A tethered membrane and a clay platelet1–20 share common features: both are floppy and possess planar morphology. A tethered membrane is modeled by a set of nodes (constituents) tethered (crosslinked) together by flexible bonds in a planar structure leading to a flexible membrane. The conformation evolves as the segmental dynamics sets in and the global motion takes place if each node of the membrane is allowed to execute its stochastic movement subject to excluded volume and bond-fluctuation constraints as in a self-avoiding sheet (SAS).1 The SAS exhibits a distinct short time segmental dynamics (different than the Rouse dynamics of polymer chains21–23) with a diffusive global motion. Despite the formation and dissipation of wrinkles during the conformational relaxation, very little conformational change is observed in regard to the scaling of the radius of gyration with the molecular weight (the size) of the SAS.1 It is well established1–16 that the conformation of a flexible membrane with self-avoiding constraints remains nearly flat as the entropy dissipates through its open edges.
How flexible can such conformational-invariant membranes be in a solvent? Can a sheet (membrane) crumple or collapse as a result of external variables such as the interaction between constituents of the membranes and the solvent,11 induced interaction between the constituents,17 or external shear?18 The answer is yes, as evidenced by a long list of studies reported in the literature (refs.1–18, and references therein). Wen et al.11 reported the experimental observation of the crumpled conformation of graphite oxide membrane in an aqueous suspension. They found that membranes collapse to a compact configuration as the intramembrane affinity is enhanced by changing the composition of the solvent.
Knowing the conformation and dynamics of pinned membranes in solvent is important to the understanding of protein diffusion on the surface of red blood cells,17 which are known to undergo enormous conformational changes. Thus, it is of interest to explore the conformation and dynamics of interacting sheets and tethered membranes in solvent. To this end, we have extended our previous investigation of the SAS1 by incorporating the effect of solvent and temperature. This study of an interacting sheet in solvent is also highly useful for understanding the conformation and dynamics of layered silicate (clay) platelets.1, 20 The intercalation of solute between clay platelets and their exfoliation are key issues in understanding the morphological characteristics of polymer–clay composites.19 Understanding the dynamics of a clay platelet in different solvent environments at various temperatures will help in studying multiple platelets, the next step in our on-going investigations.
In this paper, we incorporate the node–node and node–solvent interactions in our bond-fluctuation computer simulation model for SAS.1 The power of the bond-fluctuation model24 to study conformation and dynamics of polymer chains is well-known for the simplicity and efficiency of the discrete lattice with ample degrees of freedom for bond fluctuations and node movements. The effectiveness of the bond-fluctuation model for SAS or tethered membrane is demonstrated in our recent study.1 The bond-fluctuation model becomes more important here in a more complex environment as more nodal interactions are included. This study is focused on the effect of temperature on the radius of gyration of a membrane in a solvent. We are not aware of any other such simulation work. In the following, we show how the stiffness, and therefore the crumpling, of a membrane11 depend on temperature, node–node interactions, and node–solvent interactions.
We consider a cubic lattice of size L3 with L = 64, 100, 200. A sheet of size Ls2 is placed on the lattice. The initial configuration consists of nodes connected on a square grid (lattice) of size Ls2, with the lattice constant of the sheet twice that of the host lattice as in our previous study. As with a polymer chain,24 a node of the sheet occupies the unit cube (eight sites) of the host lattice. A lattice site cannot be occupied by more than one node to incorporate the excluded volume effects. The size of the node, the unit cube with eight lattice sites, is selected primarily to impose constraints via excluded volume effects on the changes in length of the bond to which it is connected.24 It is convenient and efficient to implement variations (fluctuations) in bond lengths and enhance the degrees of freedom for movement of each node with such a nodal description. For all other purposes such as evaluating distances between nodes and nodes and solvent sites, a node is treated as a point at the center of the unit cube. The initial distance between the nearest neighbor nodes is 2 in units of the host lattice constant. Each interior node is connected with its four nearest neighbor nodes and the boundary nodes are connected with three, as on an open square grid. The bond length (the possible distance between two consecutively connected neighboring nodes) can vary between 2 and , with the exception of as in the bond-fluctuation model for a polymer chain.24 The configuration of the sheet, initially a two-dimensional square plane of cubic nodes, changes as the interacting nodes execute their stochastic motion in an effective solvent medium while maintaining the bond lengths within their fluctuation limit.
The empty host lattice sites constitute the effective solvent medium with one solvent component per site. A solvent and a node site can be as close as one unit lattice distance unlike the minimum distance (two) between nearest neighbor nodes. The interaction energy of a node is
where i runs over each node and j over neighboring sites within a range r. Different node–node interaction (J(n, n) = nn) and node–solvent (J(n, s) = ns) interactions are considered within a range (r), which can be varied. Obviously, there are 6 nearest neighbor sites at r2 = 1, 12 neighboring sites at r2 = 2, and so on. Note that the number of neighboring sites increases rapidly as we extend the range of interactions. For example, for a relatively long range interaction, with r2 = 1–12, the number of neighboring sites is 178. In this study, we consider the range of interaction r2 up to 4, 6, and 8 with the same interaction strength (nn, ns), i.e. with the number neighboring sites 32, 56, and 80, respectively.
The Monte Carlo (MC) method25 to move each node is implemented as follows. A node (cube-centered) at a site i and one of its neighboring sites j (out of 26) are selected randomly. If site j is empty, then the change in length of each of its connected bond 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 for the attempted move, then we evaluate the change in energy ΔE = Ej − Ei, where Ei is the interaction energy in the original configuration and Ej in the new configuration. The node is actually moved from site i to site j with a Boltzmann probability exp(−ΔE/kBT), that is we use the Metropolis algorithm to move the interacting nodes. Periodic boundary conditions are used along each direction. Attempts to move each node once define one unit Monte Carlo step (MCS) time.25 The simulation is performed for a sufficiently long time to reach the asymptotic regime. During the simulation, we keep track of the mean square displacements of the center node (Rn2) of the sheet, 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. Different lattice sizes are used to test for the finite size effects on the qualitative results. We restrict the discussion here to a limited number of interaction strengths (nn, ns) at T = 1–100 (in arbitrary units of interaction energy and Boltzmann constant). Our main goal here is to see the effect of solvent (interaction) and temperature on the conformation and dynamics of the sheet.
RESULTS AND DISCUSSION
Computer simulations are performed at different lattice sizes for a range of parameters mentioned here. No finite size effect is detected on the qualitative nature of the variations of the physical quantities. Figure 1 shows snapshots of the sheet at the initial stage (top) and end of a simulation run. The change in the conformation of the sheet is clear: after a million time steps, the sheet is wrinkled and somewhat crumpled. The wrinkles propagate and dissipate from the open edges during the simulation and are better seen in our visual analysis of their animation. The magnitudes of wrinkling and crumpling depend on the temperature and interaction (strength as well as range, see below). The snapshots of Figure 1 are generated at temperature T = 2 for attractive sheet node–node and sheet node–solvent interactions.
In Figure 2, we present a typical variation of the mean square displacement of the center node (Rn2) of the sheet and that of its center of mass (Rc2) with the timestep (t) on a log–log scale at a temperature T = 1. We have tried to identify the power-law dependence, e.g.,
where ν1 and ν2 are short and long time (asymptotic) exponents and Ac and Bc are corresponding prefactors. A power-law exponent ν2 ≃ 1/2 implies that the sheet has reached its asymptotic diffusive limit rather well. The short time collective dynamics is evidently slow with well-defined power-laws, ν1 ∼ 3/8–1/4 (see the inset Fig. 2 for R). A crossover from slow dynamics to diffusive motion is obviously expected for the collective motion of such a sheet with tethered nodes. The relaxation and corresponding crossover time for reaching different dynamics depend on the size of the sheet, its interactions, and the temperature.
The dynamics of a node is somewhat more complex due to its strong correlation with the neighboring nodes and solvent sites. The data for the variation of R with t are more fluctuating for the obvious reasons, i.e. a relatively smaller number of data sets were used for averaging. However, one may estimate the power-law dependence,
where exponents γi with i = 1–4 refers to increasing time regimes with corresponding prefactors (An to Dn). Estimates of these exponents are γ1 = 0.15 ± 0.01, γ2 = 0.10 ± 0.01, γ3 = 0.25 ± 0.01, and γ4 = 0.28 ± 0.05. This example illustrates that the motion of the node has not reached presumably its asymptotic diffusive regime. The trend is however clear even at such a low temperature T = 1. Note that the data in long time regimes are not as reliable as those in the short time regime. It goes without saying that the dynamics of an internal (individual) node is more complex than that of the center of mass of the sheet. At this stage we can only identify different time regimes. Further, note that the dynamics of a node becomes much more complex for an interacting sheet in a solvent here than that of the SAS.1
To show that the motion of the center node becomes diffusive, similar to the center of mass of the membrane in the asymptotic (long time) regime, it is practical to consider smaller sheets. Raising the temperature may also help in enhancing the mobility and therefore help in reaching the asymptotic regime. The main drawback with the small sheets is the lack of appropriate short time dynamics (e.g., analog of Rouse dynamics for a chain) as demonstrated in our previous study1 with a simpler system (SAS). In Figure 3, we present such a study, that is the variations of the mean square displacement of the center node of the sheet and that of its center of mass with a relatively small sheet of size 162 at temperatures T = 2 and 10. Data points at the low temperature are more fluctuating, especially for the center node. However, within the range of fluctuations, it is clear (T = 10) that the motion of the center node becomes diffusive similar to that of its center of mass in the asymptotic time regime.
Variation of the radius of gyration (R) with the timestep (t) is presented in Figure 4 for two temperatures T = 1, 2. The sheet node–node interaction is repulsive (nn = 0.5) whereas the sheet node–solvent interaction is attractive (ns = −0.5) for this set of data points. As before, data for longer times have lower number of independent runs, and therefore are more fluctuating. Despite large fluctuations, the radius of gyration (Rg) seems to relax in about t = 106 steps. The radius of gyration at the higher temperature (T = 2) approaches a lower value than that at the lower temperature (T = 1). Further, the sheet seems to relax faster at a higher temperature. A closer look at the radius of gyration in the relaxation time regime may help clarify the pattern. The temporal evolution of the radius of gyration and its components is presented in Figure 5. It is clear that the Rg relaxes faster at higher temperature and tends to reach a lower magnitude. Variations of the longitudinal (x, z) and transverse (y) components with MCS time show a clear difference in their relaxation patterns (Fig. 5). The longitudinal components are much larger than the transverse component, which implies that the sheet retains its planar morphology at temperatures T = 1 and 2.
To quantify the temperature dependence of the equilibrated radius of gyration, simulations are carried out at different temperatures. Figure 6 shows the variation of the equilibrium values of Rg2 with the temperature for two types of node–node interactions (nn = 0.0, 0.5) in an attractive solvent environment (ns = −0.5). The radius of gyration decays on increasing the temperature. Attempts are made to find the best fit with different regression analysis of which two are shown in Figure 6. The decay of the radius of gyration with the temperature is neither exponential nor the power-law. Perhaps at low temperatures (T ≤ 3), it is a power-law for nn = 0.5 (i.e. stiffer sheet). However, the dependence of Rg in entire temperature range is more complex, possibly following a stretched exponential. It is interesting to note that the decay of Rg depends on stiffness as the two sheets seem to show different decay patterns.
Finally, variations of the equilibrated radius of gyration with the temperature over a range of interactions for two types of sheets, a relatively “stiff” (nn = 1) and a “flexible” (nn = −1) membrane are presented in Figure 7. Although the sheet size (Ls2 = 162) is relatively small, the trend is clear. The radius of gyration decreases with the temperature for the stiff sheet whereas it increases for the flexible sheet. It is difficult to quantify the overall quality of the solvent as both stiffness determined by the node–node interaction and quality of the solvent (node–solvent interaction) orchestrates the variation of the radius of gyration with temperature. However, differences in the effects of temperature on the radius of gyration for stiff and flexible sheets are clear. Further, note the opposite effects of the range of interaction. Increasing the range increases the radius of gyration for stiff chains whereas it decreases for the flexible sheet, particularly at lower temperatures.
SUMMARY AND CONCLUSIONS
MC simulations are used to study the conformational dynamics of an interacting sheet (tethered membrane) in effective solvent media. Node–node and node–solvent interactions are used to demonstrate how the membrane moves, relaxes, and conforms as a function of temperature and interaction. Tethered nodes continue to execute their stochastic movements at each set of parameters. From the analysis of the mean square displacement of the center node (Rn2) and that of their center of mass (R) as a function of timestep (t) one can gain some insight about the segmental mobility and global motion of the membrane. The variation of R with t exhibits different power-laws in different time regimes (short to long). As expected, we observe subdiffusive short time dynamics as for the SASs. However, the intermediate time dynamics is more complex with different power-law exponents. Data are too fluctuating to identify universal power-laws in this regime. Segmental dynamics related to an individual node with the larger sheets have not reached diffusive motion during the course of our simulation. The center of mass of the membrane shows diffusion in the asymptotic time regime. Thus, the global dynamics of the membrane has reached the asymptotic regime but we are unable to identify with good quality data whether the local segmental movement in solvent will reach such a diffusive motion. Note that the motion of a node in a SAS reaches the diffusive asymptotic regime similar to that of its center of mass.1 The segmental mobility is dependent on the interaction variables (nn, ns, r) and temperature (T).
How the local segmental dynamical mode propagates through the membrane depends on its stiffness and flexibility, which are controlled by temperature, nodal and solvent interactions apart from the excluded volume constraints. We observe that a repulsive interaction between the nodes leads to a stiffer sheet while an attractive interaction enhances flexibility with more wrinkles and crumpling to the sheet, particularly at relatively low temperatures T = 1–4. For a stiff sheet (positive nn), in attractive solvent (negative ns), we find that the radius of gyration decreases on increasing the temperature. Generally, Rg increases on increasing the range of interaction and the rate of increase is more pronounced at low temperatures. In contrast, the trends are opposite for the flexible sheet (attractive nn), that is the radius of gyration increases on increasing the temperature. The magnitude of Rg decreases on increasing the range of interaction with larger changes at lower temperatures. Dependence of Rg on temperature and the solvent seem consistent with the experimental observation on graphite oxide membrane in aqueous suspension.11 The dependence of Rg on the temperature is perhaps more complex than simple exponential or power-law behavior. In the absence of analytic form of such function, our quantitative (graphical) predictions (Figs. 4 and 5) may help the understanding of the laboratory data when they become available. Our simulations have clearly shown that wrinkling and crumpling can be achieved by designing membranes with the appropriate constituent (nn) and solvent (ns) interactions at appropriate temperatures.
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. K.L.A. acknowledges support from the AFOSR for his National Research Council associateship. We thank Ray Seyfarth for discussion.