TY - JOUR
AU - Fujinami,, So
AB - Abstract We review nano-palpation atomic force microscopy, which offers quantitative mechanical property mapping especially for soft materials. The method measures force–deformation curves on the surfaces of soft materials. The emphasis is placed on how both Hertzian and Derjaguin–Muller–Toporov contact mechanics fail to reproduce the experimental curves and, alternatively, how the Johnson–Kendall–Roberts model does. We also describe the force–volume technique for obtaining a two-dimensional map of mechanical properties, such as the elastic modulus and adhesive energy, based on the above-mentioned analysis. Finally, we conclude with several counterpart measurements, which describe the viscoelastic nature of soft materials, and give examples, including vulcanized isoprene rubber and the current status of ISO standardization. atomic force microscopy, soft materials, quantitative mechanical property mapping, contact mechanics Introduction Atomic force microscopy (AFM) has become standard in materials science in the roughly 30 years since its invention [1]. The main reasons for this are its high resolution (equal to electron microscopy) and its easy handling, especially its ability to operate in air, as well as gas and liquid environments, which made the instrument a unique tool in many different research fields, including industrial engineering and life sciences. However, we believe that the essential advantage of AFM is its ability to map the materials properties rather than simple topography. There have been a number of review articles [2–4] and books [5–8] on AFM to date, focusing on its imaging process. The AFM probe tip has physical contact with the specimens during the scan, which allows the detection of materials properties. For example, tapping mode (also called ‘dynamic mode’ or ‘intermittent-contact mode’), which is regarded as the standard mode of operation in commercially available AFM instruments, produces a high-frequency (from kHz to MHz) intermittent contact, during which the interactions between probe tip and specimen surface are recorded. The experimentally obtained changes in the tip's vibrational amplitude and phase-shift are then interpreted as elastic, viscous or adhesive responses. Although we will not provide a detailed description of this mode here, it is well known that the phase image in the tapping mode reflects energy dissipative processes [9]. Although most authors suggest that the energy dissipation is the result of viscous response, our experience reveals that the contribution of adhesive hysteresis should not be ignored and sometimes becomes more dominant [10]. This is the main challenge of the interpretation of phase imaging. Needless to say, the energy dissipation map [11,12] and loss tangent mapping [13], which are calculated directly from the phase image, are also affected by this problem. This phenomenon provides an excellent control experiment for our nano-palpation AFM, which enables us to realize nanometer-scale quantitative mechanical property mapping especially for soft materials as medical palpation does for human body. There have been several attempts to address the above-mentioned challenge in the tapping mode, to reproduce the high-speed motion of the vibrating cantilever, to which the AFM probe is attached. The vibration in kHz can be fully recorded in real time using a fast analog to digital conversion board [14]. The detection of higher harmonic modes, which is followed by Fourier analysis, can reproduce the so-called force–distance curve in kHz to several hundred kHz, which contains full information on the probe tip–specimen interaction [15]. These are excellent methods for quick acquisition of materials properties mappings, but it is important to note that the frequency of the scanning affects the results [10]. This is because viscoelastic soft materials show different mechanical responses depending on the frequency inputs. It may be possible to utilize the time–temperature superposition (TTS) principle in order to convert the high-frequency responses obtained to ones at lower frequencies of interest [16]. However, we note that the TTS principle is a semi-empirical rule. It is unknown whether the ‘dispersion map’ obtained by bulk measurements is valid for surface measurements or nanometer-scale measurements. High-speed measurements have gained a great deal of attention in materials science. However, without solving the above-mentioned problems, it is impossible to make AFM be a useful tool in soft materials science. To this end, the indirect methods, which utilize TTS, must be incorporated with direct methods. The main requirements are: (i) low-frequency response, for ease of comparison with macroscopic mechanical testing, and (ii) a way to separate viscous and adhesive contributions. In this article, we introduce our efforts in these directions, with a special emphasis on item (i). The discussion of viscosity, however, remains challenging, and research in this direction is still in progress. We will show one attempt to realize viscoelastic measurements below [17]. The readers can refer to our earlier work [18,19], where we extended our methods to viscoelastic response. In order to realize the quantitative evaluation of elasticity, we must compensate for adhesive interactions. The recent progress in commercial instruments is rapid, with some machines offering elasticity evaluation, but the analysis provided often does not take the adhesive interaction into account fully. The purpose of this review article is to help AFM users who perform more precise elasticity measurements. This article is partially the result of ongoing ISO standardization in ISO/TC201/SC9: scanning probe microscopy. Although the realization of ISO documentation needs several more years, the solutions described here are likely to remain unmodified. We also present here the results of the round robin testing (RRT) under the name of VAMAS/TWA2: surface chemical analysis. Contact mechanics Due to the strong repulsive interaction between the AFM probe tip and the surface of an elastic specimen, the deformation of the indented area is unavoidable in all AFM modes, except for true non-contact modes. Consider a punch probe with a radius of 10 nm. The force of 30 nN, a typical value in contact-mode AFM, corresponds to the stress of ∼100 MPa. This order of stress goes beyond the yield stress of many plastic specimens. In case of elastic specimens such as rubbers, the resulting deformation is in the range from 10 to 100 nm. One of the most interesting features of AFM is atomic or molecular-level structural observation. At such small scales, the use of contact mechanics, which is a continuum theory, is insufficient. However, as discussed below, the contact at the scale of 10–100 nm can be treated by contact mechanics, which gives information about materials elasticity. Such microscopic quantities, possibly incorporated with the understanding of their spatial distribution, are very important when compared with macroscopic properties. Thus, although it should not be forgotten to remind the future discussion for the validity of contact mechanics, we discuss here how contact mechanics can be utilized when AFM is aimed to determine mechanical properties [20–22]. Hertzian contact Although we will not go into the details of contact mechanics [23], here we briefly review contact mechanics definitions necessary for the following sections. The original work was done by Sneddon [24], who described the contact mechanics between an axisymmetric probe and an elastic body. Figure 1 shows the schematics of the situation. The axisymmetric probe is defined as the probe whose tip at the origin with having the shape expressed by an arbitrary axisymmetric function, z = f(r), where r is the distance from the symmetry axis. The boundary condition for the function is f(0) = 0. The maximum deformation at the z-axis and the radius of the contact line are denoted by δ and a, respectively. The force exerted on the probe tip, which is detected from the deflection of the cantilever of AFM, is denoted by p. According to Sneddon, the following relationships hold among these quantities. $$p = \displaystyle{{3aK} \over 2}\left[ {\delta - \int_0^1 {\displaystyle{{xf\,(x)} \over {\sqrt {1 - {x^2}} }}\hbox{d}x} } \right],$$ (1) $$\delta = \int_0^1 {\displaystyle{{\,f^{\prime}(x)} \over {\sqrt {1 - {x^2}} }}\hbox{d}x + \displaystyle{\pi \over 2}\chi (1)} ,$$ (2) $$\chi (t) = \displaystyle{2 \over \pi }\left( {\delta - t\int_0^t {\displaystyle{{\,f^{\prime}(x)} \over {\sqrt {{t^2} - {x^2}} }}\hbox{d}x} } \right),$$ (3) where x = r/a, and K is as the elastic coefficient, which is expressed using Young's modulus, E, and Poisson's ratio, ν, as follows: $$K = \displaystyle{4 \over 3}\displaystyle{E \over {1 - {\nu ^2}}}.$$ (4) As an example, let's assume the spherical probe with the radius of R. The shape function becomes $$f\,(x) = R - R\sqrt {1 - \displaystyle{{{a^2}} \over {{R^2}}}{x^2}} \cong \displaystyle{{{a^2}} \over {2R}}{x^2},$$ (5) if we adopt the parabolic approximation (⁠|$R \gg a$|⁠). By substituting Eq. (5) into Eqs. (1)–(3), we obtain $$\delta = \displaystyle{{{a^2}} \over R}, \quad\hbox{}p = \displaystyle{K \over R}{a^3}.$$ (6) Since a is not directly measurable in AFM, we eliminate it and obtain $$p = K\sqrt R {\delta ^{3/2}},$$ (7) which is the well-known Hertzian contact solution [25]. If we ignore the adhesive interaction, it is possible to obtain the Young's modulus of the specimen by this equation, as is often done in the literature. Fig. 1. Open in new tabDownload slide The contact between an axisymmetric probe and elastic surface expressed by the Sneddon equations. Fig. 1. Open in new tabDownload slide The contact between an axisymmetric probe and elastic surface expressed by the Sneddon equations. Adhesive interaction Let's consider an example of a Lennard–Jones (LJ) type interaction potential. The force, p, per unit area, between two half-spaces separated by a distance, z, is written as $$p(z) = - \displaystyle{{8w} \over {3{z_0}}} \left [ {{{\left( {\displaystyle{{{z_0}} \over z}} \right)}^3} - {{\left( {\displaystyle{{{z_0}} \over z}} \right)}^9}} \right],$$ (8) where z0 is the equilibrium distance and p(z0) = 0. The work of adhesion (adhesive energy), w, is the work to move the two half-spaces from z0 to infinity with the following definition: $$w = \int_{{z_0}}^\infty {\,p(z)\hbox{d}z} .$$ (9) The relationship between normalized distance, z/z0, and normalized force, p/(w/z0), is shown in Fig. 2a. The integral of the force–distance curve in the attractive region corresponds to w. For comparison, the interaction expressed in the Hertzian model is shown in Fig. 2b. There is no attractive force in the Hertzian model, only hard wall repulsion at contact. Fig. 2. Open in new tabDownload slide The interaction forces (per unit area) for (a) Lennard–Jones, (b) Hertzian, (c) DMT, (d) JKR and (e) MD models. Fig. 2. Open in new tabDownload slide The interaction forces (per unit area) for (a) Lennard–Jones, (b) Hertzian, (c) DMT, (d) JKR and (e) MD models. Derjaguin, Muller and Toporov contact This model was derived by Derjaguin, Muller and Toporov (DMT) [26]; it applies to rigid systems with low adhesive energy, w, and/or small radii of curvature, R. The long-range adhesive forces are taken into account, but only in the outer space of contact, as shown in Fig. 2c. The tip–sample contact geometry is constrained to be Hertzian. The adhesive force acts like an additional external load. It follows that $$\delta = \displaystyle{{{a^2}} \over R}, \quad\hbox{}p = \displaystyle{K \over R}{a^3} - 2\pi wR,$$ (10) where −2πwR is the adhesive force, which is always constant. Therefore, the model cannot express an increasing adhesive force caused by the increase in the contact area during indentation. According to Maugis [27], it is convenient to introduce the following dimensionless parameters: $$\bar a = a{\left( {\displaystyle{K \over {\pi w{R^2}}}} \right)^{1/3}},$$ (11) $$\bar \delta = \delta {\left( {\displaystyle{{{K^2}} \over {{\pi^2}{w^2}R}}} \right)^{1/3}},$$ (12) $$\bar p = \displaystyle{\,p \over {\pi wR}}.$$ (13) These parameters simplify both the Hertzian contact, Eq. (6), and the DMT one, Eq. (10), as follows, $$\bar \delta = {\bar a^2},\hbox{}\bar p = {\bar a^3},$$ (6′) $$\bar \delta = {\bar a^2},\hbox{}\bar p = {\bar a^3} - 2.$$ (10′) Johnson, Kendall and Roberts contact A model first derived by Johnson, Kendall and Roberts (JKR) [28] includes short-range adhesion, which is essentially a delta function with strength w, acting only within the contact zone, as shown schematically in Fig. 2d. The model was reinterpreted by Maugis using the idea of fracture mechanics [29]. The movement of contact line can be analysed analogous with crack propagation. The JKR model is given by $$\delta = \displaystyle{{{a^2}} \over R} - {\left( {\displaystyle{{8\pi wa} \over {3K}}} \right)^{1/2}},$$ (14a) $$p = \displaystyle{K \over R}{a^3} - {(6\pi wK{a^3})^{1/2}}.$$ (14b) If we use Maugis's dimensionless parameters, the above equations become $$\bar \delta = {\bar a^2} - \displaystyle{2 \over 3}\sqrt {6\bar a} ,$$ (14a′) $$\bar p = {\bar a^3} - \sqrt {6{{\bar a}^3}} .$$ (14b′) Maugis–Dugdale contact and adhesion map Maugis introduced a Dugdale potential to describe the contact, as shown in Fig. 2e [27]. Maugis–Dugdale (MD) model is then expressed by the following equations: $$\displaystyle{{\lambda {{\bar a}^2}} \over 2}\left\{ {({m^2} - 2){{\rm s}^{ - 1}}m + \sqrt {{m^2} - 1} } \right\} + \displaystyle{{4{\lambda ^2}\bar a} \over 3}\left\{ {\sqrt {{m^2} - 1} {{\rm s}^{ - 1}}m - m + 1} \right\} = 1,$$ (15a) $$\bar \delta = {\bar a^2} - \displaystyle{4 \over 3}\lambda \bar a\sqrt {{m^2} - 1} ,$$ (15b) $$\bar p = {\bar a^3} - \lambda {\bar a^2}\left\{ {\sqrt {{m^2} - 1} + {m^2}{{\rm s}^{ - 1}}m} \right\},$$ (15c) where m = 1 + h0/a is a dimensionless parameter. He also introduced another dimensionless parameter, λ, which becomes the elasticity parameter, written as follows: $$\lambda = 2{\,p_{\rm c}}{\left( {\displaystyle{R \over {\pi w{K^2}}}} \right)^{1/3}},$$ (16) where pc is defined to hold w = h0pc (see Fig. 2e). If we set the value of pc to the minimum force for an LJ-type interaction in Eq. (8), then $${\left. {\displaystyle{{\hbox{d}p} \over {\hbox{d}z}}} \right|_{z = {z_{\rm c}}}} = 0 \Rightarrow {z_{\rm c}} = \root 6 \of 3 {z_0},\hbox{}{\, \quad p_{\rm c}} = - p({z_{\rm c}}) = \displaystyle{{16w} \over {9\sqrt 3 {z_0}}}.$$ (17) In this case, we obtain $$\lambda = \displaystyle{{16} \over {9\sqrt 3 }}\root 3 \of {\displaystyle{9 \over {2\pi }}} {\left( {\displaystyle{{16R{w^2}} \over {9{K^2}z_0^3 }}} \right)^{1/3}} \cong 1.16\mu ,$$ (18) where μ is the Tabor parameter [30]. As shown above, the MD model is rather complicated, making it challenging to apply to the analysis of experimental data. The importance of this model lies in the fact that it contains both the DMT and JKR models as its extreme cases. When λ approaches infinity, the MD model coincides with the JKR one. When λ approaches zero, the MD model coincides with the DMT one. Based on this idea, Johnson and Greenwood built the so-called ‘adhesion map’, where two parameters, |$\bar p$| and λ, are used to determine which model is well suited to actual experimental situations [31]. The adhesion map is shown in Fig. 3. The ratio between the applied and adhesive forces is denoted by |$\bar p$| (−2πwR in the DMT case and −1.5 πwR in the JKR one, the latter of which will be derived later). Thus, if the adhesive force is negligible or the applied force is sufficiently large, the Hertzian model is valid. Other cases need either the DMT, MD or JKR model. Four specific cases are added in the adhesion map, where open square, triangle and circle denote K = 20 MPa, 200 MPa and 2.0 GPa, respectively. Other parameters are identical (p = 10 nN, w = 0.2 J m−2, R = 10 nm and z0 = 0.2 nm). The K value of 2.0 MPa, typical of rubbery specimens, gives λ of 325, located outside of the map. Thus, in many polymeric materials studied by AFM, the JKR model best describes the phenomena. The filled circle, in Fig. 3, denote K = 2.0 GPa and R = 1.0 nm, which is in the MD model region. However, from the practical standpoint, the JKR model is manageable, while it is rather difficult to find a parameter set in good agreement with the DMT model. Therefore, we will utilize the JKR model hereafter in the following chapters. It is worth mentioning that Carpick et al. [32] derived a more realistic formula based on the MD model, though we will not consider it here. Fig. 3. Open in new tabDownload slide The adhesion map with four possible parameter sets suitable for AFM measurements. Fig. 3. Open in new tabDownload slide The adhesion map with four possible parameter sets suitable for AFM measurements. Comparison between the DMT and JKR models The gray solid lines, in Fig. 4, are the theoretical JKR curves (Eq. 14), where E = 100 MPa, ν = 0.35 and R = 10 nm. The adhesive energies, w, are set to 0.00, 0.05, 0.10, 0.15 and 0.20 J m−2. We note several important characteristics of the JKR curve. First, the surface of a specimen is deformed even for p = 0, where the adhesive force balances with the elastic restoring force. Secondly, there is a region where δ < 0. This indicates that the specimen is pulled up by a strong adhesive force. Even at the maximum adhesive force, the contact area (πa2) is not equal to zero. A fit to these curves using the DMT model results in the black dashed lines. The curve with w = 0.00 J m−2 is fitted well by the DMT model, giving EDMT = 100 MPa. However, the rest of the curves shows EDMT between 84 and 89 MPa, showing that the DMT fitting underestimates the theoretical curve. Fig. 4. Open in new tabDownload slide The DTM fit against the JKR theoretical curves with different adhesive energies, w. Fig. 4. Open in new tabDownload slide The DTM fit against the JKR theoretical curves with different adhesive energies, w. In experiments, the force–distance curve measurement or simply the force curve measurement for the AFM cantilever–probe tip assembly is used to move forward and backward along the surface normal and to measure the relationship between force and distance. The process consists of approaching and withdrawing, the former of which is often used for DMT fitting (one of the commercially available operation modes, PeakForceQNM by Bruker, uses the latter). When the assembly approaches the surface, it first feels long-range attractive force. Then, at the instant when the gradient of the force field and the spring constant of the cantilever coincide, the assembly undergoes sudden adhesion to the surface. Often, this point is regarded as the origin of the contact, while it is not simple to determine this point experimentally. Figure 4 shows that the DMT fitting is rather sensitive to the choice of the contact point, which is one of the disadvantages of this model. In this figure, we have superimposed two black solid lines, which indicate the stiffness at maximum deformation. Among nano-indentation researchers, it is widely accepted to use the withdrawal curve, since the approach curve contains contributions from both elastic and plastic deformations. Another drawback is the difficulty in precisely determining the zero point in a nano-indenter. To overcome this, a method developed by Oliver and Pharr (OP) is mostly used [33], where measured stiffness at the deformation maximum is converted into Young's modulus based on Sneddon's (or Hertz's) equation. This method is applicable in this case, since usually |$\bar p$| is very high. However, the application of the OP model to AFM force–distance curves is not as simple as for a nano-indenter. The Hertzian stiffness, sHertz, can be derived from Eq. (7) as follows $${s_{{\rm Hertz}}} = \displaystyle{{\hbox{d}p} \over {\hbox{d}\delta }} = \displaystyle{\hbox{d} \over {\hbox{d}\delta }}(K\sqrt R {\delta ^{3/2}}) = \displaystyle{3 \over 2}K\sqrt {R\delta } = \displaystyle{3 \over 2}Ka.$$ (19) Accordingly, sHertz is proportional to the elastic coefficient. In Fig. 4, the curves with w = 0.00 and 0.20 J m−2 show apparent stiffness of 1.54 and 2.13 N m−1, respectively, which indicates that the existence of adhesive interaction overestimates the modulus by 40%. There are other potential drawbacks in applying the OP model to AFM measurements. First, the turning point between the approach and withdrawal curves is unreliable, due to the hysteresis in the piezoelectric scanner that controls the relative distance between the cantilever–probe tip assembly and the specimen. The second and more essential problem is attributed to the motion of the contact line, which differs between approach and withdrawal. This is analogous to advancing and receding contact angles. Thus, the phenomena occurring at the turning point are rather complicated, possibly creating an additional energy dissipation path. As discussed above, it is preferable not to use the data near the turning point. Because of this, some commercial instruments leave the choice of the fitting region up to the user. At a glance, this is a nice idea, while it fails. If we convert the JKR curves into log–log plots (with appropriate shifts in both force and deformation to make them positive), the slope of each curve must be 1.5 if the DMT model applies. However, the slope depends on the choice of fitting region. The calculation of the DMT modulus based on this method thus does not work. It is obvious that the analysis based on the DMT model fails to estimate elasticity, either underestimating or overestimating it, depending on the calculation method, if the experiment is well represented by the JKR model. Force–distance curves and JKR analysis In the previous section, we have considered several contact mechanics models. We found that: Many soft materials can be modeled well by the JKR model rather than the DMT one. DMT-based analysis of force–distance curve is unreliable. In this section, we describe how analysis using the JKR model works. Conversion of the force–distance curve Figure 5a shows a schematic force–distance curve, which must be analysed by the JKR model. The horizontal axis denotes the displacement of the piezoelectric scanner, z, to control the distance between the AFM probe tip and the specimen's surface. The vertical axis is cantilever deflection, Δ, which is easily converted to the force, p, by Hooke's law, $$p = k\Delta ,$$ (20) where k is the spring constant of the cantilever. During the approach, we observe the sudden jump-to-contact from the point (zc, 0) to the contact point (zc, Δc). Further approach increases p by crossing the zero-force point (z0, 0) from negative to positive. At this point, adhesive and elastic restoring forces balance to each other. Therefore, we will refer to this point as the ‘balance point’. During the withdrawal, a much larger adhesive force (negative in sign) than that observed at the jump-to-contact is usually observed. Negative deformation (pull-up) is incorporated in this region. After the encounter of the maximum adhesive force at the ‘max-adhesion point (z1, Δ1)’, the rapid and unstable decrease in the contact radius, a, happens, which is followed by the detachment of the probe from the surface (jump-out). If the specimen surface is sufficiently hard, as shown by the dashed line in the figure, a relative displacement measured from the contact point (zc, Δc), z − zc, always coincides with Δ − Δc. On the other hand, when the specimen has elastic deformation, the deformation is defined as $$\delta = (z - {z_{\rm c}}) - (\Delta - {\Delta _{\rm c}})\quad ({\Delta _{\rm c}} \lt 0).$$ (21) Figure 5b shows the force–deformation curve obtained from Fig. 5a using Eqs. (20) and (21). This plot is the target of JKR analysis. We note that the contact point (zc, Δc) is defined during the approach. The dashed line in Fig. 5b is for the response of a hard surface. Fig. 5. Open in new tabDownload slide (a) A schematic of force–distance curve for a soft material with adhesive contact. (b) The force–deformation curve obtained from (a). Fig. 5. Open in new tabDownload slide (a) A schematic of force–distance curve for a soft material with adhesive contact. (b) The force–deformation curve obtained from (a). JKR two-point method As shown in from Eq. (14), it is difficult, though not impossible, to perform implicit-function curve fitting by eliminating a to obtain a (δ, p) relationship. Instead, a simpler algebraic method based on the JKR model has been reported, which is known as the JKR two-point method [34]. The force at the max-adhesion point (δ1, p1) is calculated from the following conditions: $${\left. {\displaystyle{{\partial p} \over {\partial a}}} \right|_{\,p = {\,p_1}}} = \displaystyle{\partial \over {\partial a}}\left[ {\displaystyle{K \over R}{a^3} - {{(6\pi wK{a^3})}^{1/2}}} \right] = 0.$$ (22) By combining Eqs. (14a) and (14b), we obtain $${\,p_1} = - \displaystyle{3 \over 2}\pi wR,$$ (23) $${\delta _1} = - \displaystyle{1 \over 3}{\left( {\displaystyle{{\,p_1^2 } \over {{K^2}R}}} \right)^{1/3}}.$$ (24) Note that the maximum adhesive force p1 is slightly different from that of the DMT model (−2πwR). Eq. (23) gives the estimation of adhesive energy, w. Since p = 0 at the balance point (δ0, 0), we obtain $${\delta _0} = \displaystyle{{\root 3 \of {16} } \over 3}{\left( {\displaystyle{{\,p_1^2 } \over {{K^2}R}}} \right)^{1/3}}.$$ (25) By solving Eqs. (24) and (25), the algebraic formula to determine the elastic constant, K, is acquired as follows: $$K = - {\left( {\displaystyle{{1 + \root 3 \of {16} } \over 3}} \right)^{3/2}}\displaystyle{{{\,p_1}} \over {\sqrt {R{{({\delta _0} - {\delta _1})}^3}} }}\quad \hbox{(}{\,p_1} \lt 0\hbox{)}.$$ (26) In short, just the information from the two points, the balance point and the max-adhesion point, gives the estimate of Young's modulus and adhesive energy. It is worth noting that the advantage of this two-point method compared with the DMT analysis discussed above. First, there is no need for precision in determining the contact point as shown in Eq. (26), where only the remainder between δ0 and δ1 is necessary. We can also avoid the problem of the turning point. Thus, this method is most reliable in determining Young's modulus for soft materials [35–37]. Figure 6a shows a typical force–distance curve measured on vulcanized isoprene rubber (IR). The dashed and solid lines correspond to the approach and withdrawal curves, respectively. The relevant force–deformation curve is shown in Fig. 6b as obtained from Eqs. (20) and (21). The contact point was determined by the approaching jump-in-contact. The JKR analysis revealed E = 2.66 MPa and w = 0.205 J m−2 for the measured values of δ0 = 81.2 nm, δ1 = −26.6 nm and p1 = −18.0 nN. We assumed ν = 0.5, a typical value for rubbery specimens. The JKR theoretical curve was also superimposed in the figure and was in good agreement with the experimental withdrawal curve. Fig. 6. Open in new tabDownload slide (a) The force–distance curve for IR vulcanizate. (b) Force–deformation curve converted from (a). The JKR theoretical curve is also superimposed. Fig. 6. Open in new tabDownload slide (a) The force–distance curve for IR vulcanizate. (b) Force–deformation curve converted from (a). The JKR theoretical curve is also superimposed. The measurement was performed with a NanoScope V with a MultiMode8 AFM system (Bruker, USA), using a cantilever whose nominal spring constant was k = 0.76 N m−1 and radius of curvature was R = 15 nm (OMCL- RC800PSA-WS, Olympus, Japan). However, the use of the nominal value is not recommended for quantitative measurements. As of the writing of this paper, there is no ISO documentation available for determining these parameters. Thus, we adopted the most representative methods to experimentally measure these values. The spring constant was measured with a thermal noise spectrum using the NanoScope software [38] to be k = 0.616 N m−1. The radius of curvature was determined with a so-called blind reconstruction method [39] using a widely used probe characterizer, TipCheck (Aurora NanoDevices, Canada), and found to be 18.6 nm. On several occasions, the determination of δ1 becomes erroneous, resulting in unreliable Young's modulus calculations. In such cases, we can adopt another two-point method. First, let us calculate the JKR stiffness, sJKR, as follows: $$S_{{\rm JKR}} = \displaystyle{{dp} \over {d\delta }} = {{\left( {\displaystyle{{\partial p} \over {\partial a}}} \right)} / {\left( {\displaystyle{{\partial \delta } \over {\partial a}}} \right) = \displaystyle{{1 - \displaystyle{1 \over 2}\left( {\displaystyle{{a_0 } \over a}} \right)^{3/2} } \over {1 - \displaystyle{1 \over 6}\left( {\displaystyle{{a_0 } \over a}} \right)^{3/2} }}\displaystyle{{3Ka} \over 2} = Cs_{{\rm Hertz}} }},$$ (27) where a0 is the contact radius at the balance point. Thus, $${s_{{\rm JKR}}}({a_0}) = 0.6{s_{{\rm Hertz}}}.$$ (27′) In place of Eq. (26), we can use the following equation: $$K = {\left( { - \displaystyle{{2s_{{\rm JKR}}^3 } \over {27{C^3}R{\,p_1}}}} \right)^{1/2}} = {\left( { - 0.343\displaystyle{{2s_{{\rm JKR}}^3 ({a_0})} \over {R{\,p_1}}}} \right)^{1/2}}$$ (28) The JKR stiffness, sJKR, was measured as the slope of the dashed line in Fig. 6b to be 0.383, which resulted in E = 2.85 MPa. The values overlapped each other within 7% error. We conclude this section by proposing a method to determine the contact point from the withdrawal curve. From Eqs. (24) and (25), we obtain $${\delta _0}:{\delta _1} = \displaystyle{{\root 3 \of {16} } \over 3}: - \displaystyle{1 \over 3} \cong 2.52: - 1.$$ (29) Since the jump-in-contact phenomenon is frequently disturbed by the surface (or probe tip) dirtiness, the contact-point determination sometimes becomes unreliable. We may be able to use Eq. (29) as an alternative. In the case of Fig. 6b, the difference between the contact points determined by the approach curve and the withdrawal curves was 4.0 nm. Standardization activity The international standardization of scanning probe microscopy is in progress for the framework of ISO/TC201/SC9. The 21st plenary meeting of ISO/TC201 was held in Tampa, Florida, in 25–27 October 2012. A resolution made there related to the subject treated in this article (Resolution 12 in Doc ISO/TC201/SC9 N213) reads as follows: ‘ISO/TC201/SC9 requests Prof. K. Nakajima to start an international RRT on guidelines for the determination of elastic modulus for compliant materials using atomic force microscopy and circulate the technical protocol for the RRT by July 31, 2013, preferably under VAMAS’. According to the request, the protocol for an international inter-laboratory comparison has been prepared and sent out as VAMAS/TWA2/A16 activity to AFM experts with two specimens: vulcanized IR (E ∼ 3 MPa, ν = 0.5) and low-density polyethylene (LDPE, E ∼ 0.3 GPa, ν = 0.45). Eighteen data points for IR and 15 data points for LDPE have been collected up to now. Here, we present the results of these data. Supplementary Table 1 shows the specifications of experimental conditions. Each group used different AFM systems, scanners (open-loop or close-loop) and cantilevers. The methods to determine k and R are also different. However, the trimmed data showed a reasonable overlap, as shown in Fig. 7. (The original data before the trimming are shown in Supplementary Fig. 1.) The averaging result shows E = (2.77 ± 0.44) MPa for the data in Fig. 7. The error of 16% contains the uncertainty of k and R as well as the spatial distribution of the specimen modulus. The modulus distribution revealed by the modulus map, explained below, is <10%. We believe that the accuracy will be improved after appropriate standard methods for determining k and R are developed, which will be the goal of our future work. The uncertainty in quantification of the force–distance curve will also be considered [40]. The LDPE results are also provided in the supplementary data online. Fig. 7. Open in new tabDownload slide A summary of round robin tests for determining the Young's modulus of a vulcanized IR specimen (trimmed). Fig. 7. Open in new tabDownload slide A summary of round robin tests for determining the Young's modulus of a vulcanized IR specimen (trimmed). Nano-palpation AFM Principle and applications To map the local mechanical properties of soft materials, force–volume (FV) measurement equipped with a Bruker NanoScope AFM (or the recently developed PeakForceQNM by Bruker [41], QI-mode by JPK) is the most appropriate method. In this mode, force–distance curve data are recorded until a specified cantilever deflection value (trigger set-point), Δtrig, is attained for n × n points (128 × 128 or larger) over a two-dimensional surface. At the same time, z-displacement values, ztrig, corresponding to the trigger set-point deflection are recorded to build an ‘apparent height’ image. As an example, the image of vulcanized styrene–butadiene rubber (SBR)/IR 7 : 3 blend specimen is shown in Fig. 8a. The image was taken in the FV mode with a ramp rate of 5.58 Hz. The cantilever used (OMCL-RC800PSA-WS, Olympus, Japan) had k = 0.72 N m−1 and R = 28 nm. The topographic height image taken in this mode is virtually the same as obtained by the conventional contact mode if the contact force set-point and the trigger set-point are identical. If all the points over the surface are sufficiently rigid, the set of recorded displacements, ztrig, represents the true height of the specimen. However, if the surface deforms as discussed above, the obtained data are not true topographic information. However, since we have a force–distance curve for each point, we can estimate the maximum specimen deformation value for each point, referring to Eq. (21), $${\delta _{\max }} = ({z_{{\rm trig}}} - {z_{\rm c}}) - ({\Delta _{{\rm trig}}} - {\Delta _{\rm c}}).$$ (30) Fig. 8. Open in new tabDownload slide Nano-palpation AFM images of vulcanized SBR/IR blend specimen. The scan size was 3.0 µm. (a) Apparent height, (b) deformation, (c) true height, (d) Young's modulus (log-scale) and (e) adhesive energy. Fig. 8. Open in new tabDownload slide Nano-palpation AFM images of vulcanized SBR/IR blend specimen. The scan size was 3.0 µm. (a) Apparent height, (b) deformation, (c) true height, (d) Young's modulus (log-scale) and (e) adhesive energy. Consequently, two-dimensional arrays of specimen deformation values can be regarded as the specimen deformation image as shown in Fig. 8b. In this image, the deformation at hard ZnO particles added to help vulcanization was almost zero. The IR region, seen as the deeper islands in the apparent height image, had larger deformation, which is attributed to lower Young's modulus. The force–distance curve analyses for 16 384 (=128 × 128) data points yield the Young's modulus distribution (Fig. 8d) and adhesive energy distribution (Fig. 8e) images at the same time and the same location. As for the Young's modulus distribution, it was observed that E = (1.40 ± 0.39) MPa for IR and E = (6.45 ± 0.86) MPa for SBR. We now have apparent height (ztrig) and sample deformation (δmax) images taken at the same time, and Δtrig is the preset value and constant for all the force–distance curves. Then, if the appropriate determination is performed for the contact point [the array of (zc, Δc)], this realizes the reconstruction of the ‘true height’ image, free from sample deformation as shown in Fig. 8c [42,43]. In this figure, deep islands of the IR region contain no more depletions. The cryo-microtomed surface has such flatness. In other words, the height variation in the ‘apparent height’ image is merely the result of the difference in Young's modulus. Note that the reconstruction of a ‘true height’ image is impossible by any other modes of operation. Like medical palpation, nano-palpation AFM enables us to obtain the true height, Young's modulus images, and other properties. To date, we have applied the technique to a wide variety of specimens, including carbon black (CB) reinforced natural rubber (NR) [44], carbon nanotube (CNT)-reinforced NR [45,46], reactive polymer blend [47], block copolymers [11,43,48,49], deformed plastics [50,51], human hair [52,53], honeycomb-patterned polymer films [54–56], and CNT-reinforced hydrogel [57]. As an example of nano-palpation AFM, the Young's modulus image of CB-reinforced IR vulcanizate is given in Fig. 9, using 10 phr (parts per hundred rubber) of high abrasion furnace-grade CB as a filler. The modulus image clearly shows the CB part, the IR matrix and the interfacial (IF) region. The tensile testing revealed that the effective volume fraction f was about 4 in the modified Guth-Gold equation [58], $${E_{\rm f}} = {E_{\rm m}}(1 + 2.5f\phi + 14.1{\,f^2}{\phi ^2}),$$ (31) where Ef and Em denote Young's modulus of filled system and rubber matrix, respectively. ϕ is the volume fraction of filler. In Fig. 9, by the threshold analysis, we found ϕIR: ϕCB: ϕIF = 0.61 : 0.08 : 0.31. The ϕCB value of 0.08 does not contradict the original filler content of 10 phr. Consequently, (ϕCB + ϕIF)/ϕCB = 4.9 provides another way to estimate the effective volume fraction. Nano-palpation AFM can be used to check the pre-existing reinforcement theories. Furthermore, since we know the modulus of the IF region, it might be possible to build a more sophisticated theory of reinforcement in the future. Fig. 9. Open in new tabDownload slide Young's modulus image of HAF-CB reinforced IR obtained by nano-palpation AFM. The scan size was 2.0 µm. Fig. 9. Open in new tabDownload slide Young's modulus image of HAF-CB reinforced IR obtained by nano-palpation AFM. The scan size was 2.0 µm. Viscoelastic measurement Viscoelastic soft materials often show speed- and temperature-dependent responses. The force–distance curve measurement is not the exception [59]. In Fig. 8 obtained by the FV mode, the ramp rate of 5.58 Hz was used with the triangular motion of z-piezoelectric scanner. On the contrary, PeakForceQNM is operated with the sinusoidal motion with much higher frequencies. How about the difference seen in the resultant modulus values? In order to check the point, PeakForceQNM was employed for the same specimen and the same cantilever with three different ramp rates, 250 Hz, 500 Hz and 1.0 kHz. The Young's modulus map for 1.0-kHz ramp rate as an example is shown in Fig. 10. Figure 10a was obtained using the NanoScope software, based on the DMT calculation. Since the pixel number (256 × 256) was larger than that of the FV, the image quality seemed higher. However, because the image was based on the DMT model, it is highly possible to underestimate the modulus values. Thus, we performed the recalculation of the original data using the JKR two-point method [60], which is shown in Fig. 10b. The elasticity parameters, λ, for the IR and SBR regions calculated from Fig. 10b were 180 and 10.3, respectively. Therefore, both were in the JKR regime in the adhesion map. The ratios of the underestimation of EDMT against EJKR were 22 and 25%, respectively, where EJKR(IR) = 2.70 ± 0.42 MPa and EJKR(SBR) = 159 ± 25 MPa. Fig. 10. Open in new tabDownload slide PeakForceQNM modulus images of vulcanized SBR/IR blend specimen. The scan size was 3.0 µm. (a) Young's modulus calculated by the NanoScope software using the DMT model (b) calculated based on the JKR two-point method. Fig. 10. Open in new tabDownload slide PeakForceQNM modulus images of vulcanized SBR/IR blend specimen. The scan size was 3.0 µm. (a) Young's modulus calculated by the NanoScope software using the DMT model (b) calculated based on the JKR two-point method. The modulus for the IR region was similar to that obtained by FV, while that for the SBR region was very different. In order to explain this deviation, we return to the TTS principle, which can be expressed by the Williams–Landel–Ferry (WLF) equation [61]: $$\log {a_{\rm T}} = - \displaystyle{{{\hbox{C}_1}(T - {T_{\rm r}})} \over {{\hbox{C}_2} + (T - {T_{\rm r}})}},$$ (32) where aT is a shift factor, T is the temperature of the specimen system and Tr is the reference temperature. It is related to the glass-transition temperature, Tg, as Tr = Tg + 50°C. The constants C1 and C2 are called ‘universal constants’ and C1 = 8.86 and C2 = 101.6. The Tg of SBR is −5°C. Although there is a need for further validation to check whether the WLF equation can be applied to such nanometer-scale measurements, we adopted Eq. (32) and shifted the measurement frequency, f, to aTf, where T was measured to be 22°C. The result is shown in Fig. 11a. The leftmost filled square is the modulus of FV, while the rest are for PeakForceQNM. All the data were in good agreement with the bulk master curve obtained for a pure SBR specimen by a shear-mode rheometer (after the conversion from shear modulus G′ to E′). Thus, in interpreting the modulus measured by any modes of AFM, it is important to consider the frequency and the temperature used in the experiment. Fig. 11. Open in new tabDownload slide (a) FV and PeakForceQNM Young's modulus data superimposed on the bulk master curve for storage modulus of vulcanized SBR. Each data were reproduced from Figs. 8d and 10b. (b) The comparison between bulk master curve (storage modulus, E′, loss modulus, E″, and loss tangent, tan δ) with the data by nano-rheology AFM [17]. Fig. 11. Open in new tabDownload slide (a) FV and PeakForceQNM Young's modulus data superimposed on the bulk master curve for storage modulus of vulcanized SBR. Each data were reproduced from Figs. 8d and 10b. (b) The comparison between bulk master curve (storage modulus, E′, loss modulus, E″, and loss tangent, tan δ) with the data by nano-rheology AFM [17]. In order to further elucidate the validity of TTS for nanometer-scale surface measurements, we have developed nano-rheology AFM, which enables us to perform wide-range frequency sweep measurement at a fixed temperature. The experimental details can be found in the literature [17], the key was the use of a tiny (and therefore high resonant frequency) piezoelectric actuator, operated independently from any AFM electronic circuit. In addition, we fully utilized the knowledge of contact mechanics, especially the JKR stiffness in Eq. (27), which provided the quantitative estimation of dynamic moduli. The result is shown in Fig. 11b. Once again, we found good agreement between the bulk data and the data obtained by this method. Note that the measurement was performed at room temperature only. In most cases, it is necessary to have multiple data sets with different temperatures to produce the master curve. The nano-rheology AFM is also capable of obtaining nanometer-scale resolution images. To conclude, we present the results of a tapping-mode phase image of SBR/IR vulcanizate, shown in Fig. 12 [22]. The phase image in Fig. 12b allows distinguishing among three regions, namely, SBR, IR and ZnO. However, the energy dissipation (Edis) image in Fig. 12c calculated from the phase, ϕ, and amplitude ratio, A/A0, in Fig. 12a does not have good contrast. Edis was calculated using [2,62], $${E_{{\rm dis}}} = \displaystyle{{\pi k} \over Q}\left( {\sin \phi - \displaystyle{\,f \over {{\,f_0}}}\displaystyle{A \over {{A_0}}}} \right),$$ (33) where the cantilever specifications were as follows: k = 4.0 N m−1, quality factor, Q = 191, resonant frequency and f0 = 77.9 kHz. The drive frequency, f, was identical to f0. This happened due to the fact that ϕ in the SBR region has its peak at 81° and in the IR region at 96°. Since Edis depends on sin ϕ, there is a lower contrast (sin 81° = 0.987 and sin 96° = 0.994). An interesting question here is the change in direction in phase from 90° (π/2) obtained when the cantilever–probe tip assembly vibrates freely without any interaction with the specimen surface. The phase in the SBR region changed in the same direction as that on the ZnO hard particles. This may seem counterintuitive, since SBR is a rubber. The explanation is that the drive frequency of 77.9 kHz corresponds to aTf of 61 MHz, at which SBR behaves as glassy material by crossing the glass-transition region (see Fig. 11). In conclusion, the tapping-mode phase contrast was obtained due to adhesive interaction for both ZnO and SBR regions, while viscous interaction dominated in the IR region. Fig. 12. Open in new tabDownload slide A tapping-mode image of vulcanized SBR/IR blend specimen. The scan size was 3.3 µm. (a) Amplitude ratio, (b) phase and (c) energy dissipation images. (d) A histogram of phase obtained from (b) [22]. Fig. 12. Open in new tabDownload slide A tapping-mode image of vulcanized SBR/IR blend specimen. The scan size was 3.3 µm. (a) Amplitude ratio, (b) phase and (c) energy dissipation images. (d) A histogram of phase obtained from (b) [22]. Conclusion In this review article, we have described the pitfalls in the widely used Hertzian and DMT models in analysing AFM data, especially for soft materials. As an alternative, the JKR model can be applied to measure Young's modulus quantitatively, in the range from MPa to GPa. The ISO standardization activity on the force–deformation curve analysis is in progress. Our nano-palpation AFM can provide more fruitful information, such as quantitative materials properties mapping. We also propose a new two-point method and a new way to determine the contact point. The contact mechanics is basically elastic theory, which cannot be applied to viscoelastic materials. Nevertheless, we found a degree of correlation between speed-dependent modulus values by AFM methods and bulk dynamic properties, by considering the TTS principle. The reproduction of the force–deformation curve on viscoelastic materials is a more delicate issue, which will be an important future direction. Supplementary data Supplementary data are available at http://jmicro.oxfordjournals.org/. Funding This work was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan. Acknowledgement The authors are grateful to Prof. Toshio Nishi, who guided us as the Principal Investigator until March 2013. The rubbery specimens were provided by our collaborators, Dr H. Iwabuki of Industrial Technology Center of Okayama Prefecture, Japan, Dr T. Igarashi of Bridgestone Corporation, Japan, and Prof. Y. Sato of Sophia University, Japan. We also thank them for fruitful comments. References 1 Binning G , Quate C F , Gerber Ch , Weibel E . Atomic Force Microscope , Phys. Rev. Let. , 1986 , vol. 56 pg. 930 Google Scholar Crossref Search ADS WorldCat 2 García R , Pérez R . Dynamic atomic force microscopy methods , Surf. Sci. Rept. , 2002 , vol. 47 pg. 197 Google Scholar Crossref Search ADS WorldCat 3 Giessibl F J . Advances in atomic force microscopy , Rev. Mod. 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TI - Nano-palpation AFM and its quantitative mechanical property mapping
JF - Microscopy
DO - 10.1093/jmicro/dfu009
DA - 2014-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/nano-palpation-afm-and-its-quantitative-mechanical-property-mapping-AFja0Z9NAS
SP - 193
EP - 208
VL - 63
IS - 3
DP - DeepDyve
ER -