Colloidal Stability Dependence on Polymer Adsorption through

Langmuir 2009, 25, 3529-3537

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Colloidal Stability Dependence on Polymer Adsorption through Disjoining Pressure Isotherms A. Gama Goicochea,*,† E. Nahmad-Achar,‡ and E. Pe´rez§ Centro de InVestigacioń en Polı´meros (Grupo COMEX), Marcos Achar Lobatoń No. 2, Tepexpan, 55885 Acolman, Estado de Me´xico, Mexico, Instituto de Ciencias Nucleares, UniVersidad Nacional Autońoma de Me´xico, Circuito Exterior, C.U., Apartado Postal 70-543, 04510 Me´xico D. F., Mexico, and Instituto de Fı´sica, UniVersidad Autońoma de San Luis Potosı´, A´lVaro Obregoń 64, 78000 San Luis Potosı´, Mexico ReceiVed August 8, 2008. ReVised Manuscript ReceiVed January 5, 2009 The disjoining pressure of polymers confined by colloidal walls was computed using dissipative particle dynamics simulations at constant chemical potential, volume, and temperature. The polymers are able to adsorb on the surfaces according to two models. In the so-called surface-modifying polymers, all monomers composing the chains have the same affinity for the substrate, whereas for the end-grafted polymer only the monomer at one of the ends of the polymer molecule adsorbs on the colloidal surface, resembling the behavior of dispersing agents. We find that these adsorption models yield markedly different disjoining pressure isotherms, which in turn predict different stability conditions for the colloidal dispersion. Our results show that for end-grafted polymers, a larger degree of polymerization at the same monomer concentration leads to better stability than for the surface-modifying ones. But also the unbound monomers of the surface-modifying type dominate over both kinds of polymers at large surface distances. The origin of these differences when the chemical nature of monomers is the same, and molecular weight and polymer concentration are used to characterize colloidal stability, is found to be mainly entropic.

I. Introduction The stability of a colloidal dispersion is governed by the balance of forces that arise from the subtle competition between shortrange attraction and long-range repulsion. The physical origin of these forces in most colloidal dispersions of interest is van der Waals (attraction) and electrostatic (repulsive), as in the DLVO model (see ref 1), or some other repulsive mechanism, such as steric forces.2 Understanding these forces and their interplay is of paramount importance to predict the conditions for colloidal stability, which is a topic of major interest in basic research and for industrial applications. Among the forces that take place in colloidal suspensions one finds also (many-body) interactions mediated by the fluid surrounding particles, which become evident at close interparticle distance, adding a contribution to the total effective interparticle force. Such complex fluids are composed typically of the solvent and stabilizing agents, such as polymers or surfactants that may be adsorbed on the particles, modifying the colloidal stability.3 Neutral polymer adsorption has been extensively studied using the mean field approximation that divides the configuration of the adsorbed polymers into loops and tails.4 The loops are energetically favorable for the attraction and the tails are entropically favorable for repulsion between two close surfaces with adsorbed polymers. The balance between these forces may produce repulsion for saturated surfaces and attraction for unsaturated surfaces at * To whom correspondence should be addressed. E-mail: [email protected]. † Centro de Investigacioń en Polı´meros (Grupo COMEX). ‡ Universidad Nacional Autońoma de Me´xico. § Universidad Autońoma de San Luis Potosı´. (1) Israelachvili, J. N. Intermolecular and Surfaces Forces, 2nd Ed.; Academic Press: New York, 1992. (2) Zhulina, E. B.; Borisov, O. V.; Priamitsyn, V. A. J. Colloid Interface Sci. 1990, 137, 495. (3) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983. (4) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.

large distance.5 However, a numerical solution of the equations of the mean field approximation predicts a change from repulsion to attraction when the distance between the walls decreases in the lower limit of saturation.6 The variety of scenarios for polyelectrolyte adsorption is even richer and less understood because of the presence of long-range electrostatic forces.7 Moreover, the charge inversion produced by polyelectrolyte adsorption is used to build up multilayer films on colloidal particles by a self-assembled process,8 giving rise to the possibility of modifying surface properties with selective control. Most experimental efforts to measure forces in confined complex fluids involve the use of the surface force apparatus (SFA),1 or atomic force microscopy,9 where the confined liquid is in thermodynamic equilibrium with its surrounding bulk. When discussing forces in colloidal systems it is then more appropriate to employ the concept of disjoining pressure (DP),10 which is the difference between the force (per colloidal particle unit area) normal to the confining surfaces, and the fluid’s bulk pressure, with which the confined liquid is in equilibrium. DP is a direct method for determining the free energy of interaction, hence its importance. On the theoretical side, most approaches involve density functional theory (see, for example, refs 11 and 12) and mean-field calculations (see, for example, 13). As for computer simulations of confined fluids, there are several reports in the literature (see, for example, ref (5) (a) de Gennes, P. G. Macromolecules 1981, 14, 1637. (b) de Gennes, P. G. Macromolecules 1982, 15, 492. (6) Meńdez, J. M.; Johner, A.; Joanny, J. F. Macromolecules 1998, 31, 8297. (7) Joanny, J. F. Eur. Phys. J. B 1999, 9, 117. (8) Kato, N.; Schuetz, P.; Fery, A.; Caruso, F. Macromolecules 2002, 35, 9780. (9) McNamee, C. E.; Tsujii, Y.; Ohshima, H.; Matsumoto, M. Langmuir 2004, 20, 1953. (10) Derjaguin B. V. and Churaev, N. V. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; John Wiley and Sons: New York, 1986. (11) Pollard, M. L.; Radke, C. J. J. Chem. Phys. 1994, 101, 6979. (12) Balbuena, P. B.; Berry, D.; Gubbins, K. E. J. Phys. Chem. 1993, 97, 937. (13) Yi, T.; Wong, H. J. Colloid Interface Sci. 2007, 313, 579.

10.1021/la802585h CCC: $40.75  2009 American Chemical Society Published on Web 02/19/2009

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14 and references therein), but those most closely relevant to SFA experiments, for example, are simulations carried out in the Grand Canonical (GC) thermodynamic ensemble, where the chemical potential (µ), volume (V), and temperature (T) are fixed to ensure equilibrium of the confined fluid with the bulk. Recent experiments and simulation studies have also proposed the stabilization of colloidal particles by smaller, charged particles. The latter may decorate the former, creating “halos” around them.15 Evidence for this kind of stabilization has been observed experimentally using uncharged micrometersized silica and highly charged, nanometer-sized zirconia particles;16 this experiment has been explained using computer simulations.17 Others have investigated the influence of the polymer’s molecular architecture, such as spherical, linear, and branched molecules.14,18 One of us19 has recently implemented a methodology that combines a mesoscopic type of molecular dynamics, known as dissipative particle dynamics (DPD),20 with Monte Carlo sampling (using the Metropolis algorithm) in the GC ensemble (MCGC)21 in order to predict adsorption and DP isotherms of polymers on surfaces. Excellent agreement was found between the experimental22 and predicted19 adsorption isotherms of polyethylene glycol on alumina and silica surfaces and between the DP isotherm of a linear, nonionic, fluorocarbon polymer on Si(100) wafers,23 with their respective counterparts, predicted using MCGC-DPD.19 The stability was then evaluated from DP isotherms. We chose DPD because it has proved to be a successful method to predict thermodynamic properties of systems with large time and length scales that are not easily accessible by classical simulation.24 DPD can be used to simulate solvent and polymer molecules explicitly, a difficult combination that requires a good solvent model and a large number of monomers and chains to make appropriate comparisons with experimental results. In the present work we seek to understand how different polymer adsorption mechanisms affect the global stability of a model colloidal dispersion by means of MCGC combined with DPD. Our main interest is to identify key aspects of theses confined systems as they are displayed on the DP. To this end, we simulate two kinds of polymeric systems: one where all monomers have equal preference to be adsorbed and another where only the anchor monomer has this preference. Different molecular weights and polymer concentrations are analyzed. After this Introduction, in Section II we introduce the simulation methods and the polymer adsorption models, while the details pertaining to the actual simulations are found in Section III; then in Section IV we present the results and their discussion and, finally, we draw some concluding remarks in Section V.

II. Methods and Models A. DPD. DPD20 is a traditional molecular dynamics algorithm in the sense that one must integrate in time Newton’s (14) Gao, J.; Luedtke, W. D.; Landman, U. J. Phys. Chem. B 1997, 101, 4013. (15) Cha´vez-Pae´z, M.; Gonza´lez-Mozuelos, P.; Medina-Noyola, M.; MeńdezAlcaraz, J. M. Physica A 2004, 341, 1. (16) Tohver, V.; Smay, J. E.; Braem, A.; Braun, P. V.; Lewis, J. A. Proc. Nat. Acad. Sci. 2001, 98, 8950. (17) Liu, J.; Luijten, E. Phys. ReV. Lett. 2004, 93, 247802. (18) Dijkstra, M. Europhys. Lett. 1997, 37, 281. (19) Gama Goicochea, A. Langmuir 2007, 23, 11656. (20) Hoogerbrugge, P. J.; Koelman, J. M. V. A. Europhys. Lett. 1992, 19, 155. (21) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: New York, 2002. (22) Esumi, K.; Nakaie, Y.; Sakai, K.; Torigoe, K. Colloids Surf., A 2001, 194, 7. (23) Mate, C. M.; Novotny, V. J. J. Chem. Phys. 1991, 94, 8420. (24) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1987.

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second law of motion using finite time steps to obtain the particles’ positions and momenta from the total force. A difference from atomistic molecular dynamics24 is that the DPD model involves not only a conservative force (FC), but also random (FR) and dissipative (FD) components acting between any two particles i and j, placed a distance rij apart. These particles are mesoscopic by construction, and with no internal structure. The total force is the sum of these three components, Fij ) FCij + FDij + FRij, and their explicit expressions are as follows

FijC ) aijω(rij)eîj FijD ) -γω2(rij)[eîj · b v ij]eîj

(1)

FijR ) σω(rij)eîjξij where eîj ) rij/rij, rij ) ri-rj, rij ) |rij|, vij ) vi-vj, ri, being the position, and vi the velocity of particle i, respectively. The variable ξij is a random number uniformly distributed between 0 and 1 with Gaussian distribution and unit variance; aij, γ, and σ are the strength of the conservative, dissipative, and random forces, respectively. The ω(rij) factor is a weight function that carries the explicit distance dependence of the forces and is given by

ω(rij) )

{

1 - rij ⁄ Rc, rij e Rc 0, rij g Rc

(2)

where Rc is a cut-off distance. At interparticle distances larger than Rc, all forces are equal to zero. This simple distance dependence of the forces, which is a good approximation to the one obtained by spatially averaging a van der Waals-type interaction, allows one to use relatively large integration time steps. The strengths of the dissipative and random forces are related in a way that keeps the temperature internally fixed, kBT ) σ2/2 γ; kB is Boltzmann’s constant. The natural probability distribution function of the DPD model is that of the canonical ensemble, where N (the total particle number), V, and T are kept constant. For more details and applications of the DPD technique see, for example, refs 25-28. B. MCGC-DPD. We use a hybrid29 MCGC-DPD algorithm because we have found19 it is advantageous to combine the mesoscopic reach of DPD with the appropriate thermodynamic conditions necessary to model SFA experiments, which is how the DP is typically measured. This means that one must ensure that the confined fluid is in chemical equilibrium with the unconfined bulk. The hybrid MCGC-DPD algorithm is briefly described in what follows, and the reader is referred to ref 19 for full details. The basic structure of the algorithm is that of the usual MC, i.e., the Metropolis algorithm,21 except that the particles positions are not chosen randomly but rather from the positions obtained from integrating the equation of motion (using the velocity-Verlet algorithm, see ref 30) with the forces given by eq 1. Starting with a given configuration one performs a number of DPD moves (10 in our case), that is, the positions and momenta of all particles are updated after each integration of the equation of motion, until 10 moves have been completed. The choice of number of DPD moves was based on computational efficiency. Once this is accomplished, the total energy of the system is calculated and (25) Espanõl, P.; Warren, P. Europhys. Lett. 1995, 30, 191. (26) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423. (27) Maiti, A.; McGrother, S. J. Chem. Phys. 2004, 120, 1594. (28) Groot, R. D.; Rabone, K. L. Biophys. J. 2001) , 81, 725. (29) Mehlig, B.; Heermann, D. W.; Forrest, B. M. Phys. ReV. B 1992, 45, 679. (30) Vattulainen, I.; Karttunen, M.; Besold, G.; Polson, J. M. J. Chem. Phys. 2002, 116, 3967.

Colloidal Stability Dependence on Polymer Adsorption

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its value is compared with the energy of the first step (out of the 10-step cycle). The Metropolis criterion is then applied to decide which configuration is kept and which is discarded. When this has been carried out, a (monomeric solvent) particle insertion or removal is tried a certain number of times, according to the usual GC selection rules21 using the potential obtained from the conservative DPD force, and at the end averages are computed of the properties of interest. Only monomeric solvent particles are interchanged with a virtual bulk, so that it is the chemical potential of the solvent what remains fixed. C. Solvent and Polymer Models. The model solvent used in this work is a liquid made up of monomeric DPD particles, whose interaction parameter aij (see eq 1) is chosen so as to represent water at room temperature. Each solvent particle represents three water molecules, which is the so-called coarsegraining degree.26 The actual number of solvent particles fluctuates from one simulation to another, to keep the solvent’s chemical potential fixed. We consider that the solvent is at the θ-temperature or higher, so that attractive interactions are negligible. Our model for the nonionic polymer is a linear chain of DPD particles joined by freely rotating springs; the number of beads will depend on the degree of polymerization one wants to model. We model a dilute polymer solution, so that depletion forces due to polymer concentration lowering do not take place.3 Following ref 19, we use interaction parameters between the polymer’s beads and between them and the solvent particles so that the model reproduces the solubility of polyethylene glycol (PEG) in water. In each simulation the number of polymer molecules is kept fixed. All DPD particles have the same size, whether they are solvent or polymer beads. D. Surface Model. For the confining surfaces we use a DPDlike force that is a function only of the distance perpendicular to the plane of the walls, which is taken here as the z-direction. This constitutes a soft wall, as shown by eq 3

[ ]

Fw(z) ) aw_spe 1 -

z ; zc

z < zc

(3)

and it is zero for z > zc. There appears a constant, aw_spe, that defines the repulsive interaction between the walls (w) and the confined fluid particle (spe), which may be a solvent particle or a polymer’s monomer. Affinity of a particle (aw_i) for the substrate over another (aw_j) is modeled by a smaller repulsive interaction (aw_i < aw_j).

III. Simulation Details We use dimensionless units throughout. The masses are all equal to 1, as are the cutoff distances Rc and zc. The values of the force constants in eq 1, σ and γ, are chosen as 3 and 4.5, respectively, so that kBT* ) 1. The time step chosen for the dynamics part of the code was set to ∆t* ) 0.03. The chemical potential imposed for the solvent so that the total average density is F*Tot ) 3.0 was found to be µ* ) 37.7. By choosing this average density one has the smallest system where the DPD quadratic equation of state obeys a scaling law that makes it independent of the choice of particle-particle interaction parameters.26 The transversal area of the simulation box is kept constant and it is equal to Lx* × Ly* ) 5 × 5, whereas the longitudinal direction, Lz*, was varied from the smallest possible compression while maintaining equilibrium (∼1-2 DPD particle diameters) to Lz* ) 5. We use periodic boundary conditions in xy plane but not in the z-direction since we placed the effective walls perpendicular to it. To reach equilibrium, for each system at least 105 Monte Carlo configurations were generated, followed by another 105

Figure 1. Two adsorption models considered in this work. (a) Surfacemodifying polymers, where all monomers in the polymer are equally likely to adsorb; (b) end-grafted polymers, where only the monomer at the end of the chain (the anchor monomer) adsorbs over the substrate.

configurations for the calculation of averages of the properties of interest. The repulsive interaction parameter, aij, which appears in the conservative force (see eq 1) is chosen as aii ) 78.0 for particles of the same species and aij ) 79.3 for particles of different kind. This choice of parameters implies that every DPD particle has a volume equal to 90 Å3, which means that one can group 3 water molecules into a DPD particle. This coarse-graining degree has been shown26 to reproduce the isothermal compressibility of water at room temperature, and from it one can extract a dimensionalized value for Rc ) 6.46 Å which is the characteristic length scale for our simulations. The repulsive wall interaction parameter, aw_spe (see eq 3), was chosen as aw_mono ) 60 when the particle interacting with the wall was a monomer of the polymer molecule. For solvent molecules, such interaction was set at aw_solV ) 120; this choice of parameters promoted polymer adsorption on the substrates over solvent molecules that led to specific stability conditions of the colloidal dispersion, which was the purpose of this study. It has been suggested19 that one could obtain reasonable phenomenological values of the wall interaction parameter from the value of the interfacial tension between the confined fluid and the solid substrate, when that information is available experimentally. Monomers of a polymer were joined by Hookean springs

b Fijspring ) -K(rij - req)eîj

(4)

where K is the spring constant and req its equilibrium distance. We chose K ) 100.0 and req ) 0.7 as did Rekvig et al.,31 a choice that has proved to be adequate to represent polymer molecules. For a coarse-graining degree equal to 3, a 7-bead polymer corresponds to mapping PEG with molecular weight Mw ) 400.19 We also performed simulations for other degrees of polymerization, following this mapping. In regards to the adsorption mechanism, we choose two models, depicted in Figure 1. In one of them, termed “surface modifying” (Figure 1a), all monomers belonging to a polymer molecule have the same affinity for the substrate. Therefore, the wall-monomer interaction parameter is aw_mono ) 60 for all monomers in the (31) Rekvig, L.; Kranenburg, M.; Vreede, J.; Hafskjold, B.; Smit, B. Langmuir 2003, 19, 8195.

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Figure 2. (a) Disjoining pressure isotherms of surface-modifying polymers at a fixed degree of polymerization (Mw ) 400), for increasing polymer concentration. (b) Same as in (a) but for end-grafted polymers. (c) Disjoining pressure isotherms for surface-modifying polymers at fixed concentration and varying degree of polymerization. (d) Same as in (c) but for end-grafted polymers. The cartoons in each figure merely illustrate the type of polymer used. Lines are guides to the eye.

polymer, while for solvent monomers aw_solV ) 120, as stated above. For the case displayed in Figure 1b only the monomer at one end of the polymer has affinity for the wall (the anchor monomer), with aw_mono ) 60 as in the previous case, but the rest of the polymer’s monomers (the tail) interact with the wall as if they were solvent monomers, i.e., with aw_mono ) 120. We call this an “end-grafted” polymer. In both cases the polymer-solvent interaction remains equal to aij ) 79.3. A key property of interest for the present work, which is crucial for the evaluation of the state of stability of the colloidal dispersion, is the so-called disjoining pressure.10 The pressure of a fluid confined by walls, in particular the component perpendicular to them, PN, is in general different from the (unconfined) fluid’s bulk pressure, PBulk. The difference between them is known as the disjoining pressure, Π, and it is a function of the separation between the parallel walls, Lz*

Π(L*z ) ) PN(L*z ) - PBulk.

(5)

The bulk pressure is obtained from the average of the diagonal components of the pressure tensor, while the normal pressure is

Figure 3. Comparison of disjoining pressure isotherms of 20 molecules of short-chain (called Mw ) 400) vs 10 molecules of long-chain (Mw ) 800), surface-modifying polymers. The cartoons in the figure merely illustrate the type of polymer used and do not represent the actual polymer lengths used. Lines are guides to the eye.


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Figure 4. Comparison of disjoining pressure isotherms of 10 molecules of short-chain (called Mw ) 400) vs 5 molecules of long-chain (Mw ) 800) end-grafted polymers. Inset: entire range over which Π* was obtained. The cartoons in the figure merely illustrate the type of polymer used and do not represent the actual polymer lengths used. Lines are guides to the eye.

calculated from the zz-component, averaged over the length of the simulation box in the direction perpendicular to the walls. The components of the pressure tensor are calculated using the virial route,24 as shown below N

Pzz )

N

C , ∑ mibv i · bv i + ∑ ∑ zijFij,z

i_)1

(6)

i)1 j>i

where the first and second terms represent the kinetic and (conservative) interaction contributions, respectively. In eq 6, mi represents the mass of each DPD bead, which is equal to 1, zij ) zi - zj is the component of the positions of particles i and j in the z-direction, and Fij,zC is the z-component of the total conservative force between particles i and j. Equivalent expressions are used for Pxx and Pyy, simply by replacing z for x and y, respectively.

IV. Results and Discussion We begin by considering the surface-modifying polymer shown in Figure 1a. We first obtained the DP of a fluid with a fluctuating number of solvent particles and a given concentration of molecules of polymer (PEG) with molecular weight Mw ) 400. Starting from a wall-to-wall separation of Lz* ) 5.0, and then reducing it by ∆Lz* ) 0.1 until the maximum compression allowed while maintaining equilibrium was reached, we obtained the DP isotherms shown in Figure 2a. For the smallest concentration (10 polymer molecules per volume), one observes oscillations in the DP, which are typical of confined simple fluids (see ref 1), of approximately constant period but decaying amplitude. The maxima in DP correspond to the most stable thermodynamic conditions, while the negative minima represent regions where the fluid is unstable. As the number of polymer molecules is increased (say, from 10 to 20 molecules per volume), the negative minima disappear, and the overall DP curve becomes more repulsive, leading to better colloidal stability, especially for the case of 30 molecules at short distances. For grafted-type polymers, displayed in Figure 1b, also with Mw ) 400, and confined at the same concentrations as the surfacemodifying polymers, we obtained their DP isotherms which are shown in Figure 2b. For a given polymer concentration, the force on the colloidal walls produced by the confined fluid is

considerably larger when grafted polymers are used. Even at the smallest concentration the DP isotherm is repulsive over the entire range, making this type of polymers more efficient as dispersants than their surface-modifying counterparts, in agreement with experience. To determine the influence of the molecular weight on the dispersion stability, we performed a series of simulations for three different Mw’s at a fixed polymer concentration, for both kinds of polymer. The results are shown in Figure 2c and d for the surface-modifying and grafted polymers, respectively. The efficiency of the latter over the former type of polymers is evident; for example, one has to double the molecular weight of the surfacemodifying polymers to obtain a completely stable fluid (no negative minima in DP), whereas for the grafted kind, even at the smallest molecular weight one gets stability. Also, as expected, as the polymer concentration is increased, better colloidal stability is obtained in both cases. Notice too that for a given molecular weight (say, Mw ) 400) the maximum attainable compression is smaller when surface-modifying polymers are used than for end-grafted ones. This arises from the thick layer formed by the end-grafted polymers, which have less freedom to move once the anchor monomer is attached to the surface. That the greater stability attained is not a consequence of greater Mw itself, as so often misinterpreted, but rather of greater monomer mobility, can be seen by keeping the concentration of polymeric monomers constant while varying the molecular weight. Consider the case when Mw ) 400, which corresponds to molecules made up of 7 DPD beads connected by springs. When 20 of such molecules are present in the solvent, one has a total of 140 polymeric monomers confined by walls. This concentration can also be achieved when 10 molecules of polymers with Mw ) 800 are used, and the question now arises as to which case provides the best results, from the thermodynamic stability point of view. In Figure 3 we present the DP isotherms obtained for surface-modifying polymers for the case just described, showing that the shorter polymers are better dispersants, when compared with larger ones, at the same monomer concentration. Multiplying the dimensionless Π* by kBT/Rc3 we obtain that the DP for short polymers can be up to 4.5 × 105 Pa larger than that for the larger polymers at certain wall separations, for T ) 300 K. Stability via surface modification is then much better attained through the use of monomeric species, than through polymer chains. The same behavior is found for the grafted polymers, displayed in Figure 4, only that in this case the difference in DP between small and large polymers is of the order of 10 MPa, even at a smaller monomer concentration than that of Figure 3 (70 for Figure 4 vs 140 for Figure 3), which is a manifestation of the efficiency of this type of dispersants. In Figure 5 we compare the DP isotherms of grafted and surfacemodifying polymers of the same molecular weight and at the same concentration. As could be expected from the discussion above, the former dominates over the latter type of polymers, over the entire range of the simulations, although at certain distances between the confining walls they are almost equal. The phenomenon is greater at shorter distances between the surfaces, as shown in the inset. Given that the trend seemed to be that small polymers dominated over large ones at a given monomer concentration, we carried out simulations where the harmonic force between polymeric monomers was removed. The confined fluid for such a case is composed of a (fluctuating) number of solvent particles and a fixed number of unbound polymeric monomers. In Figure 6 we show the comparison between the DP for 20 molecules of surface-modifying polymers with Mw ) 400 as in

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Figure 5. Disjoining pressure isotherms of grafted (solid line, circles) and surface-modifying (dotted line, squares) polymers. Both types of polymer chains had the same length (Mw ) 400, 7 DPD beads) and were used at the same concentration (20 polymer molecules per volume). Inset: entire range over which Π* was obtained. Lines are guides to the eye.

examples above, i.e., when there are 140 bound polymeric monomers in the confined fluid, with the DP of 140 unbound surface-modifying monomers in water. The DP isotherm for the unbound monomers is larger than that of the polymers over the entire range of the simulations, becoming up to about 30 MPa larger at certain wall-to-wall separations. Clearly, unbound monomers have more mobility which leads to better efficiency when it comes to covering the surface of the colloids. We now turn to the grafted polymers, which we have shown to be more efficient dispersants than the surface-modifying ones. Figure 7 shows the DP of 20 molecules of grafted polymers with Mw ) 400 vs the DP of 140 unbound, surface-modifying monomers, the same as in Figure 6. At short distances between walls, the fluid with solvent and grafted polymers displays larger values of the DP, leading to better colloidal stability. However, there is a crossing point (∼Lz* ) 3.0 in Figure 7) after which the roles are reversed and the fluid with solvent and unbound monomers displays larger DP values. Hence, at the largest surfaceto-surface distances, the unbound polymeric monomers are the best dispersants, which seemed rather unexpected. It would appear that the entropic gain of having monomers with more mobility to sample the configurational space than polymers is the leading mechanism responsible for the higher values of disjoining pressure, as shown in Figures 6 and 7. The same argument should hold when comparing large with small polymers at the same monomer concentration. To test this hypothesis we calculated the Helmholtz free energy change, ∆F, from the DP data, according to the following prescription

∆F )A

∫ Π(h) dh.

(7)

In eq 7, A is the transversal area of the confining walls, h is the separation between them (also called Lz*), and Π is the DP (see eq 5). Using the data presented in Figure 3 we obtained the difference between ∆F of 10 molecules of surface-modifying polymers with Mw ) 800 (14 DPD beads) and 20 molecules of polymers of the same kind but with Mw ) 400. The resulting ∆(∆F), shown as the continuous line in Figure 8a, is positive over the entire range of distances between walls, indicating not

only that the shorter polymer case is thermodynamically more favorable, but that the origin of this phenomenon should be mostly entropic. Indeed, since ∆(∆F) ) ∆(∆U) - T∆(∆S), where U is the internal energy, T the absolute temperature and S the entropy, and ∆(∆U) cannot grow with h, since we are comparing states with the same volume, then ∆(∆U) ≈ 0 (as h increases) and T∆(∆S) must be negative, i.e., the change in entropy of the fluid with short polymers is larger, as expected. The change in internal energy is approximately zero (due to particle number fluctuations, characteristic of MCGC) because there are no new interactions when the bonds between monomers are cut in the polymer chains. The systems compared are thus almost identical (within very small fluctuations in the number of solvent particles) except for the reduced degrees of freedom present when one has long as opposed to short chains, keeping the polymeric monomer density constant. To prove this, we show in the inset the internal energy difference (normalized by the averaged particle number) between these two systems, obtained from the simulations, which shows very small fluctuations about zero as h* increases. In Figure 8b we show the free energy difference between a fluid with five molecules of grafted polymers with Mw ) 800 and 10 molecules of the same kind of polymers, with Mw ) 400, obtained from Figure 4, so that the polymeric monomer concentration is the same for both, i.e., 70 polymeric monomers in solution. Once again the fluid with shorter polymers is more favorable thermodynamically, and the difference is larger than that of Figure 8a. The solid curve flattens out relatively quickly as a function of h*, indicating a topped entropy gain for the short grafted polymers. Note also that ∆(∆F) < 0 for very small h*. It is only at these very short distances between colloid surfaces that the real dispersing function of the end-grafted polymers is not overcome by entropic considerations. The inset shows the internal energy difference (normalized by the averaged particle number) between the two cases compared, which proves this difference does not increase with growing h*. From Figure 5 we obtained the free energy difference between 20 molecules of surface-modifying polymers and 20 molecules of grafted polymers, both with Mw ) 400, and show the result in Figure 8c. The inset shows the internal energy difference


Figure 6. Comparison between the disjoining pressure isotherm of 20 molecules of 7-bead, surface-modifying polymers (solid line, circles) vs the one corresponding to unpolymerized monomers of the same kind and at the same concentration as in the polymer case (dotted line, squares). The only difference between these two systems is the absence of springs joining the polymer monomers for the isotherm shown in dotted line. Lines are guides to the eye. See the text for full details.

(normalized by the averaged particle number) between these two cases. Since the only difference between the two systems compared is the adsorption mechanism of the polymers, one must ascribe the relatively large free energy difference to the entropic gain of the grafted polymers, arising from the mobility of the polymeric monomers not adsorbed on the surfaces. Integrating the difference (see eq 7) between the DP of 20 molecules of 7-bead, surface-modifying polymers, and the equivalent number of unbound polymeric monomers (from Figure 6), one obtains the ∆(∆F) shown in Figure 9. Once again this difference is always positive, indicating the unbound monomeric fluid has a smaller free energy and thus is thermodynamically more favorable. Except for the small fluctuations in the number of solvent molecules, Figure 9 shows the comparison of two systems whose only difference is the presence of springs holding

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together DPD atoms of the polymer molecules. Notice how the internal energy difference, shown in the inset, does not increase monotonically, while the free energy difference does. The configurational entropic gain of the unbound polymeric monomers (with respect to that of the polymers) therefore grows with increasing volume, as one would expect, which is why they are better stabilizers than the polymers. Finally, carrying out the calculation of ∆(∆F) from the data in Figure 7, we obtain the free energy difference between a confined fluid made up of solvent and 20 molecules of 7-bead, end-grafted polymers and the equivalent number of unbound polymeric monomers (of the surface-modifying kind). The results are presented in Figure 10, where one observes that ∆(∆F) is now negative; as the volume (or h*) increases ∆(∆F) becomes more negative, which is due to the larger DP obtained for endgrafted polymers (see Figure 7). This trend is however reversed at some wall-to-wall separation (h* ≈ 3.5), where ∆(∆F) starts growing. By contrast, the internal energy difference (inset) fluctuates about zero, as a function of the wall-to-wall separation. In the end, the entropic gain of the loose monomers always wins, at least within the context of the interactions considered in the DPD fluid model. A comparison of our main findings with experimental results might be useful at this point. As mentioned in the Introduction, from SFA experiments we can obtain directly the interaction between two surfaces, mediated by polymers in solutions, in conditions similar to those of the simulations reported here. Therefore, we shall focus our comparisons on SFA experiments. One possible example of a surface-modifying polymer is poly(ethylene oxide) (PEO) in water, a good solvent for this polymer at 0.1 M of KNO3, when it is adsorbed on mica. Monomers of the PEO have the same preference to be adsorbed on mica in these conditions, and with SFA it is possible to measure the force exerted by the surfaces. Klein and Luckham32 found a strong dependence of the surface forces on the compressiondecompression conditions and hysteretic effects. However, at quasi-equilibrium conditions they found that (i) the interaction between the surfaces mediated by polymers was repulsive for separate experiments, with PEO of Mw ) 4 × 104 and 1.6 × 105. At the same separating distance and experimental conditions,

Figure 7. Same as in Figure 6, but for grafted polymers. Twenty molecules of 7-bead grafted polymers (solid line, circles) were confined by the walls to obtain the DP isotherm (full range shown in the Inset). The number of unpolymerized, surface-modifying monomers (dotted line, squares) was the same as in Figure 6. The cartoons in the figure merely illustrate the type of polymer used and do not represent the actual polymer length or number of monomers used. Lines are guides to the eye.

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Goicochea et al.

Figure 9. Change in Helmholtz free energy per unit area, ∆(∆F/A), as a function of box length in the z-direction (solid line, right axis), called here h*, obtained from the difference between the free energy of 20 molecules of surface-modifying polymers with Mw ) 400, minus the equivalent concentration of unpolymerized monomer units. The data integrated (dotted line, left axis) to obtain this free energy difference were taken from Figure 6. The inset shows the internal energy difference, normalized by the averaged particle number, between the systems compared. The uncertainties are smaller than the symbols’ size. Lines are guides to the eye.

Figure 8. (a) Helmholtz free energy difference ∆(∆F) per unit area A (right axis, solid line) as a function of box length in z-direction, called here h*, obtained from integration of the negative difference of Π* for 10 molecules of surface-modifying polymers with Mw ) 800 minus that of 20 molecules of the same kind but with Mw ) 400 (left axis, dotted line). Under these circumstances, the number of polymer monomers per volume is the same for both cases, i.e., 140 monomers. The DP data were taken from Figure 3. (b) Free energy difference (solid line, right axis) between a fluid with five molecules of end-grafted polymers with Mw ) 800 and 10 molecules with length Mw ) 400 of the same kind of polymers. The data to obtain -∆Π* (dotted line, left axis) were taken from Figure 4. (c) Free energy difference (solid line, right axis) between a fluid with 20 molecules of surface-modifying polymers minus one with also 20 molecules of the grafted kind of polymer, both with Mw ) 400. DP data taken from Figure 5. The insets show the internal energy difference, normalized by the averaged particle number, between the systems compared. The uncertainties are smaller than the symbols’ size. Lines are guides to the eye.

Figure 10. Change in Helmholtz free energy per unit area, ∆(∆F/A), as a function of box length in the z-direction (solid line, right axis), called here h*, obtained from the difference between the free energy of a fluid with 20 molecules of grafted polymers with Mw ) 400, minus the equivalent concentration of unpolymerized monomer units of the surface-modifying kind. The data integrated (dotted line, left axis) to obtain this free energy difference were taken from Figure 7. The inset shows the internal energy difference, normalized by the averaged particle number, between the systems compared. The uncertainties are smaller than the symbols’ size. Lines are guides to the eye.

(ii) the magnitude of the repulsive force increased 2-fold when the polymer concentration was increased from 10 to 150 µg mL-1 for the PEO of Mw ) 1.6 × 105; (iii) the repulsive force mediated by the polymer with Mw ) 4 × 104 was lower than that produced by the Mw ) 1.6 × 105 polymer. Observation (i) does not agree exactly with the results reported in Figure 2a and c, where some oscillations between attraction and repulsion appear for polymers of low molecular weight. (32) Klein, J.; Luckham, P. F. Macromolecules 1984, 17, 1041.


This disagreement is due to the high molecular weight of the polymers used in the experiments, in comparison with those of the simulations. However, our results do show the same trend seen in experiments,32 as the molecular weight is increased. Likewise, observations (ii) and (iii) have the same trends as those of the simulated DP isotherms presented in Figure 2a and c: the repulsion increases with the polymers’ concentration and molecular weight. Contrary to the surface-modifying polymers, the end-grafted polymers do not present hysteretic effects in the experiments.33 These latter systems are experimentally modeled by diblock copolymers with a short block that is affine to the surface (mica), and a block of polystyrene or polyisoprene that prefers the solution because it is a good solvent for this moiety.33,34 The forces measured for these systems show that the interaction between surfaces mediated by these polymers was always repulsive and that the interaction increased with the molecular weight of the polymer.34 The surface force increases also with the concentration of the adsorbed polymer, as is shown by the scaling model used to fit the experimental data.33 All these experimental results are qualitatively reproduced by the simulations presented in Figure 2b and d. The disjoining pressure isotherms presented in Figures 3 and 4 and the corresponding free energy differences presented in Figure 8a and b have an underlying difference. While the DP for the end-grafted polymers depends strongly on molecular weight, the DP for the surface-modifying polymers shows a weaker dependence. This behavior for the end-grafted polymers is expected because they tend to assemble normal to the surface, therefore surface forces depend on the extension of the polymers perpendicular to the surface (see model in ref 33). However, the fact that the disjoining pressure displays a considerably weaker dependence on molecular weight for surface-modifying polymers seems surprising. The difference between the DP isotherms in Figure 3 and free energy difference reported in Figure 8a for the two cases compared were the smallest among all systems we studied. This feature has also an experimental counterpart that is related to scaling properties of the polymers at relatively high concentrations, like those reached in a confined system. Indeed, in a semi-dilute regime, the osmotic pressure is effectively independent of the molecular weight of the polymer and it depends only on the monomer concentration.35 The set of experimental results discussed above shows that the predictions from our simulations are consistent with the available experimental reports. There appear to be no experimental results in the literature with which to compare our predictions shown in Figures 5, 6, and 7, which makes it all the more interesting to prove that in some situations the unpolymerized monomers can be better stabilizers than the corresponding polymers. It is our hope this last prediction is followed up by experimental groups.

V. Conclusions The fact that a polymer with low molecular weight may act as a better dispersant than one with larger Mw (at the same monomer concentration) might appear at first as somewhat surprising. This is a consequence, however, of the adsorption model chosen because the substrate tends to be covered up more easily if the surface-modifying polymer chains (Figure 1a) are (33) Taunton, H. J.; Toprakcioglu, Ch.; Fetters, L. J.; Klein, J. Macromolecules 1990, 23, 571. (34) Watanabe, H.; Tirrell, M. Macromolecules 1993, 26, 6455. (35) Daoud, M.; Cotton, J. P.; Farnoux, B.; Jannink, G.; Sarma, G.; Benoit, H.; Duplessix, R.; Picot, C.; de Gennes, P. G. Macromolecules 1975, 8, 804.

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short than if they are large. A qualitatively different scenario is obtained when considering polymers that adsorb at one of the ends of the molecule only, as shown in Figure 1b. In such a case, the stability mechanism acts differently, since the surface of the colloid is not covered up by the grafted polymers, but these act as good dispersants because they form polymer brushes, with only one end physically attached to the surface. The rest of the monomers in the polymer molecule form a disordered layer that protects the surface against touching another since this would require the polymer brushes of opposite surfaces to align, which is entropically unfavorable. We have shown that, although end-grafted polymers are better dispersants than their surface-modifying counterparts, at the same polymer concentration, short polymers are better stabilizers than large ones, also at the same monomer concentration. This holds true for both kinds of dispersants discussed in this work; see Figures 3 and 4. Additionally, we showed that monomers of the surface-modifying polymers are the best dispersants, leading at large surface-to-surface distances to higher DP than even the grafted polymers. We argued that the gain in configurational entropy of monomers (or short polymers) is mainly responsible for this phenomenon, which we supported with calculations of the Helmholtz free energy difference. Since we compared systems whose only difference was the presence or absence of harmonic forces between adjacent monomers in polymers, and the only interactions are short-range repulsive ones, the difference in free energy traces changes in the entropy of the system. The calculations presented here were all carried out for a fluid modeled and confined by short-range, repulsive forces (except for the springs joining monomers in polymer molecules). Therefore, the disjoining pressure obtained does not include electrostatic or van der Waals contributions,10 the competition of which may give rise to different trends from those predicted here. However, for typical ionic strengths, say greater than 0.1 M, the Debye length (characteristic length of electrostatic interaction) is less than 1 nm,3 which is well within the range of our DPD forces. As for the two adsorption models introduced, they represent a wide variety of real-world polymeric dispersants,36 so we expect our conclusions to be relevant for a large class of materials. When designing and testing grafted dispersants experimentally, one should keep in mind that the anchor part of the polymer must be insoluble in the solvent, while the tail should have high affinity for it. One important aspect which should not be overlooked is the fact that when unpolymerized monomers are used instead of large polymer molecules, there may not be a need to spend the energy necessary to grow the polymer chains. The work presented here is another example of the usefulness of DPD as a modeling tool to help in the design of materials with specifically tailored characteristics. Acknowledgements This work was supported by the Centro de Investigacioń en Polı´meros (CIP, Grupo COMEX). A.G.G. acknowledges enlightening discussions with A. Grafmu¨ller, H. Hernandez, V. Knecht, K. Tauer, and in particular, C. Seidel, and thanks R. Lipowsky for the hospitality of the Max Planck Institute of Colloids and Interfaces, where final parts of the manuscript were completed. Discussions with R. Castillo, E. Mayoral, J. M. Meńdez-Alcaraz, and L. M. Rıós are gratefully acknowledged also. E.P. thanks CIP for its support and hospitality during his sabbatical year. E.N.-A. thanks DGAPA-UNAM for financial support during this research project. LA802585H (36) Schofield, J. D. Prog. Org. Coat. 2002, 45, 249.

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