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Progress of Theoretical and Experimental Physics
, Volume 2018 (3) – Mar 1, 2018

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27 pages

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- Oxford University Press
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- © The Author 2018. Published by Oxford University Press on behalf of the Physical Society of Japan.
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- 2050-3911
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- 2050-3911
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- 10.1093/ptep/pty014
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Prog. Theor. Exp. Phys. 2018, 033J01 (27 pages) DOI: 10.1093/ptep/pty014 On adiabatic pair potentials of highly charged colloid particles 1,2,∗ Ikuo S. Sogami Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan E-mail: sogami@cc.kyoto-su.ac.jp Received April 25, 2017; Revised January 27, 2018; Accepted January 29, 2018; Published March 5, 2018 ................................................................................................................... Generalizing the Debye–Hückel formalism, we develop a new mean ﬁeld theory for adiabatic pair potentials of highly charged particles in colloid dispersions. The unoccupied volume and the osmotic pressure are the key concepts to describe the chemical and thermodynamical equilibrium of the gas of small ions in the outside region of all of the colloid particles. To deﬁne the proper thermodynamic quantities, it is postulated to take an ensemble averaging with respect to the particle conﬁgurations in the integrals for their densities consisting of the electric potential satisfying a set of equations that are derived by linearizing the Poisson–Boltzmann equation. With the Fourier integral representation of the electric potential, we calculate ﬁrst the internal electric energy of the system from which the Helmholtz free energy is obtained through the Legendre transformation. Then, the Gibbs free energy is calculated using both ways of the Legendre transformation with respect to the unoccupied volume and the summation of chemical potentials. The thermodynamic functions provide three types of pair potentials, all of which are inversely proportional to the fraction of the unoccupied volume. At the limit when the fraction factor reduces to unity, the Helmholtz pair potential turns exactly into the well known Derjaguin–Landau–Verwey–Overbeek repulsive potential. The Gibbs pair potential possessing a medium-range strong repulsive part and a long-range weak attractive tail can explain the Schulze–Hardy rule for coagulation in combination with the van der Waals–London potential and describes a rich variety of phenomena of phase transitions observed in the dilute dispersions of highly charged particles. ................................................................................................................... Subject Index A70, I08, J32, J36, J40 1. Introduction In the dilute dispersions of colloid particles with large surface charges, we can observe a rich variety of phenomena, such as salt-induced melting of colloid crystals [1], coexisting ordered and disordered states [2–4], phase transitions from gas to liquid and liquid to solid [5,6], crystallization through multi- stage phase transitions [7], re-entrant order–disorder transition [8–10], and stable voids coexisting with ordered or disordered states [11]. To describe such phenomena, it is required to develop a theory for the adiabatic pair potentials of particles that depend on the variables and parameters characterizing the dispersions. Recognizing that the strong electrolytes form effectively homogeneous systems, Debye and Hückel (DH) [12–15] calculated three thermodynamic functions, i.e., the internal electric energy and the Helmholtz and Gibbs free energies, by using the average electric potential satisfying the linearized Poisson–Boltzmann (PB) equation. The success of the DH theory of strong electrolytes [12–17] comes from the approximate equality between anions and cations in the system. In contrast, such features are hardly discernible in colloid dispersions. The masses, radii, and valences of the particles © The Author 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami are incomparably larger than those of small ions, and the timescale of motion of the particles is very large compared with that of small ions. Therefore, it is relevant to assume that the small ions are in a state of thermodynamic equilibrium with the particles taking a temporarily stationary conﬁguration in the dispersion. Accepting such an adiabatic hypothesis, Levine and Dube (LD) [18–21] calculated the internal electric energy of the system and obtained for the ﬁrst time the adiabatic pair potential possessing medium-range repulsion and long-range attraction of intermediate strength. Then, Derjaguin and Landau [22] and, independently, Verwey and Overbeek [23] calculated the Helmholtz free energy of the dispersion and derived the purely repulsive adiabatic pair potential. Combining the pair potential with the van der Waals–London attractive potential, they succeeded in proving the Schulze–Hardy rule, claiming that the coagulation takes place in proportion to the sixth power of the valence of added salt. The Derjaguin–Landau–Verwey–Overbeek (DLVO) scheme, which can describe the stability and instability of the concentrated dispersions, has long been accepted as the standard theory of colloid science. In monodisperse colloid dispersions, we can observe iridescence, which indicates the formation of colloid crystals. The DLVO theory predicted (see pp. 182–5 of Ref. [23]) that the increase of the salt concentration deepens the so-called secondary minimum of the potential and works to stabilize the colloid crystal. To conﬁrm this prediction, Hachisu et al. [1] made careful observations of iridescence in dispersions of polystyrene latexes with different particle fractions and salt concentrations. They found that, in contradiction to the DLVO prediction, an increase in the salt concentration causes the colloid crystals to melt into disordered states. Their discovery of the salt-induced melting of colloid crystals has shown that the DLVO theory has faults in the description of long-range electric phenomena. By taking the summation of chemical potentials, the Gibbs free energy can be derived from the Helmholtz free energy. Using this recipe, the author formulated a mean ﬁeld theory based on the linearized inhomogeneous PB equation [24–28] and derived the Gibbs free energy with the pair potential possessing medium-range strong repulsion and long-range weak attraction, which enables us to describe long-range phenomena and also prove the Schulze–Hardy rule by combining the van der Waals–London potential [27]. The inhomogeneous PB equation was chosen to bring in the form factors for the surface charges of particles so that thermodynamic functions can be calculated without divergence difﬁculties. In the present article, we develop another mean ﬁeld theory for the interaction of colloid particles by following the traditional line of the DH, LD, and DLVO formalisms based on the homogeneous PB equation. The key ingredient of the theory is an ensemble averaging (EA) with respect to the conﬁgurations of particles, which is applied to the coordinate integrations deﬁning the thermody- namic quantities of the dispersion. By choosing such an EA that works to average over all probable particle conﬁgurations with equal weight and adopting the unoccupied volume as the thermodynamic variable of the system, we carry out the linear approximation of the PB equation and ﬁnd a set of a renewed linearized PB equation and an additional constraint equation. These equations include effectively the excluded volume effects. The constraint equation, which has been overlooked in all past theories, is the necessary condition for the consistency of the electric potential of the whole dispersion. With a Fourier integral representation of the electric potential, the internal electric energy and the Helmholtz free energy are computed and then the Gibbs free energy is calculated using both ways of the Legendre transformation with respect to the unoccupied volume and the summation of chemical potentials. The three types of pair potentials, including the excluded volume effects, are 2/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami derived in a uniﬁed way on the new footing formulated by the EA scheme. In particular, we obtain a repulsive potential of the Helmholtz type, being identical in form to the DLVO potential, and a new pair potential of the Gibbs type with a medium-range strong repulsive part and a long-range weak attractive tail similar to the one derived in the previous theory [24,25]. In Sect. 2, we introduce the concept of ensemble averaging and obtain a renewed form of the linearized PB equation with a constraint condition. A Fourier integral representation of the electric potential for the whole dispersion is derived in Sect. 3. From the internal electric energy and the Helmholtz free energy, the respective pair potentials are obtained in Sect. 4. Examining the two routes relating the two free energies of the system, a new adiabatic pair potential of the Gibbs type is derived in Sect. 5. The meanings of thermodynamic variables and functions of colloid dispersions are discussed in Sect. 6. The theory based on the inhomogeneous PB equation is outlined in Appendix A and the Schulze–Hardy rule is proved in Appendix B. 2. The electric potential of the whole dispersion The positive and negative ions in strong electrolytes have a tendency to attract the opposite charges around them. Debye and Hückel visualized such correlations among ions as the dressing of clouds of opposite charges and called it the ionic atmosphere. Considering that the formation of the ionic atmosphere acts practically to decrease the strength of the electric ﬁeld in the solutions, they applied the linear approximation to the PB equation for the electric potential and interpreted the strong electrolyte as a homogeneous thermodynamic system consisting of a gas of effective ions with ionic atmospheres. Just like the formation of ionic atmospheres among small ions, the surface charges of the colloid particle work to attract small ions with opposite charge and form a unique electric structure, which has been called the electric double layer (see pp. 4–5 in Ref. [23]) in colloid science. Its inner layer consists of the intrinsic surface charges of the particle and small ions with opposite charges attracted strongly by the intrinsic charges so as to condense on the surface. This is a process known by the technical terms counterion condensation [29–31] (see also Ref. [32] and references therein), counterion association [33], and counterion ﬁxation [34]. The outer layer of the ionic double layer is formed by an excess of counterions and has a diffuse dressing structure with a thickness proportional to the inverse of Debye’s screening parameter κ, deﬁned by 4πe 2 2 κ = z N , (1) k TV where N /V is the concentration of the jth species of small ions with valence z , and and T are, j j respectively, the dielectric constant and the temperature of the dispersion. Henceforth, we postulate that the particles in the colloid dispersion are effective particles with effective surface charges Z e and effective radius a affected by counterion condensation. The electric double layer of the particle has a ﬂexible dynamical structure determined by the electric potential (r) of the dispersion. 2.1. The PB equation and ensemble averaging for thermodynamic quantities To formulate the adiabatic hypothesis, we consider a colloid dispersion with temperature T and volume V . Suppose that the distributions of the small ions are described by the density functions n (r) and the particles take tentatively the conﬁguration {R}={R , R , ... , R }, where R is the center- 1 2 N n of-mass coordinate of the nth particle. The electric potential of the dispersion (r) is determined by 3/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami solving the Poisson equation ∇ (r) =−4π z en (r) (2) j j under the Neumann boundary condition 4πZ e ν ·∇(r) =− σ (r) (3) n n for the nth particle with the surface charge Z e, where ν is the normal unit vector to its surface S n n n and σ (r) is its surface charge density obeying σ (r)dS = 1. (4) n n Following the convention in colloid science, we postulate that the particles possess continuous charge distributions after a coarse graining [35], which works to smooth nano-level details of their surfaces and that the small ions take the Boltzmann distribution as z e n (r) = n exp − (r) . (5) j j0 k T In this setting, the information on the particle conﬁguration {R} and their spatial extent is presumed to be implemented in the potential (r). In the dispersion with a deﬁnite particle conﬁguration {R}, the small ions are distributed in the unoccupied region V ({R}) that is outside of all particles, i.e., V ({R}) ={r ∈ V : r ∈ / v (R ) ∀n}, (6) n n where v (R ) is the region occupied by the nth particle. n n Let us consider a quantity created by integrating its density depending on the potential (r) over the unoccupied region V ({R}). It takes over marks of the particle conﬁguration through the potential and also via the integral domain. The two types of these marks are essentially different. While the former comes inherently from the adiabatic hypothesis formulated in the boundary value problem, the latter arises additionally by restricting the integral to the region V ({R}). Is it possible to qualify this integral as a proper thermodynamic quantity? To answer this question, we have to notice the points that the probability of the particles existing in a speciﬁc conﬁguration is extremely low in the actual system and that there are overwhelming numbers of particle conﬁgurations that result in the same thermodynamic state of the dispersion. Hence, it is not relevant to qualify the integral calculated over the region V ({R}) with the speciﬁcally ﬁxed particle conﬁguration as a proper thermodynamic quantity. One practical measure to deﬁne the thermodynamic quantity for the dispersion is to accept an ensemble averaging (EA) with respect to the sets of integral domains associated with all possible particle conﬁgurations. An appropriate EA can work to average away the unnecessary marks of particle conﬁgurations that are brought about by the process of coordinate integration. For a physical density Q[(r)], let us deﬁne the thermodynamic quantity Q[] by applying a supplementary procedure of the EA to the integral over the unoccupied region V ({R}) as follows: Q[]= Q[(r)]dV . (7) V ({R}) EA 4/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami The EA procedure that concerns exclusively the integral domain does not affect the information stored in the potential (r). Applying the formula (7) to the density function n (r), the number N of small ions of the jth j j species is expressed as N = n (r)dV . (8) j j V ({R}) EA Let us integrate the Poisson equation (2) over the unoccupied region and apply the EA procedure. Then, the Gauss integral theorem, the boundary condition (3), and Eq. (8) lead us to the condition of charge neutrality, namely, z N + Z = 0. (9) j j n j n Equations (2)–(5) and (8) constitute the framework for the Poisson–Boltzmann (PB) equation. The PB equation is invariant under the following weak gauge transformation as (r) → (r) + c, (10) where c is an arbitrary constant. By setting Q[(r)]= 1 in Eq. (7), the volume V of the unoccupied region is determined in the EA scheme as V ≡ dV = V − v ({R }) = V − v , (11) n n n V ({R}) EA n n EA where v is an excluded volume of the nth particle, which possesses the valence Z and the electric n n double layer. So far, a concrete scheme for EA has not yet been chosen. Taking an analogy with the micro- canonical ensemble in statistical mechanics, we postulate here that details of particle conﬁgurations in the integral region are irrelevant to the thermodynamics of the dispersion and that the thermodynamic quantities are constructed by averaging contributions from all patterns of unoccupied regions V ({R}) with equal weight. With this averaging, called ﬂat ensemble averaging (FEA), the thermodynamic quantity Q[] is determined by the integral of its density Q[(r)] over the whole dispersion with the volume V as follows: Q[]= Q[(r)] dV = f Q[(r)] dV (12) V ({R}) V FEA ¯ ¯ with a proportionality factor f . Putting Q[(r)]= 1 in this equation, f is ﬁxed by V 1 f = = 1 − φ = 1 − v . (13) V V Accordingly, we can interpret f and φ = v /V , respectively, as the fraction of the unoccupied volume and the volume fraction of the particles of the system. These factors play crucial roles in the present theory. 5/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami In the previous theory [24,25], the thermodynamic functions were deﬁned naively by the integrals over the whole volume of the dispersion (f = 1). Roij et al. [36] criticized it by insisting that the particles were treated as spheres permeable to the small ions, and Knott and Ford [37] made laborious calculations to estimate the excluded volume effects. Our idea of the FEA is found in efforts to reply to their criticisms. 2.2. Linearization of the PB equation The PB equation is known to be soluble exactly only in the 1D problem such as charged plates immersed in an electrolyte [23,38,39]. To obtain analytical solutions in 3D problems, the Boltzmann distribution in Eq. (5) must be linearized. Retaining terms up to the ﬁrst order with respect to the potential (r) in the expansion of the Boltzmann distribution, we ﬁnd the linearized inhomogeneous equation 1 1 2 2 (∇ −¯ κ )(r) =−4πe z N − κ ¯ (r)dV , (14) j j where the quantity κ ¯ is the new screening parameter deﬁned by 4πe 2 2 −1 2 κ ¯ = z N = f κ . (15) k T V This linearized equation is still invariant under the weak gauge transformation (10). By applying the FEA procedure to the integral of Eq. (14) over the unoccupied region, the neutrality condition (9)is conﬁrmed to hold. The weak gauge symmetry of Eq. (14) can be ﬁxed in a natural way. Expressing the right-hand side of Eq. (14) by a constant R, we can rewrite it into two relations. Then the replacement (r) → (r) − R/(κ ¯ ) in both relations leads us to the linearized Poisson–Boltzmann (LPB) equation with the square of the new screening parameter κ ¯ as 2 2 (∇ −¯ κ )(r) = 0 (16) and additionally the following nonlinear integral equation as 4πe z N + κ ¯ f (r)dV = 0. (17) j j The local structure of the electric potential around the particle is determined by solving the LPB equation (16) with the boundary condition (3). The additional equation (17) represents the global characteristics and the consistency of the potential in the whole dispersion. This nonlinear equation works to restrict the magnitude of the electric potential (r). In the literature, the linearized Poisson–Boltzmann equation refers exclusively to the well known form of the differential equation with the square of Debye’s original screening parameter κ in Eq. (1). The constraint equation (17), which has been found for the ﬁrst time in the present theory, is an inevitable consequence of the choice of the FEA scheme and the linearization approximation. 6/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami 2.3. Local electric potential and electric double layer of an effective particle The structure of the electric double layer of the effective particle in the colloid dispersion is deter- mined by the local electric potential derived as a solution of the LPB equation (16) under the boundary condition (3). Let us focus our attention on a single particle with the effective surface charge Z e and radius a n n being stationary at a position R and derive the local electric potential ψ (r) by solving the LPB n n equation 2 2 (∇ −¯ κ )ψ (r) = 0 (18) under the Neumann boundary condition ∂ψ Z e n n =−4π σ (19) ∂r |r−R |=a n n with σ = 1/(4πa ). It is readily conﬁrmed that the solution takes the following form: κ ¯ a Z e e 1 −¯ κ|r−R | ψ (r) = e . (20) 1 +¯ κa |r − R | n n Note that the linear approximation allows us to solve the LPB equation locally around the particle without receiving any direct inﬂuence from other particles except for the excluded volume effect −1/2 included in the screening parameter κ ¯ = f κ. This potential ψ (r) outside of the particle embodies the structure of its electric double layer. The inner layer possesses the effective surface charge Z e with the correction factor κ ¯ a g (κ) ¯ = (21) 1 +¯ κa due to the extent of the particle, and the outer layer has a diffuse structure with thickness 1/κ ¯ decaying through the exponential function. The continuous distribution of surface charge on the particle works to shield out the effect of its interior electric ﬁeld. It is conventional to assume that the electric ﬁeld inside of the particle has no inﬂuence on the physicochemical property of the dispersion. To show clearly the region where the solution ψ (r) is effective, let us deﬁne the potential (r) = ψ (r)θ (|r − R | − a ), (22) n n n n where the Heaviside function θ(|r − R | − a ) works to restrict the effect of ψ (r) in the outside n n n region of the nth particle. We call (r) the external potential of the nth particle. 2.4. A superposed electric potential for the dispersion To examine the restrictive condition (17), it is necessary to construct the following potential by summing up the external potentials (r) over all particles and multiplying the overall factor −1 f as −1 (r) = f (r), (23) 7/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami which we call the superposed electric potential and interpret as the electric potential over the whole dispersion. The integral of the potential (r) over the whole dispersion brings forth Z e 4πe −1 −¯ κr −1 ¯ ¯ (r)dV = f g (κ) ¯ 4π re dr = f Z (24) n n κ ¯ V a n n at the inﬁnite volume limit (V →∞). This limit is acceptable since the rapidly decreasing function exp(−¯ κr) ensures the convergence of coordinate integrals. It is the lower limit a of the integration that works to cancel the factor g (κ) ¯ . With this result and the neutrality condition (9), the potential (r) is conﬁrmed to satisfy the constraint (17) as follows: ⎛ ⎞ ¯ ⎝ ⎠ 4πe z N + κ ¯ f (r)dV = 4πe z N + Z = 0. (25) j j j j n j j The fraction of the unoccupied region f in the nonlinear integral equation (17) is canceled with the −1 multiplied factor f in the superposed electric potential. In this way, the potential (r) deﬁned as the superposition of all of the component external potentials is qualiﬁed as the electric potential over the whole dispersion and the FEA is conﬁrmed to be compatible with the linear approximation to the PB equation. If the FEA were not adopted, the sum of extra terms depending on the occupied volumes v (R) of other particles (m = n) emerge [37] necessarily in Eq. (24), and consequently the restrictive condition (17) cannot hold. The acceptance of the FEA that works to average away the superﬂuous marks of the particle conﬁgurations has solved approximately the problem [36,37] of the permeable or impermeable particles. 3. Fourier integral representation for the electric potential of the dispersion It is not relevant to use the superposed electric potential (r) as it stands for computations of thermodynamic functions. As shown in Sects. 4.1 and 4.2, there appears the difﬁculty of divergences. To make up such a practical shortcoming, we have to ﬁnd a Fourier integral representation for the superposed electric potential by adopting the mathematical expedient of inﬁnite volume as the thermodynamic limit. 3.1. Fourier integral representation for the external potential The component electric potential of the nth particle ψ (r) in Eq. (20) is readily conﬁrmed to have the integral representation ik·r ψ (r) = ψ (k)e dV (26) n n k (2π) with the Fourier transform ψ (k) given by Z e 1 −ik·R ψ (k) = 4π g (κ) ¯ e . (27) n n 2 2 k +¯ κ 8/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami Taking this result into account, we express the Fourier integral representation of the external potential of the nth particle in the form reg ik·r (r) = (k)e dV , (28) n n k (2π) reg where the Fourier transform (k) is postulated to be given by Z e 1 reg n reg −ik·R (k) = 4π f (k) e . (29) n n k +¯ κ Notice here that, in place of the constant factor g (κ) ¯ in Eq. (27), the rotationally invariant function reg reg of the wave-vector f (k) = f (k)(k · k = k ) is presumed to exist. n n reg We must determine the functional form of f (k) so that it can reproduce the Heaviside function in the external potential in Eq. (22) and remove all divergences otherwise arising in the integrals of thermodynamic functions. To show the effect of regularization to get rid of divergences in the reg wave-vector integrations, the superscript “reg” is attached to the quantities. Hereafter, (k) and reg f (k) are called, respectively, the regularized Fourier transform and the regularization factor. 3.2. Determination of the regularization factor Taking the angular integration in the k space in Eq. (28), we ﬁnd that the external potential (r) has reg 2 2 a contribution from the pole k = iκ ¯ of the part 1/(k +¯ κ ) in the kernel (k). For the external potential with the constant factor g (κ) ¯ to be reproduced, the regularization factor has to obey the reg condition f (k)| = g (κ) ¯ . n k=iκ ¯ reg To make the argument deﬁnite, let us impose here the postulate that the regularization factor f (k) is a holomorphic function. The κ ¯ dependence of g (κ) ¯ in Eq. (21) throws out a hint to focus our attention on the holomorphic function −ia w (30) 1 − ia w w =z reg of the variable z on the double sheet Riemann surface. Taking the condition f (iκ) ¯ = g (κ) ¯ into n n account, we deﬁne the regularization factor to be −ia k −ia w n n e e reg reg f (k) = f (k) = √ ≡ . (31) n n 1 − ia w 2 1 − ia k w =k·k 3 7 reg 1 i π 1 i π 2 2 Note that the function f (k) has simple poles at e and e on the double sheets of the a a n n Riemann surface. 3.3. Reproduction of the external potential reg To conﬁrm that the integral representation (28) with the regularization factor f (k) in Eq. (31) can reproduce the external potential (r) in Eq. (22), it is necessary to recognize the point that the reg poles of f (k) do not contribute to the k integral taken along the real axis on the ﬁrst sheet of the Riemann surface. 9/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami After the angular integration of the representation (28)inthe k space, the contour integral taken along the real axis in the complex k space leads to the residue calculus around the pole k = iκ ¯ as follows: Z e 1 k n reg i|r−R |k (r) = f (k) e dk n n πi|r − R | (k + iκ) ¯ (k − iκ) ¯ n −∞ Z e 1 k n reg i|r−R |k (32) = 2πi f (k) e θ(|r − R |− a ) n n n πi|r − R | k + iκ ¯ k=iκ ¯ Z e 1 −¯ κ|r−R | = g (κ) ¯ e θ(|r − R |− a ), n n n |r − R | where the Heaviside step function appears in the k integral so that the exponential function exp[i(|r − R |−a )k] vanishes at the limit of |k|→∞ in the upper half plane on the ﬁrst sheet of the n n Riemann surface for the contour integral. Consequently, the Fourier integral representation (28) has been proved to reproduce the external potential after the wave-vector integration. If the regularization reg factor f (k) were replaced by the factor g (κ) ¯ in Eq. (32), the Heaviside step function would not n n reg appear in the result. This shows that the factor f (k) correctly carries the role of expressing the spatial extent of the nth particle. It is the effect of the extent of the particle that works to remove the divergence from the thermodynamic functions, as shown below. It is instructive to inquire “In which region does the external potential (r) in Eq. (28) satisfy the 2 2 LPB equation (16)?” Applying the operator (∇ −¯ κ ) to the potential and carrying out the angular integration, we ﬁnd reg 2 2 ir·k 2 2 (∇ −¯ κ ) (r) =− (k +¯ κ ) (k)e dV n n k (2π) (33) 1 Z e k i(|r−R |−a )k n n =− e dk, −1 π a n −∞ k + ia where the exponential function vanishes at the limit |k|→∞ in the unoccupied region of the particle. As a matter of course, Eq. (33) does not hold in the inside region {r : |r − R |≤ a } of n n the particle. Consequently, (r) and (r) satisfy the LPB equation in the unoccupied region of the dispersion. 3.4. The electric potential of the dispersion and rule for the order of integrations −1 Summing up the external potentials in Eq. (28) over all particles and multiplying the factor f ,we obtain the Fourier integral representation of the electric potential over the dispersion as follows: reg ik·r (r) = (k)e dV , (34) (2π) reg where the Fourier transform (k) is given by Z e 1 reg n reg reg −1 −1 −ik·R ˜ ¯ ˜ ¯ (k) = f (k) = 4πf f (k) e . (35) n n k +¯ κ n n The electric potential of the dispersion (r) has two representations in Eqs. (23) and (34). While the superposed potential is adequate to show its direct mathematical and physical meanings, the Fourier integral potential is relevant for the computation of thermodynamic functions in an analytical way. 10/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami Not to lose the strength of the Fourier integral representation in Eq. (34), we have to pay attention to the order of integrations with respect to the coordinate and wave-vector variables. For a consistent calculation of thermodynamic functions of the dispersion, we impose the following rule for the order of integrations as follows: Rule 1. The coordinate integration must always be taken prior to the integration with respect to the wave-vector in the calculation of thermodynamic functions of the dispersion by using the Fourier integral representation in Eq. (34). If the rule were broken, the advantage of the integral representation (34) is lost in calculating thermodynamic functions. Namely, if the wave-vector integration is made ﬁrst, the Fourier inte- gral is pulled back to the superposed combination of the component potentials with the Heaviside function. As a result, we are confronted with complexities and, far worse, divergence difﬁculties in computations of physical quantities. 4. Thermodynamic functions 1: Internal energy and Helmholtz free energy For the sake of brevity, the monodisperse dispersion of the particles possessing effective surface charge Ze and radius a is investigated in Sects. 4 and 5 to calculate three kinds of thermodynamic functions F , i.e., the internal energy F = E, the Helmholtz free energy F, and the Gibbs free energy G. Putting the parts of thermodynamic functions that depend on the effective valence Z of the particle as F = E , F , and G , we express the thermodynamic functions of the system as follows: Z Z Z Z F = F + F , (36) ss Z where F is the contribution coming from the solvent and the small ions. The adiabatic pair potentials ss of particles are derived from the part F . The contribution F , which has no direct inﬂuence on the Z ss analysis of the part F , will be examined in Sect. 6. The electric energy of the system is derived from the quadratic form of the electric ﬁeld. Accord- ingly, in the linear approximation, the Z-dependent parts F = E , F , and G , which are related Z Z Z Z to each other by some thermodynamic relations, possess generically the decompositions as F F F = U (R ) + V , (37) Z mn m=n where U (R ) is the adiabatic pair potential of F -type between the mth and nth particles with the mn center-to-center distance R =|R − R | and V is the single potential of F -type of the particle mn m n in the dispersion. 4.1. Internal electric energy The Z-dependent part of the internal electric energy of the dispersion has two contributions. One is given by the integral of the electric ﬁeld energy density [∇(r)] /8π over the whole dispersion and the other comes from the self-energies of the surface charges of the particles. The self-energy of a spherical particle with surface charge Ze and radius a is readily calculated to 2 2 be V = Z e /2a. Consequently, the Z-dependent part of the internal electric energy is given by Sph 2 2 E = [∇(r)] dV + E = f [∇(r)] dV + E , (38) Z Sph Sph 8π 8π FEA 11/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami where the FEA deﬁned in Eq. (12) has justly been taken at the inﬁnite volume limit and E is the Sph sum of the self-energies of the particles given by 2 2 Z e E = V = . (39) Sph Sph 2a n n Substituting the Fourier integral representation of (r) in Eq. (34) into Eq. (38) and carrying out the r integral ﬁrst in obedience to the rule of integration, we obtain 2 reg reg ¯ ˜ ˜ E = f k (−k) (k)dV + E Z k Sph 8π (2π) (40) 2 2 2 1 Z e k −1 reg 2 i R −R ·k ( m n) = f [f (k)] e dV + E , k Sph 2 2 2 2 4π (k +¯ κ ) m,n reg where f (k) is the regularization factor in Eq. (31) with a = a for the monodisperse system. This provides the decomposition of Eq. (37) for F = E . Then the angular integrations in the wave-vector space result in the electric pair potential 2 2 ∞ 3 1 Z e k E −1 reg 2 iR k mn U (R ) = f [f (k)] e dk (41) mn 2 2 2 πi R (k +¯ κ ) mn −∞ and the electric single potential 2 2 ∞ 4 1 Z e k E −1 reg 2 V = f [f (k)] dk + V . (42) Sph 2 2 2 2π (k +¯ κ ) −∞ Using the Goursat theorem, the integrals in Eqs. (41) and (42) are computed from the contributions of the double pole at k = iκ ¯ . The particles with the effective valence Z and radius a have the adiabatic electric pair potential 2 2 2 Z e 1 +¯ κa + (κ ¯ a) 1 1 E −1 −¯ κR U (R) = f − κ ¯ e (43) 1 +¯ κa R 2 with the center-to-center distance R, and the electric single potential 2 2 2 Z e (κ ¯ a) 3 E −1 V =−f + κ ¯ + V , (44) Sph 2 1 +¯ κa 2 where reg Z = Zf (iκ) ¯ = Zg(κ) ¯ (45) is the valence including the constant factor g(κ) ¯ in Eq. (21) with a = a. reg Notice that if the factor f (k) were replaced back to the constant g(κ) ¯ in the integral in Eq. (42), we would be confronted with a linearly divergent result for the electric single potential V .Itisthe reg regularization factor f (k) depending on the spatial extent of the particle that resolves the difﬁculty of divergence. The electric pair potential U (R) thus obtained possesses medium-range repulsion and long- −1 range attraction of intermediate strength. Excepting the factor f , this potential is approximately equivalent to the original LD pair potential [18–21], which had been denied by the DLVO researchers 12/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami since the entropy effects are not taken into account. As shown in Appendix B, the medium-range repulsive part of this potential is not strong enough to explain the Schulze–Hardy empirical rule for coagulation when combined with the van der Waals–London attraction. 4.2. Helmholtz free energy In the colloid dispersion, the solvent playing the role of the thermal bath works to preserve the whole system in an isothermal state. Therefore, the dispersion can be regarded as a good homogeneous thermodynamic system with respect to the temperature T , and the two thermodynamic functions E and F satisfy the Legendre transformation E = F − T (∂F /∂T ), which provides the relation ∂ F E Z Z =− (46) ∂T T T for their Z-dependent parts E and F . Z Z Let us substitute the expression for E in Eq. (40) into Eq. (46) and carry out the T integration by using the relation T 2 k 1 dT =− . (47) 2 2 2 2 2 2 (k +¯ κ ) T (k +¯ κ )T Then we obtain the integral representation for F in the form 2 2 1 Z e 1 −1 reg 2 i(R −R )·k ¯ m n F = f [f (k)] e dV + E . (48) Z k Sph 2 2 2 4π k +¯ κ m,n The decomposition of Eq. (37) for F = F provides us with the following integral representations for the pair potential and single potential of the Helmholtz type as 2 2 ∞ 1 Z e k F −1 reg 2 iRk U (R) = f [f (k)] e dk (49) 2 2 πi R k +¯ κ −∞ and 2 2 ∞ 2 1 Z e k F −1 reg 2 V = f [f (k)] dk + V , (50) Sph 2 2 2π k +¯ κ −∞ respectively. Notice here again that the single potential V diverges linearly, if the regularization reg factor f (k) is replaced by the constant g(κ) ¯ in the integral (50). The residue calculations in these integrals lead, respectively, to 2 2 Z e 1 F −1 −¯ κR U (R) = f e (51) for the Helmholtz pair potential and 2 2 Z e F −1 V =−f κ ¯ + V (52) Sph for the Helmholtz single potential. In the case where f = 1, the potential U (R) coincides exactly with the well known DLVO potential. 13/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami 5. Thermodynamic functions 2: Gibbs free energy There are two routes that relate the Helmholtz free energy F to the Gibbs free energy G: one is to follow the Legendre transformation (Route L) and the other is to take the sum of the chemical potentials of all components of the system (Route C ). There holds a theorem that states that “The two routes L and C are equivalent for the homogeneous thermodynamic system”. As emphasized by Hill in his study on the thermodynamics of small systems [40,41], however, the two routes may be inequivalent if the condition of homogeneity is broken. The colloid dispersions, which display a variety of phase transitions [1–11,42,43], are not necessarily homogeneous systems. Therefore, we must calculate the Gibbs free energies by using possible variables along both Route L and Route C without any preconceived notion and compare objectively the results to judge whether or not the proper thermodynamic variables are chosen for the description of the colloid dispersion. 5.1. Route C: Total sum of the chemical potentials For the monodisperse system, the condition of charge neutrality in Eq. (9) takes the form z N + NZ = 0, (53) j j where N is the number of colloid particles with effective surface charge Ze. This condition requires that, if we interpret the N as thermodynamic variables, the valence Z must be treated as a thermo- dynamic variable since the adiabatic hypothesis ﬁxes the number N of the colloid particles to be constant. The chemical potential of a small ion of the jth species is given by the derivative of the Helmholtz free energy F with respect to the variable N . Following this deﬁnition and utilizing the identical relation ∂Z N = Z, (54) ∂N which is deduced from Eq. (53), we can compute the Gibbs free energy by taking the total sum of all chemical potentials in conformity with the condition of neutrality as follows: ∂F ∂F ∂F G = N = N + Z . (55) j j ∂N ∂N ∂Z j j Z j j j In this expression, (∂F /∂N ) is the chemical potential of the small ion of the jth species and j Z (∂F /∂Z ) should be interpreted as the chemical potential of the effective surface valance Z of the particles. To investigate the Z-dependent part of the Gibbs free energy G given by ∂F ∂F Z Z G = N + Z , (56) Z j ∂N ∂Z Z j we have to observe the integral representation for F in Eq. (48). The second term on the right-hand side of Eq. (56) can readily be reduced to 2F . To calculate the ﬁrst term, it is necessary to recognize that F depends on N and T only through the factor κ ¯ , which satisﬁes the relation Z j 2 2 ∂κ ¯ ∂κ ¯ N =−T . (57) ∂N ∂T 14/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami Then, combining these results, we obtain the outcome ∂F G =−T + 2F = F + E . (58) Z Z Z Z ∂T 5.2. Route L: Legendre transformation There exist two candidates, V and V , for the volume of the system. It is necessary to calculate the Z-dependent part G of the Gibbs free energy along Route L for both of them. Let us suppose ﬁrst that the unoccupied volume V is the proper thermodynamic variable and V is a parameter. The part G is related to F by the transformation as Z Z −1 ¯ ¯ ∂F ∂f ∂fF Z Z −1 ¯ ¯ G = F − V = F − V fF − f V . (59) Z Z Z Z ∂V ∂V ∂V The second term on the right-hand side of this equation can be reduced to F . To calculate the third term, it is sufﬁcient to use Eq. (48) showing that fF depends on the variable V only through κ ¯ . Using the fact that κ ¯ is the homogeneous function of the same order with respect to both variables V and T , we ﬁnd readily the relation 2 2 ¯ ¯ ¯ ¯ ∂fF ∂κ ¯ ∂fF ∂κ ¯ ∂fF ∂fF ∂F Z Z Z Z Z V = V = T = T = fT . (60) 2 2 ∂κ ¯ ∂T ∂κ ¯ ∂T ∂T ∂V ∂V Accordingly, the part G can be calculated to be ∂F G = F + F − T = F + E . (61) Z Z Z Z Z ∂T Next, we execute the Legendre transformation with respect to the variable V as −1 ¯ ¯ ∂F ∂f ∂fF Z Z −1 ¯ ¯ G = F − V = F − V fF − f V . (62) Z Z Z Z ∂V ∂V ∂V Since V is the dependent function of V , the second term on the right-hand side of this equation is −1 reduced to (f − 1)F . Similar calculus in Eq. (60) brings about the relation ∂fF ∂F Z Z V = fT . (63) ∂V ∂T Then, combining these results, we obtain the part G in the form ∂F −1 −1 ¯ ¯ G = F + (f − 1)F − T = (f − 1)F + E . (64) Z Z Z Z Z ∂T In this way, the Legendre transformations with respect to the variables V and V provide quite different results in Eqs. (61) and (64). 5.3. Adiabatic pair potential of Gibbs type The goal of Route C in Eq. (58) sorts out Eq. (61) derived by way of Route L with the variable V from Eq. (64) obtained via Route L with the variable V . With this result, we accept the viewpoint that the unoccupied volume V is more suitable than the volume V of the dispersion as the proper thermodynamic variable of the system. 15/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami The coincidence of the outcomes in Eqs. (58) and (61) reveals that the complex colloid dispersion can behave effectively as a kind of homogeneous system. This remarkable result allows us to draw a picture in which the gas of small ions reaches chemical and thermal equilibrium in an arena possessing the unoccupied volume V surrounded by particles with ﬂuctuating surface valence Z. Now, information on the interactions of the colloid particles can be extracted from the result G = F + E . Using the decomposition of Eq. (37) for F = G , we obtain the adiabatic pair Z Z Z Z potential and single potential of the Gibbs type as follows: 2 2 2 Z e 2 + 2κ ¯ a + (κ ¯ a) 1 1 G F E −1 −¯ κR U (R) = U (R) + U (R) = f − κ ¯ e (65) 1 +¯ κa R 2 and 2 2 2 Z e (κ ¯ a) 5 G F E −1 V = V + V =−f + κ ¯ + 2V , (66) Sph 2 1 +¯ κa 2 respectively. The resultant pair potential U (R) possesses a medium-range strong repulsive part and a long-range weak attractive tail. 5.4. Signiﬁcance of the inequality V = V and the FEA scheme In almost all past research on colloid dispersions [18–25], the volume of the dispersion V has been implicitly adopted as the thermodynamic variable of the system. It is instructive to examine the Legendre transformation with respect to V under the condition f = 1. We ﬁnd readily the relation G = E . This is not a reasonable result, since the entropy effect Z Z is not properly taken into account in the internal energy. It is very tempting and not unnatural for us to speculate that the DLVO researchers had actually encountered this unacceptable result G = E , Z Z which had forbidden them to go to the Gibbs free energy and led them to stay at the stage of the Helmholtz free energy F . Combining the pure-repulsive potential and the van der Waals–London attractive potential, the DLVO researchers have succeeded in explaining all short-range phenomena concerning the stability and instability in concentrated colloid dispersions. However, their theory, which fails to describe the stability and instability of the colloid crystals, has faults in the description of long-range electric phenomena. To improve the deﬁcit of the DLVO scheme, the present author has noticed [24] that Route C can lead us to the Gibbs free energy as in Eq. (58) and has formulated the theory that predicts the pair potential with long-range attraction in Eq. (A.14). However, it is crucial to realize that the choice of Route C has been made as an emergency measure to explain the long-range phenomena observed in the dilute dispersions. The signiﬁcance of the inequality V = V had not been recognized when the theory was formulated in terms of the Gibbs free energy [24,25]. We have only reached full recognition of the necessity of the concept of the unoccupied volume V (= V ) at the stage when the FEA scheme is formulated in Eqs. (12) and (13) in the present article. The DLVO researchers have not left any sign or trace showing why they formulated their theory not on the Gibbs free energy but on the Helmholtz free energy. 16/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami 6. Discussion To set about querying which of the free energies, F or G, is suitable to describe a rich variety of isothermal phase behavior of the colloid dispersion, let us examine ﬁrst the specimen preparation of the dispersion consisting of small ions with concentrations N /V and monodisperse particles with the volume fraction φ = v /V . Immediately after the preparation, the specimen can be regarded as being homogeneous because both the particles and the small ions are executing violent thermal motion over the whole dispersion. The physical state of such a system is temporarily described by the Helmholtz free energy F (T , V , N , φ). (67) temp j After a short lapse of time, the equality of the movements of the particles and the small ions breaks down. While the particles quickly slow down their speed on average, the small ions preserve their rapid motion in a thermal equilibrium in the region unoccupied by the particles. Consequently, the homogeneity that existed temporarily at the stage of the specimen preparation has been lost and the function F (T , V , N , φ) is no longer able to describe the physical states of such a system. temp j Historically, however, this function has long been adopted as the Helmholtz free energy of the colloid dispersion without a deep inquiry concerning the quality of the volume V . To characterize the physical stage of the dispersion when the gas of small ions tends to reach thermal equilibrium in an arena outside of all colloid particles, we choose the numbers of small ions N , the valences of particles Z and the unoccupied volume V deﬁned in Eq. (11) as the proper j n thermodynamic variables. Then, the Helmholtz free energy of the system where the gas of small ions coexists with the particles possessing the temporarily stationary conﬁguration {R} can be expressed as follows: F (T , V ; N , Z : {R}), (68) j n where the information on the conﬁguration {R} is assumed to be retained solely in the electric potential (r). At this stage, the concepts of thermodynamics must be used carefully, since the dispersion is no longer homogeneous in the usual sense. Taking the results in Sects. 5.1 and 5.2 into account, we postulate that the quantities N , Z , V , and F can preserve extensive properties provided that their j n variations are limited to be sufﬁciently small. Accordingly, for an inﬁnitesimal , there holds the relation (1 + )F (T , V ; N , Z : {R}) = F (T , (1 + )V ; (1 + )N , (1 + )Z : {R}). (69) j n j n Expanding this with respect to and retaining its ﬁrst order, we ﬁnd the equation ∂F ∂F ∂F F − V = N + Z (70) j n ∂N ∂Z ∂V j n j n Henceforth, the semicolon is used as a delimiter to show that variables N and Z are related by the j n homogeneous relation (9) of charge neutrality. 17/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami whose left- (right-)hand side is Route L (C ) leading to the Gibbs free energy G from the Helmholtz free energy F. The Gibbs free energy for the system composed of a gas of small ions and particles with the conﬁguration {R} is expressed as G(T , P; N , Z : {R}) (71) j n in terms of the pseudo-extensive variables N and Z , and the intensive variables T and P = P +P , j n osm s where P and P are, respectively, the osmotic pressure of the small ion gas and the pressure of osm s the solvent. The chemical and thermodynamic equilibrium is realized through ﬁne tuning of the variables N , Z , and P in the arena with the unoccupied volume V surrounded by the surfaces of j n osm the particles and the inner wall of the vessel. To answer the inquiry of which of the free energies is suitable to describe the ﬁne phase behavior of colloid dispersions, the set V and F should be compared with the set P and G. The variable osm V has a somewhat unﬁnished nature since its value is ﬁxed mainly by the state of the arena and is not particularly sensitive to details of the state of the small ions. In fact, in the V and F scheme, the particles demarcating the arena interact with each other almost independently of the distribution of small ions inside the arena and bring forth the purely repulsive shielded Coulombic potential in Eq. (51). It is indispensable for the relevant free energy, however, to incorporate the inﬂuence of the small ions possessing large degrees of freedom that contribute to the osmotic pressure and interact with the particles forming the wall of the arena. As implied by the neutrality condition, many more counterions than co-ions exist inside the arena. Accordingly, the interactions between the small ions and the particles effectively produce additional attractive components. This characteristic, which can be recognized in the Gibbs pair potential (65), proves that the P and G scheme can osm describe properly the contributions of small ions inside the arena. Therefore, judging from the importance of the degrees of freedom of the system, we have to accept that the Gibbs free energy is the relevant thermodynamic function of the dispersion consisting of a gas of small ions and particles. The Gibbs free energy G consists of the Z-dependent part G and the contribution G from the Z ss solvent and the small ions as G = G + G = G + G + G , (72) ss Z 0 DH Z where we approximate G by the sum of G , which is the part from the solvent and the small ions in ss 0 the limit of vanishing electric charge (e → 0) and G , which is given by the Debye–Hückel theory. DH In terms of the chemical potential of the solvent molecules μ and those of the jth solute molecules μ , the Gibbs free energy G is written [44] (see also p. 37 in Ref. [27]) as follows: G = N μ + N μ + k T N ln , (73) 0 s s j B j eN j j where N = V /v is the number of solvent molecules expressed by the molecular volume v and the s s s last term is the entropy of mixing between the solvent and solute molecules. The contribution G DH The right-hand side is reduced to Eq. (56) for a monodisperse system consisting of particles with the effective valence Z. 18/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami is estimated [12–17]as G =− k TN v κ ¯ . (74) DH B s s 8π Taking the derivative of the Gibbs free energy G with respect to N , the chemical potential of the solvent molecules in the dispersion is calculated to be ∂G 1 1 ∂G μ = = μ − k T N + k Tv κ ¯ + . (75) s B j B s ∂N N 16π ∂N s s s To extract the osmotic pressure P from the total pressure P, we have to investigate the equilib- osm rium state between the colloid dispersion and a solution of pure solvent, which are supposed to be separated by a semi-permeable membrane through which only the solvent molecules can pass [44] (see also p. 37 in Ref. [27]). The condition that the chemical potential of the solution with the pres- sure P is equal to that of the dispersion with the pressure P + P at equilibrium is expressed as s s osm follows: μ (P ) = μ(P + P ) s s s osm 1 1 ∂G (76) = μ (P + P ) − k T N + k Tv κ ¯ + v . s s osm B j B s s N 16π s ∂V Since P P , the Taylor expansion of the term μ (P + P ) results in osm s s s osm μ (P + P ) = μ (P ) + v P , (77) s s osm s s s osm where the relation ∂μ /∂P = v is used. Consequently, we ﬁnd the equation of state s s s ⎛ ⎞ 1 ∂G ⎝ ⎠ P V = k T N − κ ¯ V − V (78) osm B j 16π ∂V for the osmotic pressure P . This is the generalization of the van ’t Hoff law for colloid dispersions. osm 3 4 The κ ¯ V term on the right-hand side of this equation expresses the Debye–Hückel effect of the ionic atmosphere among small ions. The last term is the contribution from the Z-dependent part of the Gibbs free energy. In a colloid dispersion, the leading part of the act of thermodynamic evolution is performed by the gas of small ions in the arena with the unoccupied volume V . Its scenario is written in the Gibbs free energy G and the equation of osmotic state (78). As a beneﬁcial consequence of the adiabatic description, the development of the total system consisting of the gas of small ions and the particles proceeds so as to minimize the Gibbs free energy G in which the role of the dynamical variable is granted to the coordinate R of the particles. Accordingly, it is the adiabatic pair potential U ({R}) included in the part G that should be used to solve the many-body problem of colloid particles. Taking note of the decrease in osmotic pressure, Langmuir [45] proposed an idea of “the Coulombic attraction between the particles” in his framework of colloidal dispersion formulated by applying the DH theory as it stands. Although his interpretation is ingenious and suggestive, he overlooked the importance of the asymmetry between the particles and the small ions and failed to introduce the concept of the adiabatic pair potential between the particles. 19/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami E 2 Fig. 1. Comparison of three adiabatic pair potentials. Potential curves in the reduced forms afU /Z , F 2 G 2 ¯ ¯ afU /Z , and afU /Z are drawn against R/a for the case κ ¯ a = 1. G 2 Fig. 2. The Gibbs pair potential afU /Z versus R/a for values of κ ¯ a from 0.8 to 2.0. E F G In Fig. 1, the adiabatic potentials U (R), U (R), and U (R) are compared for κ ¯ a = 1. The potential U (R) resulting from the internal energy is not qualiﬁed to describe the isothermal processes F G in the dispersion. As shown in Appendix B, both potentials U (R) and U (R) can be used to prove the Schulze–Hardy rule for coagulation. However, the purely repulsive potential U (R), which is identical with the DLVO potential at the limit f = 1, fails to explain the Hachisu phase diagram and is not able to describe other long-range phenomena in dispersions. The potential U (R) with a medium-range strong repulsion and a long-range weak attraction can explain a variety of phase transitions in the dispersions. In the dispersion with constant T and P, the colloid particles interact with each other through the pair potential U (R). Figure 2 shows the behavior of the Gibbs pair potential in the reduced form G 2 G afU /Z for different values of κ ¯ a. The potential U (R) endowed with the advantages of both E F potentials U (R) and U (R) can describe the short-range behavior of the dispersion and the rich variety of long-range phenomena of complex phase transitions observed in the dilute dispersions of highly charged particles [1–11]. To investigate the many-body problem of the particles interacting through the pair potential, we have to proceed with computer simulations [27] by using either the molecular dynamics (MD) or the Monte Carlo (MC) technique. Tata et al. [46–53] made extensive studies of the phase behavior 20/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami of colloid dispersions by MC simulations. Using the Gibbs pair potential in Eq. (A.14), which was derived in the theory based on the linearized inhomogeneous PB equation [24–28], they succeeded in describing well the main features of phase transitions observed in dilute dispersions of highly charged particles of polystyrene latex and silica colloids [1–11]. MD simulations [54–56] were also carried out by using the Gibbs pair potential in Eq. (A.14). The Gibbs potentials in Eqs. (65) and (A.14) have qualitatively similar characteristics for disper- sions with a ﬁxed particle fraction. However, the difference of the screening parameters κ ¯ and κ and the existence of the fraction factor bring about quantitative differences in these potentials. In −1 particular, the factor f in front of the new potential in Eq. (65) claims that the interaction between the particles is conspicuously enhanced in the concentrated dispersions. Namely, the valence of the particle described by the potential in Eq. (65) changes its value effectively depending on the unoc- cupied fraction as f Z. To verify such a feature of the new pair potential, it is necessary to make precise and systematic studies of the properties of the colloid dispersions over a wide range of volume fractions of particles, because the FEA scheme is a novel hypothetical procedure for calculation of the equilibrium averaging in colloid dispersions. For further veriﬁcation of the theory, it is necessary to undertake precise measurements of various properties of colloid dispersions. One promising attempt at such experiments is the careful obser- vation done by Sun et al. [57,58] for the shear moduli of colloid crystals in dispersions of highly charged particles. The Kikuchi–Kossel diffraction [59,60] for colloid crystals [3,61] is the most precise method of obtaining information on its symmetry and lattice constants. We are planning to undertake Kikuchi–Kossel diffraction analyses of the colloid crystals of titania particles in the Kibo module on the ISS satellite [62]. On the hypothesis of ﬂat ensemble averaging and the choice of the unoccupied volume V as the thermodynamic variable, we have formulated the mean ﬁeld theory of colloid dispersions, where the Gibbs free energy can be derived by way of the Legendre transformation and the total sum of the chemical potentials from the Helmholtz free energy. The theory provides the adiabatic pair potential of particles, which can describe the long-range as well as the short-range phenomena in the colloid dispersions of highly charged particles. It seems reasonable to evaluate this scheme as a milestone in trials to improve the fundamental concepts of the thermodynamics for homogeneous systems so as to be applicable to the investigation of inhomogeneous systems consisting of multi-components with different timescales. Colloid dispersions providing us with deﬁnite experimental results are ideal systems that are indispensable for general studies of complex and inhomogeneous macroionic systems. Acknowledgements The author expresses his sincere thanks to Professors N. Ise and K. Itoh for their discussions and encouragement and to Dr K. Umetsu for drawing the ﬁgures of the potential curves. The author also thanks Dr M. Smalley for a careful reading of the manuscript. Appendix A. Adiabatic pair potentials derived from the inhomogeneous PB equation We outline the theory based on the linearized PB equation [24,25,27] with the source terms expressing the distributions of the effective surface charges of particles. Following the LD and DLVO researchers, the volume V of the dispersion is adopted as the thermodynamic variable and the condition f = 1is imposed. 21/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami The electric potential (r) of the dispersion is postulated to satisfy the Poisson equation ∇ (r) =−4π z en (r) − 4π Z eρ (r), (A.1) j j n n where n (r) is the Boltzmann distribution function of the small ions in Eq. (5), which is related to the number N by the relation N = n (r)dV , (A.2) j j and ρ (r) is the charge distribution function for the nth effective particle with the effective valence Z normalized by ρ (r) dV = 1. (A.3) Integration of Eq. (A.1) over the whole dispersion with the normalizations in Eqs. (A.2) and (A.3) provides us with the condition of charge neutrality in Eq. (9). Linearizing the Boltzmann distribution in Eq. (5), we obtain the linearized inhomogeneous PB equation 2 2 (∇ − κ )(r) =−4π Z eρ (r) (A.4) n n and the restrictive condition in Eq. (17) with f = 1. Substitution of the integral representations ik·r (r) = (k)e dV (A.5) (2π) and ik·r ρ (r) = ρ˜ (k)e dV (A.6) n n (2π) into Eq. (A.4) leads readily to the relation Z e 1 (k) = 4π ρ˜ (k) (A.7) 2 2 k + κ between the Fourier transforms (k) and ρ˜ (k). The surface charge distribution of the nth particle at the position R is conveniently represented by Dirac’s δ-function as ρ (r) = δ(|r − R |− a ). (A.8) n n n 4πa Its Fourier transform is expressed by −ik·r −ik·R ρ˜ (k) = ρ (r)e dV = e f (k) (A.9) n n n 22/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami with the form factor sin ka f (k) = . (A.10) ka Substituting this Fourier transform ρ˜ (k) into Eq. (A.7), we ﬁnd the expression Z e 1 −ik·R (k) = 4π f (k) e . (A.11) 2 2 k + κ Note that the electric potential (r) represented by the Fourier integral in Eq. (A.5)isproved to satisfy the restrictive condition (17) with f = 1, which had not been found at the stage for the prescription of linearization of the inhomogeneous PB equation in the previous articles [24,25]. reg ˜ ˜ Comparison of the kernels (k) in Eq. (A.11) and (k) in Eq. (35) shows that the regularization reg factor f (k) is replaced by the form factor f (k) of the surface charge distribution in Eq. (A.10). n n Therefore, the difference in the theories based on the homogeneous and inhomogeneous PB equations reg comes down to that of the factors f (k) and f (k). Consequently, the integral representations for n n the Z-dependent parts of the internal electric energy and the Helmholtz free energy given in Eqs. reg (40) and (48) can be used in the theory of the inhomogeneous PB equation by replacing f (k) with f (k). Their sum results in the integral representation of the Z-dependent part of the Gibbs free energy. In place of the three pair potentials in Eqs. (43), (51), and (65), we obtain the respective pair potentials as follows: ∗2 2 Z e κa coth(κa) 1 E −κR U = − κ e , (A.12) R 2 ∗2 2 Z e 1 F −κR U = e , (A.13) and ∗2 2 Z e 1 + κa coth(κa) 1 G E F −κR U = U + U = − κ e , (A.14) R 2 where sinh(κa) Z = Zf (iκ) = Z (A.15) κa is the effective surface charge including the effect of the form factor. For the sake of simplicity, we use here the same symbols to represent the pair potentials without confusion. Three types of single potentials can also be obtained without divergence in the theory of the inhomogeneous PB equation [24,25]. The three potentials in Eqs. (A.12), (A.13), and (A.14) have the same functional structures possessing different coefﬁcients composed of the parameter κa with the respective poten- tials in Eqs. (43), (51), and (65). The two sets of the pair potentials show qualitatively similar behaviors for the dispersion with the ﬁxed unoccupied fraction. 23/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami Using the Fourier transform in Eq. (A.11) and performing the k integral in Eq. (A.5), we ﬁnd the explicit form for the average electric potential (r) as follows: Z e sinh κa 1 n n −κ|r−R | (r) = e θ(|r − R | − a ). (A.16) n n κa |r − R | n n This should be compared with the electric potential in Eq. (23), which is utilized in the theory for the homogeneous PB equation in the text. To calculate thermodynamic quantities without divergence, however, we have to use not this average potential but the integral representation in Eq. (A.5) with the Fourier transform in Eq. (A.11). Appendix B. Proof of the Schulze–Hardy rule For the description of the long-range phenomena in dilute dispersions, it is suitable to visualize the colloid particles as effective particles with effective surface charges. By contrast, it is adequate to consider that the particles have deﬁnite surface potentials to investigate the short-range phenomena in the concentrated colloid dispersions. In particular, the latter viewpoint has been properly used in the study of the coagulation of colloid particles. We have to derive three types of pair potentials with the electric surface potential from the respective pair potentials with the surface charge. At this stage, it should be recognized that nano-level details of the surface structure of the particle are presumed to be averaged out in the description of colloid science. With such a coarse graining, we are able to eliminate complexities of ﬁne interactions such as hydration effects [35] and to represent the colloid particles as spheres with smooth surfaces even at a close distance. Here let us investigate the short-range phenomena of coagulation of submicron-sized particles under the implicit assumption of coarse graining. Let us express the surface potential of the particle as ψ . Putting |r − R |= a in the potential in a n Eq. (20), we readily ﬁnd Ze 1 Z e −¯ κa ψ = ψ (r)| = = e . (B.1) a n |r−R |=a a 1 +¯ κa a E F G Corresponding to the pair potentials with surface charge U , U , and U , let us represent the pair potentials with the surface potential ψ as U , U , and U by using subscripts. Substitutions of a E F G the expression for ψ in Eq. (B.1) into Eqs. (43), (51), and (65) lead readily to pair potentials with surface potentials as follows: 1 +¯ κa + (κ ¯ a) 1 1 −1 2 2 −¯ κ(R−2a) U (R) = f a ψ − κ ¯ e , (B.2) 1 +¯ κa R 2 −1 2 2 −¯ κ(R−2a) U (R) = f a ψ e , (B.3) and 2 + 2κ ¯ a + (κ ¯ a) 1 1 −1 2 2 −¯ κ(R−2a) U (R) = f a ψ − κ ¯ e , (B.4) 1 +¯ κa R 2 respectively. The formula U (R) is used often and is also known as the DLVO potential. 24/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami To investigate the coagulation of spherical particles, we have to consider the situation where the surface-to-surface distance S = R − 2a is very small. For a small S, the van der Waals attraction can be approximated (see pp. 186–99 in Ref. [23] and pp. 60–1 and 241–2 in Ref. [27]) by 1 a U (S) =− H (B.5) 12 S with the Hamaker constant H [63–65]. Note that, for a small S and a large κ ¯ a, both of the pair potentials U and U take the same asymptotic form as follows: F G −1 2 −¯ κS U , U → U (S) = f aψ e . (B.6) F G R The approaching particles in the dispersion with high salt fraction feel the potential U (S) = U (S) + U (S). (B.7) A R Suppose that the potential U (S) takes a maximum at S = S . It is reasonable to interpret that the coagulation starts when the potential maximum U (S ) vanishes to zero. Therefore, the conditions for coagulation are given by dU (S) | = 0, U (S ) = 0, (B.8) S c dS which are readily solved, leading to 1 1 1 −1 2 −1 f aψ e = H κ ¯ a, S = . (B.9) 2 12 κ ¯ The surface of the spherical particles can be well approximated by plate surfaces when the particles come close to each other. Gouy [66,67] and Chapman [68] solved exactly the PB equa- tion for a charged plate immersed in an electrolyte with valences ±z and found the relation γ = tanh[zeψ /(4k T )], where ψ is the surface electric potential of the plate and γ is a constant. 0 B 0 2 2 Setting ψ = ψ and taking κ ¯ = (8π/k T )z n into account, we ﬁnd the relation a 0 B −1 3 5 4 1152 (k T ) (tanh γ) −6 n = ∝ z (B.10) 2 2 6 π exp(2) f H (ze) for the concentration n and valence z of the salt. Therefore, the Schulze–Hardy rule can be proved by using either the pair potential of the Helmholtz or Gibbs type. For a small S and a large κ ¯ a, the pair potential of internal electric energy U takes the asymptotic form 1 1 −1 2 −¯ κS U → U (S) = f aψ e . (B.11) E R 2 κ ¯ a When the above argument for the coagulation is applied to the combined potential U (S) with this asymptotic form for the pair potential of internal electric energy, we fail to prove the Schulze– Hardy rule. 25/27 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033J01/4922030 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033J01 I. S. Sogami References [1] S. Hachisu, Y. Kobayashi, and A. Kose, J. Colloid Interface Sci. 42, 342 (1973). [2] N. Ise and T. Okubo, Acc. Chem. Res. 13, 303 (1980). [3] T. Yoshiyama, I. Sogami, and N. Ise, Phys. Rev. Lett. 53, 2153 (1984). [4] S. Dosho et al., Langmuir 9, 394 (1993). [5] A. K. Arora, B. V. R. Tata, A. K. Sood, and R. Kesavamoorthy, Phys. Rev. Lett. 60, 2438 (1988). [6] B. V. R. Tata, M. Rajalakshmi, and A. K. Arora, Phys. Rev. Lett. 69, 3778 (1992). [7] I. S. Sogami and T. Yoshiyama, Phase Transitions 21, 171 (1990). [8] J. Yamanaka, H. Yoshida, T. Koga. N. Ise, and T. Hashimoto, Phys. Rev. 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Progress of Theoretical and Experimental Physics – Oxford University Press

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