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Keywords:

  • bond fluctuating chains;
  • bond-fluctuating sheet;
  • clay platelet;
  • coarse-grained model;
  • exfoliation;
  • Monte Carlo simulation

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MODEL
  5. RESULTS AND DISCUSSION
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES AND NOTES

Exfoliation of a stack of sheets (a model for clay platelets) in a dynamic matrix of polymer chains is investigated by a computer simulation model. How the interplay between the thermodynamics (interaction-driven) and conformational (structural constraints) entropy affects the exfoliation of sheets is the subject of this study. A stack of four sheets with a small initial interlayer distance constitutes the layer on a discrete lattice. The layered platelets are immersed in a matrix represented by the mobile polymer chains which occupy a fraction (concentration) of the lattice sites. Both sheets and chains are modeled by the bond-fluctuation mechanism and execute their stochastic motion via Metropolis algorithm. An attractive and a repulsive interaction between the polymer matrix and platelets are considered. Exfoliation of the sheets is examined by varying the molecular weight of the polymer chains forming a dynamic network matrix with various degrees of entanglements. At low-molecular weight of the polymer, exfoliation is achieved with repulsive interaction and the exfoliation is suppressed with attractive matrix as sheets stick together via polymer mediated interaction introduced by intercalated polymer chains. Increasing the molecular weight of the polymer matrix suppresses the exfoliation of sheets primarily due to enhanced entanglement—at high-molecular weight (with the radius of gyration of polymer chains larger than the characteristic linear dimension of the platelets), the stacked (layered) morphology is arrested via entropic trapping and exfoliation ceases to occur. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2696–2710, 2008


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MODEL
  5. RESULTS AND DISCUSSION
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES AND NOTES

Understanding the exfoliation of clay platelets in a polymer matrix is very important in designing nano-clay composites with the desirable thermal and mechanical characteristics1–9 such as reducing gas permeability and flame propagation and enhancing strength with respect to tensile response. Controlling the distribution of clay platelets in appropriate polymer matrix is crucial to achieve such thermomechanical characteristics. Despite enormous interest in recent years,1–21 technical challenges remain in achieving the uniform distribution and exfoliation of clay platelets due to their unique characteristics such as difficulty in arresting the intraplatelet deformations (wrinkles) to enhance entropy with the large surface/interface area in the exfoliated or intercalated state in a composite matrix. Investigating the multi-scale characteristics from microscopic details to macroscopic observations of the exfoliation and intercalation of the layers of clay platelets, often difficult in laboratory experiment, is however key to finding ways to control their distribution. Computer simulation experiments17–21 have become viable learning tools with idealized and relatively simple models in such systems10–18 where it is easier to invoke hypotheses and constraints and vary the range of parameters usually inaccessible in laboratory experiments.

The size of a typical clay platelet (e.g., montmorillonite) is of the order of hundreds of nanometers. Clay platelets stack together with an interlayer spacing of the order of a nanometer to form a layered structure. Much of the previous studies10, 11, 15–18 seem to deal with the intercalation of polymer in a slit or ‘gallery’ formed by two constrained surfaces/platelets with little or no dynamics. Although some of the theoretical investigations15, 16 are too crude to incorporate relevant fluctuations others lack ample mobility of platelets in appropriate host space. For example, Lee et al. have used molecular dynamics simulation to study the intercalation of polymer chains in a slit with fixed surfaces17 and surfaces with constrained movement along the longitudinal direction for variable slit spacing.18 Sinsawa et al.19 have examined the intercalation and exfoliation processes of clay sheets using an off-lattice molecular dynamics simulation. It was difficult to investigate desired structural evolution (in particular, long time exfoliation) due to relatively long relaxation time—a typical bottleneck in computer simulations of such complex systems with off-lattice methods. Thus, the utility of such a powerful tool with off-lattice simulations is severely limited despite its apparent strength in probing the microscopic structural details in depth.

We have proposed a bond-fluctuating model22 for clay platelet with ample degrees of freedom on a discrete lattice to capture some of the pertinent details including multi-scale mode dynamics.20, 21, 23–25 Unlike the kink analysis26, 27 of platelets, we have studied the exfoliation of a stack (layer) of sheets by the bond-fluctuation model. Effects of solvent on the exfoliation and dispersion of platelets are examined in a host matrix with effective solvent20 where each lattice site other than those occupied by the sheets represents the solvent. Although this study provides insight into how the exfoliation is enhanced by increasing the temperature and changing the quality of solvent, the solvent fluctuations are neglected. Very recently, we have examined28 the exfoliation of a layer of platelets in a matrix with explicit solvent particles. We find that, with a relatively high attractive interaction between sheets and the solvent particles, the layer of sheets remains intact via intercalation of solvent particles in the interstitial spacing (gallery) between the sheets. For low attractive to repulsive interaction between sheets and solvent particles, the sheets are easy to exfoliate. We extend this study by incorporating polymer chains (i.e., replacing solvent particles by chains) in presence of a stack of platelets here and examine the effect of the molecular weight of polymer chains and their interaction with sheets on exfoliation. The model is described in the next section followed by results and discussion with a conclusion at the end.

MODEL

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MODEL
  5. RESULTS AND DISCUSSION
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES AND NOTES

We consider a cubic lattice of size L3 as a host matrix. A layer of stacked platelets each of size Lmath image is placed in the center as before20, 21 but in presence of polymer chains each of length (molecular weight) Lc as shown in Figure 1. A polymer chain is a set of Lc nodes tethered together by fluctuating bonds while a platelet (sheet) is described by a set of Lmath image nodes tethered together by flexible bonds on a square grid. The minimum distance between the nearest neighbor nodes (the bond length) in both chains and sheets is twice the lattice constant due to excluded volume constraints. A node is represented by a unit cube (i.e., eight sites) of the lattice22, 29 as described before.20, 21, 23–25 The bond length between consecutive nodes (both in chains and sheets) can fluctuate between 2 and √10 with an exception of √8 as the node attempts to move stochastically (see below) to one of its 26 adjacent cubes (referred as sites in the following description).22, 29 Initially, four sheets each in a planar square grid configuration are stacked together with a small separation of four lattice constants in the center of the lattice. Attempts are made to insert chains randomly in their random configurations to occupy a fraction ϕ of the lattice sites. Note that the probability of inserting chains between the interstitial spacing of sheets is not zero but very small for long chains. However, attempts are made to move chains randomly (keeping the sheets in the same initial position) for million time steps to prepare the initial state of the sample (see below).

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Figure 1. Configuration of a layer of four platelets each of size 162 immersed in polymer chains each of length Lc = 16 at a concentration 0.2 on a 643 lattice at 103 time step. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Apart from the excluded volume effects, polymer chains and sheets interact with a short range square-well pair potential between their nodes. The interaction energy of a node is described by,

  • equation image

where i runs over each node and j over neighboring sites within a range r = √6. Polymer–platelet interaction (via their node–node interaction V(c,s) = ε) is the same for all sites within the range (r). We have used ε = −1 and 1 to represent sheets in polymer matrix with attractive and repulsive interactions, respectively. Sheet–sheet and polymer–polymer interactions are purely excluded volume. Distance r is measured in unit of lattice constant and the interaction energy in arbitrary unit as before.20, 21, 23–25

The Metropolis algorithm is used to move each particle (node of sheets or polymer chains) stochastically. A particle (cube centered) at a site i and one of its neighboring adjacent sites j (out of 26) are selected randomly. If site j is empty, then change in length of each of its connected bonds is evaluated in a configuration in which the node was to move from site i to site j. If the length of each connected bond lies within the allowed fluctuation, then we evaluate the change in energy ΔE due to a possible change in configuration. The particle is finally moved from site i to site j with a Boltzmann probability exp (−ΔE/T), i.e., we use Metropolis algorithm to move the interacting nodes. The reduced temperature T is a measure, in arbitrary units, involving the Boltzmann constant and the interaction energy strength (ε). Periodic boundary conditions are used along each direction. Attempts to move each particle once (on average) define the unit Monte Carlo step (MCS) time constant.29

Polymer chains are moved randomly while keeping the sheets in the initial stacked configuration to mix and prepare the sample. The Monte Carlo simulation is then performed for a sufficiently long time (typically of the order of million time steps here) to evaluate morphological changes in the distribution of sheets and polymer chains. We study the density profiles of sheets and chains and variations of the root mean square displacements and radius of gyration of chains and sheets with time steps. A number of independent samples are used to obtain statistical averages of these quantities. Different lattice sizes are also used to test for the finite size effects on the qualitative results; no significant finite size effects are observed to alter our conclusions. Our main goal is to examine the effect of molecular weight and the nature of interaction between the polymer chain matrix and the sheets on their exfoliation and intercalation.

RESULTS AND DISCUSSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MODEL
  5. RESULTS AND DISCUSSION
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES AND NOTES

Data are generated on a 643 sample. After a layer of four sheets each of size 162 is placed at the center of the lattice, attempts are made to insert many (Nc) chains, each of length Lc to occupy a fraction ϕ of the lattice sites. The polymer volume fraction ϕ = 0.1, 0.2 and chain length Lc = 4, 8, 16, 32, 48, 64 are considered. It should be pointed out that it is difficult to insert all possible number of chains particularly those with high-molecular weight at higher volume fraction despite a relatively large number of attempts. The polymer matrix with higher molecular weight may therefore have more fluctuations—a caution should be exercised while carrying out such simulations. Before implementing the Metropolis algorithm to move each node (sheets and polymer chains) stochastically, attempts are made to move polymer chains randomly for as long as 106 time steps to prepare a thoroughly mixed sample. These simulations become CPU intense easily with the larger samples. Therefore the number of independent runs and size of samples, chains, and sheets are carefully selected to gain as much insight as possible. Simulations are carried out for sufficiently long time to clarify the trends (see below). We restrict here to an attractive (ε = −1) and a repulsive (ε = +1) interaction between the polymer chains and the sheets within a range r = √6 (in unit of lattice constant) at a temperature T = 1 (in appropriate unit of the Boltzmann constant and the interaction energy).

A set of typical snapshots at the end of simulations are presented in Figures 2 and 3 with attractive and repulsive interaction between the polymers and the sheets with the polymer matrix density ϕ = 0.2. Let us examine these figures to probe the effect of interaction between the layered sheets and the polymer chains and their molecular weight on the exfoliation of platelets. Because of the entropic constraints on the conformations, shorter chains can move and intercalate or intersperse more easily than the chains with higher molecular weights.

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Figure 2. Snapshots of the final configurations with the attractive (ε = −1) interaction between polymer matrix and sheets respectively) after 106 time steps. Sample size 643 is used with the polymer volume fraction ϕ = 0.2 and polymer chain lengths Lc = 4 (first row, first column), 16 (first row, second column), 32 (second row, first column), and 64 (second row, second column).

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Figure 3. Snapshots of the final configurations with the repulsive ε = 1 interaction between polymer matrix and sheets respectively) after 106 time steps. Sample size 643 is used with the polymer volume fraction ϕ = 0.2 and polymer chain lengths Lc = 4 (first row, first column), 16 (first row, second column), 32 (second row, first column), and 64 (second row, second column). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Snaps with the lowest molecular weight (Lc = 4) polymer matrix show that the sheets maintain their layered configuration with ε = −1 (attractive) interaction (Fig. 2) and disperse well with ε = +1 (repulsive) interaction (Fig. 3). One may draw an immediate conclusion that the type of interaction between the polymer matrix and sheets affects their exfoliation. The concept of gallery between the sheets and intercalation of polymer chains therein are obviously more suitable for the layered configuration with ε = −1 where sheets are held together via interactions with their interstitial polymer chains. The repulsive polymer matrix (ε = +1) leads to a well-exfoliated configuration (Fig. 3 with Lc = 4). There is no well-defined layering or gallery at the end of the equilibration. The intercalation of chains in the initial stacked structure with gallery tears the sheets apart in time where the concept of gallery or intercalation becomes irrelevant.

Apart from conformational constraints, interaction between sheets sets in via interstitial polymer chains (solvent) which maintains the layered structure with ε = −1 and exfoliates (with relatively isotropic dispersion) with ε = +1. Despite enormous literature on the subject we are not aware of any computer simulation that shows intercalation and exfoliation so clearly with freely mobile stacked sheets and polymer chains concurrently. Effect of dynamic polymer matrix on the exfoliation becomes somewhat smeared at higher molecular weights as the constraints on sheets mobility are enhanced. For example, with Lc = 64, the dispersion of sheets is constrained and their stacked (layered) morphology persists due to entangled polymer matrix even with the repulsive solvent (ε = +1). Thus, the entropic trapping (cage) of sheets becomes more clear with Lc = 64 especially at higher polymer volume fraction ϕ = 0.2 (see below). Note that the size of higher molecular weight polymer chains (e.g., Lc = 64) is much larger than the linear dimension of the sheets which may not be realistic for a laboratory sample. However, it accentuates the physical mechanism, e.g., entropic trapping without increasing the concentration of the polymer matrix for a theoretical analysis.

In general, the attractive polymer matrix and its higher molecular weights suppress exfoliation of sheets i.e., it enhances the layering while the repulsive interaction causes them to exfoliate at least at these volume fractions (ϕ = 0.1, 0.2) of polymer chains. These observations seem consistent with theoretical studies by Balazs et al.30, 31 Note that the polymer entanglement becomes dominant with the high-molecular weight polymer (e.g., Lc = 64) even at the polymer volume fraction ϕ = 0.2 where the layering persists regardless of the type of interaction between the polymer matrix and platelets as we will see below. Thus, the layering is favorable in attractive matrix and further re-enforced by higher molecular weight of the polymer chains (see Fig. 2). In contrast, the exfoliation is enhanced in repulsive solvent but suppressed by higher molecular weight of the polymer matrix (see Fig. 3). Identifying the trend in exfoliation of sheets becomes complex at the intermediate molecular weight of the polymer matrix due to interplay between the interactions driven thermodynamics and the underlying structural entropy. Attempts are made to confirm these observations quantitatively in the following.

Density profiles of polymer and sheets may provide some insight into their distribution. Let us define y-axis (normal to the initial platelet planes) as the longitudinal direction; the z- and x-axes constitute transverse directions. The planar density (di) is,

  • equation image

where ρi,j,k = 1 if the lattice site i, j, k (the Cartesian coordinate x, y, z) is occupied by a particle (node) and ρi,j,k = 0 if it is empty. Figure 4 shows the longitudinal density profile of sheets in an attractive polymer solvent with molecular weight Lc = 4–64 at the volume fraction ϕ = 0.2. The oscillatory profile with four maxima in the density of sheets (location of sheets) shows that the layered structure is sustained in the long time. The interlayer spacing is however decreases on increasing the molecular weight of the polymer. Sheets are held together via attractive interaction with the interstitial (intercalated) polymer chains especially at low-molecular weights. Intercalation of polymer chains is more prevalent at lower molecular weight (LcLs) due to lower energy. As a result the intersheet distance is higher due to intercalation of shorter chains. Intercalation becomes increasingly difficult with the higher molecular weight polymer (Lc > Ls). The probability of adsorption of polymer chains to the external surfaces of the layered sheets increases with Lc while their entanglement becomes more dominant. For example at Lc = 64, polymer chains seem to form a dynamic cage surrounding the sheets via (i) adsorption at the sheets and (ii) entanglement of chains. Both, the entanglement of the polymer matrix and its attractive interaction with the sheets push the sheets closer.

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Figure 4. Variation of the density of sheets with the longitudinal distance. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = −1, 10 independent samples, 106 time steps. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Corresponding longitudinal density profile of polymer chains is presented in Figure 5. One can immediately note the complementary oscillation in polymer density in the same regions of the lattice, e.g., zx planes around y = 20–44. The complementary variation of the polymer density to that of the platelets is rather easy to see by comparing Figures 4 and 5. For example, the dominant polymer density maxima peaks for Lc = 4 are around y = 24, 30, 36 where the platelet density shows its minima. The locations of maxima peaks of the platelet density are around y = 20, 28, 34, 42 where polymer density has its minima. The polymer density is higher between the sheet layers. Increasing the molecular weight of polymer smears their distribution with larger fluctuations in contrast to well-defined and systematic shifts in density peaks for sheets (Figs. 4 and 5). Close examination of Figure 5 however reveals a systematic trend: the magnitude of the maxima peaks increases on increasing the molecular weight, e.g., from Lc = 4 to Lc = 16 followed by a decreasing (reverse) trend with Lc = 32 to Lc = 64, a nonmonotonic dependence of interlayer polymer density on the molecular weight. In the following we try to explain this trend.

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Figure 5. Variation of the polymer density with the longitudinal distance. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = −1, 10 independent samples, 106 time steps. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Because the interaction between the polymer chains and the sheets is attractive, the energy is lower with larger number of chain nodes close to the sheets surface including the interstitial spacing between the sheets. Increasing the molecular weight from Lc = 4 allows more nodes to interact with the sheets which reduces the energy until it reaches the linear dimension of the sheet, i.e., Lc = Ls = 16. A chain (in the interstitial gallery) with the conformation comparable with the two neighboring sheets has lower energy than chains with both lower molecular weight (Lc < Ls) as well as higher molecular weight (Lc > Ls). Chains with lower molecular weights have lower number of nodes surrounding the sheets as the packing of chains with lower molecular weight in the gallery is not as efficient as those with the higher molecular weight and vice versa. On increasing the molecular weight (i.e., Lc = 32, 48, 64), the probability of intercalation of the whole chain in the gallery (interstitial spacing between sheets) become lower due to larger fluctuations with larger radius of gyration. The number of sheet nodes surrounding the polymer chain within the range of interaction is therefore lower for larger chains (Lc > Ls). Chains in the gallery become less close-packed resulting in a lower density. It is extremely difficult to intercalate large chains (Lc = 64) due to comparatively large radius of gyration and its fluctuation and enhanced entanglement of the polymer matrix at ϕ = 0.2 in comparison to a lower polymer density (follows). As a result, the polymer density profile with Lc = 64 exhibits lower density in the sheet regions, platelets layer is arrested by the entangled or spanning network of the polymer cage—an entropic induced layering.

The effects of the molecular weight of the dynamic morphology of the polymer matrix and its interaction with the sheets on the exfoliation and intercalation are clearly seen with the polymer density ϕ = 0.2. It is however worth pointing out that the volume fraction (ϕ) of the polymer matrix plays a critical role as the entanglement of polymer chains and constraints due to their spanning network increases on increasing their volume fraction and vice versa. To illustrate this point, the longitudinal density profiles of the polymer chains at ϕ = 0.1 and 0.2 are presented in Figure 6. Shapes of the profiles are similar with lower polymer density in the galleries of sheets at ϕ = 0.1 except with the high-molecular weight Lc = 64. At lower volume fraction (ϕ = 0.1), polymer chains are able to intercalate into the interstitial spacing of the sheets due to their higher mobility (higher free volume and more relaxation for conformations of chains). The entanglement at ϕ = 0.1 is much lower than that at ϕ = 0.2 where polymer chains form a more sluggish spanning network surrounding the sheets and are unable to intercalate as easily.

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Figure 6. Variation of the polymer density with the longitudinal distance. Statistics: sample size 643, polymer volume fraction ϕ = 0.1 (top) 0.2 (bottom), ε = −1, 10 independent samples, 106 time steps. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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The dynamics of polymer chains and sheets is also examined as they disperse and equilibrate. Figure 7 shows the variations of the root mean square displacement Rc and its components (Xc, Yc, Zc) of the center of mass of polymer chains with the time step on a log–log scale. Linear variation of displacement with the time step on a log–log scale shows that the polymer matrix is in diffusive steady-state. Nearly similar behavior of all components (x, y, z) further shows that the stochastic motion of polymer matrix is isotropic. It is worth pointing out that linear behavior of RMS displacement with the time step is an indication of a power-law dependence, Rc = Dtν, with a power-law exponent ν = 1/2 and diffusion constant D. Although the exponent ν is nearly the same for polymer matrix at each molecular weight Lc, the diffusion constant (the intersection of the linear fit with y-axis is ln D) varies systematically with the molecular weight; D decreases on increasing the molecular weight.

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Figure 7. Variations of the root mean square displacement Rc and its components (Xc, Yc, Zc) of the center of mass of polymer chains with the time step on a log–log scale. Data points at every 2 × 104 time steps are shown. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = −1, 10 independent samples. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Effect of molecular weight on the dynamics of the polymer matrix is more transparent on analyzing the variation of the RMS displacement of the center of mass of the polymer chains with time steps on a normal scale as shown in Figure 8 for both an attractive and a repulsive interaction matrix. A systematic slowing down is clearly seen on increasing the molecular weight regardless of interaction. At high molecular weight, the motion of the polymer chains becomes so slow that the free volume for sheets to move is arrested in a nearly quasi-static state in observable time scale. The dynamics of the polymer matrix slows down considerably on increasing the molecular weight of the polymer chains. We know that the percolation of chains32 depends strongly on their molecular weight and that the percolation threshold decreases on increasing the molecular weight of the polymer chains. Because of the spanning network of polymer chains, with and without their entanglement, the relaxation of free volume becomes very slow. The contrast in relaxation is easier to see in visualization (see also Figs. 2 and 3).

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Figure 8. Variations of the root mean square displacement Rc of the center of mass of polymer chains with the time step for attractive (ε = −1, left) and repulsive (ε = +1, right) matrix. Data points at every 4 × 104 time steps are shown. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, 10 independent samples. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Corresponding variation of the RMS displacement of sheets with the time step is presented in Figure 9. These data have relatively large fluctuations in comparison to that of the polymer chains (compare Fig. 9 with Fig. 7). A close examination of the variation of displacements of sheets with the time steps may however be informative. Despite their small magnitude, the variation of the x, y, and z components of the displacement with the time step are different; especially the variation of Yc with time step is different from that of Zc and Xc. Overall, sheets seem to move more along the longitudinal direction than in transverse directions. However, the range of longitudinal motion is not large enough to make a meaningful impact on the exfoliation of the sheets. The motion of sheets is rather sporadic over the time steps which imply that their dynamics is not as cooperative as that of the polymer matrix in which they are embedded in.

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Figure 9. Variations of the root mean square displacement Rc and its components (Xc, Yc, Zc) of the center of mass of sheets with the time step on a log–log scale. Data points at every 2 × 104 time steps are shown. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = −1, 10 independent samples. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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In repulsive polymer matrix, the variations of the RMS displacements of the polymer chains and that of the sheets with the time steps are presented in Figures 10 and 11 respectively, (corresponding to Figs. 7 and 9 with attractive matrix). Data in Figure 11 are comparatively less fluctuating with the repulsive matrix than the corresponding data in Figure 9 with attractive matrix. As a result, it is possible to evaluate the respective slopes of the plots with better accuracy at least in polymer matrix with low molecular weights (Lc = 4). Effects of interaction, attractive and repulsive, of the matrix on the intercalation and exfoliation are more clear with the low molecular weight polymer (Lc = 4) as discussed above. A 0.5 slope of ln Rc versus t plot in Figure 10 suggests that the polymer chains are diffusing and dispersing well. Corresponding data for sheets in Figure 11 provides a slope of about 0.8 much higher than diffusion (0.5). The repulsive interaction between the dynamic matrix and the sheets drive them to move faster than diffusion and exfoliate well.

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Figure 10. Variations of the root mean square displacement Rc and its components (Xc, Yc, Zc) of the center of mass of polymer chains with the time step on a log–log scale. Data points at every 2 × 104 time steps are shown. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = +1, 10 independent samples. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Figure 11. Variations of the root mean square displacement Rc and its components (Xc, Yc, Zc) of the center of mass of sheets with the time step on a log–log scale. Data points at every 2 × 104 time steps are shown. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = +1, 10 independent samples. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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The exfoliation of stacked sheets and distribution of polymer chains and their dynamics discussed so far is limited to host matrix in which the interaction between the polymer chains and platelets is mostly attractive (ε = −1)—a relatively good solvent environment for sheets to stack together in a layer via polymer mediated interlayer interaction. It would be interesting to explore what happens when the type of interaction between polymer and layered sheets is changed to repulsive, ε = +1. The longitudinal (y) and transverse (x) density profiles of polymer chains and sheets in a repulsive solvent with low molecular weight (LcLs) are presented in Figure 12. Both density profiles (x, y) show that sheets are still distributed around the center of the lattice where polymer chains are sparsely populated—depletion zone for repulsive chains. Although there is some sporadic oscillation in the transverse density profile of sheets along with a complementary lower density of polymer chains, it is not a sign of persistent layered structure. By the basic nature (size and flexibility) of the sheets some fluctuations in their density is expected.

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Figure 12. Transverse (x) and longitudinal (y) density profiles of sheets (ds) and polymer chains (dc) in a repulsive polymer solvent with the molecular weight Lc = 4, 8, 16. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = −1, 10 independent samples, 106 time steps. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Increasing the molecular weight of the polymer solvent leads to entanglement with onset of localized spaces or cages discussed above (see Fig. 3). Especially with the long polymer chains Lc = 64, the layered morphology remains intact. This is due to constraints imposed by the surrounding polymer chains—entanglement-induced entropic trapping. Some layering persists even at a lower molecular weight Lc = 48. Thus, with the repulsive interaction, the exfoliation of sheets is enhanced considerable with the low-molecular weight polymer; the layered configuration is trapped by entangled chains at high-molecular weight.13

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Figure 13. Transverse (x) and longitudinal (y) density profiles of sheets (ds) and polymer chains (dc) in a repulsive polymer solvent with the molecular weight Lc = 32, 48, 64. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = −1, 10 independent samples, 106 time steps. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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Monitoring the density profiles of sheets and polymer chains along with the visual analysis of the snapshots, we are able to predict the probability of exfoliation or sustaining the initial stacked (layered) configuration. The temporal variation of the radius of gyration (Rg) of sheets may provide further insight into the exfoliation of sheets. The radius of gyration Rg of a sheet is defined by,

  • equation image

where rij is the coordinate of the (i, j) node of the sheet with linear dimension Ls, i.e., i, j = 1, 2, …, Ls. Sheets are initially in a stacked (layered) configuration with a small interlayer spacing. If sheets maintain the layered structure, their configuration remains a square plane normal to the longitudinal (y) direction (see Figs. 2, 3). The longitudinal (Rgy) component of Rg may have little variation with the time steps if sheets remain in layered configurations. The longitudinal component Rgy may show relatively larger variations if they were to exfoliate. Note that sheets may exfoliate via their transverse moves alone (i.e., via slipping out of layering), presumably with a somewhat low probability.

In Figure 14 we present the variation of the radius of gyration (Rg) and its longitudinal component (Rgy) with the time steps in polymer matrix with attractive and repulsive interactions with various molecular weights. With the repulsive matrix at low-molecular weight (LcLs), Rg relaxes rather fast to a lower value with nearly the same magnitude. With higher molecular weight matrix (LcLs), a systematic increase of Rg appears with the molecular weight of the matrix apart from trapped (entropic) configuration with Lc = 64. Sheets are exfoliated with an isotropic Rg (nearly the same value, independent of Lc) with lower molecular weight matrix. The longitudinal component (Rgy) increases with the time step as the sheets exfoliate and move away from their high concentration stacked (layered) configuration. The rate of increase of Rgy can be considered as a measure of exfoliation rate at least initially. The growth rate of Rgy decreases on increasing the molecular weight of the matrix and becomes zero with Lc = 64—entropic trapping in a layered structure.

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Figure 14. Variation of the radius of gyration (Rg) and its longitudinal component (Rgy) with the time steps in polymer solvent with repulsive (ε = cs = +1, top row) and attractive (ε = cs = −1, bottom row) interactions with a range of molecular weights. Statistics: sample size 643, polymer volume fraction ϕ = 0.2, ε = −1, 10 independent samples, 106 time steps. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

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In presence of attractive polymer matrix, Rg of sheets relaxes to higher growing values apart from sheets in presence of a relatively high molecular weight (Lc = 64) matrix. Increasing the molecular weight of the matrix leads to a slower growth pattern of the radius of gyration of the sheets with an entropic trapping at Lc = 64. The longitudinal component (Rgy) shows very little growth (compare Rgy with ε = −1 and ε = +1 in Fig. 14) which implies very little movement along the longitudinal direction for sheets to relax. Note that the variation of Rgy even in the long time is too small to alter the stacked structure significantly.

SUMMARY AND CONCLUSIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MODEL
  5. RESULTS AND DISCUSSION
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES AND NOTES

Exfoliation of a stack of flexible sheets is examined in polymer matrix (good and poor environment with attractive and repulsive interactions) with a range of molecular weights by a coarse-grained computer simulation model. Polymer chains and sheets are described by nodes tethered together by fluctuating covalent bonds on a cubic lattice. Chains and sheets interact via their nodes within a short range (r = √6); an attractive and a repulsive interactions are considered for a range of molecular weights (Lc = 4–64). Both chains and sheets are mobile via stochastic movement of their nodes with the Metropolis algorithm. A stack of four sheets are embedded in a polymer matrix with a small interlayer separation initially. Exfoliation of sheets is examined as a function of molecular weight of the polymer chain matrix with an attractive and a repulsive interaction.

In presence of low molecular weight polymer (LcLs), the stacked platelet configuration persists with attractive matrix (ε = −1) via an indirect intersheet interaction mediated by the intercalated polymer chains. The stacked platelets exfoliate in presence of repulsive polymer matrix (ε = +1) with low-molecular weights. At high molecular weight, the stacked (layered) platelet configuration is trapped in a cage via entanglement and/or percolation of surrounding polymer chains, an entropic trapping. Although the entropic cage formed by mobile chains is dynamic, the relaxation time for the cage renewal is too large to provide free volume for sheets to exfoliate. In the polymer matrix with the intermediate molecular weight, the interplay between the interactions controlled thermodynamics and structural constraints leads to a complex dispersion patterns with a varying degrees of layering and exfoliation depending on the molecular weight of the matrix and type of interaction—a subject of continued interest.

Acknowledgements

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MODEL
  5. RESULTS AND DISCUSSION
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES AND NOTES

Support from the Materials and Manufacturing Directorate of the Air Force Research Laboratory, the Air Force Office of Scientific Research (AFOSR) is gratefully acknowledged.

REFERENCES AND NOTES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MODEL
  5. RESULTS AND DISCUSSION
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. REFERENCES AND NOTES