Excluded Volume Model for the Reduction of Polymer Diffusion into

Aug 26, 2013 - Frank Lange , Patrick Judeinstein , Cornelius Franz , Brigitte Hartmann-Azanza , Salim Ok , Martin Steinhart , and Kay Saalwächter. AC...
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Excluded Volume Model for the Reduction of Polymer Diffusion into Nanocomposites Jeffrey S. Meth,*,† Sangah Gam,‡ Jihoon Choi,‡ Chia-Chun Lin,‡ Russell J. Composto,‡ and Karen I. Winey‡ †

DuPont Nanocomposite Technologies, Central Research & Development, E. I. DuPont de Nemours & Co., Inc., Route 141, Wilmington, Delaware 19880-0400, United States ‡ Department of Materials Science and Engineering, University of Pennsylvania, 3231 Walnut Street, Philadelphia, Pennsylvania 19104-6272, United States ABSTRACT: An analytic model for the slowing down of polymer chain diffusion in nanocomposites attributable to excluded volume effects is presented. The nanocomposite is modeled as an ensemble of cylinders through which the polymer chains diffuse. The reduction of polymer diffusion in each cylinder is equated with the reduction of diffusion for a sphere through a cylinder. The distribution of cylinder diameters within the ensemble is determined from statistical mechanical theories based on the packing of spherical particles. For low loadings of spherical particles in nanocomposites, this model results in a master curve for the reduced diffusion coefficient. With no adjustable parameters, the model agrees with recent data for tracer diffusion measurements in polymer nanocomposites at low loading.

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INTRODUCTION The dynamics of polymer chains in nanocomposites is an important research area because dynamics determine the behavior of these materials during processing to produce articles and their aging in use. At the molecular level, the dynamics have been probed by neutron scattering. For polyethylene oxide confined within the cylindrical pores of an anodized aluminum oxide template, Martin et al. provided data that suggested that polymer chains could disentangle under strong confinement.1 This hypothesis was reinforced by Schneider et al., who measured similar behavior in nanocomposites comprising colloidal silica (CS) dispersed into hydrogenated 1,4-polyisoprene.2 Because neutron spin echo experiments are conducted on the nanosecond time scale, that work could not draw conclusions about the ramifications of chain disentanglement on center of mass motion. Alternatively, polymer diffusion experiments are useful for measuring center of mass motion.3 Recent experiments measuring the diffusion coefficient of deuterated polymer chains into nanocomposites over length scales much larger than the radius of gyration demonstrated that the diffusion decreases with spherical particle loading.4−8 This has been shown for deuterated polystyrene chains diffusing into a nanocomposite comprising phenyl-capped CS dispersed into polystyrene,4,5 for deuterated polymethylmethacrylate (PMMA) chains diffusing into a nanocomposite comprising bare CS dispersed PMMA,6 and for deuterated polystyrene chains diffusing into a nanocomposite comprising CS grafted with polystyrene brushes.7 Furthermore, for all three of these systems, the data for the normalized diffusion coefficient collapse onto a master curve © 2013 American Chemical Society

dependent on the variable ID3D/2Rg (where ID3D is the average interparticle spacing in three dimensions attributed to the filler loading and Rg is the radius of gyration of the penetrant chain). We infer from this observation that there may be a common explanation for the physics governing polymer diffusion in nanocomposites with spherical nanoparticles. At low loadings of nanoparticles, one mechanism potentially responsible for reduced diffusion may be attributed to excluded volume, where the chains simply have less room to maneuver in the presence of obstacles; in this low loading regime, the average size of a “pore”, “channel”, or “opening” in the nanocomposite is larger than the size of the penetrant chain, that is, ID > 2Rg. At high loadings, where the average size of the “opening” is smaller than the size of the penetrant chain, a second contribution may come from confined diffusion, where chains reptate through the pores in the composite. This paper presents a model to explain the slowing down of polymer center of mass diffusion, that is, the reduction in the diffusion coefficient, in the regime of low volume loading of spherical nanoparticles. To accomplish this, we modeled the polymer chains as spheres diffusing down cylindrical pores. This is a major approximation: that a diffusing chain can be modeled as a sphere and that analyses developed for the dynamics of spheres in fluids can be extended to the present Special Issue: Michael D. Fayer Festschrift Received: June 28, 2013 Revised: August 23, 2013 Published: August 26, 2013 15675

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THEORY There are three components to the model: (1) knowing how the diffusion coefficient of a single sphere diffusing through a single cylinder is retarded as a function of their relative size; (2) converting the topology of a nanocomposite into an ensemble of cylinders with a distribution of radii; and (3) constructing a statistical average that applies to many spheres diffusing through this ensemble of cylinders. 1). Diffusion of a Single Sphere through a Single Cylinder. Knowing the reduction of diffusion of a single sphere traveling through a single cylinder is primary for the model. Using fluid mechanics, Haberman and Sayre solved this problem for the motion of a solid sphere of radius Rs suspended in a viscous liquid moving through a cylindrical pore of radius r.9 They numerically derived the solution for the effective drag force experienced by the sphere for nine values of the parameter λs = Rs/r. Through the Stokes−Einstein relation, it is known that the diffusion coefficient of a sphere in a viscous medium is inversely proportional to the frictional drag force. Therefore, a prediction of the inverse of the relative drag is equivalent to a prediction of the relative diffusion coefficient. In the experimental data presented in the literature,4−7 the relative diffusion coefficient has been plotted as a function of ID3D/2Rg, which is analogous to the radius of the cylinder divided by the radius of the diffusing sphere. Therefore, to readily enable a comparison with data, the theoretical results of Haberman and Sayre are replotted against the variable ys = r/Rs = 1/λs in Figure 2. Because of the Stokes−Einstein relationship, the

situation. In the melt, polymer chains move by the mechanism of reptation, which is very different from fluid dynamics of rigid bodies but at long times and large distances, which are the experimental conditions being considered here, the diffusion of the center of mass of the chains still obeys Fick’s law. For any one polymer chain moving along a single cylinder, the fluid dynamics analysis of a solid sphere moving through a cylinder filled with viscous liquid is extended to predict the relative diffusion coefficient for that polymer chain. To more accurately represent the nanocomposite, the statistical mechanics of hard sphere liquids is used to determine the distribution of cylinder diameters corresponding to a particular volume loading of spherical nanoparticles of a given size. The ensemble average is then calculated, and the resulting prediction compares favorably to experimental data. To illustrate the model, Figure 1 depicts the geometry under consideration. Figure 1a shows a conceptual picture of a

Figure 1. Conceptual figure of a polymer chain diffusing through a nanocomposite: (a) the spaces (light blue) between nanoparticles (dark blue spheres) are transformed into (b) a set of cylinders with a distribution of radii through which the polymer chain (red) diffuses. The interparticle distance ID is shown in panel a, and the pore radius r is shown in panel b.

polymer nanocomposite. The nanoparticles, depicted as blue spheres, are distributed throughout a polymer matrix in accordance with a single configuration of a hard sphere liquid. The red contour represents the contour of a polymer chain singled out from the melt phase. Figure 1b shows a collection of cylinders intended to replicate the distribution of pore diameters that exist in the nanocomposite. This connection is one hypothesis of this work: that the distribution of openings in a 3D nanocomposite can be mapped onto a collection of aligned cylinders with a distribution of cross sections, provided that the size distribution of the cylinders is chosen properly. The nature of a configuration of a hard sphere liquid is such that all cross sections that cut through the 3D nanocomposite will have the same distribution of openings (within the fluctuations determined by the partition function). This reduces the problem of diffusing in one direction through a 3D composite into the problem of diffusing in one direction through an ensemble of cylinders. This reduction should not work at high volume loadings, where the radius of gyration of a polymer chain is much larger than the cavity size because the chain can simultaneously occupy more than one cavity in the composite. Hence, the polymer dynamics will in addition be influenced by the correlation between the distribution of sizes of openings in one cross-sectional slice and the distribution of sizes of openings in an adjacent cross-section. At sufficiently low loadings, however, we assume this correlation is negligible, and the problem can be simplified as illustrated in Figure 1b.

Figure 2. Numerical results of Haberman and Sayre9 for the reduction of the diffusion coefficient for a sphere with radius Rs diffusing through a cylinder with radius r (black circles) compared with the partition function of eq 1 with a polymer radius of gyration Rg (black line). The partition function is a good analytic approximation to the numerical results.

ordinate axis is labeled as a relative diffusion coefficient instead of relative inverse drag. When the cylinder is infinitely large, ys = ∞, the diffusing sphere does not experience an increase in drag, and the diffusion is not slowed down. As the cylinder becomes smaller and ys decreases, the drag on the diffusing sphere increases, and consequently its diffusion coefficient decreases. When ys < 1, the diameter of the cylinder is now smaller than the diameter of the sphere, and the sphere cannot diffuse at all. 15676

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Muthukumar and Baumgartner.16,17 Thus, the concept that hydrodynamic physics can be applied to loosely confined polymer chains diffusing through cylinders is established in the literature. However, in all of these examples, the polymer was diffusing through a liquid phase. Here we extend this analysis to the melt phase. This could be justified because the length scales over which the chains are diffusing are much larger than their radius of gyration, so the diffusion follows Fick’s law. Clearly, this analogy will break down when y = r/Rg < 1 because a hard sphere cannot move through a cylinder narrower than the sphere diameter, while a polymer chain can, in principle, deform and translocate through an opening smaller than its Rg. However, the magnitude of F(y) becomes practically negligible in this range. A polymer chain could still diffuse by elongating into a nonspherical shape and moving along the pore, a process that will be referred to here as confined diffusion. The dynamics behind this confined diffusion mechanism may be described by entropic barrier and translocation theories when the medium is a liquid.16−19 Where the crossover between these mechanisms occurs in nanocomposite diffusion is not yet clear. This is especially true when one considers diffusion through a nonuniform system with a variety of pore sizes. For example, the flux can proceed through the larger pores via the excluded volume mechanism, whereas diffusion through the smaller pores would be controlled by confined diffusion. In a sample with a distribution of pore sizes, the flow will be biased toward the larger pores. 2). Transforming a Nanocomposite into a Distribution of Cylinders. Modeling a porous glass such as anodized aluminum oxide as a collection of cylindrical pores is straightforward. It is harder to intuitively make that connection for a nanocomposite. The distribution of spherical fillers in a nanocomposite may be thought of as a single configuration of a hard sphere fluid, and the statistical mechanics that has been developed to describe the structure of hard sphere fluids can then be applied to such nanocomposites. On the time scale of the experiments considered here, the filler particles were immobile. The structure of such hard sphere fluids is rigorously described by the n-point probability formalism created by Torquato and coworkers.20 For diffusion through a polymeric nanocomposite consisting of immobile CS dispersed in polymer, we hypothesize that the structure of the polymer phase can be represented by an ensemble of cylindrical passageways, openings, or pores. Also, the composite can be divided into slices perpendicular to the direction of the flux. Because of the isotropic, ergodic nature of the system, the pore diameter distribution will be the same in each slice. Furthermore, again due to ergodicity, the distribution of pore diameters in a slice will be equivalent to the distribution in pore diameters that a deuterated polymer chain “sees” in all slices as it diffuses through the nanocomposite. We use the n-point probability formalism to transform the nanoparticle size and volume loading into the pore size distribution function P(x), where x = r/; r is the radius of the cylindrical pore (see Figure 1b) and is the average diameter of the spherical particles.20 For CS dispersed into a polymer matrix and given that the CS particles are immobile, the dispersion can be equivalent to a single configuration of a polydisperse hard sphere fluid; this was demonstrated in the experimental systems of concern herein.8 The n-point probability formalism predicts the geometric properties of this composite, including the pore sizes.

An analytic expression for the numerical results is convenient for incorporating their results into the present model. In their work,9 they expressed the relative drag as a ratio of two polynomials, but they were looking at their results plotted versus λs. When replotted versus ys, however, another analytical expression becomes obvious. Casassa derived the partition function of a Gaussian polymer chain confined within a cylinder, F(y):10 ∞

F (y ) = 4 ∑ i=0

1 exp( −βi2 /y 2 ) 2 βi

(1)

where y = r/Rg, r is the radius of the cylindrical pore, and Rg is the radius of gyration of the polymer chain. In this equation, βi are the roots of J0(βi) = 0, where J0 is the Bessel function of the first kind of order zero. The two results are compared in Figure 2, and it shows that the relative diffusion coefficient of a hard sphere diffusing inside a cylinder is accurately represented by the partition function for a Gaussian polymer chain when one equates Rs with Rg. With regards to F(y), it represents the reduced configurational entropy of a polymer chain in a cylindrical pore.10 Figure 2 suggests that mathematically the relative diffusion coefficient for a hard sphere is practically identical to the reduction in the polymer partition function due to configurational entropy loss. The cylinder cuts off configurations that are highly extended and reduces configurations available to the chain. Notice that, even at large values of y, Figure 2 shows that the relative diffusion coefficient is reduced by a substantial amount, ∼20% for y = 10. This analysis demonstrates that the polymer partition function for a Gaussian chain is an accurate analytical representation for the relative diffusion coefficient of a hard sphere diffusing through a cylinder filled with viscous fluid. In the present work, we hypothesize that the partition function is also a valid expression for the relative diffusion coefficient of a polymer chain diffusing through a cylinder for time scales greater than the reptation time, when the chain diffusion is Fickian. This connection is not unprecedented. Diffusion of polymers through pores has been studied. Experimentally, Cannell and Rondelez demonstrated that polystyrene chains in ethyl acetate diffused slower through porous membranes as the pore size decreased.11 Similar results were obtained by Guillot et al.12 Davidson and Deen successfully modeled these results using a hydrodynamic approach.13 They applied the hydrodynamic physics behind hard sphere diffusion to polymer chains. In a separate work, Davidson et al. performed Monte Carlo simulations describing the partitioning of flexible macromolecules between bulk solution and cylindrical pores.14 They demonstrated that the calculated partition coefficient was in accord with the theoretical expression derived by Casassa10 in the limit of long chains. By using dynamic light scattering (DLS), Guo et al.15 measured the diffusion coefficient of polystyrene solutions that had penetrated the pores of controlled pore-size silica glasses. They compared their data to a theoretical expression13,14 and showed good agreement for low values of λh = Rh/r, the ratio of the polymer hydrodynamic radius to the pore radius. As a function of molecular weight, which served as a proxy for Rh, two regimes of diffusion hindrance were observed. The data for the lowmolecular-weight regime, wherein the polymer was loosely confined, was successfully modeled by the hydrodynamic theory, whereas the high-molecular-weight regime was attributed to chain entropic barriers, as described by 15677

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two dimensions would increase. We can add variances in this case so that for a polydisperse system with particle size variance σ2 one would have:

To know the pore sizes corresponding to a 2D slice through the 3D composite with spherical fillers, we employ the 2D pore-size distribution predicted from the 2D n-point formalism. The definition of pore size is as follows. In a 2D slice through the nanocomposite, each point in the plane will be occupied either by polymer or by filler. For any single point in the plane within the polymer phase, a circle with radius r can be constructed. One then increases r until the circle impinges on the nearest filler particle. This maximum circle radius defines the pore radius r for that location in the plane. Each and every point in the plane has its own value for the pore radius, which has a distribution given by P(x), with x = r/. For the 2D case, the normalized pore size probability density function P(x) is given by:20 P(x) = 4ϕS(2a0x + a1) exp[−4ϕS(a0x 2 + a1x)]

S=

1 32 3π 2

+ σ2

(3)

From eq 2, the average, normalized pore radius, = / disk can be calculated: ⟨x⟩ =

⎡ ϕSa12 π exp[ϕSa12 /a0] erfc⎢ ⎢⎣ 16ϕSa0 a0

⎤ ⎥ ⎥⎦

(4)

where erfc is the complement of the error function. The final step is to connect the variable x, the ratio of the pore size to the particle size, with y, the ratio of the pore size to the polymer size. Given that x = r/disk and y = r/Rg, eliminating r yields x = yRg/disk. With this, P(x) from eq 2 is transformed into P(y):

(2)

where a0 = (1 + ϕ(S − 1))/(1 − ϕ)2, a1 = 1/1 − ϕ, and S = ⟨d⟩2/⟨d2⟩. The volume fraction of CS in the composite is ϕ, and this will also be the area fraction of CS in a cross-section. The expression can be reduced to the monodisperse solution by allowing S → 1. Experimentally, S is the inverse of the polydispersity determined by DLS. In the present case, the value of S is determined by geometry. Given a monodisperse hard-sphere liquid in three dimensions that is cut with a plane to produce a 2D slice, the diameters of the disks would have a range of values, which create the polydispersity. In the 2D slice, the average diameter of the disks is related to the diameter of the spheres as disk = π/4 . One can also derive the inverse polydispersity of the disks, S = 3π2/32 = 0.925. In Figure 3, we present P(x) as a function of filler volume fraction and scaled pore size for the situation where S = 0.925. At low volume loadings, the distribution is very broad, with a maximum at r/ ≈ 3. As the volume loading increases, the distribution becomes narrower as the most probable pore size is smaller. For a polydisperse hard sphere liquid, the diameter relationship would remain the same, but the polydispersity in

⎛ ⎛ R g ⎞⎞ ⎛ ⎛ Rg ⎞ ⎞ P ⎜⎜y , ⎜ ⎟⎟⎟ = 4ϕS⎜⎜2a0⎜ ⎟y + a1⎟⎟ ⎝ ⎝ ⟨d⟩disk ⎠⎠ ⎝ ⎝ ⟨d⟩disk ⎠ ⎠ 2 ⎡ ⎛ ⎛ R ⎞ g exp⎢ −4ϕS⎜⎜a0⎜ ⎟ y2 ⎢ d ⟨ ⟩ ⎝ ⎠ disk ⎝ ⎣ ⎛ R g ⎞ ⎞⎤ + a1⎜ ⎟y ⎟⎥ ⎝ ⟨d⟩disk ⎠ ⎟⎠⎥⎦

(5)

This expression explicitly demonstrates the inclusion of the ratio Rg/disk, the key parameter in the model that relates the molecular weight of the diffusing polymer chain to the diameter of the filler particles. 3). Constructing the Statistical Average. Finally, the statistical average that represents the diffusive flow of spheres through an ensemble of cylindrical pores is constructed. The flux through the ensemble will be the flux through each cylinder, integrated over the distribution function of the radii of the cylinders. In this way, as a consequence of Fick’s first law, the reduced flux is directly proportional to the reduced diffusion coefficient, and the following equation is constructed:

∫ F(y)P(y)y 2 dy D = (1 − ϕ) D0 ∫ P(y)y 2 dy

(6)

In this expression, y2 is included within the integral because the flux is proportional to the cross-sectional area of the cylinder. We use the Gaussian polymer chain partition function F(y) given in eq 1 as the analytical form to describe the numerical results of Haberman and Sayre. In the denominator, the flux is proportional to the area of each tube multiplied by the probability of having a tube of that diameter. The ratio of the integrals calculates the average value of the relative flux, . Because this is diffusive dynamics, the ratio of the fluxes equals the ratio of the diffusion coefficients. This must be multiplied by the factor (1 − ϕ), which represents the total area of the tubes − the area available for diffusive transport − where ϕ is the volume fraction of nanoparticles in the composite. (The volume fraction and the area fraction are identical for composites.) Thus, the reduced diffusion coefficient of an ensemble of spheres diffusing through cylindrical pores can be predicted.

Figure 3. Normalized pore distribution function for a 2D slice through polydisperse (S = 0.925) hard spheres, as a function of pore radius scaled by average 2D disk diameter, shows that the pores become smaller as the loading increases. The curves represent nanoparticle volume fractions ranging from 0.01 (blue) to 0.50 (mustard). 15678

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Because the flow of spheres through cylinders has been shown to have the same mathematical form as the flow of Gaussian polymer chains through cylinders, we posit that eq 6 can be used to describe the reduced diffusion of deuterated polymer chains, in the melt phase, through an ensemble of cylindrical pores filled with protonated polymer. In the melt and at low loadings of nanoparticles, Rg ≈ N1/2, the entangled chains obey Gaussian statistics. Studies suggest that the Rg of polymers in a nanocomposite with spherical nanoparticles of the sizes used in the experimental results below remains unperturbed.21 The reduced diffusion coefficient D/D0 can now be predicted as a function of the material parameters, disk/Rg, S, and ϕ, using eq 6. The result for polymer nanocomposites with monodisperse spherical particles is presented in Figure 4 as

To compare D/D0 from the model with the experimental data, we need to transform the parameters disk/Rg and ϕ into ID/2Rg. Because the diffusive flow in one direction through an ensemble of cylinders is modeled, we use the expression for the 2D interparticle distance ID2D, defined as:5 ⎧ ⎫ ⎞1/2 2 ⎪⎛ ϕ ⎪ ID2D = ⟨d⟩disk ⎨⎜ max ⎟ e σ − 1⎬ ⎪ ⎪ ⎭ ⎩⎝ ϕ ⎠

(7)

This ID2D is related to the average 2D pore diameter, which is 2, as expressed in eq 4. This connection can be used to determine the appropriate value of ϕmax, which is the maximum packing fraction for random packing in two dimensions. Figure 5 plots eq 4 along with eq 7 (normalized by disk) for the

Figure 4. Reduced diffusion coefficient of polymer chains through an ensemble of cylindrical pores as a function of volume loading for various values of the parameter disk/Rg.

Figure 5. Comparison of the 2D and 3D pore/cavity sizes resulting from n-point probability formalism and approximate expressions. The average 3D cavity radius normalized by the average particle diameter is well-approximated by eq 8, and the average 2D pore radius, eq 4, normalized by the average particle diameter is approximated well by eq 7

function of ϕ for disk/Rg varying from 0.5 to 8. Several interesting features arise. First, as expected, the model clearly shows that the diffusion coefficient decreases with increased volume loading. This is expected because the polymer chain is diffusing in a more crowded environment. Second, at fixed ϕ, there is a reduction in diffusion as the polymer chain increases in size relative to the particle size, that is, as disk/Rg decreases. For large values of disk/Rg, corresponding to situations where the polymer chain is much smaller than the size of the filler particle, the decay is relatively slow as volume loading increases. For small values of disk/Rg, corresponding to situations where the polymer is larger than the filler particle, the decay is relatively fast. This happens because, at a constant filler loading, where ID is constant (in both two and three dimensions), increasing the polymer Rg reduces the ratio of ID/ 2Rg, and the polymer diffusion is occurring in a more confined environment. Third, the diffusion coefficient is always reduced under the range of parameters modeled. At 1 v % loading and disk/Rg = 8, the diffusion is reduced by ∼5%, even though the pore radius is four times larger than the chain size. This demonstrates that excluded volume is effective at reducing diffusion. Note that even at high volume fractions the reduced diffusion coefficient does not go to zero. Chains are still able to diffuse because of the statistical nature of the hard-sphere configuration. Although the average pore diameter can be smaller than the Rg of the polymer, there still exist some larger pores through which polymer could flow.

value of ϕmax = 0.85. This value of ϕmax is in accord with theories for the random packing of disks in two dimensions, where the maximum packing fraction has been reported to range from 0.78 to 0.89.22 Equation 7 provides an excellent approximation to the more detailed theory,20 especially at the low volume loadings of interest here. Thus, we can conclude that with ϕmax = 0.85, ID2D = 2 at low loadings, and in this range the parameter ID2D/2Rg is practically the same as /Rg. As a digression, Figure 5 also shows the relationship between the 3D average cavity radius (calculated numerically from the appropriate equations20) and the ID3D approximation5 ID3D

⎧ ⎫ ⎞1/3 σ 2 ⎪⎛ ϕ ⎪ max = ⟨d⟩sphere ⎨⎜ e − 1⎬ ⎟ ⎪ ⎪ ⎭ ⎩⎝ ϕ ⎠

(8)

Here, too, the agreement is very good when ϕmax = 0.72. These values for the maximum packing fractions in two and three dimensions should be interpreted as fitting parameters and not as values of the volume loading at maximally jammed random packings of hard particles. Note that compared at the same volume loadings, the 3D cavity is smaller than the 2D cavity, especially at low volume loadings. 15679

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REDUCED DIFFUSION COEFFICIENTS: MODEL VERSUS EXPERIMENT In this section, the model is applied to the published experiments measuring polymer diffusion into nanocomposites. The first example covers the diffusion of deuterated polystyrene of different molecular weights into polystyrene nanocomposites with various volume loadings of CS.4 The diameter of the CS was 28.7 nm, with a variance of 0.107, when a log-normal particle size distribution was used. 8 The weight-average molecular weight, M w, of the host polymer was kept constant at 265 kDa. The deuterated polystyrene Mw values were 49, 168, and 532 kDa, with Rg = 5.95, 11.0, and 19.6 nm; correspondingly, disk /Rg was 3.80, 2.06, and 1.15, respectively. Figure 6 plots D/D0 versus ID2D/2Rg for the

Figure 7. Reduced diffusion coefficient versus confinement parameter for the deuterated PMMA/PMMA couple from ref 6.

polymer chain was 100 kDa, with Rg = 8.2 nm. There were three PMMA matrices into which the polymer diffused. The first matrix had a nanoparticle diameter of 50 nm, with polydispersity of 0.069 and a PMMA Mw of 337 kDa and disk/Rg = 6.10. The second matrix had a nanoparticle diameter of 28.6 nm with polydispersity of 0.115,8 with a PMMA host Mw of 600 kDa and disk/Rg = 3.54. The third matrix had a nanoparticle diameter of 13 nm with polydispersity of 0.108, with a PMMA Mw of 337 kDa and disk/Rg = 1.59. The measurement and the model agree at low confinement, ID2D/2Rg > 5, and the data are again higher than the model for large confinement. Because dPMMA interacts favorably with silica, the role of enthalpic interaction in this study could be important. The third and fourth examples are from additional polystyrene systems. In the third example,7 the particles are 51 nm in diameter, with polydispersity of 0.22. They were functionalized with a low-molecular-weight initiator. Mw values for the penetrant deuterated PS were 49 kDa, with disk/Rg = 8.59, and 532 kDa, with disk/Rg = 2.61. The Mw of the matrix PS was 160 kDa. The results are displayed in Figure 8a. The fourth example comes from the same work.7 The nanoparticles are functionalized with polystyrene brushes with Mn = 87 kDa. By varying the molecular weight of the penetrant deuterated polystyrene tracer chain, the interaction between the penetrant and the particle’s brush varies. The penetration of the tracer into the brush decreases as the tracer Mw increases. This results in wet brushes for low-Mw penetrants and dry brushes for high-Mw penetrants. So there is a smaller effective diameter for the filler particles when a low Mw penetrant is the tracer and a larger effective diameter for the filler particles when the tracer has a larger Mw. This variation in the effective diameter of the particles was included within the model to accurately capture this effect. The results are displayed in Figure 8b. Both of these examples are interesting because of the large values of ID2D/ 2Rg, and it is observed that the model works well at these large interparticle separations.

Figure 6. Plot of reduced diffusion coefficient versus the confinement parameter for the polystyrene data from ref 1

experimental results, along with the model. Several interesting features are observed. First, the theory itself nearly collapses onto a master curve. The reduced diffusion is almost independent of the ratio disk/Rg. This was unexpected, given the complexity of the set of eqs 2−6 and because there is no way to separate out the variable ID2D/2Rg from these equations. In addition, in Figure 4, there is no scaling relationship between D/D0 for different values of disk/Rg. As a further comparison, D/D 0 is predicted from the simple exponential equation:

⎛ Rg ⎞ D = exp⎜ − ⎟ D0 ⎝ ID2D ⎠

(9)

This expression is an empirical approximation to the model results. The data still collapse onto a master curve when plotted versus ID2D/2Rg, as it did when plotted against ID3D/2Rg in the initial publication. For values of ID2D/2Rg > 5, the model and the data appear to converge. In this region, the excluded volume effect is likely responsible for the reduction in the reduced diffusion coefficient. At lower ID2D/2Rg, the theory predicts a lower diffusion coefficient than observed in experiments. The diffusion of dPMMA into PMMA nanocomposites comprises the second example, shown in Figure 7.6 In this experiment, the Mw of the deuterated PMMA penetrant

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DISCUSSION The analysis shows that excluded volume effects can account for the observed reduction in polymer diffusion for values of ID2D/2Rg > 5. In this region, the excluded volume model 15680

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by a pore.16,17 Indeed, our data scales with the predictions from this theory.4 The subsequently developed translocation theory demonstrated a more analytical approach to this problem.18,19 The remaining challenge is to map the structure of the nanocomposite onto the geometrical parameters of the model and then determine how much flux this mechanism can contribute. This might be accomplished by incorporating the results from the n-point formalism into Muthukumar’s theories, using the 2D expressions for the bottleneck sizes and the 3D expressions for the cavity sizes. Consideration of the cartoon in Figure 1 suggests that the distinction between constraints imposed by nanoparticles and those imposed by polymer entanglements may be distinguished by their respective scale sizes. Schneider et al. used these concepts to relate the apparent confinement length of a polymer chain in a nanocomposite to the geometric parameters of the filler configuration and the topological tube confinement.2 They concluded that there was strong evidence of chain disentanglement in polymer nanocomposites at high loading, which would reduce the melt viscosity and hence increase the diffusion coefficient. Direct measurements of entanglement density are not available. It is inferred from other observables. Our diffusion data, combined with this model, do not contradict the hypothesis but do not prove it either. It is also unknown when the disentanglement could occur during sample processing. In the initial formulation, the polymer chains are above the overlap threshold concentration.8 Prior to forming the diffusion couples, the samples were annealed for several days at a temperature above the polymer Tg,8 and it is here that disentanglement could occur. Measuring diffusion as a function of annealing time prior to the formation of the couple could provide further interesting results. One difference between the constraints imposed by entanglements and those imposed by filler particles is the ramification on contour length fluctuations. In a polymer melt, the polymer can leak through the reptation tube wall, decreasing the overall length of the contour and providing another relaxation mechanism. In a nanocomposite, the polymer is rigidly restricted within the cavity between nanoparticles, which would suppress contour length fluctuations. This could slow down the relaxation of the polymer and reduce the diffusion coefficient. Our model can be compared with a previous model suggested by Ogston and coworkers.23 They derived a model for sedimentation and diffusion of macromolecules in solutions of the rodlike polymer hyaluronic acid. Their theory predicted an exponential decay for these observables, with the decay term proportional to the size of the macromolecule and to the square root of the concentration of hyaluronic acid (like the CS filler here). This prompted the empirical model given by eq 9, where Rg represents the size of the macromolecule and ID2D, as seen in eq 7, is proportional to the inverse of the square root of the volume loading. Therefore, the ratio Rg/ID2D is analogous to what was derived by Ogston. As demonstrated in Figures 6−8, when disk/Rg < 3, the model is very similar to this exponential decay, which was unexpected. The model is not exactly an exponential, and we have not been successful at deriving this form from our equations. For disk/Rg > 3, the model does not predict an exponential decay, even if a proportionality constant is included in the argument of eq 9. Enthalpy, which accounts for the attractive interaction between the polymer and the filler, cannot account for the

Figure 8. (a) Data for reduced diffusion coefficient versus confinement parameter for polystyrene with hard nanoparticles from ref 7. (b) Data for reduced diffusion coefficient versus confinement parameter for polystyrene with soft nanoparticles from ref 7. The legend represents the molecular weight of the penetrant chain. The model converges with the data for large values of ID2D/2Rg.

predicts a pseudomaster curve for the reduced diffusion coefficients, in good agreement with experimental results. At lower values of ID2D/2Rg, the excluded volume model predicts slower diffusion than observed experimentally. From the data, we infer that there may be another mechanism that is enabling the flux of material through the nanocomposite. There are at least two possible concepts to explain this effect. The first is that under high confinement polymer chains can distort and wriggle through pores that are smaller than Rg. The second concept is that under high confinement the polymer chains disentangle and the diffusion coefficient of the unentangled chain is larger than that of the entangled chain. These two possibilities, taken either separately or together, would result in faster diffusion of the chains. Diffusion of polymer chains in solution through openings smaller than R g has been theoretically described by Muthukumar and collaborators.16−19 The entropic barrier diffusion model that was initially developed, and implemented numerically, modeled the diffusion between cavities separated 15681

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been developed. A clear connection was built between the excluded volume partition function for a polymer chain and the increased drag that a sphere experiences when diffusing through a cylindrical pore. The n-point probability formalism was used to convert the pore sizes within a nanocomposite with spherical fillers to a model distribution of aligned cylinders. Using these facts, an expression for the reduction in diffusion that an ensemble of hard spheres would experience while diffusing through a porous medium was derived. We extended the interpretation of hard-sphere diffusion to that of polymer chains moving through a porous system. This is acceptable because, mathematically, the model used for the diffusion of a Gaussian chain is the same partition function that was found to be a good approximation to the solution of the fluid flow problem of Haberman and Sayre. In addition, the diffusion experiment occurs over distance scales that are larger than Rg, where chain diffusion is Fickian. From this, we were able to model the reduction in diffusion that polymer chains would experience when diffusing into nanocomposites with spherical fillers. The assumption that fluid dynamics of a sphere diffusing through a cylinder is applicable to polymer chain diffusion is not obvious. It was demonstrated that this model can be favorably compared with experimental data. At values of ID2D/ 2Rg > 5, the theory converges with the data with no adjustable parameters. However, this theory fails to account for the reduced diffusion properly when ID2D/2Rg < 5. The present theory overestimates the reduction, suggesting that some other mechanism is responsible for increasing the flux through the nanocomposite, and it is suggested that chain disentanglement or non-Gaussian conformations may account for the discrepancy. In the future, we will attempt to elucidate the mechanism of diffusion in this confined regime by combining the geometric arguments for the cavity and pore sizes with reptation dynamics.

discrepancy at high loadings. One initially expects attractive interactions to slow diffusion even further. Alternatively, enthalpy may not play a major role in the interdiffusion experiment because for each enthalpic interaction site enjoined between the deuterated polymer chain and the filler particle an enthalpic interaction between the matrix polymer chain and the filler particle must be eliminated. This balance may nullify this effect. Indeed, the fact that the data for PMMA and PS matrices are similar argues against an enthalpic effect. Deviations from ideal dispersion may contribute to the increased diffusion seen in the data set at high particle loadings. We know from SAXS and TEM measurements that minor clustering of the filler particles existed in these nanocomposites.8 The effect of clustering is to shift the pore-size distribution to higher values. While contact between the particles would block diffusion, the larger open areas between the clusters would increase diffusion by reducing the effective excluded volume. Currently, we cannot translate the observed structure factor into a pore size distribution, so this effect cannot yet be quantified. At high loadings, the observed diffusion coefficient is about ten times greater than predicted by the model. The minor clustering observed in the composites should not result in such a dramatic shift in the data. Finally, the present model is unable to capture diffusion in highly loaded systems, and a more accurate description needs to be developed. In real composites, polymer chains can extend beyond the cylindrical cross sections constructed in this model. The physical boundaries created by spherical inclusions present convex surfaces, not a uniform concave curvature, to the chain. The reduction in the number of allowable configurations will differ from eq 1. A potential way to address this problem would be to use the partition function for a random walk around a sphere, similar to what has been previously done.24 At high loadings, the statistical average would need to emulate the various configurational arrangements of multiple spheres, subject to the constraint of a fixed volume loading. This could provide an alternate model to capture the reduction in configurations as a function of volume loading and chain size. As stated in the Introduction, we do not expect this model to work when a polymer chain occupies more than one cavity at a time. This transition can be described by relating the end-toend distance to the 3D cavity diameter. With appropriate values, it is found that this condition is breached when the filler loading exceeds ∼1 v %, or roughly when ID2D/2Rg ∼ 5, in accord with data presented. The correlation between pores is not addressed in this work. At high loadings, while there are some large pores that can allow diffusion, they may not be connected in any sense that will allow macroscopic polymer transport. This will only be relevant in cases of high loading. In those circumstances, there will be less flux than predicted by the model. As such, this would not account for the deviation of the data from the model. Although nanocomposites containing anisotropic particles (platelet, rods, etc.) are of great interest, extending the present analysis to systems with nonspherical particles is intractable. In particular, we are unable to account for the effects of nanoparticle orientation on the unoccupied space between particles, and any statistical averaging may ultimately be correlated to an effective sphere model.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: jeff[email protected]. Phone: +1 302-695-1129. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Nigel Clarke from the University of Sheffield and Ken Schweizer from UICU for valuable discussions. This research was supported by the National Science Foundation NSF/EPSRC Materials World Network DMR-1210379 (R.J.C., K.I.W.) and DuPont CR&D (R.J.C.). Support was also provided by the NSF/MRSEC-DMR 1120901 (K.I.W., R.J.C.) and Polymer Programs DMR09-07493 (R.J.C.).

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REFERENCES

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CONCLUSIONS In summary, a model that explains the reduction of polymer diffusion into nanocomposites with spherical nanoparticles has 15682

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