12328
J. Phys. Chem. C 2007, 111, 12328-12334
Interaction between Coated Graphite Nanoparticles by Molecular Simulation Daniel Duque* Departamento de Fı´sica Teo´ rica de la Materia Condensada, Facultad de Ciencias, UniVersidad Auto´ noma de Madrid, 28049 Madrid, Spain
Brian K. Peterson Air Products and Chemicals, Inc., 7201 Hamilton BouleVard, Allentown, PennsylVania 18195-1501
Lourdes F. Vega Institut de Ciencia de Materials de Barcelona, Consejo Superior de InVestigaciones Cientı´ficas, Campus de la UAB, 08193 Bellaterra, Spain ReceiVed: January 18, 2007; In Final Form: May 31, 2007
We present and discuss here simulation results for a realistic model of chain molecules anchored on selected nanoparticles. The nanoparticles are taken to be composed of graphite, and the anchored molecules are C12 alkane chains. The main goal is to investigate the effect of the anchored chains on the mediated forces between the coated nanoparticles. We also examine the structure of the anchored chains. The grafting density (the surface density of grafted chains) is identified as a key parameter that drastically affects the chain adsorption and the effective force between nanoparticles. At low grafting densities, the chains are basically adsorbed on the surfaces, while at high grafting densities the chains form brushes. The effect of the temperature on these mediated forces is also investigated following the same procedure. At low grafting densities the force depends weakly on the temperature, indicating a prevalence of energy over entropy. At high grafting densities the conformation of the brushes has a direct impact on the force, which is seen to be clearly temperature dependent. This has strong implications in practical applications, since the possibility to control a colloidal dispersion by means of the temperature is here shown to depend on the grafting density. These results serve both as a source of baseline results useful for comparison and as a stepping stone toward future work with systems that are more involved to simulate.
I. Introduction As explained by the classic Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory,1,2 the interaction between colloidal particles is governed by long-range van der Waals attractions and shorter range electrostatic repulsion. The latter can effectively stabilize a colloidal dispersion that would otherwise aggregate (flocculate). An alternative, or additional, mechanism of colloidal suspension stabilization is achieved by adsorbed polymers or chain molecules on the surface of the colloids.3 The idea is that steric hindrance and entropy would also prevent colloidal particles from sticking together. These polymers can be physically adsorbed or chemically bound to the surface, in which case they are called “grafted polymers”. The chemical bond is usually established by the chemical reaction of a functional end with the colloidal surface. There are several applications of this technique, including emulsion polymerization in supercritical CO2,4 emulsion stabilization,5 and microparticle formation by spray processes.6,7 One of the most exciting applications is found in biocompatibility studies, in which the coating molecules not only physically stabilize the dispersion but also add new functionality to them.8 In principle, the addition of grafted molecules can tailor the pair nanoparticle/coating for specific applications in an adequate solvent. * Corresponding author.
Experimental techniques have appeared that can directly probe the forces between colloids, such as the surface force apparatus (SFA) of Israelachvili9 or colloid-probe atomic force microscopy.10 As in many instances, given the many parameters involved, molecular simulation can be a powerful tool in fine tuning the parameters for specific applications, as long as realistic models for the nanoparticles, the coating molecules, and the solvent are used. Despite the clear interest of these systems, there are but a few simulation studies that treat them realistically. A large number of works exist that deal with the general problem of molecules grafted onto surfaces, either from theory,11-14 simulations,15-17 or both.18 These usually focus on structural information (density profiles, chiefly) and only apply to very long molecules, for which certain approximations of polymer physics apply, such as the independence of molecular details for distances beyond the correlation length. In this limit, the interaction potential can be highly simplified, and the conclusions apply for a wide variety of systems. For alkane molecules, this would mean polyethylene molecules about 100 units long, at least. In many circumstances, the anchored molecules are not long enough for these methods to be employed and more refined, specific force field models should be used. There have also appeared several works devoted to systems of confined, nongrafted molecules.19-22 In many of these, especially in those23,24 inspired by the SFA of Israelachvili,9
10.1021/jp070430c CCC: $37.00 © 2007 American Chemical Society Published on Web 07/31/2007
Interaction between Coated Graphite Nanoparticles
J. Phys. Chem. C, Vol. 111, No. 33, 2007 12329
realistic models for the molecular interactions are used. This is feasible because the molecules are much shorter than the typical grafted polymers. Apart from structural information, the works inspired by the SFA also calculate forces between colloidal particles, which is of course the main output of the SFA. The few works that directly focus on the problem of finding the interaction between surfaces coated by grafted molecules are the simulation works of M. Murat and G. S. Grest25,26 in the 1980s and of J. Carson Meredith et al. in 1999.27 These represented a clear step forward on understanding these systems from a molecular perspective; however, the simplified nature of the interaction potential used leads to conclusions that are qualitative and not directly applicable to any specific experimental system. The same applies to some theoretical approaches that have targeted the interaction force between surfaces with grafted polymers, like those by A.C. Balazs and co-workers28-31 (ref 31 is a very useful review) and other groups.32 We also mention the somewhat controversial theoretical result33 that long grafted polymers could promote attraction between colloidal particles, even if the interactions were purely repulsive. Further criticism34 and subsequent method refinements35 indicate that this claim was probably incorrect. The aim of this paper is to present results for a well-defined experimental system. The nanoparticles are taken to be composed of graphite, and the anchored molecules will be C12 alkane chains, with one of the C atoms grafted to the surface. A specific system modeled by our simulations would be a functional molecule such as dodecanoic (lauric) acid, H3C-(CH2)10COOH, anchored at a functionalized graphite surface by a reaction with the alcohol residue, thus, exposing the methyl terminus. Within this choice of surface and grafted molecule we will consider the effects of other control parameters, such as grafting density and temperature, on the structure of the grafted molecules and the interaction between coated surfaces. This paper is part of an ongoing effort to understand experimental systems of anchored chains onto surfaces; a previous article dealing with the most technical parts of these simulations has already appeared.36 The particular system investigated here was chosen because of its potential applications for stability dispersions, as a guide to obtaining key features affecting the stabilization, before experiments are performed. In section II we will provide the details of the model used and of the simulation algorithm. We will present and discuss our main results in section III and will end with some conclusions in section IV.
TABLE 1: Parameters for the Grafted Molecules stretching and bending σst l0 κb/kB θ0
0.038 Å 1.54 Å 62500 K/rad2 114°
torsion 355.03 K -68.19 K 791.32 K
c1/kB c2/kB c3/kB Lennard-Jones case CH3-CH3 CH2-CH2
artifacts that have been noticed in previous works. This is due to the fact that a rigid bond does not contribute to the virial at all, whereas a stretching potential, even a very stiff one, will. Since the force is evaluated from the virial, missing these contributions leads to incorrect results, such as the appearance of a net force for very wide separations. Since in the original TraPPE scheme the bond lengths were fixed, they should not fluctuate too much (relative to l0) in our case; hence, it is necessary that σst is small. Notice that the explicit inclusion of the temperature in the potential of eq 1 makes sure there is no temperature dependence in the bond length fluctuations (at least, for an isolated bond); this amounts to having a temperature-dependent spring constant. For this temperature range, this choice is not important, and a stiff spring constant can likewise be used. An angle bending potential applies for three consecutive sites, of the form
ub(θ) )
ust(l) )
kBT σ2st
(l - l0)2
(1)
The parameters σst and l0, together with all of the other ones used for the fluid, are listed in Table 1. This binding potential is actually a slight deviation from the TraPPE framework, which fixes all bonds to be exactly l0; we have previously discussed that the inclusion of a binding potential is important when computing the net force between the nanoparticles.36 In summary, the failure to include flexible potentials was found to cause
κb (θ - θ0)2 2
(2)
and a torsional potential applies for four consecutive sites:
ut(φ) ) c1[1 + cos φ] + c2[1 - cos(2φ)] + c3[1 + cos(3φ)] (3) All monomers that are not bound by any of these potentials, either because they belong to different molecules or because they are separated by four bonds or more, interact via a truncated and shifted Lennard-Jones potential:
II. Methods A. Molecular Model. For the grafted molecules, we have employed the TraPPE united-atom alkane parameters.37 In this framework, the CH3 and CH2 groups are modeled as interacting point-like sites. Two consecutive sites separated at a distance l are held around an average distance l0 by a harmonic stretching potential
σ 3.75 Å 3.95 Å
/kB 98 K 46 K
ua,b(r) ) ab
{( ) ( ) ( ) ( ) } σab r
12
-
σab 6 σab r rab,c
12
+
σab rab,c
6
(4)
where r is the distance between the two sites; the interaction is cut off at rab,c, which we will take as 3.5σab. The labels a and b distinguish the kind of site: there are three possibilities, corresponding to CH3-CH3, CH2-CH2, and CH3-CH2. Parameters for the first two are listed in Table 1; the last one results from the Lorentz-Berthelot combining rules
σab )
σaa + σbb ab ) xaabb 2
(5)
Regarding the surface, we will assume the colloidal particles to be very large compared with the anchored molecules, so that their surfaces can be approximated by planes. For slightly curved surfaces, our results could be improved by a Derjaguin approximation. We employ an averaged approximation for the
12330 J. Phys. Chem. C, Vol. 111, No. 33, 2007
Duque et al.
TABLE 2: Parameters for the Graphite Surfaces
B. Simulation Details. The simulations presented in this work are time-consuming not only for equilibrium and averaging purposes but also for the generation of “meaningful” initial configurations, that is, configurations that represent valid initial physical states of the system of interest. Note that the molecules under consideration are in a clearly “forced” situation, compressed between two surfaces and with several possibilities to overlap with any other of the anchored chains in either wall. Given the particular characteristics of the system, we have carried out configurational bias Monte Carlo simulations.38 This method is very useful for chain molecules at high densities, since the bonded interactions are much stronger than the other interactions. At each iteration, a monomer of the system is selected at random, and a partial regrowth of its molecule, from the monomer to the free end, is tried. This is accomplished by the generation of 20 trial segments according to the bonded stretching interactions (i.e., stretching, angle bending, and torsion.) One of these segments is then chosen at random, with a bias given by the Boltzmann factor of all other (nonbonded) interactions, and used to place the new monomer, from which the process is carried out again, until the end of the chain is reached. In order for a detailed balance to be satisfied, a similar procedure must be carried out for the “old” chain, and acceptance of the whole regrowth is finally dependent on the ratio of the new and old “Rosenbluth factors” (these factors being accumulated products of normalizing factors for each monomer, as explained in, e.g., ref 38.) As could be anticipated, a successful regrowth of a chain is less likely the closer the starting monomer is to the grafted end. The acceptance ratio for regrowth from the grafted monomer (i.e., complete regrowth), is about 15%, whereas regrowth from the end monomer (which will only change this monomer) can be about 90%. Thus, when selecting a monomer at random, an a priori bias is introduced: the probability of choosing monomer i is pi ) 1/(Zi), with a normalizing factor Z ) ∑i1/i. In this way, the monomers from which it is less likely to regrow the chain are selected more often. The system is initialized by placing the grafted ends on square lattices onto both surfaces, in registry (that is, one lattice exactly on top of the other.) In particular, we have taken n ) 2 × 100 chains, so two 10 × 10 lattices are considered. The two surfaces are squares of side L ) 79 Å (hence, a grafting density of 0.016 Å-2), which we will refer to as “high density”, or 158 Å (hence, a grafting density of 0.0040 Å-2), which we will call “low density.” The simulation cell is therefore a square box of dimensions L × L × h. Periodic boundary conditions are employed in the x and y directions. The convenient choice of squares is justified by our neglect of the surface atomic structure (otherwise, hexagons should be used in the case of graphite.) If a previous run cannot provide a suitable initial configuration, some procedure must be employed to obtain it. In this case, all chains are completely grown from their grafted ends, by “layers”; that is, each new monomer is added to all of the chains. If some chains cannot be grown, we go back to the previous layer recursively until all chains are grown to the desired length. We also impose an initial repulsive surface field in order to straighten the chains up and ease the calculation of the initial configuration. The system is equilibrated for 106 MC steps (one step being defined as one attempt to partially regrow a chain), after which production runs of 8 × 106 steps are simulated. Quantities of interest are averaged over 8 blocks of 106 steps, and error bars are determined from the mean standard deviation of these values.
σss ∆ ss/kB Fs σg
3.40 Å 3.35 Å 28 K 0.114 Å-3 0.16 Å
surface interactions, thus losing detail of the explicit atomic structure of the surfaces. This is justified for adsorbed molecules, as long as the ones consider, since the atomic details will in this case play a minor role. In particular, we have used the wellknown Steele’s 10-4-3 potential for graphite:
usa(z) ) 2πFssaσsa2∆
[( ) 2 σsa 5 z
10
-
]
( )
σsa4 σsa 4 (6) z 3∆(0.61∆ + z)3
with the parameters listed in Table 2; the interaction parameters sa and σsa represent the interaction of a solid surface with either a CH3 or a CH2 site and are also calculated from the LorentzBerthelot rules, eq 5. The only exceptions are the grafted sites, for which this potential does not apply, but rather a grafting potential of the form
ug(z) )
kB T σg2
(σ0 - z)2
(7)
with the values in Table 2. The equilibrium length σ0 corresponds to the minimum of the surface, grafted CH3 potential, which is computed according from the Lorentz-Berthelot rules, eq 5, using the σ values of CH3 and σss. The x and y coordinates are kept fixed for grafted ends; the details behind this choice of grafting potential are explained in ref 36. The explanation for this choice parallels the one for eq 1: rigid grafting potentials have been found to introduce artifacts in the force calculated. Equivalently, in order that the grafting length does not fluctuate too much, it is necessary that σg is small. The direct interaction between surfaces, which can be calculated from eq 7, is found to be negligible compared with the interaction mediated by the grafted chains for this particular system (this may not hold for other systems, specially if long-range electrostatic forces are present). Since there are two surfaces in the system, which we will take as placed at z ) -h/2 and z ) h/2, the external field that is applied to each of the sites is
uext(z) ) usa
(2h - z) + u (2h + z) sa
(8)
with Steele’s potential, eq 6; here, the label a distinguishes between CH2 sites or end terminal CH3 ones. We will refer to h simply as the “separation” of the surfaces. The grafted CH3 terminal ends are obvious exceptions: for these, only the grafted force, eq 7, applies; more precisely, ug(z - h/2) for ends at the “bottom” surface and ug(h/2 - z) for ends at the “top” one. We do not consider solvent molecules in this work. This limits the applicability of our results to cases in which the solvent density is small (high temperatures or low pressures). We nevertheless expect (as our preliminary results for solvated systems show) that the solvent would be expelled at narrow separations, so that our results in this limit can be considered quite general.
Interaction between Coated Graphite Nanoparticles
Figure 1. Snapshots at high (a) and low (b) coverage, both for a separation of 15.8 Å and for a temperature of 500 K. For the sake of clarity, only molecules grafted on one of the surfaces are shown. This surface is also plotted, endowed with a random texture. Grafted ends are colored in light gray, terminal ends are in red, and all of the other sites are in darker gray. The radii of the spheres is the Lennard-Jones value in Table 1.
III. Results and Discussion We present and discuss next some of the most relevant results obtained by simulating anchored alkane molecules into graphite surfaces. Although several other simulations were performed, we focus on the most representative results concerning grafting density, distance between nanoparticles, anchored chain length, and temperature. In Figure 1, we show snapshots corresponding to equilibrated configurations at high and low densities, both for a separation of h ) 15.8Å and a temperature of 500 K. We can already see the drastic effect of grafting density. At low densities the chains have room to adsorb on the surface of the nanoparticles, whereas at high ones steric repulsions force the molecules to leave the surface and form a thick brush. The corresponding monomer density profiles for these conditions are provided in Figure 2. Notice that h ) 15.8 Å is the separation between the two surfaces: because of the repulsive part of the surface potential, the thickness actually available to the molecules is about 2 × 3 Å smaller. These monomeric profiles are obtained by dividing the system in slabs 0.1975 Å wide and are normalized so that their integral over z provides the density per unit area. Thus, for low density, the total integral will be 2 × 100/1582 ≈ 8.0 × 10-3 Å-2, and for high density, 2 × 100/792 ≈ 0.032 Å-2. For this figure, we
J. Phys. Chem. C, Vol. 111, No. 33, 2007 12331
Figure 2. Monomer number density profiles, Fm(z), for a separation of 15.8 Å and a temperature of 500 K. Curves correspond to monomers: 1 (blue), 3 (brown), 6 (green), 9 (red), and 12 (orange). Part (a), high coverage; part (b), low coverage.
have selected from all 12 monomer profiles: the first (anchored) monomer, the third one, the sixth one, the ninth one, and the twelfth (terminal) one. For the anchored monomers, two sharp peaks are obtained (insets are provided so that its height can be visualized without losing resolution on the y axis). These monomers are subject to the special grafting potential of eq 7, that is chosen to be quite stiff, hence sharp peaks are obtained. The next monomer plotted, number three, is still close to the nanoparticles at high and low coverage. For the next monomers, the situation is different. At high coverage (Figure 2a), the next monomers show intermediate maxima corresponding to chains that leave the surface. Finally, most of the free end monomers are in the center of the space between the two surfaces. At low coverage, on the other hand, all monomers are adsorbed on the surfaces, independently of their position along the chains. Hence, for the case of high grafting density, a thick coating layer is obtained, while low grafting densities promote the formation of thin coating layers onto the nanoparticles (for the same temperature, chain length, and surface interaction). In Figure 3, we present density profiles for high coverage at different nanoparticle separations, from 15.8 to 27.65 Å. For these, we provide total molecular density profiles in Figure 3a; this means that these are calculated as the sum of monomer profiles with the resulting value divided by 12, in order that the profiles represent molecular number density. Of all 12 monomer profiles, we have plotted only the terminal free monomer profiles in Figure 3b. As we have discussed, at this coverage, the crowding at the surface causes the straightening of the grafted molecules, thus forming a brush. This is clearly reflected in the bimodal distribution of the end monomers in
12332 J. Phys. Chem. C, Vol. 111, No. 33, 2007
Figure 3. Number density profiles, F(z), for high coverage. Part (a), molecular (total) density; part (b), end monomer density. Curves correspond to separations of 15.8 (brown), 19.75 (green), 23.7 (red), and 27.65 Å (blue).
Figure 3b: the peak close to the surface corresponds to molecules that manage to stay absorbed or that bend back toward the surface; the outside peak corresponds to molecules with ends detached from the surface. We can see that at the wider separation the two particles are largely independent as far as chain structure is concerned. As the separation narrows, the two brushes entangle and the density profiles overlap. The corresponding density profiles for low grafting density are presented in Figure 4, for nanoparticle separations ranging from 9.875 to 23.7 Å. This coverage regime is completely different, as is clearly observed in Figure 4b; the monomers are always basically adsorbed on the graphite surfaces. In Figure 5, we show the force between the two surfaces (solid lines to guide the eye.) This is a force per unit area, hence a pressure, which we present in units of Kelvin per Ångstrom cubed (hence, the quantity is P/kB.) For a modest area of 1000 × 1000 Å2, a force per unit area of kB1 K/Å3 translates into a total force of about 140 nN, in the typical range for a colloidal system and in the range of forces measurable by the surface force apparatus or by colloid-probe atomic force microscopy. For both coverage values, a long range attractive regime is followed by a sharp repulsion at short separations. An analysis of the contributions to the energy shows that the attraction is mainly due to the van der Waals attraction between monomers in chains grafted in the opposite surfaces, with the monomeropposing surface interaction and the surface-surface interactions playing minor roles. This is expected, since the minimum of the surface potential, eq 6, is occupied by molecules grated on the surface, and the molecules on opposing walls are primarily attracted by the minimum of the monomer-monomer Lennard-
Duque et al.
Figure 4. Number density profiles, F(z), for low coverage. Part (a), molecular (total) density; part (b), end monomer density. Curves correspond to separations of 9.875 (magenta), 11.85 (cyan), 15.8 (brown), 19.75 Å (green).
Figure 5. Calculated force between surfaces, per unit area. Orange circles, low coverage, T ) 500 K; red squares, high coverage, T ) 500 K; green plus signs, low coverage, T ) 300 K; cross signs, high coverage, T ) 300 K. Colored lines are spline fits to the data, only to guide the eye.
Jones potential, eq 4. The repulsion is due to steric effects at close contact. All results thus far have been provided for a temperature of 500 K. We have also considered a lower temperature of 300 K, close to ambient temperature. The density profiles do not show any qualitative differences, with the peaks accentuated (as is to be expected at lower temperatures), and they are therefore not shown. The effect on the force curve is, on the other hand, quite revealing. The curves are also shown in Figure 5, with lines to
Interaction between Coated Graphite Nanoparticles guide the eye. At low coverage, the curve hardly changes. This clearly corresponds to a situation dominated by the energy, in which entropy does not play an important role. In contrast, the force curve at high coverage is markedly different at lower temperatures, as the coated surfaces approach, with a deeper minimum. This is presumably due to a subtle combination of enthalpic and entropic effects. The total free energy of the system has enthalpy and entropy contributions. The former will have a small dependence on temperature, while the later will have a large one: at higher temperatures, chains undulate, with a large associated entropy that effectively reduces the interaction between the coated nanoparticles. At lower temperatures, the undulation is reduced, with a smaller associated entropy. Moving the surfaces closer reduces the conformational freedom of the chains, which is associated with a negative entropic contribution. This contribution increases in magnitude with decreasing distance. In other words, the TS term will always cause a repulsive force, but this one will prevail at higher temperatures. IV. Conclusions We have presented and discussed forces between surfaces coated with anchored chain molecules by means of Monte Carlo simulations on a well-defined system. The technique of molecular simulation is ideally suited to explore relevant systems previous to, or in parallel with, experiments. The details of the systems under consideration (type of grafted molecule, nature of nanoparticle, etc.) are readily changed, making straightforward the systematic estimation of the force directly from these calculations. The typical central processing unit (CPU) time of our runs is 6 CPU days on a 2.80 Pentium 4 processor, for each separation. The particular system investigated here consisted of graphite nanoparticles and C12 grafted alkane chains. We have identified the grafting density as a key parameter that drastically affects the chain adsorption: at low grafting densities, the chains are basically adsorbed on the attractive surfaces, while at high grafting densities, the chains form brushes. This is reflected in the effective force, which shows a shorter range for low densities and a longer range for high ones. The effect of the temperature on these mediated forces is revealing. At low grafting densities, the force depends weakly on the temperature, indicating a prevalence of energy over entropy. At high grafting densities, the conformation of the brushes has a direct impact on the force, which is seen to be clearly temperature dependent. This has strong implications in practical applications, since the possibility to control a colloidal dispersion by means of the temperature is here shown to depend on the grafting density. The present work is relevant in itself, as a source of useful baseline results for comparison and because its methods can also be easily applied to similar systems. Furthermore, it also represents a stepping stone toward other systems that are more involved to simulate but of practical interest. In particular, two effects important in many instances can be incorporated, although both of them lead to more intensive computations. One is the possible presence of charges; although in the systems studied here they are not relevant, it is common that charges appear in either the chains or the surfaces in several experimental systems of interest. These charges can be included in a simulation by means of the Ewald summation technique, increasing the simulation time. The other important ingredient is the presence of solvent molecules, which can have an important influence whenever their concentration is significant (at low temperatures and high pressures.) In these cases, the
J. Phys. Chem. C, Vol. 111, No. 33, 2007 12333 explicit inclusion of solvent molecules is more realistic, since the coated molecules will experience forces mediated by the solvent. The solvent can be chosen to promote either aggregation or dispersion, by the subtle combination of anchored chains and adequate solvent molecules. Of course, solvent molecules may be charged themselves (as in the case of water or other common polar solvents), thus, connecting with the first point. At present, we are carrying out preliminary simulations of systems with solvent molecules (with and without charges), a matter for future publications. Acknowledgment. Partial financial support for this work has been provided by the Spanish government, under Project Nos. CTQ2004-05985-C02-01 and CTQ2005-00296/PPQ and a Ramo´n y Cajal contract, and by a contract with MATGAS 2000 A.I.E. Additional financial support from Generalitat de Catalunya under Project No. SGR2005-00288 is also acknowledged. Part of the simulations presented here were carried out using resources from the Supercomputer Center of Catalonia (CESCA). References and Notes (1) Derjaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633. (2) Verwey, E. J.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (3) Advincula, R. C.; Brittain, W. J.; Caster, K. C.; Ru¨he, J. Eds. Polymer Brushes: Synthesis, Characterization, Applications; John Wiley & Sons: New York, 2004. (4) Kendall, J. L.; Canelas, D. A.; Young, J. L.; DeSimone, J. M. Chem. ReV. 1999, 99, 543-563. (5) Dickson, J. L.; Binks, B. P.; Johnston, K. P. Langmuir 2004, 20, 7976-7983. (6) Mawson, S.; Johnston, K. P.; Betts, D. E.; McClain, J. B.; DeSimone, J. M. Macromolecules 1997, 30, 71-77. (7) Mawson, S.; Yates, M. Z.; ONeill, M. L.; Johnston, K. P. Langmuir 1997, 13, 1519-1528. (8) Meziani, M.; Rollins, H.; Allard, L.; Sun, Y.-P. J. Phys. Chem. B 2002, 106, 11178-11182. (9) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: New York, 1992. (10) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831-1836. (11) Alexander, S. J. Phys. (Paris) 1977, 38, 983. (12) de Gennes, P. G. Macromolecules 1980, 13, 1069. (13) Szleifer, I.; Carignano, M. A. In AdVances in Chemical Physics XCIV Prigogine, I., Rice, S. A., Eds.; John Wiley & Sons: New York, 1996; pp 165-260. (14) Mu¨eller, M. Phys. ReV. E 2002, 65, 030802. (15) Grest, G. S.; Murat, M., In Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Binder, K., Ed.; Oxford University Press: New York, 1995; p 476. (16) Kreer, T.; Metzger, S.; Mu¨ller, M.; Binder, K.; Baschnagel, J. J. Chem. Phys. 2004, 120, 4012-4023. (17) Milchev, A. In Computational Methods in Surface and Colloid Science. Surfactant Science Series, vol. 89; Boro´wko, M., Ed.; Marcel Dekker: New York, 2000. (18) Pattanayek, S. K.; Pham, T. T.; Pereira, G. J. Chem. Phys. 2005, 122, 214908. (19) Vega, L. F.; Panagiotopoulos, A. Z.; Gubbins, K. E. Chem. Eng. Science 1994, 49, 2921-2929. (20) Schoen, M.; Diestler, D. J. Phys. ReV. E 1997, 56, 4427-4440. (21) Schoen, M.; Gruhn, T.; Diestler, D. J. J. Chem. Phys. 1998, 109, 301-311. (22) Gruhn, T.; Schoen, M. Phys. ReV. E 1997, 55, 2861-2875. (23) Porcheron, F.; Rousseau, B.; Fuchs, A. H.; Schoen, M. Phys. Chem. Chem. Phys. 1999, 1, 4083-4090. (24) Porcheron, F.; Rousseau, B.; Fuchs, A. H. Mol. Phys. 2002, 100, 2109-2119. (25) Murat, M.; Grest, G. S. Phys. ReV. Lett. 1989, 63, 1074-1077. (26) Murat, M.; Grest, G. S. Macromolecules 1989, 22, 4054-4059. (27) Carson Meredith, J.; Sanchez, I. C.; Johnston, K. P.; de Pablo, J. J. J. Chem. Phys. 1998, 109, 6424-6434. (28) Sigh, C.; Pickett, G. T.; Balazs, A. C. Macromolecules 1996, 29, 7559-7570. (29) Singh, C.; Balazs, A. C. J. Chem. Phys. 1996, 105, 706-713.
12334 J. Phys. Chem. C, Vol. 111, No. 33, 2007 (30) Singh, C.; Pickett, G. T.; Zhulina, E.; Balazs, A. C. J. Phys. Chem. B 1997, 101, 10614-10624. (31) Balazs, A. C.; Singh, C.; Zhulina, E.; Chern, S. S.; Lyatskaya, Y.; Pickett, G. Prog. Surf. Sci. 1997, 55, 181-269. (32) Whitmore, M. D.; Baranowski, R. Macromol. Theory Sim. 2005, 14, 75-95. (33) Roan, J. Phys. ReV. Lett. 2001, 86, 1027.
Duque et al. (34) Matsen, M. W. Phys. ReV. Lett. 2005, 95, 069801. (35) Roan, J. Phys. ReV. Lett. 2001, 87, 059902. (36) Duque, D.; Vega, L. F. J. Chem. Phys. 2006, 124, 034703. (37) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1998, 102, 2569. (38) Frenkel, D.; Smit, B. Understanding Molecular Simulation, 2nd ed.; Academic Press: New York, 2002.
