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Layers of stacked platelets are common in clay nanocomposites.1 Uniform dispersion of individual platelets in a composite matrix is often desirable to achieve an optimal benefit from the layered filler. Controlling the distribution of clay platelets in solvent and appropriate nanocomposites is a challenging problem in designing materials with desirable thermal, mechanical, electrical, barrier, and accessible surface area characteristics. Understanding the exfoliation of clay platelets is highly desirable to find ways to control their distribution although difficult to achieve in laboratory samples. In computer simulation experiments, on the other hand, one may invoke constraints and vary the range of parameters generally inaccessible in direct laboratory measurements so as to identify the trends in morphological variations in observable time scales, with direct access to structural and temporal information about the system. Therefore, we would like to address how a layer of stacked clay platelets can disperse in a solvent and how their exfoliation depends on the size of the sheets, temperature, and quality of the solvent. To investigate the large scale structural changes of a layer of sheets, it is important to understand characteristics such as conformational relaxation and dynamics of a single sheet.
We have proposed a coarse-grained model for a sheet and studied its conformation and dynamics in detail in an effective solvent media.2, 3 In our coarse-grained model a sheet is described by a set of nodes tethered (crosslinked) together by flexible bonds in a planar structure. Such a coarse grained description is frequently used in modeling a linear polymer4, 5 including protein macromolecules6 where nodes are tethered together in a chain with a node characterizing a unit consisting of many atoms or molecules such as an amino acid group. Sheets with tethered nodes in a two-dimensional plane possess unique characteristics providing multiple pathways for dynamical modes to propagate and structural fluctuations, a measure of entropy, to dissipate from open edges.2, 3 Enormous efforts have been made in recent years2, 3, 5–12 to study the dynamics and morphological responses of tethered membranes and clay platelets. Using such a model for a sheet, we are focused here on the exfoliation of a layer of platelets.
Sinsawa et al.4 recently investigated the intercalation and exfoliation processes of clay sheets by an off-lattice molecular dynamics simulation. Despite the apparent strength in exploring the microscopic structural details with the large degrees of freedom (for nodes and bonds) with off-lattice computer simulations, it was difficult to explore desired changes in their structure due to long relaxation times. Consequently, the exfoliation time becomes too large to reach desirable states. We have proposed a bond-fluctuating model for the sheet on a discrete lattice to capture some of the pertinent structural details with ample degrees of freedom while accelerating segmental dynamics.2, 3 In this study, we examine the exfoliation of four bond-fluctuating sheets stacked together initially with minimal separations in an effective solvent medium. The model is presented in the next section followed by results and discussion. Conclusions from this work are summarized at the end.
We briefly describe the model already introduced2, 3 to study the structure and dynamics of an interacting tethered membrane or sheet, as mentioned earlier. The host matrix is a cubic lattice of size L3. A sheet is described by a set of nodes connected by bonds with an initial configuration on a planar square grid (Ls2); the distance between the nearest neighbor nodes is twice the lattice constant, the minimum bond length. As described before,2, 3 a node is represented by a unit cube (i.e., eight sites) of the lattice.13, 14 Since, a lattice site cannot be occupied by more than one node because of excluded volume constraints the minimum distance between consecutive tethered nodes, (i.e., the bond length) is two in units of the lattice constant. The bond length 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) similar to the bond-fluctuation model for a polymer chain.13 A layer of four sheets are stacked together with a small separation of four lattice constants (Fig. 1). The underlying lattice structure (Fig. 1) provides the simplicity and computational efficiency while providing ample degrees of freedom for nodes to move and for sheet bonds to fluctuate. The empty lattice sites represent effective solvent medium. The minimum distance between an effective solvent site and a node is unit lattice constant (i.e., half the minimum bond length between consecutive nodes in the sheet) to maximize the collective degrees of freedom of solvent (by increasing its amount and minimize its constitutive size). It is possible to use unit cube or larger blocks of empty lattices sites to represent a constitutive solvent but it will restrict the collective degrees of freedom considerably. Since the size of a unit cube with eight lattice sites is eight times larger than a single lattice site to represent a solvent site, the number of constitutive solvent will be reduced accordingly. As a result, the size of the solvent ensemble will be reduced substantially if we had to consider larger constitutive size to represent a solvent unit. We however restrict to smallest constitutive solvent size to attain a maximum resolution in statistical ensemble phase space for the effective solvent medium.
The interaction energy of a node is described by,
where i runs over each node and j over neighboring sites within a range r = √6. We would like to point out that the range r should be large enough to incorporate appropriate constituents (platelet nodes and solvent) in the local vicinity of a node. For example, the minimum value of r must be two to reach a nearest node in a sheet. The longer range interactions (i.e., with r2= 8, 10) on the other hand require such a large computational resources3 that it would not be feasible to carry out our early exploratory investigations in a reasonable time. Therefore, we restrict interactions to r2= 6 for investigating the exfoliation for now, with the interaction potential currently a distance independent constant. In terms of intraplatelet interactions, no more than the 3rd nearest neighbors interact. For interplatelet interactions, up to eight solvent layers can exist between the sheets and they will still interact. Furthermore, we know3 the effects of temperature and platelet size on the conformation and dynamics of a single sheet with r2= 6. The node–node interaction is excluded volume (hard-core) and the node–solvent interaction [J(n,s) = ε] is the same for all sites within a range (r). Different values of the node–solvent interaction ε = −2, −1, 1, 2 are used, representing sheets in solvent environments of varying affinity for the sheets. Negative and positive interactions attract and repel nodes with strength depending on the magnitude of ε and sites within the range r. Distance r is measured in unit of lattice constant and the interaction energy in arbitrary unit as before.3
Each node moves with the following Monte Carlo (MC) method. A node (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 the change in length of each of its connected bonds is evaluated in a configuration in which the node was to move 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 node is finally moved from site i to site j with a Boltzmann probability exp(−ΔE/kBT), i.e., we use Metropolis algorithm to move the interacting nodes. The reduced temperature is a measure, in arbitrary units, involving the Boltzman constant (kB) and interaction energy strength (ε). Periodic boundary conditions are used along each direction. Attempts to move each node once (on average) define the unit Monte Carlo step (MCS) time constant. The simulation is performed for a sufficiently long time (typically of the order of million time steps here) to evaluate morphological changes in the distribution of sheets. During the simulation, we track the mean square displacements of the center node (Rn2) of each sheet and that of its center of mass (Rc2), and its radius of gyration (Rg). A number of independent samples are used to obtain statistical averages of these quantities. Here, the y-axis is normal to initial sheet configuration and referred to as the transverse direction (Fig. 1), while the in-plane z and x directions are referred as longitudinal directions. Different lattice sizes are used to test for the finite size effects on the qualitative results. Our main goal is to examine the effect of temperature, sheet size, and quality of the effective solvent (i.e., the interaction between the nodes and solvent sites) on the exfoliation of the sheets.
RESULTS AND DISCUSSIONS
The computer simulations are performed at different lattice sizes (643–2003) for a set of parameters i.e., temperatures T = 2, 5, 10, platelet sizes Ls2 = 122–642, and the quality of solvent via node-solvent interactions ε = −2, −1, 1, 2. No finite size effect is detected on the qualitative nature of the variations of the physical quantities, as the cutoff and excess box size limit interactions with periodic images of the layers, and there are no explicit hydrodynamics. Figure 2 shows snapshots of a stack of 322 sheets at the end of simulations with one million time steps at T = 2, 5, 10 with ε = −1. Changes in the structure from the initial configuration (Fig. 1) of each sheet and their exfoliation are clearly visible with different degrees of wrinkles, longitudinal (z, x), and transverse (y) separations, and their distribution at different temperatures. We know that wrinkles propagate along connected pathways of each sheet and dissipate from the open edges during the simulation and are better seen in our visual analysis of their animations. Further, the magnitudes of wrinkling and crumpling depend on the temperature and interaction (strength as well as range) as pointed out in our previous studies.2, 3 For the range of parameters explored here, the overall conformation of each sheet remains nearly flat due to entropy dissipation. The longitudinal (z, x) and transverse (y) separations between the platelets are measures of the extent of exfoliation. Visual inspection of these representative snapshots suggests that the exfoliation dominated by transverse separation here is enhanced on raising the temperature, i.e., comparing snaps at T = 2 and 10 (Fig. 2).
Size Effect on Exfoliation
How does the size (i.e., the molecular weight) of sheets affect the exfoliation? Platelets are stacked together in a layer with a relatively small separation initially (Fig. 1). In a fully exfoliated configuration, the distance between the center of mass of sheets should be larger than their radius of gyration. Sheets can be separated via stochastic movements along both longitudinal and transverse directions. The transverse (y-component) motions of interior sheets are constrained by the sheets at the outer layers of the stack. Therefore, the transverse mobility of interior sheets is enhanced only after the outer sheets separate from the stack by a sufficiently large amount, nominally on the order of the transverse (y) component of the radius of gyration of a free sheet. Intraplaner stochastic motion of nodes leads to various dynamical modes in the longitudinal directions. Exfoliation of interior sheets is possible due to such longitudinal movements. However, the intraplanar (z, x) dynamical modes dissipate rather fast from the edges of the sheets and therefore, the longitudinal (z, x) modes do not seem as effective as the transverse (y) component in propelling the sheets out of their layer configuration at least in the solvent media considered here. Nevertheless, both longitudinal and transverse components of the stochastic movement of sheets in concert play important roles in exfoliation. Obviously, the relaxation time for a sheet to get out of its local environment is larger for larger sheets. The exfoliation, therefore, becomes slower on increasing the size of the sheets as seen from the snap shots presented in Figures 3 and 4. With relatively smaller sheets (Fig. 3), good exfoliation is achieved within the course of our simulations while the larger sheets still remain somewhat in their layered configurations even after a million time steps (Fig. 4). Thus, reducing the platelet size enhances the exfoliation.
Solvent Quality Effect on Exfoliation
Quality of the solvent is also important in exfoliation, particularly at low temperatures. We have considered four solvent environments in a chronological increment in quality with attractive ε = −2, −1 and repulsive ε = 1, 2 node–solvent interactions. Figure 5 shows that the exfoliation is enhanced considerably on changing the quality of solvent from repulsive (ε = 2) to attractive (ε = −2) at the temperature T = 2. The exfoliation is further enhanced on increasing the temperature (i.e., to T = 5) as seen from the snapshots in Figure 6 (also Fig. 2). At such a high temperature, the exfoliation is driven thermally more than by the quality of solvent. This is in contrast to low temperatures where solvent-induced exfoliation is more pronounced.
Beyond the visual analysis of exfoliation presented above, quantitative analysis on mobility and morphology are needed. To this end we have examined the motion of sheets and their radius of gyration. Differences in characteristics of motion of a free single sheet and that of a sheet in stack of layer in the same solvent environment should provide some insight into the exfoliation process. Variations of the mean square displacement of the center of mass (Rc2) of a free sheet and that of a sheet in the layer with the time step (t) are presented in Figure 7 for sheets of sizes 122–642. The slope of Rc2 versus t is a measure of the overall speed of the sheet. Note that the magnitude of the displacement of a sheet in the stack appears to be larger than that of a free sheet (Fig. 7) which does not imply that sheets in stack are more mobile than a free sheet. Let us assume that the mean square displacement (R2) exhibits a leading power-law dependence with the time step (t), i.e., R2= Dtν, where D is a constant and ν is a power-law exponent. The magnitude of the power-law exponent determines how fast the motion is, e.g., ν = 1 refers as diffusion and 2 as drift; the higher the value of ν the faster is the motion. The slopes of the log–log plots (Fig. 7) are estimates of the power-law exponent, which clearly show that a free sheet is more mobile than a sheet in stack particularly with the larger sheets. If stacked sheets are fully exfoliated, the average speed of each sheet should become comparable to that of a free sheet. That is, the data plotted for Rc2 versus t for a free sheet should be parallel to that of a sheet (previously) in the stack. This is indeed the case for relatively small sheets, 122, 162, and possibly for 242 particularly in the long asymptotic time regime as evident in Figure 7. The differences in slopes (i.e., the speed) of a free sheet and that in a layer are however clear (Fig. 7) for larger (322, 482, and 642) sheets: sheets in a stack show subdiffusive behavior (ν < 1). This implies that the motion of a larger sheet in the stack is slow in comparison to that of a free sheet in a similar solvent. Further, the motion of the sheet in the stack becomes slower on increasing the sheet size. Variations of the transverse (y) components of the mean square displacement with the time step should provide similar differences. Figure 8 shows the corresponding data for the variations of the transverse (y) component of the mean square displacements with the time step and is consistent with those in Figure 7. Thus, quantitative analysis of the mean square displacement of sheets in layers shows that the exfoliation is enhanced by reducing the size of the sheets, consistent with our visual analysis (Figs. 3 and 4).
As mentioned above, the exfoliation is accomplished by both transverse and longitudinal motion of the sheets. Differences in the variation of the mean square displacement of a free sheet and a sheet in a stack clearly shows that the mobility in transverse direction plays an important role in exfoliation. How important is slippage in exfoliation? Slippage of a sheet implies larger mobility along the longitudinal directions. The longitudinal (z) component of the mean square displacements of a free sheet and a sheet in stack is presented in Figure 9. Even though data for the longitudinal component (Fig. 9) are more fluctuating than that for the transverse displacements (Fig. 8), it appears that the mobility of a sheet in stack is comparable to that of a free sheet. At least in the earliest stage of exfoliation, it appears that slippage, though important, does not play a dominant role in exfoliation. Slippage also does not necessarily mean the sheets reach an exfoliated state. There are competing physical processes of thermal (Brownian) motion and the medium viscosity that impact whether or not a layer that “slipped” out of a stack will do so completely or will it likely fluctuate between having slipped out and its initial position.
Temperature Effect on Exfoliation
The effect of temperature on the exfoliation can be analyzed by examining the change in the radius of gyration of sheets at different temperatures. More precisely, the changes in the transverse (y) component of the radius of gyration could provide better evidence for exfoliation. Variations of the transverse component of the mean square radius of gyration with the time step are presented in Figure 10 at a low and a high temperature (T = 2, 10) for small to large sheets (162, 322, 642). We see that the growth of the radius of gyration at high temperature is greater than that at the low temperature. As the stack exfoliates, the transverse separation between sheets becomes larger, removing constraints for sheets to explore their conformations. Consequently, the transverse component of the gyration radius grows with exfoliation. Our data clearly show that increasing the temperature can induce exfoliation faster.
It is difficult to exfoliate clay platelets since a platelet (and a tethered membrane) is known to retain its general “flatness” or “planarity” because of dissipation of entropy. Because of the enormous complexity in studying a planar constituent we have to resort to simplifications. Although we are restricted to simplified model systems, our ultimate goal is to understand exfoliation of clay platelets in polymer nanocomposites. Using a bond-fluctuating coarse grained model for a platelet, we have presented a computer simulation study to understand how a stack of sheets exfoliates in an effective solvent medium. In particular, we have examined the effects of temperature, platelet size, and solvent quality on exfoliation. We find that the exfoliation can be enhanced by raising the temperature, reducing the platelet size, and increasing the quality of solvent by introducing more attractive interaction between solvent and the platelets.
Support from the Materials and Manufacturing Directorate of the Air Force Research Laboratory, the Air Force Office of Scientific Research (AFOSR), and the NSF-EPSCoR is gratefully acknowledged.