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On the potential of mean force of a sterically stabilized dispersion.

Abstract The potential of mean force (PMF) is the total free energy of a many-body colloidal system, and consequently it includes all the interactions the colloids experience due to collisions with themselves and with the solvent. Here, the PMF of a colloidal dispersion under various circumstances of current interest, such as varying solvent quality, polymer coating thickness, and addition of electrostatic interaction, is obtained from radial distribution functions available from the literature. They are based on implicit-solvent, computer simulation studies of a model Ti[O.sub.2] dispersion that takes into account three major components to the interaction between colloidal particles, namely van der Waals attraction, repulsion between polymer coating layers, and a hard-core particle repulsion. In addition, a screened form of the electrostatic interaction was included. It is argued that optimal conditions for dispersion stability can be derived from a comparative analysis of the PMF under the different situations studied. This thermodynamics-based analysis is believed to be more accessible to specialists working on the development of improved colloidal formulations than that based on the more abstract, radial distribution functions.

Keywords Stability of Ti[O.sub.2] dispersions, Application of molecular dynamics simulations, Polymer brush coatings for stability, Potential of mean force, Thermodynamic stability


Titanium dioxide (Ti[O.sub.2]) particles dispersed in aqueous solvent constitute perhaps the most important industrial test bed for theories of colloidal stability and are also the focus of numerous experiments designed to increase the understanding of the interaction between the Ti[O.sub.2] particles and the polymeric dispersant to improve the conditions of optimal stability. (1) Some of the most popular applications of Ti[O.sub.2] dispersions are found in consumer goods such as architectural white water-based paints, (2) toothpaste, and others. (3) There are also important environmental applications of titania dispersions. (4)

It is known that a dispersion of colloidal particles can be kinetically stabilized, i.e., stabilized through particles' collisions, by coating the particles with polymers or with polyelectrolytes. (5) Coating the particles' surface with polymers grafted onto the surfaces so as to form polymer "brushes" is an efficient mechanism of stability because there is an entropy loss if the opposite brushes overlap, which is thermodynamically unfavorable. When there are electrostatic charges on the colloids, their Coulomb repulsion tends to increase the separation between them, thereby improving the dispersion's stability. There are numerous studies available in the literature which have addressed aspects such as the role of long polymer tails in the steric stabilization of colloids, (6-8) the influence of the temperature in polymer-coated colloidal dispersions, (9) the appearance of depletion forces (10) and how they induce flocculation in polymer-coated polystyrene dispersions, (11) and the stability of colloidal dispersions in media with high salt content, (12) to name but a few.

The basic interactions that compete in the phenomenon of colloidal stability are the short-range, van der Waals attraction, and long-range electrostatic repulsion. Those are the essential ingredients of the so-called DLVO theory (after Derjaguin, Landau, Verwey, and Overbeek), (13) which has met with considerable success. However, van der Waals attraction is important only when the particles are not coated and can get in close contact with one another. When a polymer brush is grafted onto the particles' surface, other interactions come into play, not only of entropic nature, but arising also from three-body repulsion between polymer chains. (14) By three-body repulsion, we mean here the interactions a polymer chain experiences with other polymer chains and with itself, of the type polymer-polymer-polymer. They give rise to a third term in the so-called virial expansion of the pressure of the system in terms of the density, and therefore, it is named the "third virial coefficient." (15) These interactions have been used in the past as mechanisms to promote entropic (steric), electrostatic colloidal stability, or a combination of both. (5) Advances have been achieved through the application of density functionals and integral equations for cases such as varying ionic strength or solvent quality. (16,17) A relatively modern alternative to the theoretical and experimental efforts devoted to the understanding and optimization of colloidal dispersion comes from the field of computer simulations. (18) Among their advantages is the fact that one can solve the interaction model for many particles almost exactly, which most theoretical approaches cannot accomplish. Also, one has total control over the thermodynamic and physicochemical conditions of the model dispersion, something that is not easily achieved in most experiments. From computer simulations, one can obtain correlation functions that can shed light on the kinetic or thermodynamic stability conditions of the dispersion. Kinetic stability is achieved when there is a barrier in the interaction potential which is larger than the thermal energy, so that collisions due to Brownian motion (kinetic energy) are enough to keep the particles apart. Thermodynamic stability, on the other hand, requires that the system of particles is at the minimum energy. One of those correlation functions is the pair distribution function, also known as the radial distribution function, (18) which is commonly used to determine the number of nearest neighbors any particle in a fluid has as a function of the relative distance between the given particle and its neighbors (per unit volume), due to their specific interactions. The radial distribution function can be interpreted as a measure of the probability of finding a particle at a given distance away from another given reference particle, i.e., it is a function that quantifies how the local density varies in a fluid. (18) Although a considerable amount of work on colloids has been amassed during the past decades, theoretical and computational information still remains relatively inaccessible to most researchers carrying out experiments to improve the stability of formulations, because properties such as correlation functions are not as easily grasped as are thermodynamic concepts.

In this work, our focus is on illustrating how some simple guidelines can be followed to use the functions mentioned above in the search for colloidal stability criteria and apply them to a specific example taken from the literature. Most of the analysis reported here is based on the PMF, which is the total free energy of the colloidal dispersion as a function of the colloids' relative positions, and it therefore determines the thermodynamic properties of the system. It should not be confused with the pair interaction potentials; the PMF is the average potential that the colloids experience as a result of all the pair interactions (with the solvent and with other particles) present in the dispersion. The model whose PMF is analyzed here under various circumstances is based on the competition between the attractive van der Waals interaction between particles and the repulsion between the particles' polymer coating. The latter originates from the third virial coefficient, which is the third term in a series expansion of the pressure of the colloidal dispersion in terms of the density. (14,15) The first term in such expansion represents the contribution to the pressure of noninteracting particles, as in an ideal gas. The second virial coefficient arises from the interactions between pairs of particles, while the third virial coefficient is due to three-body interactions. In addition, we compare the PMF obtained from other competing theories (Alexander-de Gennes (19,20) and Milner-Witten-Cates (21)) based on different assumptions so that a criterion can be established to determine uniquely the physical basis for colloidal stability.

Models and methods

Most of the results discussed in this work were obtained for colloidal dispersions stabilized by non-electrostatic means, although we shall briefly comment on the consequences of adding a simplified model of electrostatic interactions to the stabilizing mechanism proposed here when discussing the final figure in this article. Our starting point is a mean-field theory, proposed by Zhulina et al., (14) and hereafter referred to as ZBP, for the interaction between colloids covered with polymer brushes. Such interaction has two contributions: a short-range attractive term, arising from the van der Waals interaction between colloids, which is known to depend on the inverse of the relative distance between the colloids' surfaces (13):

[U.sub.vdW] = [A.sub.H]/12 [R.sup.2]/[h.sup.2], (1)

where [A.sub.H] is Hamaker's constant, R is the colloidal particles' radius, and h is the colloids' surface to surface distance. The other term is a repulsive contribution arising from the interaction between the polymer brushes on opposite colloidal surfaces as they approach each other, for relative separation distances that are smaller than the particles' size. (22) The total interaction is shown in equation (2):

U(r) = [DELTA][F.sup.0.sub.0] [[[beta].sup.[pi]]/r + [[pi].sup.2]/12[r.sup.2] (1 - [gamma])] + [U.sub.hc], (2)

where r is the relative distance between the colloids' centers of mass (in reduced units), [beta] is the solvent's quality, [DELTA][F.sup.0.sub.0] is the free energy of the uncompressed polymer layer at the [theta]-temperature, and y is a constant that incorporates the polymer-polymer repulsive interactions through the dimensionless third virial coefficient ([omega], which represents three-body interactions), the polymer grafting density on the colloidal surface ([GAMMA] = number of grafted chains/area of the colloidal particle), and Hamaker's constant, namely:

[gamma] = [A.sub.H]/96[pi][K.sub.B]T[omega] [(1[GAMMA][Na.sup.2]).sup.3], (3)

with [k.sub.B] being Boltzmann's constant, T is the absolute temperature, N is the polymers' degree of polymerization, and a is the monomers' size. (14) All parameters within the brackets in equation (2) are dimensionless ([beta], [gamma], r), while in equation (3), [A.sub.H] has dimensions of energy as does [k.sub.B]T, and [GAMMA] has units of 1/area, making the product [GAMMA][Na.sup.2] dimensionless. The last term in equation (2), [U.sub.hc] is only a hard-core potential whose purpose is avoiding that particles completely penetrate each other. (22) The dimensionless polymer grafting density is [[GAMMA].sup.*] = [GAMMA][Na.sup.2]. In Fig. 1, we show the behavior of the interparticle potential shown in equation (2) for two cases: when the constant [gamma] in equation (3) is [gamma] < 1, indicative of weak interparticle attraction, and [gamma] > 1, which occurs for strong interparticle attraction. If the polymer grafting density ([GAMMA]) is large, or the degree of polymerization (N) is increased, then [gamma] is reduced and may be less than one because the three-body repulsions ([omega]) are increased also under such conditions [see equation (3)]. When [GAMMA] and/or N are large, each monomer that makes up the polymer chains collides with an increasing number of neighboring monomers, which in turn increases three-body repulsions. On the other hand, if there is a strong colloid-colloid attraction, then the Hamaker constant is large, and the van der Waals attraction (equation (1)) dominates, making [gamma] larger than one.

The competition of the van der Waals attraction with the repulsion between polymer layers, as defined by equation (2), without the hard-core repulsion [U.sub.hc] gives rise to the pair potentials presented in Fig. 1. As shown in Fig. 1a, for situations when [gamma] is less than 1, the colloidal dispersion is always stable (whenever U(r) is positive). Under [theta]--to good--solvent conditions, the dispersion is stable because there is a dominance of three-body repulsion ([omega]) over the van der Waals interaction, which is sufficient to make the dispersion thermodynamically stable. Pair interactions contribute to improve the stability when [beta] > 0. When [gamma] > 1 (see Fig. lb), there appears a transition from an unstable dispersion (see, for example, the black line in Fig. 1b) to a stable one as the solvent quality increases (increasing [beta], see, for example, the blue line in Fig. 1b), established by the appearance of a maximum in the interaction potential. The competition between the attractive van der Waals interaction and the repulsive three-body correlations is responsible for such maximum, which leads to a kinetically stable colloidal dispersion.

The model shown in equation (2) takes into account the three-body repulsion between monomers that makes up the polymer chains (through the third virial coefficient, [omega]), and radial distribution functions obtained through computer simulations using this model (22) have demonstrated that it can lead to colloidal stability, that is repulsion between polymer-coated colloids. Other models for the effective force between polymer brushes such as that of Alexander and de Gennes (AdG) (19,20) do not take into account chain-chain interactions. In particular, AdG's model assumes that the chains density profile is a step function with all the chains ends placed at the layer surface, that there is no interchain interaction, and that the polymer brushes are in a good solvent. It considers only two principal contributions to the many-body force in compressed polymer brushes: a short-range repulsion, due to the osmotic pressure that arises from the increased density of monomers in the compressed region, and a medium range attraction whose origin is the elastic energy of the polymer chains. Although both models (ZBP and AdG) predict repulsion between strongly compressed polymer brushes, the physical origin of such repulsion is different. While AdG attribute it to an entropy loss due to a restriction in available volume to the chains, ZBP attribute it instead to three-body interactions between the monomers making up the chains. An alternative, self-consistent field model (21) proposed by Milner, Witten, and Cates (MWC) does take into account interchain interaction in the brush but yields a PMF curve that differs very little from that of AdG's. For the sake of posterior comparison, we shall consider the following expressions for the PMF between polymer brush-coated colloidal particles in good solvent with the present model:

[W.sub.AdG](h)/[k.sub.B]T = (2[h.sub.0])A[[GAMMA].sup.3/2][4/5[(2[h.sub.0]/h).sup.5/4] + 4/7[(h/2[h.sub.0]).sup.7/4]], (4)

[W.sub.MWC](h)/[k.sub.B]T = (2[h.sub.0])A[[GAMMA].sup.3/2][1/2(2[h.sub.0]/h) + 1/2[(h/2[h.sub.0]).sup.2] - 1/10[(h/2[h.sub.0]).sup.5]]. (5)

Equations (4) and (5) represent the PMF for the AdG and MWC models, respectively, where in both cases [h.sub.0] represents the thickness of the uncompressed polymer layer, A is the colloids' surface area, [GAMMA] is the polymer grafting density, and h is the distance separating the surfaces of colloids when the polymer layers are compressed. It must be noticed that both models are valid only for compressed polymer brushes, i.e., for h [less than or equal to] < 2[h.sub.0]. Figure 2 shows a schematic diagram of the model colloidal dispersion that is the subject of this work, for illustrative purposes only.

We focus here on the PMF ([W.sub.PMF](r)), which is the average interaction the particles experience due to collisions with each other and with the solvent; hence, it provides important thermodynamic information about a many-body system. It can be obtained from the colloids' radial distribution functions, g(r), through the relation (15,23):

[W.sub.PMF](r) = - [k.sub.B]Tln[g(r)]. (6)

We shall use equation (6) to obtain the PMF for the ZBP model, using the radial distribution functions calculated in reference (22), and compare with the models shown in equations (4) and (5). The colloids' radial distribution functions on which this work is based were obtained from standard computer simulations that integrate Newton's second law of motion for a set of colloidal particles with the solvent included implicitly, using the so-called Velocity-Verlet algorithm, (18) for a potential interaction function between colloids given by equation (2), for a system with 896 colloidal particles in a box with periodic boundary conditions. (22) The solvent is included implicitly through the choice of the value of the Hamaker constant. The time step used for the integration of the equation of motion in reduced units was equal to t = 0.007[t.sub.LJ], where [t.sub.LJ] is the Lennard-Jones time step. (18) The simulations were run for 3 x [10.sup.4] cycles during the equilibrium phase and at least for 2 x [10.sup.5] cycles during the production phase. At the end of the simulations, the radial distribution functions were calculated and from them the PMF is obtained through equation (6); see full details in reference (22).

Results and discussion

First, we show the PMF for ZBP's model (14) as a function of the polymer brush thickness, then for increasing quality of the solvent, and finally for a colloidal dispersion with and without electrostatic interactions. For all cases, we chose T = 300 K; quantities expressed in reduced units are indicated with asterisks.

Figure 3 shows the PMF results for a Ti[O.sub.2] colloidal dispersion in water whose Hamaker constant is [A.sub.H] = 6 x [10.sup.-20] J, (24) with average particle size [sigma] = 200 nm. (25) For the thinnest polymer coatings (black and blue lines, with [h.sub.0] = 10 and 30 nm, respectively), the PMF barrier is not too high, less than 2[k.sub.B]T, and could be overcome in particle-particle collisions due to thermal fluctuations, leading to flocculation of some particles whose fraction is reduced as [h.sub.0] is increased, as expected, (5) (see the green and red lines in Fig. 3). A first minimum in the PMF appears at relative distances below [r.sup.*] ~1.5, which is however relatively shallow and can easily be overcome by Brownian motion, followed by a second one at [r.sup.*] ~2.7, which is even shallower. The oscillations shown by the PMF arise from the corpuscular nature of the dispersion, (13) with a period given approximately by the particle size ([sigma]), which is considered to be monodispersed throughout this work. In actual water-based paints, Ti[O.sub.2] is known to have a distribution of sizes, and in such case, the oscillations shown in Fig. 3 are expected to be much less pronounced, yielding an almost monotonically decreasing PMF.

Increasing the parameter [beta] is equivalent to improving the solvent quality (see Fig. 1), with [beta] = 0 representing a theta-solvent. (14) In Fig. 4, we present the PMF for a colloidal dispersion with [gamma] fixed at 0.5, which means stability is always obtained through three-body repulsion overcoming the van der Waals attraction (Fig. la). As Fig. 4 shows, when the compression is stronger, i.e., for [r.sup.*] close to 1, the contribution from the third virial coefficient dominates over the van der Waals attraction due to polymer-polymer repulsion, as the layer is brought closer into contact. Within this range, increasing the binary interactions through the solvent quality parameter, [beta], improves the stability even more, leading to increasingly large potential barriers, well above the thermal energy, [k.sub.B]T. The depth of the short range (1 < [r.sup.*] < 1.5) and larger range (2 < [r.sup.*] < 2.5) wells increases also with [beta], but their depth is less than the thermal energy and therefore does not yield permanent particle flocculation. We have included in Fig. 4 the PMF curves that were obtained from the AdG (dotted purple line) (19,20) and MWC (dashed orange line) (21) models, using equations (4) and (5), respectively, for comparison. To do so one has to properly normalize these equations, which is done when the length is reduced with 2[h.sub.0], yielding a reduced colloid area [A.sup.*] = [GAMMA][(2[h.sub.0]).sup.2] and a reduced grafting polymer density [[GAMMA].sup.*] = [GAMMA](2[h.sub.0]). In the "Appendix", we show in detail how the expressions for the AdG and MWC models are reduced. Those models are defined only for compressed polymer brushes; therefore, they become identically zero when the brushes do not overlap. Regarding the oscillations shown by the PMF (and their absence in the AdG and MWC models) in Fig. 4, the same analysis as that of Fig. 3 applies here. At the strongest compression of the polymer layers (for values of [r.sup.*] close to 1) and for good solvent conditions ([beta] = 1), the PMF (blue line in Fig. 4) obtained from the ZBP model (14,22) is more repulsive than those corresponding to the AdG and MWC models, indicating that ternary interactions should not be neglected at large compression because they can be the leading repulsive mechanism. Recent explicit-solvent computer simulations of planar surfaces coated with relatively short polymer brushes (26) have confirmed that both AdG and MWC models reproduce fairly well the ZBP PMF at intermediate compression of the brush, but they are not as good at very strong compression, as Fig. 4 shows. Although ZBP was designed for colloidal particles coated with polymer brushes, it can also be applied to colloids coated with layers of adsorbed polymers (sometimes called "surface modifying" polymers) because ternary interactions play the same role in both situations.

Lastly, we consider briefly the influence of weak electrostatic interactions in the stability of the ZBP model. To do so in a way that is consistent with the mean-field nature of the electrically neutral model, one adds a screened electrostatic contribution to the total particle-particle interaction potential [see equation (2)] of the so-called Yukawa type given by the following expression:

[U.sub.e](r) = [([Z.sub.effec]).sup.2]exp(-[[kappa].sub.D]r)/(4[pi][[epsilon].sub.0][[epsilon].sub.r])r. (7)

In equation (7), [Z.sub.effec] is the effective charge on the colloidal particle surface, e is the electron charge, [[kappa].sub.D] is the inverse of the Debye length, and [[epsilon].sub.0][[epsilon].sub.r] is the medium permittivity. We chose values appropriate for a Ti[O.sub.2] dispersion, namely [Z.sub.effec] = [100.sup.27] and [[kappa].sup.-1.sub.D] 20 [nm.sup.2] which is roughly equivalent to a solvent's ionic strength of about 1 mM. (5) (See reference 22 for full details.) The potential [U.sub.e](r) in equation (7) is then added to the potential given by equation (2), and the new full potential is used to carry out simulations under the same conditions as those without electrostatic charges. (22) Figure 5 shows the PMF (red line) obtained from the radial distribution function [see equation (6)] after computer simulations are run for a system of particles where equation (7) is added to the ZBP interaction potential, equation (2). For comparison, the PMF of an electrically neutral dispersion in a theta solvent is included in the same figure. Two features are notable in Fig. 5; on the one hand, the range of the first shallow attractive well (where the PMF is negative) is reduced by about half when the electrostatic interaction is included, namely the range for the neutral dispersion (blue line in Fig. 5) comprises values of [r.sup.*] that go from about [r.sup.*] ~1.2 up to [r.sup.*] ~2.0, while for the electrostatic case such range goes from [r.sup.*] ~1.1 up to [r.sup.*] ~1.5. On the other hand, a potential barrier appears at [r.sup.*] ~1.8 when electrostatics are included although it is relatively weak, amounting to about 0.17[k.sub.B]T; nevertheless, this may be enough for kinetic stabilization of the dispersion. These features and the shape of the red curve in Fig. 5 are reminiscent of the DLVO potential. (13) Although the PMF for the charged ZBP model shown in Fig. 5 (red curve) is similar in shape to that typical of the DLVO model, they arise from competition of forces of different origin. While DLVO is the result of attractive van der Waals interactions coexisting with repulsive double-layer electrostatic interactions, the charged version of the ZBP model produces a barrier in the PMF whose origin is the competition between attractive van der Waals interactions, ternary repulsion between polymer brushes, and repulsive electrostatic interactions between the colloidal surfaces. For large grafting density and polymerization degree, the amplitude of the oscillations seen in the curves in Fig. 5 is expected to be reduced. However, the PMF does have a strong dependence on the electrostatic properties of the dispersion, such as the polyelectrolytes pH or ionization degree, becoming more repulsive as the ionization degree increases. (28) Although it is not the purpose of this work to carry out a systematic study of the influence of such factors on the PMF, it is important to emphasize that the inclusion of electrostatics in the model leads to trends in the PMF that can be interpreted as those of increasing the polymer layer thickness or the quality of the solvent.


Optimizing the stability of colloidal dispersions for current applications requires the use of sophisticated methods that go beyond the traditional trial and error experimental tests. One such method is the application of computer simulations, which can be run on modern computers to yield important physicochemical information directly comparable with experiments, in a relatively short time. Although such methods have met with success when applied to paints and coatings, (29) they still remain relatively inaccessible to these communities due to in part to a lack of familiarity with properties routinely obtained from computer simulations, such as the radial distribution functions. Here, we show that a clearer understanding of the conditions for stability of a colloidal dispersion can be obtained from analysis of the PMF, which can be obtained directly from radial distribution functions and reduces to analytical thermodynamics calculations. Some simple guidelines can be used when comparing PMF curves to establish trends in colloidal stability, such as the height of the repulsive (positive) barriers. The larger the barrier, the more stable the dispersion is expected to be. If the PMF has attractive (negative) wells, which in principle can induce flocculation, one must opt for the PMF with the shallowest wells, since any flocculation produced by such wells is likely to be reversible and easily overcome by thermal motion. Lastly, the PMF with the steepest hard core, short-range repulsion, which occurs for relative distances [r.sup.*] [approximately equal to] 1 in the present work (see, for example, red line in Fig. 3), is to be preferred because it represents the largest repulsion between the particle surfaces, thereby inducing better stability.

The colloidal stability of the sterically stabilized dispersions is clearly dependent on the solvent quality, polymer coating thickness, and strength of the electrostatic interaction. The latter is important but not necessary for the colloidal stability of these systems because polymer brushes are able to produce a ternary repulsion among these polymers. The van der Waals attraction between particles is then balanced by this repulsion, producing a PMF that is very similar in shape to the DLVO potential but without the presence of the electrostatic repulsion. Analysis of the PMF showed that increasing the thickness of the polymer layer that coats the colloidal particles, improving the quality of the solvent (which can be done raising the temperature) or adding charges to the system have the same effect, that is, improving the kinetic stability of the colloidal dispersion. It must be kept in mind that there are a number of factors which are known to influence the stability of colloidal dispersions that have not been taken into account here. One such factor is the configuration of the adsorbed polymer, e.g., in loops, trains, or trails (30); however, this can to some extent be incorporated into our model through the appropriate choice of free energy for the uncompressed polymer layer [see equation (2) and for the constant [gamma] equation (3)]. Another is a large particle size distribution, which is almost always present in real dispersions and would lead to a smearing of the structure seen in the PMF (see, for example, Fig. 4), but this is an aspect of secondary importance when compared with the solvent quality, the thickness of the polymer layer, or three-body interactions. Also, electrostatic effects such as adsorption of strongly charged polyelectrolytes, multivalent ions in the solution, or varying pH are not accounted for explicitly by our model. Taking those electrostatic effects fully into account goes beyond the scope of the present work. Nevertheless, we believe that our contribution is a useful tool for a first approximation to the problem of colloidal stability because it is relatively easy to grasp and implement and has sufficient versatility for applications to colloid science.

DOI 10.1007/s11998-014-9600-0

Acknowledgments AGG would like to thank Universidad Autonoma de San Luis Potosi for the hospitality and necessary support for this project. The authors acknowledge M. A. Balderas Altamirano and J. P. Lopez Neria for discussions. Lastly, the authors are indebted to the reviewers of the manuscript for their criticisms, suggestions, and insightful comments.


Here, we show how equations (4) and (5) can be expressed in reduced units (indicated by an asterisk) so that they can be drawn on the same scale as the other PMF in Fig. 3. Let us start by changing variables so that the spatial coordinate is not the compressed polymer layer thickness (h) but rather the relative distance between the colloidal particles' centers of mass (r), as follows: r = h + [sigma], where [sigma] is the particle diameter (see Fig. 2). Reducing all lengths with 2[h.sub.0], we obtain for the AdG model


and for MWC model


where [A.sup.*] = [pi][[delta].sup.*2] A; [[GAMMA].sup.*] = Np/[A.sup.*], with [N.sub.p] equal to the number of polymer chains grafted onto the colloidal surface, and [[delta].sup.*] = [sigma]/2[h.sub.0]. The constants subtracted [48/ 35 in equation (8), and 9/10 in equation (9)] are chosen so that the PMF can be equal to zero when the opposing polymer brushes separate enough that they do not overlap ([r.sup.*] = 2[[delta].sup.*]), since both models (AdG (19,20) and MWC (21)) are defined only for polymer brush compression. To compare both models with our predictions for the PMF on the same scale we chose the value of [A.sup.*] [[GAMMA].sup.*3/2] = 0.05 and [[delta].sup.*] = 1, for both cases [equations (8) and (9)].


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R. Catarino Centeno, E. Perez, A. Gama Goicochea ([mail])

Instituto de Fisica, Universidad Autonoma de San Luis Potosi, Alvaro Obregon 64, 78000 San Luis Potosi, SLP, Mexico



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Author:Centeno, R. Catarino; Perez, E.; Goicochea, A. Gama
Publication:Journal of Coatings Technology and Research
Geographic Code:1USA
Date:Nov 1, 2014
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