Langmuir 1999, 15, 8037-8044
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Density Dependence of Homopolymer Adsorption and Colloidal Interaction Forces in a Supercritical Solvent: Monte Carlo Simulation J. Carson Meredith† and Keith P. Johnston* Department of Chemical Engineering, University of Texas, Austin, Texas 78712 Received March 19, 1999. In Final Form: July 14, 1999 The complex relationship among polymer solution density, adsorption, and colloidal interactions in a supercritical solvent is investigated by means of Monte Carlo simulation. The expanded grand canonical method is applied to simulate homopolymer adsorption from solution on two impenetrable flat surfaces. Adsorption isotherms indicate that chain adsorption increases as bulk density is lowered toward the bulk solution phase boundary because chains adsorb to escape an increasingly poor bulk solvent. The force between two surfaces coated with adsorbed chains changes dramatically as a function of bulk solution density. Bridging attraction is observed above the bulk upper critical solution density (UCSD). As adsorption increases, this bridging attraction decreases as the bulk solution density is decreased due to chain crowding in the pore. At the UCSD the force becomes repulsive due to entropically unfavorable chain overlaps, and bridging attraction is no longer present. At densities below the UCSD, the force becomes attractive again, due to LCST-type (entropically driven) phase separation driven by the increase in entropy of solvent expelled from the pore. Lattice-fluid self-consistent field theory agrees with our simulation results for the density dependence of adsorption isotherms and force profiles.
Introduction In recent years polymers adsorbed at interfaces have been used to stabilize emulsions and latexes in supercritical CO2,1,2 for example in dispersion and emulsion polymerization.3-7 Steric stabilization is important in spray processes such as precipitation with a compressed fluid antisolvent.8,9 Such processes are capable of producing uniform micrometer- or submicrometer-sized polymer particles. Stabilization is also important in processing of colloidal metals and crystals for photonic materials. In addition, polymer adsorption plays a key role in supercritical fluid-based coating and film formation processes. These applications reveal the need to describe the fundamental properties of polymer adsorption at interfaces in a supercritical solvent. Although polymer adsorption in liquid solvents has been studied extensively with experiment, simulation, and theory, polymer adsorption from supercritical fluids is not well characterized or understood. In these highly compressible fluids, the adsorbed amount, structure of adsorbed polymer layers, and colloidal forces are likely to vary significantly with density. † Current address: Polymers Division, National Institute of Standards and Technology, Gaithersburg, MD 20899.
(1) O’Neill, M. L.; Yates, M. Z.; Johnston, K. P.; Wilkinson, S. P.; Canelas, D. A.; Betts, D. E.; DeSimone, J. M. Macromolecules 1997, 30, 5050. (2) Yates, M. Z.; O’Neill, M. L.; Johnston, K. P.; Webber, S.; Canales, D. A.; Betts, D. A.; DeSimone, J. M. Macromolecules 1997, 30, 5060. (3) Canelas, D. A.; Betts, D. E.; DeSimone, J. M. Macromolecules 1996, 29, 2818. (4) DeSimone, J. M.; Maury, E. E.; Menceloglu, Y. Z.; McClain, J. B.; Romack, T. J.; Combes, J. R. Science 1994, 265, 356. (5) Lepilleur, C.; Beckman, E. J. Macromolecules 1997, 30, 745. (6) O’Neill, M. L.; Yates, M. Z.; Johnston, K. P.; Smith, C. D.; Wilkinson, S. P. Macromolecules 1998, 31, 2838. (7) O’Neill, M. L.; Yates, M. Z.; Johnston, K. P.; Smith, C. D.; Wilkinson, S. P. Macromolecules 1998, 31, 2848. (8) Mawson, S.; Johnston, K. P.; DeSimone, J. M.; Betts, D. E.; McClain, J. B. Macromolecules 1997, 30, 71. (9) Mawson, S.; Yates, M. Z.; O’Neill, M. L.; Johnston, K. P. Langmuir 1997, 13, 1519.
Figure 1. Pressure versus polymer concentration phase diagram (at constant temperature) illustrating an upper critical solution pressure or density (UCSD) phase boundary. Inset: Analogous lower critical solution temperature (LCST) phase boundary pertaining to increasing temperature at constant pressure. Data were taken from simulations of bulk LJ chainsolvent mixtures.11-13 Values are given in LJ reduced units, explained in Simulation section.
In a near-critical or supercritical fluid, solvent quality is a strong function of solvent density. A binary polymersolvent mixture phase separates into polymer-rich and polymer-lean phases as temperature is increased beyond the lower critical solution temperature (LCST) phase boundary.10 (Figure 1 inset) Figure 1 also indicates that phase separation occurs at an upper critical solution density (UCSD) when the solution pressure (or density) is lowered at constant temperature. For both of these LCST-type phase transitions (induced by temperature increase or density decrease), differences in the compressibility of solvent and polymer lead to an increase in entropy when solvent and chains separate. Of course, as (10) Sanchez, I. C. In Encyclopedia of Physical Science and Technology; Academic Press: New York, 1987; Vol. XI, p 1.
10.1021/la990327h CCC: $18.00 © 1999 American Chemical Society Published on Web 09/01/1999
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density is decreased the number of solvent-solvent and solvent-chain interactions decreases, leading to a positive enthalpy change. LCST (or UCSD) phase separation can occur when the increase in entropy offsets the positive gain in enthalpy, leading to a negative free energy (i.e., ∆G ) ∆H - T∆S). The LCST phase behavior and conformational properties of model LJ polymers in bulk supercritical solution have been reported recently.11-13 These studies showed that the LCST of a polymer solution is equal to the coil-to-globule transition temperature of a single chain. As demonstrated by recent simulations14 and lattice-fluid self-consistent field theory (LFSCF),15,16 the critical flocculation density of colloids stabilized with grafted chains in a supercritical solvent is closely tied to the LCST phase behavior of the stabilizing chains. In particular the critical flocculation density is equal to both the UCSD of the stabilizer in bulk solution and the singlechain coil-to-globule transition density.14 These studies have been conducted on symmetric systems where the polymer segment and solvent volume and interaction energy are equal. LFSCF theory15,16 and experiments1,2 indicate that the correspondence between bulk solution and colloidal phase behavior holds not only for grafted chains but also for adsorbing stabilizers. When polymers adsorb from supercritical solution, adsorbed amount and layer structure are sensitive functions of the bulk solution density. LFSCF theory15,16 shows that the adsorbed amount of polymer increases as solvent density is lowered toward the LCST phase boundary. As solvent density is decreased from the good solvent regime, the adsorbed polymer layer first collapses, due to a decrease in solvation of the tails of adsorbed polymer. At lower densities, near the UCSD, the layer thickness increases again, due to crowding caused by a large increase of adsorbed polymer. In LFSCF theory fluctuations are neglected to simplify calculations; note, however, that very close to the critical density and temperature fluctuations become important. In addition, mean field theories such as LFSCF theory do not treat correctly the excluded volume of a single chain. The mean field approach is therefore quantitatively correct only when chains overlap on the surface and the single-chain excluded volume is screened. To examine densities closer to the solvent critical density and to test the predictions of LFSCF theory, a more rigorous approach is needed. In this work we turn to simulation to study this problem. Previous simulation studies have considered the adsorption of chains from a melt17,18 and from solution.19 Numerous simulations have considered the conformations of adsorbed chains confined to a pore in a vacuum.20-23 Most of these previous studies do not include solvent molecules; instead, changes in the magnitude of energetic (11) Luna-Ba´rcenas, G.; Gromov, D.; Meredith, J. C.; Sanchez, I. C.; Johnston, K. P.; dePablo, J. J. Chem. Phys. Lett. 1997, 278, 302. (12) Luna-Ba´rcenas, G.; Meredith, J. C.; Gromov, D. G.; Sanchez, I. C.; de Pablo, J. J.; Johnston, K. P. J. Chem. Phys. 1997, 107, 1. (13) Gromov, D. G.; de Pablo, J. J.; Luna-Barcenas, G.; Sanchez, I. C.; Johnston, K. P. J. Chem. Phys. 1998, 108, 4647. (14) Meredith, J. C.; Sanchez, I. C.; Johnston, K. P.; de Pablo, J. J. J. Chem. Phys. 1998, 109, 6424. (15) Meredith, J. C.; Johnston, K. P. Macromolecules 1998, 31, 5507. (16) Meredith, J. C.; Johnston, K. P. Macromolecules 1998, 31, 5518. (17) Vega, L. F.; Panagiotopoulos, A. Z.; Gubbins, K. E. Chem. Eng. Sci. 1994, 49, 2921. (18) Yoon, D. Y.; Vacatello, M.; Smith, G. D. In Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Binder, K., Ed.; Oxford: New York, 1995; p 433. (19) Yethiraj, A.; Hall, C. K. J. Chem. Phys. 1989, 91, 4827. (20) Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Binder, K., Ed.; Oxford University: New York, 1995. (21) Yethiraj, A.; Hall, C. K. Macromolecules 1990, 23, 1865. (22) Shaffer, J. S. Macromolecules 1994, 27, 2987. (23) Milchev, A.; Landau, D. P. J. Chem. Phys. 1996, 104, 9161.
Meredith and Johnston
interactions between solvent and polymer are analyzed by changing the temperature. These studies are therefore relevant to UCST phenomena. Solvent must be included explicitly in order to study the solvent entropy driven phase behavior that occurs at the LCST phase boundary. Unfortunately, the presence of solvent renders simulation more demanding, because of frequent overlaps between solvent and chain. To partially alleviate the chain overlap problem, a continuum configurational bias (CCB) algorithm can be used to move chains efficiently by searching for nonoverlapping conformations.24 Gibbs (NPT) or grand canonical (µVT) ensembles may be used to simulate the equilibrium partitioning of chains and solvent molecules between a pore and a bulk reservoir directly. With these methods, the equilibrium between two distinct phases is maintained through trial insertion and deletions of solvent and chain molecules. However, even with the CCB algorithm, the acceptance rate for insertion of a chain with greater than ∼10 segments is low above the solvent critical density. To circumvent this problem, an expanded ensemble technique was developed for the simulation of polymer chemical potentials25 and phase behavior.26 In an expanded ensemble, a gradual insertion or deletion of an entire chain with trial increments and decrements in chain length replaces a full chain insertion. For athermal chains, the expanded ensemble method has been used to study gels,27 chain confinement, and nematic-isotropic phase transitions in a slit.28 In this work the expanded grand canonical Monte Carlo method is used to simulate the equilibrium of both solvent and polymer between a pore and a bulk supercritical polymer solution. Our main objective is to characterize the density dependence of polymer adsorption and colloidal interactions from a bulk supercritical solution. In the following section, we first describe the LJ bead chain model which was adopted to facilitate comparison to previous simulations of bulk solutions11-13 (see phase diagram in Figure 1) and grafted chains.14 In addition the concept of the expanded ensemble is reviewed and we describe its application to adsorption from a supercritical polymer solution. In the Results section segment density profiles, adsorbed amount, and conformational properties are presented as a function of solvent density. The root-meansquare radius of gyration, Rg, and end-to-end distance, R, of adsorbed polymer are compared to those of polymers in bulk solution. For the case of equal chain and solvent adsorption energy, the force between two surfaces coated with adsorbed chains and solvent is calculated at bulk solution densities above, equal to, and below the LCST density. The mechanisms of attraction and repulsion are discussed in terms of bridging attraction and the LCST phase transition and are compared to results from LFSCF theory and experiment. The colloidal interaction forces are also related to the chain conformation and adsorption isotherms. Simulation Method Figure 2 gives a schematic of the bulk (Figure 2a) and pore (Figure 2b) simulation boxes. The bulk solution contains nc polymer chains, each of length N, and ns solvent segments. The fraction of chain segments is φc ) Nnc/(Nnc + ns), and the chain and solvent densities are given by Fc ) ncN/V and Fs ) ns/V, where V is the total volume. The (24) de Pablo, J. J.; Laso, M.; Suter, U. W.; Cochran, H. D. Fluid Phase Equilib. 1993, 83, 323. (25) Escobedo, F.; de Pablo, J. J. J. Chem. Phys. 1995, 103, 2703. (26) Escobedo, F.; de Pablo, J. J. J. Chem. Phys. 1996, 105, 4391. (27) Escobedo, F. A.; de Pablo, J. J. J. Chem. Phys. 1997, 106, 793. (28) Escobedo, F. A.; de Pablo, J. J. J. Chem. Phys. 1997, 106, 9858.
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have a relatively uniform distribution of states. In this way the end states (chain fully removed or added) are visited with the same frequency as the other states. For a uniform distribution of chain states (i.e., uniform sampling of chain lengths) the probability of visiting each chain length x and y must be equal. Under this condition it can be shown that ex ψx - ψy ) βµex c (x) - βµc (y)
Figure 2. Schematic of the bulk solution and pore simulation boxes, which are in equilibrium. The bulk box has periodic boundaries, but the pore box has only four periodic boundary walls and two impenetrable walls, separated by a distance H.
bulk simulation box is surrounded by periodic boundaries. In the pore phase, the two walls normal to the z-axis are impenetrable surfaces. The temperature, density, and pressure are reduced with respect to the solvent LJ parameters in the usual way, i.e., T* ) TkB/s, F* ) Fσs3, and P* ) Pσs3/s, where kB is the Boltzmann constant. To ensure chemical equilibrium, we use a grand canonical ensemble to hold the chemical potential of each species constant in the bulk and pore phases. Expanded Grand Canonical Simulations. At moderate to high densities (F* > 0.3), trial insertions and deletions of a polymeric molecule can be carried out relatively efficiently in an expanded ensemble. At any given time in the simulation one chain molecule is “tagged”. The length of this tagged chain is variable, and all lengths from 0 to N are sampled as the simulation proceeds. Thus, the chain is not inserted or deleted all at once but rather one segment (or several) at a time. A general derivation of the expanded ensemble for the calculation of chemical potentials25 and phase equilibria26 of polymeric systems can be found in the literature. Here the relevant details of the expanded ensemble method as applied to polymer adsorption from a binary solution are briefly discussed. The grand canonical partition function, Ξ, for a homopolymer-solvent mixture in an expanded ensemble is given by ∞
Ξ(µc,µs,V,T) )
∞
M
∑ ∑ ∑en βµ en βµ eψ Q(nc,ns,y,V,T) n )0n )0y)1 c
s
c
c
s
s
y
(1)
where nc and ns are the total number of chain and solvent molecules, µc and µs are the chain and solvent chemical potentials, V is the volume, and y is the state of the tagged chain molecule. Chain lengths of 0 and N correspond to y ) 0 and M, respectively. During the expanded ensemble simulation, the state or length of the tagged chain changes by (1 with increment and decrement moves. The chain increments and decrements are accepted with a probability that holds the chemical potential constant. The efficiency of the simulation is controlled by the ψy preweighting factors, which set the frequency at which each chain length is visited. An efficient simulation should
(2)
Since the excess chemical potential is given by βµex c (y) ) βµc - ln(nc/V) and because nc varies during the simulation, ψy is also variable and is given by ψy ) ωy[βµc - ln(nc/V)]. Here βµc is the total chain chemical potential that is specified at the beginning of the simulation, and ωy is a chain length dependent factor. A short initial simulation can be used to calculate values of ωy that yield a uniform distribution of states. The ωy factors must satisfy ω0 ) 0 and ωM ) 1. Because µex c (y) becomes essentially constant after the first few segments, the ωy factors for a short chain can be used to generate values for longer chains. During the course of the expanded grand canonical ensemble simulation, chain increments (y f y + 1) and decrements (y f y - 1) are attempted with equal frequency. To further augment the acceptance rate of chain increments and decrements, the appended segments are regrown with the CCB algorithm. The increments and decrements are accepted with the following criterion:
P(y f y + ∆) ) min[1,(Rw)∆ exp(ψy+∆ - ψy)] (3) In eq 3, ∆ ) +1 for an increment and -1 for a decrement and Rw is the Rosenbluth weight, or bias, that is introduced with the CCB algorithm, given by
Rw )
1
Nsp
exp(-βUj(y)) ∑ j)1
Nsp
(4)
where Nsp is the number of random trial positions and Uj(y) is the interaction energy between the jth trial segment and the system. When an end-state is reached (y ) 0 or y ) M), the tagged chain is either fully removed or fully added. At this point a new tagged chain is chosen from the existing full-length chains (y ) M) or a new tagged chain of length zero (y ) 0) is added to the system. During the course of the simulation values of the incremental ex , associated with appending each chemical potential, βµc,i chain segment i are stored and summed at the end of the run to calculate the total chain chemical potential. In our simulations both solvent and chain insertion and deletions are accepted with the probability given by eq 3. However, for a mixture, the subscripts in the equations refer only to the species being transferred. For solvent monomer insertion and deletion, the expanded ensemble is not necessary, the Rosenbluth weight, Rw, is 1, and eq 3 reduces to the standard grand canonical acceptance criterion.29 To achieve thermal equilibrium within the simulation box, chain displacements are attempted with the CCB algorithm, in which a portion of the chain is regrown in a low-energy conformation that avoids overlaps with neighboring segments. The position of each appended chain segment is chosen with an energetic bias from Nsp samples. Nsp is set to 12-16 for N ) 20, and Nsp is 6 for N ) 8. The details of the CCB algorithm are given (29) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U.K., 1987.
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elsewhere.24,30-32 Solvent moves are proposed by random displacement and accepted with the Metropolis Monte Carlo criterion. Parameters. Chain and solvent segments interact through the Lennard-Jones potential, cut and shifted at rc ) 2.5σ, given in eq 5.
u(rij) )
{
[( ) ( ) ( ) ( ) ]
4ij
σij rij
12
-
σij rij
6
-
σij rc
12
+
σij rc
6
r e rc
Table 1. Conditions and Results of NVT Simulations of Bulk Polymer Solutions T*
F*
N
nc
ns
βµex c
βµex s
1.30 1.15 1.15 1.15 1.15a 1.15
0.4 0.6 0.5 0.47 0.44a 0.40
8 20 20 20 20 20
9 10 10 10 10 10
1216 3800 3800 3800 3800 3800
-9.40 ( 0.03 -32.33 ( 0.12 -34.05 ( 0.06 -33.81 ( 0.04 -33.32 ( 0.04 -32.62 ( 0.08
-1.20 ( 0.04 -1.68 ( 0.11 -1.77 ( 0.04 -1.74 ( 0.04 -1.69 ( 0.03 -1.63 ( 0.04
r > rc (5)
a This F* value is the critical solution density at T* ) 1.15, from previous simulations of bulk solutions.13 The next lowest density, F* ) 0.4, is close to the phase boundary at φc ) 0.05.
The LJ energetic and size parameters for solvent and chain are chosen to be symmetric, in keeping with previous studies of polymers in bulk solution (phase diagram given in Figure 1).11-14 For consistency we choose the same values here, so that σss ) σcc ) σcs ) 1 and ss ) cc ) cs ) 1. These parameters correspond to equal solvent and chain dispersion interactions. Thus the focus is phase behavior driven by differences in solvent and chain compressibility (LCST type) and not differences in the energetic parameters (UCST type). The surfaces are treated as semiinfinite solids composed of Lennard-Jones atoms. By integration over coordinate space, the interaction between a chain or solvent segment and a surface can be represented as a function of distance normal to the surface, z. This integrated wall-segment potential has been used previously in many adsorption simulations17,18,33 and is given by
< F* < 0.6, although a few results are given at T* ) 1.3. At T* ) 1.15 previous simulations of bulk solutions gave an upper critical solution density (UCSD) of 0.44 (UCSD/ Fc* ) 1.42) at a critical concentration of about φc ) 0.24.13 The lowest density examined in this work, F* ) 0.4, is very close to the phase boundary at the bulk solution concentration φc ) 0.05.13 Lower densities were not examined to avoid the added complication of phase separation of the bulk solution. The pore phase was simulated with the expanded grand canonical ensemble described above, with the chain and solvent chemical potential values from the bulk simulations as set points (Table 1). The difference between the set point and the average chemical potential was no greater than 0.05 (0.15%) for the chains and 0.01 (0.5%) for the solvents. The expanded grand canonical ensemble code was verified by comparing average chemical potentials, densities, and chain conformational properties to results from NVT simulations. In most cases (20-30) × 106 equilibrium steps and (60-120) × 106 averaging steps were required to obtain statistically significant averaged values of chemical potential, end-to-end distance, segment density distributions, and forces. In this paper, one MC step is one attempted move. Moves were attempted for solvent and chain according to the following prescription that gave good convergence to equilibrium: 10% solvent moves, 10% solvent insertion, 10% solvent deletion, and 60% tagged chain moves. The typical box side was 15σ, more than adequate to prevent artificial chain selfinteractions. In addition, the correlation length of the solvent at the critical density (F* ) 0.31) and T* ) 1.15 is 2σ.35 Because this correlation length is well within the 15σ box size, finite-size effects are not a problem.
0
[(
) ( )
2 σwi uwall(z) ) A 5 z
10
σwi z
4
-
(σwi)4
]
3δ(z0.61δ)3
(6)
where A ) 2πFwwi(σwi)2d, Fw is the density of the solid, and δ is the spacing between atoms in the surface. Subscript w refers to the wall and subscript i refers to either chain or solvent. There is considerably flexibility in choosing LJ parameters for the wall-fluid interaction. Consistent with previous studies of adsorption, σww ) 1, δ ) σww, and Fw ) σww-3. In this study wc:ws values of 0:0, 1:0, and 1:1 were examined to study the effect of selectively altering the wall-polymer and wall-solvent interactions. Computational Method. NVT simulations of bulk polymer solutions were first used to generate the chain and solvent chemical potentials at a given temperature (T*), density (F*), and polymer concentration (φc). Widom particle insertion was used to obtain the solvent chemical potential.34 The chain chemical potential was calculated with the expanded variable-length chain ensemble.25 In this case the expanded ensemble is not used to vary the number of chains but to increase the efficiency of chemical potential measurement through incremental chain insertions. The accuracy of our expanded ensemble code was verified by comparing chemical potentials to the results of previous workers for hard-core chains.25 Table 1 lists and βµex values for bulk solutions at the the βµex s c conditions studied in this work. The pure solvent critical temperature and density are Tc* ) 1.08 and Fc* ) 0.31 for our model. Most results for adsorption from solution are presented at a temperature of T* ) 1.15 and for solution densities ranging from 0.4 (30) Rosenbluth, M. N.; Rosenbluth, A. W. Trans. Faraday Soc. 1955, 62, 3319. (31) Siepmann, J. I. Mol. Phys. 1990, 70, 1145. (32) Mooij, G. C. A. M.; Frenkel, D. Mol. Phys. 1991, 74, 41. (33) Mu¨ller, E. A.; Vega, L. F.; Gubbins, K. E.; Rull, L. F. Mol. Phys. 1996, 85, 9. (34) Widom, B. J. Phys. Chem. 1982, 86, 869.
Results Adsorption from Lennard-Jones Chain-Solvent Mixtures. Figure 3 explores the effect of LJ wall-segment well depth for solvent and chain segments for short chains (N ) 8) at T* ) 1.3, F* ) 0.4, and φc ) 0.05. When wc ) ws ) 0 and for a pore width of 15σ, the chain and solvent density profiles are depleted near the walls but are close to the bulk value (F*φc ) 0.4(0.05) ) 0.02) near the center of the box. (The solvent distribution is not shown for this case.) When the wall-chain attraction is turned on (wc ) 1) but the wall-solvent LJ parameter ws is 0, only the chains adsorb, resulting in a high-density peak in Fc*(z) of 0.85 near each surface. The solvent density is depleted below the bulk value near the surfaces, as expected. The peak in the chain distribution occurs 1.4σ from each surface. Since the wall position corresponds to the center of the surface atoms, the external layer of surface atoms extends 0.5σ into the fluid. Thus, the first layer of chain segments is located about 1.4σ - 0.5σ ) 0.9σ from the (35) Sengers, J. V.; Levelt Sengers, J. M. H. Annu. Rev. Phys. Chem. 1986, 37, 189.
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Figure 3. Chain and solvent density profiles within a pore for various adsorption energies, N ) 8. The pore is in equilibrium with a bulk solution at T* ) 1.3, F* ) 0.4, and φc ) 0.05. z(σ) denotes z/σ. Figure 5. Excess and absolute adsorption isotherms, as defined in text, versus bulk solution density for N ) 20 and T* ) 1.15.
the results for N ) 8, the solvent and chain segments pack into layers separated by about 0.9σ near the surface. At the highest bulk density of F* ) 0.6 the chain density (Figure 4a) at the midpoint of the pore (z ) 0) is equal to the bulk chain concentration of 0.03 ()F*φc). This is expected since bulk solution conditions should be approached far from the surface. However, as bulk solution density is decreased toward the phase boundary (UCSD ) 0.44 from previous studies13), the chain segment density in the middle of the pore increases above the chain concentration in bulk solution. At the same time Figure 4b shows that the solvent density in the pore falls with a decrease in bulk density. Thus as the bulk solution approaches the phase boundary, chains begin to displace solvent within the pore, leading to an excess of chains relative to the bulk concentration. The increase in chain density at the surface as bulk density is lowered can also be seen in the relative values of the first peak maximum in Figure 4a,b. Changes in chain and solvent densities in the pore can be examined by integrating the segment density profiles to yield adsorption isotherms with Figure 4. Chain and solvent distributions for chains with N ) 20, T* ) 1.15, and a surface separation of 15σ. wc ) ws ) 1.
outer layer of surface atoms. This characteristic packing of adsorbed chains 0.9σ-1σ from the surface has been observed in previous simulations.18 When both solvent and chain experience attractive interactions with the surface (wc ) ws ) 1), both the chain and solvent density profiles exhibit excess adsorption. The peak in chain density is reduced to Fc*(z) ) 0.4, compared to 0.85 for the wc ) 1, ws ) 0 case, since solvent displaces about half the chain density near the surface. The solvent distribution is rather structured near each wall, with about 3 peaks separated by 0.9σ each. The remainder of this paper explores the case where LJ adsorption energies are balanced, wc ) ws ) 1, for chains of length N ) 20. This choice of parameters allows a focus on the effect of changes in bulk solution density and excludes UCST phase separation driven by differences in LJ energy parameters. In addition we also avoid the high chain densities that occur at the surface when wc is greater than ws (Figure 3). Figure 4 shows the effect of changing the bulk density for LJ chains with N ) 20 at T* ) 1.15, just above the solvent critical temperature of 1.08. As in
Γi )
1 H-σ
∫σ/2H-σ/2(Fi*(z) - Fi,b*) dz
(7)
where Γi is the excess adsorption of component i relative to the bulk phase and H is the distance between the centers of surface (wall) atoms. The integration limits in eq 7 reflect the fact that the surface atoms create an impenetrable boundary at 0.5σ from each surface. Absolute adsorption is calculated by removing the bulk density, Fi,b*, from eq 7, and integrating the total density in the pore. Figure 5 presents both excess and absolute adsorption isotherms at T* ) 1.15 calculated with eq 7. The excess and absolute density of chains increases by a factor of 6 as bulk density decreases from 0.6 to 0.4, just below the UCSD of 0.44. There is a small excess in solvent density in the pore, although this excess amount of solvent does not change appreciably as density is lowered. The absolute amount of solvent mirrors the decrease in bulk density. At all densities, the excess of chain segments is considerably greater than for solvent. These results agree with the predictions of LFSCF theory of homopolymer adsorption,15 where adsorption increases significantly near the bulk solution UCSD. In Figure 5 the adsorption isotherm is almost linear in shape with a rather continuous increase in adsorption. However LFSCF theory predicts a nonlinear
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adsorption isotherm that shows most of the increase in adsorption near the UCSD. These differences may reflect the difference in chain models, chain length, and limitations in the LFSCF theory discussed above. To understand the increase in chain adsorption as solution density is lowered, consider the LCST phase behavior of bulk polymer solutions. Phase separation in supercritical fluid polymer solutions10 is driven by the increase in entropy as solvent expands as density is decreased. With less solvent present, chains aggregate together under the influence of intra- and intermolecular interactions and form a polymer-rich phase. In our simulations, chains adsorb at the attractive surfaces as solvation in the bulk solution diminishes, analogous to the formation of a second high-density polymer-rich bulk phase. The 6-fold increase in chain adsorption below the UCSD is interesting considering the equal adsorption and LJ energy parameters for chain and solvent and the constant volume fraction of chain in the bulk solution (5%). In addition, as the bulk solution density decreases, the density of chains in the bulk available for adsorption ()F*φc) also decreases. Thus there are fewer chains per unit volume available to adsorb as bulk solution density decreases. Nevertheless, the excess amount of chains in the pore increases considerably. These observations underscore the profound effects that solvent density and entropy have on the adsorption of chains into a pore, even in the absence of chain and solvent energetic differences. There is a small excess of solvent in the pore that appears to grow as bulk density is lowered. This excess probably aids in supporting the increasing amount of chains in the pore through chain-solvent interactions. Excess solvent adsorption is also observed in simulations of grafted chains14 and pure supercritical solvents in pores.36 The excess has at least two origins. First, the density of surface adsorption sites is generally higher than bulk solvent densities, particularly near and below the solvent critical density. Thus adsorbing segments find more opportunity for short-ranged attractive interactions (within one or two segment diameters) near the surface than in the bulk. In addition, the increasing correlation length of the fluid near the critical density of the solvent leads to an increase in the range of surface-solvent interactions.37 Confinement of near-critical fluids in pores can also depress the pure fluid critical temperature, as well as enhance solvent excess adsorption. The excess solvent adsorption is small ( UCSD, and LCST-type attraction occurs when F* < UCSD. For completeness, we must point out that this result was only observed in the LFSCF theory for homopolymers at low concentrations or for short chains.15 At high concentrations and for long chains, the LFSCF theory indicates that bridging is inhibited by thick and crowded layers. In this case the only mode of attraction was the LCST type below the UCSD. Although the concentration used in this study is high (5%), the chains are short, and the surface is not covered well enough to prevent bridging. The present simulation results are applicable to experiments involving relatively short homopolymer chains. However, most experiments involving steric stabilization of colloids in supercritical fluids involve the use of relatively high molecular weight homopolymers. For example turbidimetry1 and dynamic light scattering2 have been used to study stabilization of poly(ethylhexyl acrylate) (PEHA) emulsions stabilized with poly(1,1-dihydroperfluoroctyl acrylate) (PFOA) homopolymer in supercritical CO2. In these experiments bridging is absent at bulk densities above the UCSD, and the colloid is stable at high densities (repulsive interaction forces). This is in contrast with our result of bridging attraction at densities above the UCSD. But the chains in our simulations are short compared to the PFOA used in the experiments (Mw ≈ 1 × 106). The PFOA chains were evidently long enough and had a high enough concentration to prevent bridging. For example, LFSCF theory indicates that as chain molecular weight is increased, bridging is diminished due to increased adsorption and surface coverage.15 As density is decreased, the PFOA stabilized emulsion flocculates at the UCSD in agreement with the present results and with LFSCF theory.15 (43) Peck, D. G.; Johnston, K. P. Macromolecules 1993, 26, 1537.
Meredith and Johnston
Conclusions Most previous adsorption simulations do not include the solvent explicitly or simulate equilibrium between the pore and bulk phases directly, which are both necessary for realistic simulations of adsorption from supercritical solutions. Efficient simulation of polymer adsorption between a bulk solution and a pore is accomplished by applying an expanded grand canonical ensemble to break the chain insertion process into a series of single segment insertions. The results underscore a central theme: even in the absence of energetic differences between solvent and chain segments, bulk solution density and solvent entropy have profound effects on homopolymer adsorption and colloidal interaction forces in supercritical fluids. Adsorbed amount and layer thickness increase as density is decreased, due to the diminishing solvent quality. LFSCF theory also predicts an increase in adsorption as density is lowered; however, the shape of the curve is not linear as in the simulations. Both simulation and LFSCF theory show the same normalized thickness (h/Rg) values and density trend in the vicinity of the UCSD. The agreement between simulation and LFSCF theory for h/Rg suggests that experiments may yield a universal density trend similar to Figure 6 for adsorption of homopolymers from supercritical fluids. The results reveal a marked density dependence of the interaction force over a small density range centered about the bulk solution UCSD. At densities above the UCSD, the surfaces are attractive, due to bridging of homopolymers. At the UCSD, the amount of bridging diminishes significantly and the force is repulsive, presumably due to chain crowding and surface saturation which produce entropically unfavorable chain overlaps and prevent bridging. Below the UCSD, the surfaces become attractive again, this time due to LCST-type attraction that is analogous to LCST phase separation in bulk solutions. Thus the density dependence of adsorption controls the interaction mechanism that operates between the colloidal surfaces, in agreement with LFSCF theory. Indeed for short chains and at low concentrations, LFSCF theory shows the same transition from bridging attraction to LCST-type attraction as density is reduced. Homopolymer adsorption and the resulting colloid stabilization are very sensitive to small changes in density (
