers in melt with nanoparticles of radius R = 1, 2 normalized with its value in the bulk for polymer chains of N = 200 and N = 100 (inset) repeat units (monomers): (i) polymer melt (blue filled circles), (ii) nanocomposite: attractive mon- omer-nanoparticle (R = 2) interactions (red filled circles), (iii) nano- n Fig. 4 Root mean squared radius of gyration of the coarse-grained composite: repulsive monomer-nanoparticle (R = 2) interactions n chains of neat and nanocomposite polystyrene systems as a function (black filled diamonds), (iv) nanocomposite: attractive monomer- of molar mass, M, in the melt at 500 K (red, green and magenta nanoparticle (R = 1) interactions (red open circles), (v) nanocom- n rhomboid symbols ). The systems contain one nanoparticle of diame- posite: repulsive monomer-nanoparticle (R = 1) interactions (black n ter 3 nm ( ≃ 1%) and 6 nm ( ≃ 6%). Neutron scattering measure- −1∕3 open diamonds). The black dashed line shows R ∕R ∝ (1 − ) . g g,0 ments [152] for high molar mass PS are also included (blue rhomboid (Color figure online) (Reprinted from [135]—Published by The Royal symbols). The black dotted line is a linear least-squares fit to a rela- 1∕2 Society of Chemistry.) 2 1∕2 tion of the form R ∝ M in the loglog coordinates of the plot. (Color figure online) (Reproduced from Ref. [153] with permission from The Royal Society of Chemistry.) the nanoparticles, assuming no change in density on mix- ing. Finally, Karatrantos et al. [135] reported that polymer chains, in all cases considered, did not depart from Gauss- nanocomposite polystyrene systems are concerned, the ian statistics. presence of the nanoparticles affected the root- mean- Mathioudakis et al. [153] applied a hierarchical simula- square radius of gyration only slightly. tion approach in order to study the behavior of PS–SiO nanocomposites. Two interconnected levels of representa- 5.2 Polymer Structure in the Vicinity of the Filler tion were employed. (a) A coarse-grained one [154], Particles wherein each polystyrene repeat unit was mapped into a single “superatom” and each silica nanoparticle into a 5.2.1 Experimental Findings sphere. The smoother effective potential energy hypersur - face of the coarse-grained representation permitted its equi- SANS measurements show a clear scattering signature of libration at all length scales by using powerful connectiv- a polymer bound layer around the particles, which arises ity-altering Monte Carlo algorithms [155]. (b) A united due to a scattering length density different from the bulk atom representation, wherein polymer chains were repre- polymer matrix material, either due to H or D enrichment sented by a united-atom model and a silica nanoparticle or a modification of the polymer density in the bound layer was represented in full atomistic detail. Coarse-graining compared to the surrounding polymer matrix [132, 133]. and reverse-mapping between the two levels of representa- The measurements of Jouault et al. [133] revealed that the tion was accomplished in a manner that preserved tacticity bound layer is independent of the particles’ volume frac- and respected the detailed conformational distribution of tion. Then, as observed by Jiang et al. [157], the bound chains [156]. At the coarser level, these authors estimated 1∕2 layer volume fraction is larger at the surface (that region the root-mean-square radius of gyration R as a func- being mostly composed of loops) and decreases at larger distances as the bound layer becomes more diffuse due to tion of the molecular weight for neat and nanocomposite the contribution from the tails. One can estimate the thick- polystyrene systems. Their results are presented in Fig. 4 ness of the bound polymer layer around 2 nm. However, along with neutron scattering results for bulk monodisperse this thickness value is a simplification because it does not PS [152] from 21 to 1100 kg/mol. As far as the 1 3 608 G. G. Vogiatzis, D. N. Theodorou Fig. 5 a Density distribution of a polyethylene melt as a function of profiles for selected systems. b Surface concentration together with distance from the surface of a filler (graphite slab, silica nanoparti- predictions based upon geometrical arguments for ideal spheres and cle or fullerene C ). The decomposition into tails, trains and loops is surface monomer density in the proximity of silica slabs. (Reprinted carried out following Scheutjens and Fleer [164]. The inset provides from [159], with the permission of AIP Publishing.) completely describe the complex chain behavior, some loop segments are constituted by the monomers in-between aspects of which will be analyzed below. two train segments, which are not adsorbed on the surface. Figure 5 exhibits three regimes for adsorbed chains: a first layer of adsorbed monomers constituting train segments, 5.2.2 Insight Obtained from Simulations a second layer where a decay is dominated by a decrease of loop segments while tail density is constant and a third The local density of the polymer in the proximity of the layer where tail segments extend in the bulk melt. As surface of a filler is often employed as an indication of the shown in the inset to Fig. 5(a) the area under the first peak strength of polymer–surface interactions and a decrease of broadens as we move to smaller particles. the first peak of the radial density distribution is expected An interesting feature of the interfacial systems to study with curvature [158]. At this point we resort to the detailed is the number of monomers that are in contact with the sur- analysis of Pandey and Doxastakis [159] concerning a face. To this end, Pandey and Doxastakis [159], defined a polyethylene layer close to a filler surface (Fig. 5). These surface concentration by integrating the density profile of authors coupled the application of preferential sampling train segments: techniques [160] with connectivity-altering Monte Carlo algorithms [161, 162] in order to explore the configura- ∞ ∫ (r)4r dr train tional characteristics of a polyethylene melt in proximity to (60) Φ = , a silica surface or around a nanoparticles and the changes where R is the radius of the nanoparticle, is the den- induced by high curvature when the particle radius is com- n train sity profile of train segments, and A is the accessible sur - parable to the polymer Kuhn segment length. face area to the polymer. If we assume that nanoparticles The inset to Fig. 5 shows that indeed as we move from a are spheres surrounded by a constant density of polymer, flat surface to a smaller nanoparticle a decrease is observed , in a layer of Δr thickness, the surface concentration is with the exception of the fullerene where a significantly given by: higher density is found. To investigate further the concen- tration of adsorbed monomers, these authors followed the 3 3 (r +Δr) − r (61) use of a simple distance criterion (adsorbed polymer chains Φ = , s 0 3r have an atom within 0.6 nm of an atom of the surface; where a constant density is multiplied with the ratio of the introduced by Daoulas et al. [163]) to decompose polymer volume of a spherical shell representing the first adsorbed segments according to the Scheutjens-Fleer theory [164] monolayer to the surface of the sphere. The geometric pre- into trains, tails and loops. Tails are the segments which are dictions following the above line of reasoning, are shown hinged to the surface at one end while the other end is dan- for different chain lengths by the dashed lines in Fig. 5(b). gling freely into the bulk polymer. Train segments consist It is apparent that a modest increase and ultimate leveling of monomers consecutively adsorbed on the surface. The 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 609 off of the surface concentration with decreasing nanopar - ticle radius is observed in sharp contrast to the estimations based on the geometric arguments, which predict a con- tinuous increase. The lower than anticipated increase of surface concentration with curvature suggests that collec- tive properties beyond the enthalpic interactions appear to play a crucial role on surface concentration. At the extreme limit where particles are comparable to the polymer Kuhn segment length, curvature penalizes the formation of long train segments. As a result, an increased number of shorter contacts belonging to different chains were made, compet- ing with the anticipated decrease of the bound layer thick- ness with particle size if polymer adsorbed per unit area remained constant. Starr et al. [165] carried out extensive molecular dynam- ics simulations of a single nanoscopic filler particle sur - rounded by a dense polymer melt. The polymers were mod- eled as chains of monomers connected via a finitely Fig. 6 Radius of gyration, R , in reduced Lennard-Jones units, of the extensible nonlinear elastic (FENE) anharmonic potential polymer chains as a function of the distance d∕ R of the center of mass of a chain from the filler surface (d is normalized by the average and interacting via a Lennard-Jones potential. That type of R of all chains). Results are presented for a attractive and b nonat- “coarse-grained” model is frequently used to study general tractive interactions. The component of R perpendicular to the sur- trends of polymer systems but does not provide information ⟂ 2 face, R is resolved. The dotted line shows R for the pure system. g g for a specific polymer. The filler particle shape was icosa- The increase of R , coupled with the decrease of R , indicates that the hedral with interaction sites assigned at the vertices, at four chains become increasingly elongated and flattened as the surface of equidistant sites along each edge, and at six symmetric sites the particle is approached. (Reprinted figure with permission from on the interior of each face of the icosahedron. These [165]. Copyright 2001 by the American Physical Society) authors considered a filler particle with an excluded volume interaction only, as well as one with excluded volume plus eigenvalues of the the radius of gyration tensor, which attractive interactions in the dilute nanoparticle regime serve as a measure for characterizing the shape of the poly- (where bulk chain dimensions are unlikely to be affected by mer chains. In the polymer melt, the intrinsic shape of the confinement between nanoparticles). By focusing on chains is that of a flattened ellipsoid or soap bar [166]. Fol- the dependence of R on the distance d from the filler sur - lowing ref. [166], Mathioudakis et al. diagonalized the face, these authors were among the first to report a change instantaneous radius of gyration tensor of every chain to in the overall polymer structure near the surface. In Fig. 6, 2 2 2 determine the eigenvalues L ≥ L ≥ L (squared lengths of 3 2 1 R , as well as the radial component of from the filler center the principal semiaxes of the ellipsoid representing the seg- ⟂2 ment cloud of the chain) and the corresponding eigenvec- R (approximately the component perpendicular to the tors (directions of the principal semiaxes). The three semi- filler surface) for both attractive and nonattractive poly - 2 2 2 axes are generally unequal. The sum L + L + L equals 1 2 3 mer–filler interactions at one temperature. A striking fea- the squared radius of gyration of the chain. These authors ture of Fig. 6 is that R increases by about 30% on observed that, when the distance of the center of mass of ⟂2 approaching the filler surface, while at the same time R the chain from the center of the nanoparticle was shorter than the mean size of the chain, the chains expanded along decreases by more than a factor of 2 for both (attractive and their principal semiaxis, L . That led to an increase of the neutral) systems. The combination of these results indicates 2 2 2 2 radius of gyration, R = L + L + L near the nanoparticle. that the chains become slightly elongated near the surface g 1 2 3 and flatten significantly. The range of the flattening effect The deformation of the molecules was smaller for chains roughly spans a distance of an unperturbed radius of gyra- whose dimensions exceed by far the radius of the nanopar- tion, R , from the surface and depends only weakly on the g,0 ticle. Far from the surface of the nanoparticle, the sum of simulation temperature, T. Moreover, the chains retain a the squares of the principal semiaxes (sum of the eigenval- Gaussian structure near the surface. ues of the radius of gyration tensor) reaches the bulk aver- Mathioudakis et al. [153] studied the shape of chains in age value of the squared radius of gyration of PS, since the the presence of a silica nanoparticle by employing coarse- molecules were not affected by the presence of the nano- grained MC simulations. These authors analyzed the particle. These results are shown in Fig. 7. Changes in the 1 3 610 G. G. Vogiatzis, D. N. Theodorou 2 2 Fig. 7 Ratio L ∕L of the largest to the smallest eigenvalue of the 3 1 radius of gyration tensor of the chain as a function of the distance of the center of mass of the chain from the center of mass of a silica nanoparticle. The systems consist of chains of molar mass M = 208 kg/mol and one nanoparticle of radius either 3 nm (silica volume fraction = 1%) or 6 nm ( = 3 and = 6%). The expected SiO SiO SiO 2 2 2 value from the random walk model for bulk PS is also included (black dotted line) [148]. (Reproduced from Ref. [153] with permission from The Royal Society of Chemistry.) intrinsic shape of chains were quantified as a function of distance of the center of mass of the chain from the center of mass of the silica particle by calculating the ratio of larg- est to smallest eigenvalue of the radius of gyration tensor. This local anisotropy of the chains as a function of distance from the centre of mass of the nanoparticle is also shown in Fig. 7. The presence of the filler surface also influences the ori- entation of the chains. Ndoro et al. [158] studied the dis- Fig. 8 a Schematic representation of the orientational angle tance dependence of the angle between the longest axis of between the longest axis of the radius of gyration tensor and the the radius of gyration tensor and the surface normal of bare surface normal (simulation snapshot). (Reprinted with permission silica nanoparticle. Their results are presented in Fig. 8. from Ref. [158]. Copyright (2011) American Chemical Society.) The observation that the free polymer chains generally pre- b PS Chain orientation as a function of the chain (center-of-mass) distance from the silica nanoparticle surface. The considered nano- fer to align tangentially to the ungrafted surface is in agree- particle diameters were 3, 4, and 5 nm. The orientation angle is cal- ment with conclusions from other researchers [148, 167]. culated between the longest axis of the squared radius of gyration In their coarse-grained model using Monte Carlo simula- tensor (eigenvector corresponding to its largest eigenvalue, disregard- tions, Vogiatzis et al. [148] studied the orientational angles ing the sign) and the surface normal (c.f. (a)). The horizontal line at 57.3 marks the average orientational angle for a random distribution. of local chain segments. They also concluded that chain (Reprinted with permission from Ref. [158]. Copyright (2011) Amer- segments in the vicinity of the nanoparticle surface were ican Chemical Society.) structured and oriented tangentially to the interface. Bačová et al. [168] conducted atomistic molecular dynamics simulations of graphene-based polymer nano- square end-to-end distance R , and the radius of gyra- composites composed of hydrogenated and carboxylated tion, R , for the chains, whose center of mass is placed graphene sheets dispersed in polar and nonpolar short poly- within a given layer. The five layers employed are set up in mer matrices, in order to gain insight into the effects of the accordance with the positions of the minima in the density edge group functionalization of graphene sheets on the profiles (cf. Fig. 9). The results for all five parallel layers properties of hybrid graphene-based materials. and both polymer matrices are plotted in Fig. 9. The data Poly(ethylene oxide) and polyethylene serve as the polar are normalized by the bulk values. The error bars corre- and nonpolar matrix, respectively. In Fig. 9 the structural spond to the standard deviation and were obtained through properties of the short polymer chains, i.e. their mean 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 611 Fig. 9 Mean square end-to-end distance of polymer chains located at Fig. 10 End-to-end distance and radius of gyration of poly- different layers parallel to graphene, normalized by the same quan- mer chains of different molecular weight of a polymer/SWCNT tity measured in bulk. In the inset to the figure, the normalized radius (r = 0.66) nanocomposite system from molecular dynamics SWCNT of gyration is plotted for the same set of data. The first layer extends simulations: (i) end-to-end distance of a polymer melt (blue open dia- within distances 0.0 and 0.6 nm from the graphene sheet, the sec- 1∕2 monds), (ii) fitting of the scaling law R ∼ N on the simulation data ond between 0.6 and 1.0 nm, the third between 1.0 and 1.5 nm, the (blue line), (iii) end-to-end distance of polymer chains in contact with fourth between 1.5 and 2.0 nm and the last one, fifth, between 2.0 the SWCNT, interacting with k T energy with the SWCNT (blue and approximately half the edge length of the simulation box, 5.0 nm. filled diamonds), (iv) radius of gyration of a polymer melt (red open (Reprinted with permission from Ref. [168]. Copyright (2011) Amer- 1∕2 circles), (v) fitting of the scaling law R ∼ N on the simulation ican Chemical Society.) data (red line), and (vi) radius of gyration of polymer chains in con- tact with the SWCNT, interacting with k T energy with the SWCNT (red filled circles). (Color figure online) (Reprinted with permission typical block averaging techniques. In both matrices (PEO from Ref. [170]. Copyright (2011) American Chemical Society.) and PE), chains in the first layer appear to be slightly swol- 2 2 len with R and R higher than the bulk values. In the e g those values) are shown. As can be clearly seen, the dimen- case of PE, the difference is larger, which can be caused by sions of polymer chains in contact with the SCWCNT its tendency to lie flat on the surface [169]. Small devia- almost overlap with those in the polymer melt for all the tions from the bulk values are observed also for the second polymer molecular weight when interacting with the and the third layer, while beyond the fourth layer the vales SWCNT with energy in the k T range. 2 2 of R and R are consistent with those in the bulk e g 5.3 End Grafted Polymers onto Nanoparticles within error bars. Karatrantos et al. [170] investigated the static proper- Controlling the spatial dispersion of nanoparticles is criti- ties of monodisperse polymer/single wall carbon nanotube cal to the ultimate goal of producing polymer nanocompos- (SWCNT) nanocomposites by molecular dynamics simula- ites with desired macroscale properties. Experimental stud- tions of a polymer coarse grained model [171, 172]. The ies [14, 73, 150] have shown that, often, nanoparticles tend SWCNT studied had a large aspect ratio and radius smaller to aggregate into clusters, with the property improvements than the polymer radius of gyration (e.g. in a well dis- connected to their nanoscale dimension being lost. One persed PS/SWCNT nanocomposite the radius of the nano- commonly used technique for controllably dispersing them tube is of the order of the Kuhn length of PS). Polymer is end grafting polymer chains to the nanoparticle surface, chains are composed of bead-spring chains of Lennard- so that nanoparticles become “brush coated” [14]. When Jones monomers m, of diameter = 1 and mass m = 1. m m the coverage is high enough, the nanoparticles are sterically Three different SWCNTs ((12,0), (17,0), (22,0) of radius stabilized, which results in good spatial dispersion [173, r = 0.46 , 0.66 , 0.85 , respectively) are con- SWCNT m m m 174]. Moreover, spherical nanoparticles uniformly grafted sidered and span the simulation cell with their atoms held with macromolecules robustly self-assemble into a variety fixed in a centered position in the simulation cell along the of anisotropic superstructures when they are dispersed in z-axis. In Fig. 10, root mean squared average ⟨R ⟩ and R e g the corresponding homopolymer matrix [14]. of the polymer chains that remained in contact with the A simpler system that is useful for understanding poly- SWCNT (so polymers in the polymer/SWCNT simulations mer brushes grafted on spherical nanoparticles immersed that do not always contact the SWCNT were omitted from in melts is that of a brush grafted to a planar surface in 1 3 612 G. G. Vogiatzis, D. N. Theodorou contact with a melt of chemically identical chains. Impor- the five regions of the work of Aubouy et al. can still be tant molecular parameters for this system are the Kuhn located, but they also provide the scaling of the exclusion segment length of the chains, b, the lengths (in Kuhn seg- zone, where matrix chains are not present. Such exclu- ments) of the grafted, N and free, N , chains, and the sur- sion zones have been observed in a special limiting case g f face grafting density (chains per unit area), . The case of of grafted polymers, namely, star shaped polymers. planar polymer brushes exposed to low molecular weight Daoud and Cotton [181] showed that, in the poor-solvent solvent was studied theoretically by de Gennes [175] and limit, the free ends of the chains are pushed outward, Alexander [176]. These authors used a scaling approach because of high densities near the center of the star. The in which a constant density was assumed throughout the Daoud-Cotton model assumes that all chain ends are a brush: all the brush chains were assumed to be equally uniform distance away, while the Wijmans-Zhulina model stretched to a distance from the substrate equal to the [178] has a well-defined exclusion zone. For Θ solvents, thickness of the brush. Aubouy et al. [177] extracted the in the limit of large curvature (small particle radius, R ), phase diagram of a planar brush exposed to a high molec- the segment density profile, (r), decreases with the 1∕2 ular weight chemically identical matrix. Their scaling radius as [178] (r)∝ R ∕r . It must be noted that analysis is based on the assumption of a steplike concen- is not linear in because the brush height depends on . tration profile and on imposing the condition that all In the limit of small curvature (large R ), a distribution of chain ends lie at the same distance from the planar sur- chain ends must be accounted for [182], leading the seg- face. Five regions with different scaling laws for the ment density profile to a parabolic form: [178] 2 2 3N b height, h, of the brush were identified. For low enough g h r (r )= 1 − where b is the statistical −1 −2 h h h 0 0 grafting densities, 𝜎< N a (with a being the mono- 1∕2 segment length, r = r − R is the radial distance from the mer size) and short free chains, N < N , the brush 3∕5 particle surface, h is the effective brush height for a flat behaves as a swollen mushroom, with h ∼ N . By keep- −1 −2 surface and h is the brush thickness. For large nanoparti- ing the grafting density below N a and increasing N , cles, the above form asymptotically recovers the planar 1∕2 1∕2 so that N > N , the brush becomes ideal with h ∼ N . f g g result (h → h ). In the case of intermediate particle radii, −1∕2 −1 For intermediate grafting densities, N <𝜎 < N , g a combination of large and small curvature behaviors is 1∕2 anticipated: [178] the segment density profile exhibits high molecular weight free chains, N > N , can pene- f g large curvature behavior near the surface of the particle, trate the brush, thus ideally wetting it and leading to 1∕2 followed by a small curvature behavior away from it. h ∼ N . Increasing the grafting density while keeping 1∕2 Finally, following Daoud and Cotton, the brush height is N < N forces the chains to stretch, thus leading to a 1∕2 1∕4 expected to scale as h ∝ N . Recently, Chen et al. brush height scaling as h ∼ N . [183] revisited the scaling laws for spherical polymer Wijmans and Zhulina [178] employed similar ideas in brushes and identified significant assumptions overlooked order to understand the configurations of polymer brushes by Daoud and Cotton. grafted to spherical nanoparticles dispersed in a polymer melt. Here the radius of the nanoparticle, R , enters as 5.3.1 Experimental Findings an additional parameter. Long polymers grafted to a sur- face at fixed grafting density, , are strongly perturbed Hasegawa et al. [173] used rheological measurements and from their ideal random-walk conformation [179]. Planar SCF calculations to show that particles are dispersed opti- geometry scaling (infinite radius of curvature) is inade- mally when chains from the melt interpenetrate, or wet, a quate to explain the case when the particle size is reduced grafted polymer brush (“complete wetting”). This occurs to a level comparable with the size of the brushes. The at a critical grafting density, which coincides with the for- SCF theory has been applied to convex (cylindrical and mation of a stretched polymer brush on the particle sur- spherical) surfaces by Ball et al. [179]. For the cylindri- face [175, 176]. Grafting just below this critical density cal case, under melt conditions, it was found that the free produces aggregates of particles due to attractive van der ends of grafted chains are excluded from a zone near the Waals forces between them. The results of these authors grafting surface. The thickness of this dead zone varies suggest that mushrooms of nonoverlapping grafted poly- between zero for a flat surface to a finite fraction of the mer chains have no ability to stabilize the particles against brush height, h, in the limit of strong curvature, when aggregation (“allophobic dewetting”). Grafting just above R ∕h is of order unity, with R being the radius of curva- n n this critical density also results in suboptimal dispersions, ture of the surface. the aggregation of the particles now being induced by an Borukhov and Leibler [180] presented a phase dia- attraction between the grafted brushes. For high curvature gram for brushes grafted to spherical particles, in which 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 613 (small radius) nanoparticles, the polymer brush chains can explore more space, resulting in less entropic loss to penetrate the brush, reducing the tendency for autophobic dewetting. Until recently, the experimental verification of theoreti- cal and simulation predictions was mostly limited to global information concerning the brush, such as its average height, but not its profile [184]. Recently Chevigny et al. [185] used a combination of X-ray and Small Angle Neutron Scattering (SANS) techniques to measure the conformation of chains in polystyrene polymer brushes grafted to silica nanopar- ticles with an average radius of 13 nm dispersed in polysty- rene matrix. They found that, if the molecular weight of the melt chains becomes large enough, the polymer brushes are compressed by a factor of two in thickness compared to their stretched conformation in solution. Also, polymer brushes exposed to a high molecular weight matrix are slightly com- pressed in comparison to brushes exposed to a low molecu- lar weight matrix environment. This observation implies a wet to dry conformational transition. The low molar mass free chains can penetrate into the grafted brush and swell it (“wet” brush). Conversely, when grafted and free chain molar masses are comparable, free entities are expelled from the corona (“dry” brush). Later, they examined the dispersion of these grafted particles in melts of different molar masses, M [186]. They showed that for M ∕M < 0.24, the nano- f g f particles formed a series of compact aggregates, whereas for M ∕M > 0.24, the nanoparticles were dispersed within the g f Fig. 11 a Concentration of the free ends of grafted chains, (z) as fe polymer host. a function of the distance z from the sphere surface for chains with N = 50 grafted to the a sphere of radius R = 3 for various lengths of free chains, N, and grafting densities. b Same as in (a) except R = 10. 5.3.2 Insight Obtained from Simulations c Same as in (a) except N = 80, R → ∞. (Reprinted with permission from Ref. [187]. Copyright (2004) American Chemical Society.) Klos and Pakula [187] simulated linear flexible polymers of N repeat units, end-grafted at density onto a spherical surface of radius R (“hairy nanoparticle”), including the case also the case for chains grafted to a flat surface, as presented in Fig. 11(c). Furthermore, for N = N , R = 10, and R → ∞, of flat impenetrable wall ( R → ∞) using their cooperative motion algorithm [188, 189]. The simulations were carried the concentration of the free ends is finite even at the surface, which means that a small fraction of the ends concentrate in out for a wide range of parameters characterizing the hairy surfaces (N , , and R) and concerned in detail the influence that region, creating grafted chain loops, in agreement with earlier Molecular Dynamics simulations of brushes on flat of length of matrix chains on the anchored ones. That was achieved by gradually varying the polymerization degree N surfaces by Grest [190]. Spatial integration of the radial mass density profiles of the matrix chains between the two extremes of a dense melt of identical chains (N = N ) and a simple solvent con- around the nanoparticle allows for estimating the height of the grafted polymer brush, which is usually defined as the sisting of single beads (N = 1). Their analysis of free grafted chain-ends concentrations, (z), is shown in Fig. 11 (a, b) second moment of the segment density distribution, (r), as fe [178, 191]: for R =3, 10 and for =1, 0.2, respectively. The length unit used in their work was c / 2 with c being the lattice con- 2 2 1 ∫ 4r dr(r − R ) (r) stant of the employed lattice Monte Carlo simulations. The n (62) h = curves indicate how the medium in which the hairy sphere is ∫ 4r dr(r) immersed influences the profiles of free ends of the grafted with respect to the height h ≡ r − R . However, comparison chains. For both sizes of the spheres, the observed tendency n with experimental brush heights requires a measurement of is such that the longer the matrix chains become, the closer where the major part of the grafted material is found. To to the surface the free ends concentrate. In particular, this is 1 3 614 G. G. Vogiatzis, D. N. Theodorou Vogiatzis and Theodorou [192] investigated the struc- tural features of polystyrene brushes grafted on spherical silica nanoparticles immersed in polystyrene by means of a Monte Carlo methodology based on polymer mean field theory. The nanoparticle radii (either 8 or 13 nm) were held constant, while the grafting density and the lengths of grafted and matrix chains were varied system- atically in a series of simulations. The primary objective of that work was to simulate realistic nanocomposite sys- tems of specific chemistry at experimentally accessible length scales and study the structure and scaling of the grafted brush. In Fig. 12 the average thickness is plotted 1∕2 1∕4 versus N . N is measured in Kuhn segments per chain and in chains per nm . The scaling prediction of Daoud and Cotton seems to be fullfilled for both the rms height h and the height containing 99% of the brush material, h . The dashed lines are linear fits, confirm- 99% ing the good agreement of the simulation data with the theoretical scaling behavior. The agreement seems to be better for the h data points. This was expected, since 99% 2 2 the average brush thickness, as defined in Eq. (62), is h h Fig. 12 The calculated brush thickness (either or ) is plot- 99% ted versus the degree of polymerization of grafted chains, N , times more sensitive to the discretization of the model and 1∕4 the grafting density, . Points correspond to systems containing an to the post processing of the data, than the straight- 8-nm-radius silica particle grafted with PS chains and dispersed in PS forward definition of the shell in which the 99% of the matrix. Linear behavior is predicted by the model proposed by Daoud brush material can be found. Moreover the results for and Cotton for star shaped polymers [181]. (Reprinted with permis- sion from Ref. [192]. Copyright (2011) American Chemical Society.) the h estimate and the scattering pattern of the whole 99% grafted corona were in favorable agreement with SANS measurements. Voyiatzis et al. [193] studied the confinement induced this effect, the brush height can also be arbitrarily defined effects on the accessible volume and the cavity size dis- as the radius marking the location of a spherical Gibbs tribution in polystyrene-silica nanocomposites by atom- dividing surface, in which 99% of the grafted material is istic Molecular Dynamics simulations. The composite included. The theory of spherical polymer brushes was systems contained a single -quartz silica nanoparticle, pioneered by Daoud and Cotton [181]. In analogy to the either bare or grafted with atactic PS chains, which was scaling model developed by Alexander and de Gennes for embedded into an unentangled atactic PS matrix [158]. planar interfaces, Daoud and Cotton developed a model for Both free and grafted chains consisted of 20 monomers. spherical surfaces through geometric considerations based The considered nanoparticle diameters were 3.0, 4.0 and on starlike polymers. The spherical brush is divided into 5.0 nm and three different grafting densities were studied: three regions, an inner meltlike core region, an intermedi- 0.0, 0.5 and 1.0 chains / nm . Those authors investigated ate concentrated region (dense brush), and an outer semi- the cavity distribution size by employing four spherical dilute region (swollen brush). Daoud and Cotton predicted probe particles. Apart from the limiting case of a dot-like for star shaped polymers in the matrix a change in the scal- probe particle (zero radius), the considered probe parti- ing behavior as the blobs of the chains become non-ideal. cles had radii, r , of 0.128, 0.209 and 0.250 nm, corre- The density profile is directly related to the average brush sponding to hard-sphere representations of helium, meth- height, h, or the extension of the corona chains. Neglect- ane and ethane. The “unoccupied” volume was defined as ing the contribution of the core to the radius of the star, the volume accessible to a probe particle of r = 0. 1∕2 1∕4 they found that h ∝ N b. Although the former rela- The influence of the grafting density on the spatial dis- tion exhibits “ideal” scaling with respect to the chain length tribution of the unoccupied volume fraction, v , and the Un 1∕4 dependence, the presence of the factor shows that the specific volume, v , for a nanoparticle with a diameter r Sp radius is in fact larger than it would be for a single linear of 3 nm is presented in Fig. 13. The black horizontal line chain. Thus, although we are in a regime where the chain corresponds to the bulk value of v . The greatest reduc- Un seems to be ideal, the structure is actually stretched. tions of v relative to the bulk value occur very close to Un the surface, at distances smaller than 1 nm. The variation 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 615 have been the focus of much recent attention [197–207]. Central problems in the area are the diffusion of nanopar - ticles and polymers through the nanocomposite melts, as well as the local polymer dynamics in the proximity of the filler particles. For example, the incorporation of nanopar - ticles can strongly modify the viscosity of polymer melts [208], and the center-of-mass mobility of polymer chains can be strongly retarded depending on nanoparticle size and concentration [209, 210]. 6.1 Nanoparticle Diffusion in Polymers 6.1.1 Experimental Findings There is good understanding of the motion of very large or very small colloidal particles of radius R in a polymer melt. The nanoparticle diffusion coefficient, D, in the large particle limit follows the classic continuum Stokes-Einstein Fig. 13 Distribution of unoccupied volume fraction, v , (lefty-axis) relation [211]. For a large and massive solute molecule Un and the specific volume, v , (righty-axis) in the vicinity of a nano- Sp of radius R in a solvent consisting of much smaller and particle with a diameter of 3 nm at a termperature of 590 K. The- lighter molecules, the diffusion coefficient, D, of the solute horizontal line corresponds to the PS bulk value of v . The grafting Un is given by [212]: density is varying from 0.0 (ungrafted) to 0.5 and 1.0 grafted chains / nm . (Reprinted from [193] with permission from Elsevier.) k T D = (63) f R where k is Boltzmann’s constant, T is the absolute tem- in the v distribution of a grafted and an ungrafted nan- Un perature, is the solvent viscosity and f takes the values oparticle is different. The separation from the particle of 4 or 6 for slip or stick boundary conditions at the solute for which v is below the bulk value for the grafted 1.0 Un surface, respectively [203]. The corresponding behavior of chains/nm system exceeds by approximately 50% the dis- small nanoparticles, comparable to the size of a monomer, tance for the ungrafted system. The v profile for a graft- Un is also described by the Stokes-Einstein relationship but ing density of 0.5 chains / nm lies in between the two with a length-scale dependent viscosity that is smaller than extremes. Unlike v , the v spatial distributions exhibit Un Sp the macroscopic bulk value [213]. The relevant apparent a strong dependence on the grafting density. An increase viscosity is controlled by the relaxation of subsections of of the grafting density leads to increased v values close Un chains with an end-to-end distance comparable to the nano- to the particle’s surface. This behavior was attributed to particle size, as has been verified by Molecular Dynamics the (i) the chemistry of the employed linker molecule and simulations [214]. Understanding nanoparticle diffusion in (ii) the expulsion of the free chains from adsorbing on the the polymer matrix is of fundamental importance. nanoparticle surface. Contrary to intuitive expectations, Despite the rather good understanding of the diffusion of variations of the accessible volume were not directly the particles in the two limits (very large and very small), related to changes of the specific volume. the dynamical behavior of nanoparticles of size compara- ble to the entanglement mesh size of the polymer is conten- tious [205, 215–217]. Brochard-Wyart and de Gennes [213] 6 Dynamics argued that the particle diffusion constant follows normal Stokes-Einstein behavior essentially when its size becomes A complete understanding of PNC dynamics requires con- larger than the entanglement mesh size. Such a sharp size- fronting the difficult many-body problem associated with dependent crossover to Stokes-Einstein scaling has been non-dilute particle concentrations and coupled nanoparticle observed by Szymanski et al. [218]. On the contrary, Cai and polymer motions over many time- and length- scales et al. [201] speculated that the motion of these intermedi- [194–196]. Simulations are a valuable option, but are ate sized nanoparticles should be faster than Stokes-Ein- computationally very intensive, resulting in only a limited stein behavior, since diffusion can be facilitated by hoplike parameter range being tractable to study, typically involv- motions trough the polymer’s entanglement mesh. The lat- ing rather small particles and weakly entangled polymers. ter is also supported by a microscopic force-level theory, The transport properties of nanoparticle-polymer mixtures 1 3 616 G. G. Vogiatzis, D. N. Theodorou wherein chain relaxation and local entanglement mesh fluc- tuations, i.e. “constraint release”, dominate over hopping [219]. Somoza et al. [220] studied experimentally the anthracene rotation in poly(dimethylosiloxane) and poly(isobutylene) by gradually increasing the chain length. These authors reported that the diffusivity of the particles exhibited a sharp transition with the increase of the poly- mer radius of gyration, R , becoming dependent on the “nanoviscosity” (rotation time of dissolved anthracene was used as a measure of the viscosity on a nanometer-sized object) rather than the macroscopic viscosity for small R ∕R ratios (with R being the particle size). Narayaman n g n et al. [221] used X-ray photon correlation spectroscopy in conjunction with resonance-enhanced grazing-incidence small-angle X-ray scattering to probe the particle dynamics Fig. 14 The diffusion coefficient, D, of a spherical particle as a in thin films, and also found that the particle dynamics dif- function of the ratio R ∕R . The particle mass is proportional to its n g fer from the Stokes-Einstein Brownian motion, the differ - volume. Open squares represent Molecular Dynamics data, while full dots represent the Stokes–Einstein relation predictions with slip ence being caused by the viscoelastic effects and interpar - boundary condition. (Reprinted with permission from Ref. [214]. ticle interactions. Meanwhile, Tuteja et al. [198] reported Copyright (2008) American Chemical Society.) that the nanoparticles diffuse two orders of magnitude faster in a polymer liquid than the prediction of the Stokes- Einstein relation, an observation possibly attributable to the for nanoparticles with diameters larger than the entangle- ment mesh size it appears that the competition of full chain nanoparticles being smaller than the entanglement mesh. Later, Grabowski et al. [199] also observed strong enhance- relaxation versus the nanoparticle hopping through entan- glement gates controls nanoparticle diffusion [222]. ment (250 times) of the diffusion of gold nanoparticles in poly(butyl methacrylate), under conditions where the nano- 6.1.2 Insight Obtained from Simulations particle dimensions were smaller than the entanglement mesh length of the polymer. Liu et al. [214] employed Molecular Dynamics simulations Cai et al. [201] have carried out an extensive study of nanoparticle diffusion by employing scaling theory to pre- of the Kremer-Grest model [149] in order to investigate the diffusion process of spherical nanoparticles in poly - dict the motion of a probe nanoparticle of size R experi- encing thermal motion in polymer solutions and melts. mer melts. Their results indicated that the radius of gyra- tion of the polymer chains was the key factor determining Particles with size smaller than the solution correlation length, , undergo ordinary diffusion with a diffusion coef- the validity of the Stokes-Einstein relation in describing the particle diffusion at infinite dilution. In Fig. 14 the diffusion ficient similar to that in pure solvent. The motion of par - ticles of intermediate size (𝜉< d <𝛼 ), where is the coefficient estimated by the MD simulations is presented pp pp alongside the predictions of the Stokes-Einstein formula. It tube diameter for entangled polymer liquids, is subdiffusive at short time scales, since their motion is affected by sub- was found that, with the increase of the size ratio of R ∕R , n g the Stokes-Einstein diffusion coefficient gradually approxi- sections of polymer chains. At long time scales the motion of these particles is diffusive, and their diffusion coefficient mates the MD data under the slip (dotted curve) boundary condition. The use of purely repulsive non-bonded interac- is determined by the effective viscosity of a polymer liq- uid with chains of size comparable to the particle diameter tions fully justifies the use of the slip ( f = 4), instead of the stick ( f = 6), boundary condition. As the size ratio, R . The motion of particles larger than the tube diameter at time scales shorter than the relaxation time of an R ∕R increases to 1, the predicted diffusion coefficients n g pp e agree reasonably well with those extracted from the simu- entanglement strand is similar to the motion of particles of intermediate size. At longer time scales (t >𝜏 ) large parti- lations. However, in the small size ratio, large deviation is observed which qualitatively agrees with the experimental cles (d >𝛼 ) are trapped by the entanglement mesh, and to pp move further they have to wait for the surrounding polymer results [198]. It seems like the local viscosity experienced by nanoparticles is much smaller than the macroscopic vis- chains to relax at the reptation time scale . At longer rep times t >𝜏 , the motion of such large particles (d >𝛼 ) cosity, as speculated by Wyart and de Gennes [213] and rep pp other researchers [223–226]. Finally, it should be noted is diffusive with diffusion coefficient determined by the bulk viscosity of the entangled polymer liquids. Finally, that in the experiments chains are strongly entangled, in 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 617 not self-consistent since it assumed that the constrain- ing forces on a particle relax entirely due to the length- scale dependent motions of the polymer melt (constraint release regime), which is an accurate simplification when particles are larger than d . Figure 15 exhibits several interesting trends. First, the relative diffusivity approaches the asymptotic value (self) D ∕D at h∕2R >> 1, verifying the “isolated” limit SE n (two particles at infinite dilution). Note that it does not necessarily approach unity if a Stokes-Einstein violation is present at the single-particle level. The overall deviation of (rel) D from the hydrodynamic result is enhanced as 2R ∕d n T decreases or N∕N increases, in analogy with a single- particle Stokes-Einstein violation effect. The underlying physical mechanism can be understood as small nanopar- ticles acquiring high mobility due to a weaker coupling Fig. 15 Relative diffusivity normalized by the single-particle Stokes- Einstein self-diffusion coefficient as a function of h∕2R (with h being to the slow relaxation of entangled melts compared to the the interparticle surface-to-surface separation distance), based on the continuum theory. As a consequence, the overestimate of structural continuum model of Yamamoto and Schweizer [219, 228]. friction by a hydrodynamic approach grows as particle size Calculations are presented for 2R ∕d = 10 (with d representing the n T T decreases and/or chain length increases. The second gen- tube diameter) and N∕N = 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line), with N being the eral feature in Fig. 15 concerns the role of the number of number of chain segments per entangled strand. The hydrodynamic entanglements, N∕N . Deviations cannot be discerned from result (solid curve) is also included as a reference. In the inset to the the hydrodynamic behavior for weakly entangled cases figure, the same results as the main frame are presented, for larger (N∕N ∼ 1) for either particle size. One may physically particle size, 2R ∕d = 40. (Reprinted from [228], with the permis- n T sion of AIP Publishing.) rationalize this result by recalling that the Rouse- like col- lective relaxation is diffusive above the segmental length- scale. Finally, the most important feature of Fig. 15 is the the reptation regime, while the polymer length used by Liu predicted non-trivial mobility enhancement compared et al. is smaller than the entanglement length of the poly- to the hydrodynamic result over a wide range of h∕2R . mer chain [227]. Hydrodynamics predicts zero relative diffusivity as h → 0 Yamamoto and Schweizer [219, 228] have formulated , whereas the results of Yamamoto and Schweizer remain and applied a microscopic statistical-mechanical theory, non-zero down to small separations. based on the Polymer Reference Interaction Site Model Kalathi et al. [230] have employed large-scale molecu- (PRISM) integral equation theory [229], for the non- lar dynamics simulations in order to examine the role of hydrodynamic relative diffusion coefficient of a pair of entanglements on nanoparticle dynamics in the crosso- spherical nanoparticles in entangled polymer melts. Their ver regime, where the diameter of the particles, , is NP work was based on a combination of Brownian motion, larger than 2 − 10d with d being the entanglement tube T T mode-coupling, and polymer physics ideas. They focused diameter. The transport behavior of nanoparticles in the on the mesoscopic regime, where particles are larger than crossover size limit appears to be complicated by hopping the entanglement spacing. The overall magnitude of the effects, length-scale dependent entanglement forces and relative diffusivity was controlled by the ratio of the par - dynamics, and the interactions of polymers and nanoparti- ticle to tube diameter and the number of entanglements cles. These authors simulated weakly interacting mixtures per chain. Figure 15 presents model calculations of the of nanoparticles and bead-spring polymer melts. For the (rel) total relative diffusivity, D (h) as a function of h∕2R polymer melts considered in that work, the entanglement (with h being the interparticle surface-to-surface sepa- chain length is approximately 45, N ~45, and d in units of e T ration distance) for two reduced particle diameters. The monomer diameter, , is around 7 (the nanoparticle diam- ordinate of the figure is normalized by the single particle eters were = 1 − 15, in units of polymer ). The dif- NP Stokes-Einstein result, D , while the abscissa extends fusion coefficients of nanoparticles smaller than d ∼7 − 10 SE T up to the point where the theory is argued to be reason- (i.e. = 1, 3, and 5, respectively, Fig. 16(a)) in long NP able. That theory is based on the mode-coupling idea chain melts show that the relevant viscosity corresponds that the relevant slow dynamical variable is the bilinear to a section of the chain with N monomers that satisfies NP 2 2 coupling of the nanoparticle and the collective polymer = N . For shorter chains the data can be described NP NP density fluctuations. The original approach [219] was 1 3 618 G. G. Vogiatzis, D. N. Theodorou Fig. 16 Terminal diffusion coefficient of nanoparticles of different size, ≥ d . In all cases, a slip boundary condition was assumed. NP T size, D , in melts of different N plotted in scaled form. The combi- Predictions of the theory of Yamamoto and Schweizer [228] are NP nation of the Stokes-Einstein equation with the Rouse model viscos- presented in solid lines. Moreover, if we estimate the Rouse viscos- ∗ 3 2 −1 ity for a melt of chain length N , = N (where is the viscos- ity based on the actual chain length N, we get D ∼(N∕ ) , NP 1 NP 1 NP NP NP ity of a monomer fluid at the same density) yields that the quantity which corresponds to the decaying curves on the left-hand side of (a). ∗ 3 D should be a constant, independent of chain length [201, 214]. (Reprinted figure with permission from [230]. Copyright 2014 by the NP NP a Nanoparticles that are smaller than the entanglement mesh size, American Physical Society) 𝜎 < d . b Nanoparticles that are larger than the entanglement mesh NP T by the Stokes-Einstein relation with the macroscopic vis- to entropy losses associated with impenetrable obstacles. A cosity of the chain fluid, i.e., natural choice of a parameter for quantifying the ability of a nanocomposite to slow down polymer diffusion is the spac- k T k T ∗ B B D = = ing between the surfaces of neighboring nanoparticles [234]. (64) NP f f N NP 1 NP For monodisperse nanoparticles, this spacing can be defined where = N (where is the viscosity of a mono- 1 1 as the interparticle distance, d , given by [235]: inter mer fluid at the same density) and f∼4 (asymptote in Fig. 16(a)). The results of Kalathi et al. [230] for smaller max d = d − 1 (65) nanoparticles are in good agreement with the predictions inter of the theory of Yamamoto and Schweizer [228]. Fig- where d and are the nanoparticle diameter and nanopar- ure 16(b) presents the results for nanoparticles with sizes ticle volume fraction, respectively.The maximum packing larger than the entanglement mesh length, d ∼7 − 10. The density of the nanoparticles, , depends on the packing diffusion of these particles in the longer chain melts does max type, such as simple cubic ( = 0.524), face-centered not follow the “universal” plateau seen for small nanopar- max cubic ( = 0.740), body-centered cubic ( = 0.680), ticles. These results suggest that the chain-scale dynamics max max and random dense packing ( = 0.637). Independently does not control nanoparticle diffusion (no Stokes-Einstein max of , d decreases as nanoparticle size decreases at scaling). However, the fact that the diffusivity at high N of max inter fixed , suggesting that smaller nanoparticles slow down these intermediate-sized particles does not reach the same polymer diffusion more effectively than larger ones [231, plateau as the small nanoparticles (Fig. 16) suggests that 232]. another effect, probably entanglements, plays an impor - tant role. Despite the fact that no conclusive evidence of 6.2.1 Experimental Findings hopping-controlled transport was found, the spontaneous fluctuations of the entanglement mesh (constraint release) Gam et al. [210] have measured the tracer diffusion of deu- in the moderately long chain melts, may be the most impor- terated polystyrene (dPS) in a polystyrene nanocomposite tant mode of nanoparticle transport through the bulk of the containing silica nanoparticles, with number average diam- material, in agreement with theoretical predictions [228]. eters, d , of 28.8 and 12.8 nm, using elastic recoil detec- tion. The corresponding volume fractions of the large and 6.2 Polymer Diffusion and Dynamics small nanoparticles, , ranged from 0 to 0.5, and 0 to 0.1, respectively. At the same volume fraction of nanoparticles, Early theoretical studies [231–233] have shown that polymer the tracer diffusion of dPS is reduced as nanoparticle size diffusion through heterogeneous media is slowed down due 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 619 Fig. 18 The normalized (by its bulk value) overall diffusion coeffi- cient of bead-spring polymer chains as a function of % volume frac- tion of nanoparticles for repulsive and strongly attractive systems. Fig. 17 Reduced diffusion coefficient of a dPS polymer tracer (Reprinted from [237], with the permission of AIP Publishing.) (D∕D ) in a silica-polystyrene nanocomposite, plotted against the confinement parameter, namely the interparticle distance, ID = d , inter relative to the tracer size, R . Open and closed squares represent chain relaxation below the reptation time. The following experimental data for nanoparticles with number average diameters conclusions were drawn from their study: (i) the mono- of 12.8 and 28.8 nm. Employing the interparticle distance estimated meric relaxation rates were not unaffected by the addition from the average number nanoparticle diameter for monodisperse particles, the scaling of D∕D seems reasonable, although the values of nanoparticles, even at high particle loadings; (ii) chain for the smaller particles are higher than those for the larger particles. conformations remain Gaussian for all loadings considered; (Reprinted from [210]—Published by The Royal Society of Chemis- and (iii) the tube diameter determined from analysis of neu- try.) tron spin echo data decreases monotonically upon adding nanoparticles. Two contributions to overall chain dynam- decreases because the interparticle distance between nan- ics were speculated. On the one hand, the number of topo- oparticles, d , decreases. The reduced diffusion coeffi- logical chain-chain entanglements decreases with increased inter cient, defined as the tracer diffusion coefficient in the nano- nanoparticle loading, i.e., the chains disentangle from each composite relative to pure PS (D∕D ) is plotted against the other since a part of the system volume is occupied by the confinement parameter divided by the tracer size in Fig. 17. NPs. On the other hand, the chain acceleration caused by All measurements nearly collapse onto a master curve the reduction of entanglements is (more than) compensated [234], although D∕D is slightly higher for the smaller by the geometric constraints that nanoparticles present to particles. For d = ID < 2R , D∕D decreases rapidly as chain dynamics. Since that second factor dominates at large inter g 0 ID∕2R decreases. For ID > 2R , D∕D remains less than loadings, the neutron scattering experiments suggested an g g 0 1 indicating that entropy loss reduces diffusion even when increase in chain relaxation time, while at the same time a nanoparticles are far apart relative to the tracer size. The reduction of chain-chain entanglements and an increase of dashed line is an empirical fit, because a theory relating D particle-chain entanglements take place. to the fundamental system parameters is lacking. Schneider et al. [236] studied experimentally the relaxa- 6.2.2 Insight Obtained from Simulations tion of entangled poly(ethylene-alt-propylene) (PEP) chains (tube diameter ~5 nm) filled with silica nanoparticles (aver - Desai et al. [237] investigated the chain dynamics of Kre- age diameter ~17 nm). The silica volume fraction was var- mer-Grest polymer melts, composed of chains with a rela- ied between 0.0 and 0.6 (as that was estimated from the tively high degree of polymerization (N = 80) filled with measured weight fraction of silica in the nanocomposite). solid nanoparticles using molecular dynamics simulations. Neutron spin echo spectroscopy (NSE) was empoloyed in These authors found that chain diffusivity is enhanced rela- order to explore chain dynamics in these nanocomposites, tive to its bulk value when polymer–particle interactions characterized by non-attractive interactions. The result- are repulsive and is reduced when polymer–particle inter- ing collective dynamic scattering function data were ana- actions are strongly attractive (Fig. 18). In both cases chain lyzed by employing the idea of a tube-like confinement for diffusivity assumes its bulk value when the chain center 1 3 620 G. G. Vogiatzis, D. N. Theodorou of mass is about one radius of gyration, R , away from the modes directly yield information on the monomer friction eﬀ eﬀ particle surface. As shown in Fig. 18 for the particle vol- and in the limit of p = 1, the plateau of ∕ is directly p p,neat ume fraction of 10%, the average chain diffusion coeffi- proportional to the ratio of ∕N in the PNC compared to 0 e cient is reduced by a factor of 2 in the presence of strongly that in the pure melt. For small nanoparticles, which act as attractive particles. The case of repulsive particles appears a diluent, there was an additional speedup, which was to be even more interesting, where the diffusion coefficient attributed to a reduction in entanglements, quantified by initially increases with increasing particle concentration, N ∕N ∼0.9 for long chains, which can also be e, melt e, PNC but then reaches a maximum before decreasing with further extracted by the stretching exponents (Fig. 19(e)). increase of the particle concentration. While the initial par- ticle concentration dependence of the diffusion coefficient 6.3 Local Polymer Dynamics reflects the polymer–particle interactions, higher concen- trations always lead to a reduction of the diffusion coeffi- 6.3.1 Insight Obtained from Simulations cient, which may be attributed to geometrical reasons (i.e. the presence of tortuous paths in systems with high particle Brown et al. [167, 241] were among the first to study the loadings). local dynamics of a model nanocomposite system. They Kalathi et al. [239] employed large-scale molecular examined the structure and dynamics of a system contain- dynamics simulations in order to study the internal relaxa- ing an inorganic (silica) nanoparticle embedded in a poly- tions of chains in nanoparticle/polymer composites. They mer (polyethylene-like) matrix. They thoroughly discussed examined the Rouse modes of the chains, which resem- the variation of structure and dynamics with increasing ble the observables of the self-intermediate scattering distance from the polymer–particle interface and as a func- function, typically determined in an (incoherent) inelas- tion of pressure. A clear structuring of the linear polymer tic neutron scattering experiment. The Rouse modes, chains around the silica nanoparticle was observed, with p = 0, 1, 2, ..., N − 1, of a chain of length N are defined as prominent first and second peaks in the radial density func- [240]: tion and concurrent development of preferred chain orien- N tation. Evidence of chain immobilization was less obvious 2 1 overall, although dynamic properties were more sensitive = cos i − . (66) p i N N 2 i=1 to changes in the pressure. Long simulations were carried The time autocorrelation of the Rouse modes is predicted out to determine the variation in the glass transition of the to decay exponentially and independently for each node p filled polymers as compared to the pure systems. In Fig. 20 for an ideal chain, with relaxation time, . The p = 0 mode the average relaxation times of the torsional autocorrelation describes the motion of the chain center-of-mass, while the function are presented, resolved into concentric shells of 5 modes with p ≥ 1 describe internal relaxations with a mode Å thickness around the nanoparticle center of mass. This number p corresponding to a sub-chain of (N − 1)∕p seg- assignment was based on the position of the center of mass ments. The comparison of the relaxation times of the differ - of the four atoms involved in the torsion at the time ori- ent modes for chains in the PNCs for three different degrees gin of the observation. Although this means that at some of polymerization, N, filled with nanoparticles of different later time an angle may belong to a different shell, it avoids sizes for = 0.1 to neat melt is presented in Fig. 19(a)–(c). the bias that would result from selecting only angles that Their results (Fig. 19) showed that, for weakly interacting remain in a particular shell (in any case diffusion of chains mixtures of nanoparticles and polymers, the effective mon- is relatively slow so that should not be a problem). Based omeric relaxation rates are faster than in neat melt when the on Fig. 20, there is some indication that the decreased nanoparticles are smaller that the entanglement mesh size. translational mobility near the interface increases the relax- In this case, the nanoparticles serve to reduce both the ation time associated with torsional equilibration (estab- monomeric friction and the entanglements in the polymer lishment of the trans-gauche equilibrium in the systems melt, as in the case of polymer-solvent mixtures. On the containing the nanoparticles with diameters of 3 and 6 nm contrary, for nanoparticles larger than half the entangle- (R30L and R60L, respectively), but otherwise most of the ment mesh size, the effective monomer relaxation remains characteristic times are very close to those obtained for the unaffected for low nanoparticle concentrations. Even in this neat system, in agreement with previous studies [241, 242] case, strong reduction of chain entanglements was of the same authors. It was concluded that, within errors, observed. These authors concluded that the role of nano- the interphase thickness was independent of the size of the particles is to always reduce the number of entanglements. nanoparticle for the range of particle systems analyzed. By assuming that the relaxation time for a chain follows the Vogiatzis and Theodorou [156] produced atomistic con- crossover bridging Rouse to reptation dynamics, the large p figurations of fullerene-filled polystyrene melts by reverse 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 621 Fig. 19 Normalized effective relaxation times of the p-th Rouse to = 10 in N = 500 at similar nanoparticle loading as in ref. NP mode for chains in nanocomposites for different nanoparticle sizes [238]). e Corresponding plot for the stretching exponent . (Reprinted at = 0.1: a N = 40; b N = 100; c N = 400. d Effect of nanoparti- from [239]—Published by The Royal Society of Chemistry.) cle loading for N = 400, = 10 (closed triangles correspond NP mapping well-equilibrated coarse-grained melt configura- suggesting an increase of the bulk glass transition tempera- tions, sampled by connectivity altering Monte Carlo, to the ture, T , by about 1 K, upon the addition of C at a concen- g 60 atomistic level via a rigorous quasi Metropolis reconstruc- tration of 1% by weight (in favorable agreement with dif- tion. The main goal of their work, i.e., the study of PS-C ferential scanning calorimetry measurements [243]). They dynamics at the segmental and local levels, has been then employed a space discretization in order to study the accomplished by analyzing long MD trajectories of their effect of C on segmental dynamics at a local level. Each well-equilibrated reverse-mapped structures. Their simula- fullerene served as the center of a Voronoi cell, whose vol- tion results generally indicate that the addition of C to PS ume was determined by the neighboring fullerenes. Back- leads to slower segmental dynamics (as estimated by char- bone carbons lying in every cell were tracked throughout acteristic times extracted from the decay of orientational the atomistic Molecular Dynamics trajectory and their time-autocorrelation functions of suitably chosen vectors), mean-square displacement (MSD) was measured for the 1 3 622 G. G. Vogiatzis, D. N. Theodorou moving in smaller (more confined) Voronoi cells exhibited faster motion than the atoms moving inside larger Voro- noi cells. Figure 21 presents the MSD of backbone carbon atoms as a function of time at a temperature of 480 K for both the filled and unfilled systems. As can be seen, nano- composite systems exhibit lower mobility when compared to their neat counterparts. The MSD of backbone carbons is depressed upon the addition of fullerenes, in good agree- ment with the neutron scattering observations of Kropka et al. [243]. In the inset to Fig. 21, a logarithmic plot of the 1∕2 MSD is presented. The scaling of t is expected for the very short time behavior studied, as Likhtman and McLeish [244] have estimated that the time marking the onset of the effect of topological constraints on segmental motion, , is Fig. 20 The radial dependence of the relaxation time of the torsional 3.36 × 10 s for polystyrene. autocorrelation function of polyethylene around a silica nanoparticle. Moreover, Vogiatzis and Theodorou [156] estimated the All values are averages taken in 5 Åshells around the nanoparticle center of mass. The points have been offset slightly for the three sys- local mean-square displacement (MSD) of backbone car- tems along the x axis for clarity. The dotted line simply indicates the bon atoms of PS, for the timespan an atom spends inside a value obtained for the 30-chain pure polymer system at the same tem- particular cell of the Voronoi tessellation. In their analysis perature (400 K). (Reprinted with permission from Ref. [167]. Copy- they used the average MSD from the three most confined right (2008) American Chemical Society.) and three least confined cells. They observed that the vol- ume of the Voronoi cells did not change significantly as a function of time. Based on that analysis for the nanocom- posite system, the degree of depression was found to be a function of the confinement induced by the fullerenes. The diffusion of chains was spatially inhomogeneous, as observed by Desai et al. [237] earlier. Small Voronoi cells lead to higher mobility of the polymer segments within them. Despite the fact that the addition of fullerenes limited the diffusion of polymeric chains, there existed regions in space, where the polymer could recover part of its dynam- ics due to the high level of confinement. This finding was then correlated with the increased rotational diffusion of fullerenes, as the volume of the Voronoi cells became smaller. These authors envisioned fullerenes as nanoscopic millstones, forcing the polymeric chains to diffuse faster, when close to them, while the geometrical constraints imposed by their presence force the chains to diffuse more slowly. Pandey et al. [245] have extensively studied the local dynamics and the conformational properties of polyiso- prene next to a smooth graphite surface constructed by Fig. 21 Mean-squared atomic displacements (MSD) of backbone graphene layers, via a multiscale simulation methodology. carbon atoms as a function of time for filled and unfilled polysty - rene systems at T = 480 K. In the case of fullerene nanocomposites, These authors first performed fully atomistic molecular an analysis of the dependence of backbone MSD on confinement is dynamics simulations of isoprene oligomers, next to the also presented for most and least confined Voronoi cells (indicative surface. Subsequently, Monte Carlo simulations of a sys- error bars also included). In the inset to the figure, the same data are tematically derived coarse-grained model were employed presented in logarithmic axes. (Reprinted with permission from Ref. [156]. Copyright (2014) American Chemical Society.) in order to create several uncorrelated structures for poly- mer systems. A reverse mapping strategy was developed in order to reintroduce atomistic detail into the coarse- time they resided in the same cell. Overall mean-square grained configurations. Finally, multiple extensive fully displacement of backbone atoms was found to be smaller in atomistic simulations with large systems of long macro- the presence of fullerenes, than in bulk PS. However, atoms molecules were conducted to examine local dynamics in 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 623 individually for every c-CH vector and colored each seg- ment from blue to red signifying lower relaxation times and higher mobility. As shown, segments in proximity to the surface were found to be slower. They then separated all train segments based on their length and calculated cor- relation times for each position along the length of the train segment (Fig 22(b)), which was an extremely challenging task. Despite a significant statistical error, several features are evident in Fig.22(b). Specifically, when a chain makes a single contact, dynamics is only decelerated to a small extent. The second important finding was that, as train segments grow in length along the surface, the dynam- ics of the repeat units becomes progressively slower, with the findings implying that, similar to chain-ends in bulk dynamics, [246, 247] the ends of train segments contribute to increased dynamic heterogeneities on the surface. How- ever, the former are only significant for short chains, the latter were present for any chain length studied. In addition, short train segments can be more pronounced around sur- faces with higher curvature [16, 159]. Finally, a PI specific result was the asymmetry present along a train segment originating from the methyl group, much alike findings on bulk dynamics along the chain backbone [247]. Rissanou and Harmandaris [248] presented a detailed analysis of the dynamics of three different polymer-gra- phene systems, through atomistic Molecular Dynamics simulations. In more detail they studied (a) PS-graphene, (b) PMMA-graphene and (c) PE-graphene interfacial sys- Fig. 22 a Visual representation of the distribution of − log along tems, as well as the corresponding bulk polymer systems. the normal to the surface and on the surface of graphene planes ( is average per slab relaxation time of the c–CH bonds). Each repeat For PS and PMMA polymer chains were 10-mers while PE unit is colored with a scheme where red corresponds to the faster seg- chains were 20-mer, in order to ensure that the backbone ments and blue to the slowest. b Average dynamics of repeat units consisted of almost the same number of CH , and/or CH on graphite along a train segment as a function of the length of it. groups for all systems (i.e. approximately 20 in all cases). (Color figure online) (Reprinted from [245], with the permission of AIP Publishing.) A characteristic quantity of the molecular level is the end- to-end vector (t), whose autocorrelation function pro- ee vides information for the orientational dynamics at the proximity to graphite. Their findings supported the pres- entire chain level. Rissanou and Harmandaris performed ence of increased dynamic heterogeneity emerging from an analysis of end-to-end autocorrelation function at dif- both intermolecular interactions with the flat surface and ferent adsorption layers and fit the corresponding curves intramolecular cooperativity. For each system, Pandey et al. for all chains to the Kohlrausch-Williams-Watts (KWW) extracted bond orientation autocorrelation functions and function [249–251]. At the entire chain level, the integral sorted them in intervals of 0.03 nm based on the position below the KWW curves defines the molecular chain end- ee of the midpoint of the c–CH bond throughout the simula- to-end relaxation time, . The molecular relaxation times mol tion trajectory. For each interval, the autocorrelation curves together with the stretching exponent, , of the KWW fits were averaged weighting by the population of c–CH vec- are presented in Fig. 23 (a) and (b) as functions of the dis- tors found in a specific interval from each run. The mean tance from the surface. Data in Fig. 23(a) reveal the dra- ee correlation times increased substantially in the proxim- matic increase of close to the graphene layer, compared mol ity of the surface, with dynamics at the surface almost 20 to the corresponding bulk values, shown with dashed lines. times slower (independently of the molecular weight of the Furthermore, a slight difference in the distance at which ee chains) than in bulk PI. the reaches the plateau distance-independent bulk mol Figure 22 presents a qualitative visual inspection of value was observed: for PE it is about ∼2 nm, whereas for the distribution of times for a specific configuration. Pan- PMMA and PS is about 3 − 4 nm. The extreme difference dey et al. [245] evaluated an autocorrelation function in relaxation times between PE and the other two systems is 1 3 624 G. G. Vogiatzis, D. N. Theodorou Fig. 24 Nanoparticle volume fraction at spinodal phase separation predicted by the Polymer Reference Interaction Site Model (PRISM) theory for hard spheres of D∕d = 10 (with D and d being the diam- eters of the nanoparticle and the polymeric beads, respectively) in a freely jointed chain polymer system of length N = 100, as a func- tion of the strength of exponential interfacial attraction at fixed spa- tial range. Total mixture packing is 0.4. The depletion and bridging induced phase separated regime bracket a window of miscibility at intermediate interfacial cohesion strength. The type of polymer-medi- ated nanoparticle organization is schematically indicated. (Reprinted from [256] with permission from Elsevier.) achievement of uniform dispersion being a long-standing challenge [1, 14, 15, 33, 150, 252, 253]. Significant progress towards the development of micro- scopic predictive theories of the equilibrium structure and Fig. 23 a Molecular relaxation time of the end-to-end orientational phase behavior of polymer nanocomposites has been made decorrelation function for PS, PMMA and PE hybrid polymer-gra- recently based on liquid state integral equation formula- phene systems as a function of the distance from graphene. Dashed lines represent the values of the molecular relaxation times of the cor- tions, density functional calculations and self-consistent responding bulk systems. b The stretching exponent, , as extracted mean field approaches. All these methods can complement from the fit with KWW functions for the three systems. (Reprinted or surpass the explicit atom methods like Monte Carlo and from [248]—Published by The Royal Society of Chemistry.) Molecular Dynamics, which have the potential to quantita- tively predict structural correlations, thermodynamics and phase behavior. obvious. The exponent of the KWW relation for PE and PS reaches a constant value in the bulk region, while the Chatterjee and Schweizer were the first to develop an analytical integral equation theory for treating polymer- same does not apply for PMMA. These values show that in the bulk region PMMA has the wider distribution of relax- induced effects on the structure and thermodynamics of dilute suspensions of hard spheres [254]. Results were pre- ation times, PS follows and PE has the narrowest one. sented for the potential of mean force, free energy of inser- tion per particle into a polymer solution, and the second 7 Phase Behavior virial coefficient between spheres. Later, Hoopper et al. [255] employed the Polymer Reference Interaction Site Polymer-nanoparticle blends exhibit a rich phase behav- Model (PRISM) theory to investigate structure, effective forces, and thermodynamics in dense polymer-particle mix- ior which is directly tied to the thermal, mechanical, and optical properties of the composite system, with the tures in the one and two particle limit [144, 145]. 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 625 7.1 Bare Nanoparticles Hall et al. [256, 257] employed Polymer Reference Interaction Site Model (PRISM) liquid state theory to study phase transitions and structure of dense mixtures of hard nanoparticles and flexible polymer coils. Their calculations were performed over the entire composi- tional range from the polymer melt to the hard sphere fluid, with the focus being on polymers that adsorb on nanoparticles. Many body correlation effects were fully accounted for in the determination of the spinodal phase separation instabilities. An example phase diagram is presented in Fig. 24. It can be discerned that depletion and bridging phase separation occur at low and high attraction strengths, respectively. Quantitatively, many particle effects are found to always reduce miscibil- Fig. 25 Second virial coefficient as a function of particle size ity. Depletion phase separation was similar for differ - for silica particles dispersed in PS matrix at volume fraction 5 %. ent attraction ranges, with a critical point at rather low (Reprinted with permission from Ref. [258]. Copyright (2015) Amer- ican Chemical Society.) filler volume fractions. This is in contrast to the bridg- ing induced demixing transition where the critical point is located at very high nanoparticle volume fractions. Moreover, increasing the attraction range increases the function of distance between particles, making it a critical thickness of the bound layer and the importance of many link to particle microstructure. The pair correlation func- body effects, which further decreases miscibility in the tion approaches asymptotically ()∕ = 0 when r < 2R n n n high filler volume fraction regime relative to what was (R being the radius of the particles) as particles cannot r → ∞ predicted by a two particle virial analysis [145]. How- interpenetrate and ()∕ ≃ 1 as as the likelihood n n ever, when bridging effects are very strong and phase of finding a particle becomes proportional to the average separation occurs at low volume fractions, decreasing particle density. Figure 25 shows the second virial coeffi- the attraction range can lead to a stronger, shorter range cients for different particle sizes at constant particle volume bridging attraction that reduces miscibility. Increasing fraction = 5%. Positive values of B indicate stable parti- particle size generally disfavors miscibility on both the cle dispersion (effective particle-particle repulsion), while depletion and bridging sides of the spinodal phase dia- a negative value signifies unstable dispersion. The results gram, though the effect on depletion is more significant. are in agreement with previous theoretical studies: the ten- Wei et al. [258] have investigated silica nanoparticle dency to dispersion increases as the particle size increases dispersions in polystyrene, poly(methyl methacrylate), [259]. It can be seen that B becomes independent of the and poly(ethylene oxide) melts by means of a density particle size after a threshold value, that meaning the effect functional approach. The polymer chains were regarded of particle size on the pair correlation function becomes as coarse-grained semi-flexible coils whose segment size insignificant. Before that critical value, B increases but its matched the Kuhn length of the polymer under investi- increasing amplitude declines as the particle size increases. gation. The particle-particle and particle–polymer inter- Density functional theory confirms that large particles actions were calculated in the grounds of the Hamaker are more likely to achieve stable dispersion than small theory, following Vogiatzis and Theodorou [148, 192]. particles. In order to characterize nanoparticle dispersion, Wei et al. employed the second virial coefficient, B , defined 7.2 Polymer Grafted Nanoparticles as: One approach for controlling the particle dispersion in the () 2 polymer matrix is to alter the particle surface chemistry 3 2 B = + 2 1 − r dr (67) 3 through the attachment of polymer chains. The composi- tion, architecture, and distribution of the grafted chains where the first term accounts for the particle contribu- can be carefully designed to tailor interparticle interac- tion, and the second one is the polymer mediated contri- tions, thereby controlling the dispersion state [33]. In the bution. The local density of particles is denoted as (), special case where the chemical composition of the graft while the average particle density as . varies as a ( ) n n and matrix chains are identical, the entropic contributions 1 3 626 G. G. Vogiatzis, D. N. Theodorou behave similarly to block copolymers, aggregating into dis- tinct morphologies [14, 268, 269]. As increases above , the matrix and grafted chains interpenetrate, resulting in a “wet” brush and repulsive interactions between nano- particles, thus stabilizing their dispersion. The autophobic dewetting line corresponds to a continuous, second-order transition, resulting from the expulsion of the melt from 1∕2 the brush for densely grafted chains (𝜎 N > 1) which should lead to nanoparticle aggregation through the attrac- tion between graft layers [270]. Larger values of R , , or P/N result in a larger entropic penalty for intermixing due to crowding of the grafted layer. As the entropic penalty grows, the interpenetration width between the matrix and grafted chains decreases until the matrix chains are com- pletely expelled, resulting in attractive interactions and par- ticle aggregation. This occurs above the autophobic phase transition at , a discontinuous, first-order transition at low grafting densities. Bansal et al. [27] have experimentally observed and Fig. 26 Illustration of the phase diagram for nanoparticle stability as modeled the anisotropic self-assembly of small PS-g- a function of grafting density () and the ratio of the lengths of the silica nanoparticles with cores of R ∼10 − 13nm in the free over the grafted chains (P/N). Particles at low grafting densities allophobic dewetting region at lower grafting densities encounter the allophobic dewetting transition at . Increasing leads to complete wetting of the brush by the melt, stabilizing the nano- ( = 0.01 − 0.10 chains∕nm ) [14]. Using slightly higher particle dispersion. Increasing the graft density further leads to the grafting densities (∼0.2 − 0.7 chains∕nm ), Chevi- autophobic dewetting transition at and unstable dispersion. Particle gny et al. [186] used similar-sized PS-g-silica NPs with dispersion is unstable at all grafting densities when P∕N > (P∕N) . N = 5 − 50 kg/mol in P = 140 kg/mol where particles (Reprinted with permission from Ref. [174]. Copyright (2012) Amer- ican Chemical Society.) with the longest grafts (P∕N = 2.8) dispersed uniformly, whereas those with the shortest grafts (P∕N = 28) phase separated from the bulk, forming spherical aggregates. dominate the thermodynamics [175, 260], and uniform dis- The dispersion of silica NPs with higher graft densities has persion can be achieved both for the case of spherical nano- been investigated in which two sets of PS-g-silica NPs, the particles [174, 261], and the case of nanorods [262]. first with N = 110 kg/mol and = 0.27 chains∕nm and the second with N = 160 kg/mol and = 0.57 chains∕nm 7.2.1 Experimental Findings have been shown to disperse at least up to P∕N = 2.3 [271] and 1.6 [27], respectively. In a number of studies, empirical phase diagrams have been Another way of tuning the mechanical properties of developed for this special case, where the particle misci- composite materials is by dispersing hydrophilic nanofill- bility is a function of grafting density, , the ratio between ers in highly hydrophobic polymer matrices [272]. Martin the molecular weight of the grafted chains, N, and matrix et al. [273] have performed simulations and experiments chains, P, i.e. P / N, as well as the particle radius, R [14, on mixtures containing polymer grafted nanoparticles in 71, 72, 173, 174, 261, 263–266]. Sunday et al. [174, 267] a chemically distinct polymer matrix, where the graft and quantified the stability of polystyrene-grafted silica nano- matrix polymers exhibit attractive enthalpic interactions at particles in PS matrices with ultrasmall angle X-ray scat- low temperatures that become progressively repulsive as tering (USAXS) and transmission electron microscopy temperature is increased. (TEM). They developed the phase diagram presented in Fig. 26 to predict nanoparticle dispersion based on the graft 7.2.2 Insight Obtained from Simulations polymer density, , and the graft and free polymer molecu- lar weights, or N and P, respectively. Trombly and Ganesan [264] have calculated the potential of The phase diagram of Fig. 26 shows three distinct mean force (PMF) between grafted nanoparticles immersed regions. When is below the allophobic limit ( ), the in a chemically identical polymer melt using a numeri- surface coverage of grafted chains is low enough that the cal implementation of polymer mean-field theory. These interactions between the particle and the matrix chains authors focused on the interpenetration width between are not screened out sufficiently and the grafted particles 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 627 Fig. 27 Potential of mean force between two grafted nanoparticles in two cases, Σ= 0.38 chains/ and matrix chain length M = 10 shown in solid black lines; Σ= 0.76 chains∕ and M = 70 shown in solid red lines. The dotted lines show the potential of mean force obtained from the corresponding cases of the work of Smith and Bedrov [274]. (Color figure online) (Reprinted from [266]—Published by The Royal Society of Chemistry.) the grafted and free chains and its relationship to the pol- ymer-mediated interparticle interactions. To this end, they quantified the interpenetration width as a function of par - ticle curvature, grafting density, and the relative molecular weights of the grafted and free chains. Meng et al. [266] used Molecular Dynamics simulations to delineate the separation dependent forces between two polymer-grafted nanoparticles in a homopolymer melt and the associated potential of mean force (PMF). The nano- particle radius (=5 in units of the chain monomers) and grafted brush length (=10) were held constant, while the grafting density and the polymer matrix length were var- ied systematically in a series of simulations. At first, it was shown that simulations of a single nanoparticle did not reveal any signatures of the expected autophobic dewet- ting of the brush with increasing polymer matrix length (in agreement with Monte Carlo simulations of Vogiatzis and Theodorou [192]). In fact, density distributions of the matrix and grafted chains around a single nanoparticle Fig. 28 Potential of Mean Force (PMF) in units of k T versus inter- appeared to only depend on the grafting density, but not particle distance, r − D (in units of monomer diameter, d), between on the matrix chain length. However, the calculated forces grafted nanoparticles (D = 5d) at grafting denisties of a 0.1, b 0.25 between two nanoparticles in a melt, presented in Fig. 27, and c 0.65 chains∕d and polydispersity indices 1.0 (circles), 1.5 (squares), 2.0 (upward facing triangles), and 2.5 (downward facing showed that increasing the matrix chain length, M, from 10 triangles) with average grafted chain length of 20 in a dense solution to 70 causes the interparticle PMF to go from purely repul- of monodisperse homopolymer matrix chains of lenght 10 (solid sym- sive to attractive with a well depth of the order of k T (with bols) and 40 (open symbols). The insets have the same axis labels as k being the Boltzmann constant). It was speculated that the main plots. (Reprinted figure with permission from [275]. Copy - right 2013 by the American Physical Society) these results were purely entropic in origin and arise from a competition between brush-brush repulsion and an attrac- tive inter-particle interaction caused by matrix depletion 1 3 628 G. G. Vogiatzis, D. N. Theodorou from the inter-nanoparticle zone (i.e. an Asakura-Oosawa two ends as a result of topological constraints [149]. The type inter-particle attraction). Figure 27 compares the PMF advent of computational algorithms enabled direct observa- with the results from Smith and Bedrov’ simulations [274] tion of entanglements that arise in polymeric melts [276–278]. of a similar coarse-grained system using the umbrella sam- Anogiannakis et al. [38] have examined microscopically at pling method for an apparently identical chain length and what level topological constraints can be described as a col- coverage. The two studies are in qualitative agreement to lective entanglement effect, as in tube model theories, or as each other, with the PMF of Meng et al. consistently shifted certain pairwise uncrossability interactions, as in slip-link toward smaller separations. models. They employed a novel methodology, which ana- Martin et al. [275] presented an integrated theory and lyzes entanglement constraints into a complete set of pairwise simulation study of polydisperse polymer grafted nano- interactions (links), characterized by a spectrum of confine- particles in a polymer matrix to demonstrate the effect of ment strengths. As a measure of the entanglement strength, polydisperisty in graft length on the potential of mean force these authors used the fraction of time for which the links are between the grafted particles. It is evident from Fig. 28 active. The confinement was found to be mainly imposed by that increasing polydispersity in graft length reduces the the strongest links. The weak, trapped, uncrossability interac- strength of repulsion at contact and weakens the attrac- tions cannot contribute to the low frequency modulus of an tive well at intermediate interparticle distances, completely elastomer, or the plateau modulus of a melt. eliminating the latter at high polydispersity index. The reduction in contact repulsion was attributed to polydisper- 8.1.1 Insight Obtained from Simulations sity relieving monomer crowding near the particle surface, especially at high grafting densities. The elimination of the Riggleman et al. [279] have carried out a detailed examina- midrange attractive well could be attributed to the longer tion of entanglements in a nanocomposite glass. They have grafts in the polydisperse graft length distribution that in conducted Molecular Dynamics simulations of an ideal turn introduced longer range steric repulsion and altered bead-spring polymer [149] nanocomposite model in which the wetting of the grafted layer by matrix chains. That the nanoparticles were dispersed throughout the polymeric work demonstrated that at high grafting densities, polydis- matrix. After equilibration in the melt state, all configura- persity in graft length can be used to stabilize dispersions tions were cooled below their glass transition temperature, of grafted nanoparticles in a polymer matrix at conditions where they were subsequently aged using MD for a short where monodisperse brushes would cause aggregation. period. Finally, a simulation of the creep response of each sample was performed, where tensile and compressive stresses were applied to the glassy specimens. In order to 8 Rheology reduce the chains to their primitive paths, these authors employed the CReTA algorithm of Tzoumanekas and The- 8.1 Polymer Entanglements odorou [278]. During the reduction process, the diameter of the particles is reduced to facilitate slippage of entangled One of the fundamental concepts in the molecular descrip- chains past each other, up to the point that further decrease tion of structure—property relations of polymer melts is in the diameter of the particles no longer has an appreci- chain entanglement. As the molecular weight of the mol- able effect on primitive path statistics. The nanoparticles ecules in a polymer melt is increased, the spatial domain are necessarily frozen in space as the algorithm proceeds. spanned by any given chain increasingly overlaps with By examining all particles, one can calculate the distri- those occupied by its neighbors. When macromolecules bution of the number of primitive path contacts per particle interpentrate, the term entanglements intends to describe in the system, shown in Fig. 29(a). The majority of the par- the topological interactions resulting from the uncross- ticles trap at least one primitive path. The entanglements ability of chains. The fact that two polymer chains cannot due to polymer chains crossing each other are expected to go though each other in the course of their motion changes exhibit little (if any) change during deformation. The only their dynamical behavior dramatically, without altering mechanism for the entanglements to disappear is through their equilibrium properties. Entanglements play a key role chain ends slipping past an entanglement junction; such in the viscoelastic properties of polymers, as evidenced, for effects are anticipated to be minimal in the glassy state. example, by the emergence of a plateau region in measure- However, Fig. 29(a) reveals appreciable changes in the ments of the storage modulus as a function of frequency. distribution of the number of primitive path contacts per Molecular simulations have confirmed that the overall particle; the number of contacts per particle increases sig- motion of the chains in a polymer melt is restricted to diffu- nificantly upon deformation. Figure 29(b) shows how the sion along their “primitive paths”, which represent the dif- average number of contacts per particle increases with fusive paths that linear chain molecules follow between their time for both tensile and compressive deformations. An 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 629 particle that exhibits the largest nonaffine displacements begins with two primitive path contacts, and as the defor- mation proceeds loses its primitive path contacts. Since that particle was not hindered by any primitive paths, it was able to move throughout the system more easily. In con- trast, the particle with the smallest nonaffine displacements (plotted using up triangles) experienced two or more primi- tive path contacts during the entire deformation. Those entanglements served to trap the particle and forced it to move along them, in an affine manner. Nanoparticles were found to serve as entanglement attractors, particularly at large deformations, altering the topological constraint net- work that arises in the composite material. Hoy and Grest [280] performed primitive path analysis [276] of polymer brushes embedded in long-chain melts. All simulations were for a coarse-grained model [149] in which monomers were represented by beads (of size ) connected by springs. The systems studied consisted of long grafted chains of length N = 501 beads, whereas the entangle- ment length in a melt is approximately N = 70 [281]. The polymeric matrix studied consisted of melt chains of length P = 1000 beads. As expected, the brush-brush entanglement bb density, (z), increases rapidly with the grafting density for overlapping brushes. The brush-melt entanglement density, bm (z), increases also with the grafting density, but even at low grafting densities there is considerable brush-melt entan- glement. Moreover, there is clear crossover from dominance of brush-melt entanglements to brush-brush entanglements as coverage increases. Figure 30 depicts brush-brush, brush- melt and melt-melt entanglement densities for three differ - −2 ent grafted densities 0.008, 0.03 and 0.07 (in units of bm ). The peak of (z) is always at z ≃ 15, but the width of the first peak increases dramatically with increasing grafting density. At low z, the crossover between a preponderance of brush-melt entanglements and preponderance of brush- brush −2 entanglements clearly occurs at 0.03 grafting density. At this coverage, the peaks of the brush-brush and brush- melt entanglement densities are of nearly equal height. For Fig. 29 a Probability that a nanoparticle has a given number of con- higher coverages, the peak of the brush-brush entanglement tacts in the initial state (open circles), after compressive deformation density is higher, the reverse of the situation for lower cov- (open diamonds), and after tensile deformation (open squares). The erages. Summarizing, when surrounded by melt, the brushes errors are approximately the size of the symbols. b The total number entangle predominantly with the melt at low coverage and of primitive path contacts per nanoparticle as the system deforms in tension (solid line) or compression (dashed line). c Number of primi- with themselves at high coverage. The peak of the brush-melt tive path contacts plotted against the instantaneous strain for three entanglement density is highest at an intermediate coverage, chosen particles as the nanocomposite system deforms. The particle but the integrated areal brush-melt entanglement density con- that exhibited the largest nonaffine displacements is represented by tinues to increase with coverage for the studied systems. left triangles while the one with the smallest nonaffine displacements is plotted using up triangles. (Reprinted from [279], with the permis- sion of AIP Publishing.) 8.2 Viscosity intriguing, overall physical picture emerges from the heter- 8.2.1 Experimental Findings ogenous nonaffine displacements and the particle-induced nucleation of entanglements (Fig. 29(b)). Figure 29(c) pro- Nanoparticles have been shown to influence mechanical vides the number of entanglements for three particles. The properties, as well as transport properties, such as viscosity. 1 3 630 G. G. Vogiatzis, D. N. Theodorou Fig. 31 Viscosity of polymer nanocomposites as a function of the polymer radius of gyration and nanoparticle diameter. Square symbols correspond to 𝜂 ∕𝜂 < 1, diamonds to 𝜂 ∕𝜂 > 1, circles to p p ∕ ≃ 1 at low nanoparticle loading, and triangles to the case where an initial increase of viscosity with nanoparticle loading is followed by a decrease. a Experimental data for athermal systems are from [208, 284]. Systems above the solid orange line should be miscible. The black “viscosity” line is extrapolated from the simulation find- ings. b Corresponding plot for dissimilar mixtures. Only the viscosity line is shown, Data are from [287–293] (Reprinted figure with per - mission from [196]. Copyright 2001 by the American Physical Soci- ety) magnetite particles were found to produce the same non-Ein- stein viscosity decrease effect. Micron-sized spherical fillers increase the viscosity of a pure polymer melt from to a value of predicted by the Einstein–Batchelor law: = 1 + 2.5 + 6.2 (68) where is the particle volume fraction [201, 219, 286]. However, for nanosized fillers, can be reduced or increased relative to the pure polymer [208, 284, 287–293]. While there have been extensive simulations on nanocom- posites, a few of them have focused on the importance of nanoparticle addition on flow behavior [290, 294]. 8.2.2 Insight Obtained from Simulations Fig. 30 Brush-brush (dashed), brush-melt (solid) and melt-melt bb bm mm (dash-dotted) entanglement densities, (z), (z), and (z) for e e e three different grafting densities of a 0.008, b 0.03 and c 0.08 chains Kalathi et al. [196] employed nonequilibrium Molecu- −2 . (Reprinted with permission from Ref. [280]. Copyright (2007) lar Dynamics simulations in order to find out whether American Chemical Society.) the shear viscosity of a polymer melt can be significantly reduced when filled with small energetically neutral nano- particles. That proved to be the case, apparently indepen- Until recently, the commonly held opinion was that particle dently of the polymer’s chain length. Analogous to solvent addition to liquids, including polymeric liquids, produces an molecules, small nanoparticles seem to act as plasticizers increase in viscosity, as predicted by Einstein a century ago and reduce the viscosity of a polymer melt. Their simula- [282, 283]. However, it was recently found by Mackay and tions allowed them to organize the viscosity data of filled coworkers [208, 284, 285] that the viscosity of polystyrene polymer melts as a function of the dimensions of the matrix melts blended with crosslinked polystyrene particles (and chains and the particles. Figure 31(a) plots simulation data later also with fullerenes and other particles) decreases and for athermal (with respect to the strength of polymer–parti- scales with the change in free volume (due to introduction of cle interactions) polymer nanocomposite melts, which cor- athermal excluded volume regions in the melt) and not with respond to the experiments where the nanoparticles and the the decrease in entanglement. Later, [285] fullerenes and 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 631 predicted by the model of Stephanou et al. for an unentan- gled PEO melt with molecular weight M = 1000 g/mol filled with silica nanoparticles of diameter D = 43 nm [296]. Due to the large nanoparticle volume fractions covered in the measurements (up to 50%), neither the Einstein equation nor the Einstein- Berthelot-Green one are applicable [296]. A better choice is the empirical equation proposed by Krieger and Dougherty [297] for dense Newtonian suspensions. It can also be observed that for ≥ 0.27 the data in Fig. 32 exhibit a plateau in the limit of infinitely high shear rates. At those shear rates, flow is so fast that thermal motion cannot destroy the imposed structure (fully aligned molecules) on polymer chains. 9 Mechanical Properties Fig. 32 Comparison of the model predictions of Stephanou et al. [295] for the relative viscosity of a polymer nanocomposite as a 9.1 Moduli of PNCs function of nanoparticle volume fraction and imposed shear with the experimental measurements of Anderson and Zukoski [296]. (Reprinted with permission from Ref. [295]. Copyright (2014) Amer- Three different models [298] have been proposed by Ein- ican Chemical Society.) stein [282, 283], Eilers [299], and Guth [300] for estimat- ing the enhancement of the shear modulus of composites melts have the same chemical structure [208, 284], while incorporating spherical particles: chemically dissimilar mixtures are considered in Fig. 31(b) ⎧ 1 + 2.5 Einstein [287–293]. In Fig. 31(a), Kalathi et al. have also included 1 + 2.5 + 14.1 Guth the miscibility line from ref. [150] suggesting that the (69) ⎨ � � 1.25 experiments correspond to miscible nanoparticle-polymer 1 + 1.25 + Eilers ⎩ 1−1.35 mixtures. The “viscosity” line drawn from the simulations with being the volume fraction of particles dispersed separates regions where the nanocomposite’s viscosity is and G and G the shear moduli of the pure polymer and smaller from those where viscosity is larger than that of the p the composite, respectively. Einstein derived his model for pure melt. For short chains, the viscosity crossover occurs small volume fractions of particles, where the enhance- when the nanoparticle size is comparable to R . In contrast, ment in the shear modulus (or viscosity increase) can be the limited data for large R suggest that the line is nearly estimated by a linear superposition of the shear distortions vertical. arising from individual particles; though this relation- Stephanou et al. [295] introduced a continuum model for ship was originally derived for shear viscosity of particle polymer melts filled with nanoparticles capable of describ- suspensions, it is also applicable to a host of other prop- ing in a unified way their microstructure, phase behavior, erties, including the shear modulus of composites. Later, and rheology in both the linear and nonlinear regimes. That Guth extended this model to higher by accounting for model was based on the Hamiltonian formulation of transport additional shear distortion arising from the interactions phenomena for fluids with a complex microstructure with between the distortions arising from neighboring particles. the final dynamical equations derived by means of a general- Eilers made empirical corrections to Einstein model to ized (Poisson plus dissipative) bracket. The model describes account for the dramatic rise in the viscosity of suspensions the polymer nanocomposite melt at a mesoscopic level by observed when the volume fraction approaches the close- using three fields (state variables): a vectorial (the momen- packing sphere density limit. tum density) and two tensorial ones (the conformation tensor Surve et al. [312] employed a combination of polymer for polymer chains and the orientation tensor for nanoparti- mean field theory and Monte Carlo simulations to study the cles). A key ingredient of the model is the expression for the polymer-bridged gelation, clustering behavior, and elastic Helmholtz energy, A, of the polymer nanocomposite. Beyond moduli of polymer-nanoparticle mixtures. Polymer self- equilibrium, A contains additional terms that account for the consistent field theory was first numerically implemented coupling between microstructure and flow. In the absence in order to quantify both the polymer induced interaction of chain elasticity, the proposed evolution equations capture potentials and the conformational statistics of polymer known models for the hydrodynamics of a Newtonian sus- chains between two spherical particles. Subsequently, the pension of particles. Figure 32 presents the relative viscosity 1 3 632 G. G. Vogiatzis, D. N. Theodorou Fig. 33 a Master curve for the scaled elastic modulus obtained from tems. The sources of experimental data are listed in refs [301–311]. simulations as a function of the particle volume fraction − . b (Reprinted from [312], with the permission of AIP Publishing.) Scaled elastic modulus for experimental polymer-particle sys- formation and structure of polymer-bridged nanoparticle at infinite frequency, G , through the Zwanzig-Mountain gels were examined using Monte Carlo simulations. These equation for isotropic molecular fluids [316]: authors used the number distribution of bridges, obtained � � from their simulations, to quantify the elastic properties of ⎛ 2 ∞ ⎡ U (r)∕ k T ⎤ ⎞ k T 3 d ⋅ B c c � 4 ⎜ ⎢ ⎥ ⎟ the polymer nanocomposites in the postgel regime. Simi- G = + g(r) r dr ⎜ ⎢ ⎥ ⎟ R 4 40 dr dr lar to classical network theories, they assumed that the only ⎝ ⎣ ⎦ ⎠ (70) contribution to the elastic response of the system comes where the first term within the parentheses represents an from the backbone of the percolated network and that the ideal contribution to the modulus due to the presence of the “sol” fraction and the dangling ends of the network do not particles and the second term accounts for the contribution impact elasticity to the system. They defined the backbone of the percolated network as the percolated cluster, exclud- ing the dangling tails and dangling loops, that can be iden- tified as the largest biconnected component of a percolated cluster. Since, for the case of bridging induced percola- tion, the interparticle bridges served as the stress bearing bonds between the particles, the enhancement in the elastic modulus was assumed to be proportional to the number of such bridges, at a given volume fraction of particles. Fig- ure 33(a) displays the elastic moduli scaled by a constant factor as a function of particle volume fraction, expressed as − , with and being the volume fraction and c c the percolation volume fraction of the particles, respec- tively. As observed from the figure, the elastic moduli fol- low a universal power law scaling, G ∝ − , with ≃ 1.79. If energetic contributions to elasticity are taken into account, higher elastic exponents appear from ∼2.1 Fig. 34 Correspondence between the measured and the predicted to 3.8 depending on the relative influence of stretching � −1 moduli, G , of radius R ≃ 100 nm silica particles in 2 kg mol entropy and bending energy [313, 314] (Fig. 33(b)). −1 PDMS (filled circles), R ≃ 100 nm particles in 13 kg mol PDMS McEwan et al. [315] predicted the storage modulus by −1 (filled light gray circles) and R ≃ 600 nm particles in 8 kg mol employing a Zwanzig–Mountain relation and Monte Carlo PDMS (open dark gray circles). Modulus predictions are also shown: MC simulations (matching colored “x” symbols), the analytical simulations. In parallel, these authors measured the modu- Zwanzig-Mountain relation (matching colored solid lines), and Hall lus from rheology experiments on samples well character- equation for close packed hard spheres (dark gray dashed line). ized with ultra-small angle X-ray scattering. These authors (Reprinted from [315]—Published by The Royal Society of Chemis- connected particle microstructure to the storage modulus try.) 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 633 upon the addition of particles, following an almost univer- sal scaling with the volume fraction of the particles. In all cases, the Zwanzing-Mountain predictions are very close to the experiments and to the predictions of the Hall equation of state for solids [318] (at high volume fractions where the materials behave in a solid-like manner). To provide insights into how polymer-grafted nanopar- ticles (NPs) enhance the viscoelastic properties of poly- mers, Hattemer and Arya [319] computed the frequency- dependent storage and loss moduli of coarse-grained models of polymer nanocomposites by employing Molec- Fig. 35 Ratio of storage (a) and loss modulus (b) of the three bare- and the three grafted-nanoparticle systems to that of pure polymer ular Dynamics simulations. Figures 35(a) and (b) present plotted as a function of the effective nanoparticle volume fraction. � �� � �� the computed G ∕G and G ∕G ratios plotted against 0 0 The moduli ratios obtained from simulations at low and high frequen- for the six polymer nanocomposites containing bare and cies are shown as blue circles and red squares, respectively. Open grafted nanoparticles at low- and high-frequency regimes symbols represent bare nanoparticles, and filled symbols represent grafted nanoparticles. The black solid, dotted, and dashed lines repre- along with the the moduli predicted by using the theo- sent predictions from the models of Einstein [282, 283], Eilers [299], retical models of Einstein, Eilers and Guth. It is evident and Guth [300], respectively. (Color figure online) (Reprinted with � � �� �� that both G ∕G and G ∕G increase with increasing vol- permission from Ref. [319]. Copyright (2015) American Chemical 0 0 ume fraction, consistent with the trend obtained from Society.) the strain distortion models, though the different models differ somewhat from each other. Moreover, the moduli due to their interaction and microstructure. The radial dis- ratio computed from simulations at high frequencies are tribution of nanoparticles, g(r), was obtained from Monte of comparable magnitude to those predicted by the mod- Carlo simulations of the particles obeying a Mewis-Russel els, especially the model proposed by Guth, whereas the [317] potential for polymer-grafted spheres. Their results ratios computed at low frequencies tend to exceed all are presented in Fig. 34. Silica particles of radii R ≃ 100 model predictions. At low frequencies, the computed nm and 600 nm were synthesized, grafted with hydroxyl- � G ∕G ratios are more strongly affected compared to �� �� terminated polydimethylosiloxane (PDMS) chains and G ∕G , which is consistent with the expectation that G is finally dispersed in PDMS matrices at volume fractions, , c more strongly affected by changes in the relaxation times ranging from 0.02 to 0.65. The experimental measurements of the polymer chains (quadratic dependence with the are presented alongside the theoretical results in Fig. 34. Rouse time) as compared to G (linear dependence). It can be clearly seen that the storage modulus increases Fig. 36 a Distribution of local C with respect to the distance r with respect to the distance r from the center of the nanoparticle for from the center of a nanoparticle for three different types of interac- the attractive particle in the melt and glass regime. (Reprinted figure tion considered at temperature T = 0.5. b Distribution of the local C with permission from [320]. Copyright 2005 by the American Physi- cal Society) 1 3 634 G. G. Vogiatzis, D. N. Theodorou composite material. Following Yoshimoto et al. [322], the local mechanical properties of the system were determined by discretizing the simulation box into small cubic ele- ments and measuring the internal stress fluctuations within each cubic subdomain [323]. Figure 36(a) shows the local shear modulus as a function of the distance from the surface of the filler. An increase of the local C is observed for the attractive systems in the vicinity of the particle. This pronounced increase may be indicative of the existence of a glassy layer around the par- ticles, even at temperatures above the glass transition tem- perature (T∕T = 1.16), which was hypothesized by Berriot et al. [324]. The results of the neutral and repulsive system are more intriguing. The nanoparticle is surrounded by a region of negative modulus which is followed by a second region of higher than the bulk modulus. Figure 36(b) shows the local shear modulus as a function of temperature for the attractive particle. It can be seen that, as the temper- ature decreases, the shear modulus of the solid-like layer around the particle increases. The thickness of that glassy layer, which is comparable to the radius of gyration of the polymer, also increases. In all cases, far from the particle, the shear modulus decays to the value corresponding to the pure polymer at the given temperature, as expected. Sum- marizing, it seems that, even above the glass transition tem- perature, nanoparticles induce the formation of a solid-like layer, whose existence has been invoked to explain experi- mentally observed increases of the storage modulus in nanocomposites. 9.3 Deformation Simulations Fig. 37 Stress-strain curves for both the pure polymer and the nano- composite at two true strain rates in both tension and compression. Riggleman et al. [325] have examined the response of a The strain rates are indicated in each figure. Error bars are indica- tive. (Reprinted with permission from Ref. [325]. Copyright (2009) polymer and a polymer nanocomposite glass to creep and American Chemical Society.) constant strain rate deformations using Monte Carlo and Molecular Dynamics simulations. These authors found that nanoparticles stiffened the polymer glass, as evidenced 9.2 Local Moduli by an increase in the initial slope of the stress-strain curve and a suppression of the creep response. Figure 37 shows It is now generally accepted that a nanoparticle will per- the stress-strain curves obtained by Riggleman et al. [325] turb the conformation of the polymer around it. However, for both the neat and the nanocomposite polymer for both the question still remains open whether such conforma- tension and compression at two different strain rates. All tional changes are directly responsible for the mechanical curves exhibit similar features: it can be discerned an ini- behavior of the polymer, i.e. whether there are any rein- tial elastic response followed by yield and strain softening forcement or weakening effects of a polymer by a nanom- when the strain ≃ 0.05. For strains beyond ≃ 0.10 the eter-sized particles, whether such effects are localized, and stress rises again as strain hardening begins. The Young’s if so, what is the extent and the magnitude of that localiza- modulus, E, was obtained by fitting the linear part of the tion. Papakonstantopoulos et al. [320, 321] have developed elastic response ( ≤ 0.02) = E. Both under tension and a formalism and applied it to calculate the local mechani- compression, the nanocomposite system was found to be cal properties of a nanocomposite system in detail. Their stiffer. Moreover, these authors reported that constant strain coarse-grained, bead-spring Monte Carlo simulations rate and constant stress deformations had different effect revealed that a glassy layer is formed in the vicinity of the on the material’s position on its energy landscape, in a attractive filler, contributing to the increased stiffness of the 1 3 Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites 635 Fig. 38 Different contributions to the stress tensor plotted against the grafted chains for the systems with R = 3 and grafting densities g(). a Stress contributions from the non-bonded interaction between 0.05 (black), 0.1 (red), 0.2 (green), and 0.4 (blue). All the stress val- particles and the grafted chains for the systems with grafting density ues are normalized by the total number of grafted chains, M. (Color 0.2, and R = 1.5 (black), 3.0 (red), and 4.5 (green). b Stress contri- figure online) (Reprinted from [327] with permission from Elsevier.) butions from the non-bonded interactions between the monomers of deformation. They found that the elastic constants and yield properties were enhanced nearly uniformly for all nanocomposite systems studied, while the strain hardening modulus depended weakly on the grafted density and the nanoparticle size. Figure 38 shows the mechanical response of the systems studied under compressive deformation at a constant rate, where the measured stress is plotted against the ideal rubber elasticity factor, g() = 1∕ − , with being the macroscopic stretch imposed on the speci- mens. Early in the deformation, the polymer nanocompos- ites exhibit an elastic response (g(𝜆 ) < 0.15) followed by yielding and strain softening. Finally, at larger stretches (g(𝜆 ) > 0.3), the polymer glasses enter the strain harden- ing regime, and the stress resumes an increasing trend as strain continues to grow. These authors decomposed the stress calculated in their simulations into its components. The normalized contribution from the non-bonded interac- Fig. 39 Stress–strain relations of cross-linked polymer nanocompos- ite networks with dispersed nanoparticles. Values of stress depend on tions between the nanoparticles and the grafted chains is the strength of polymer–nanoparticle interactions. In the high strain presented in Fig. 38(a). It can be seen that the interaction region, a rapid increase of the stress can be seen due to extended between the nanoparticle and the grafted chains increases subchains of the crosslinked network. (Reprinted with permission with the particle size, and its effect is only observed in the from Ref. [328]. Copyright (2016) American Chemical Society.) strain hardening region. This finding is coherent with the expected depletion of the matrix from particle surfaces way that neither the stress nor the strain rate were uniquely with increasing particle size. Similarly, the stress contribu- indicative of the relaxation times in the material [326]. tions from the non-bonded interactions among the grafted Chao and Riggleman [327] studied the effect of nano- chains is presented in Fig. 38(b). For low grafting densities particle curvature and grafting density on the mechanical (0.05 and 0.1) the non-bonded interaction between beads properties of polymer nanocomposites. In their study, they belonging to grafted chains does not contribute signifi- developed a coarse-grained model of a polymer glass con- cantly to the stress increase during strain hardening. How- taining grafted nanoparticles and examined the resulting ever, for higher grafting densities (0.2 and 0.4), the non- effects on the elastic constants, strain hardening modulus, bonded interactions between the grafted chains contribute as well as the mobility of the polymer segments during significantly to strain hardening. In contrast to the obvious 1 3 636 G. G. Vogiatzis, D. N. Theodorou Acknowledgements Both authors have received funding by the dependence of the non-bonded component of the stress ten- European Union through the projects NanoModel and COMPNANO- sor to the grafting density and particle radius, the bonded COMP under the grant numbers 211778 and 295355, respectively. component of the stress tensor did not exhibit significant G.G.V. thanks the Alexander S. Onassis Public Benefit Foundation changes in behavior across the various nanocomposite sys- for his doctoral scholarship (under contract number GZH 005/2011- 2012) and dr. sc. nat. Marcus Hütter (Technische Universiteit Eind- tems investigated by Chao and Riggleman [327]. hoven), dr. ir. Lambèrt C.A. van Breemen (Technische Universiteit Hagita et al. [328] performed coarse-grained Molecu- Eindhoven) and prof. dr. ir. Patrick D. Anderson (Technische Univer- lar Dynamics simulations of nanocomposite rubbers with siteit Eindhoven) for their patience and understanding during the final spherical nanoparticles on the basis of the Kremer-Grest stages of the manuscript preparation. [149] model. Figure 39 shows the stress–strain relations Funding Both authors have received funding by the European of a small mesh cross-linked polymer network for three Union through the projects NanoModel and COMPNANOCOMP nanoparticle–polymer interactions (repulsive, slightly under the Grant Numbers 211778 and 295355, respectively. attractive and attractive). There are clear differences between the repulsive and the attractive cases. However, Compliance with Ethical Standards both cases with attractive nanoparticle–polymer interac- tions seem to behave similarly, probably due to the few Conflict of interest The authors declare that they have no conflict of interest. contacts existing between the nanoparticles. Thus, the effect of the exact interaction strength on the stress–strain Open Access This article is distributed under the terms of the relations is minor. When the nanoparticles are trapped Creative Commons Attribution 4.0 International License (http:// and fixed in a cross-linked polymer network, the num- creativecommons.org/licenses/by/4.0/), which permits unrestricted ber of contacts between them is expected to increase for use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a a larger elongation ratio due to the compression in the link to the Creative Commons license, and indicate if changes were directions perpendicular to the elongation axis. These made. authors [328] have also calculated the two-dimensional scattering patterns of nanoparticles during the elonga- tion of the network. For strain levels >50% they observed References a spot pattern in the structure factor and a two-point bar pattern in the scattering intensity. 1. Balazs AC, Emrick T, Russell TP (2006) Nanoparticle pol- ymer composites: where two small worlds meet. 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