Langmuir 1994,10,4514-4516
4514
Pair Potentials between Polymer-Coated Mesoscopic Particles C. M. Wijmans,?F. A. M. Leermakers,*and G. J. Fleer Department of Physical and Colloid Chemistry, Wageningen Agricultural University, P.O. Box 8038, 6700 EK Wageningen, The Netherlands Received June 16, 1994@ A self-consistent-field model is developed which enables the calculation of the interaction potential between two polymer-coated mesoscopic particles. We use a cylindrical coordinate system, in which the polymer density profile can vary in both the radial and axial direction. Data are presented for the free energy of interaction for two particles with end-attached polymer chains in a good (athermal) solvent. The repulsion is considerably weaker than predicted by converting the pair potential between flat plates to that between spheres, using Derjaguin’s approximation. This is explained by the greater freedom of the polymer chains to move laterally out of the widening gap between the particles as compared to polymer chains between flat surfaces.
Introduction Polymers greatly influence the surface properties of colloidal dispersions and thus affect the stability of these dispersions. Up to now, most theoretical investigations have been concerned with the interaction between two infinitely large, flat surfaces bearing adsorbed polymer layers. In this paper we propose a model to study the interaction between two polymer-coated particles whose radii of curvature are of the same order of magnitude as the polymer layer thickness. We calculate the free energy change when two such spherical particles are brought together along the axis connecting their centers. We limit ourselves to the case of particles with end-tethered polymer chains, which is interesting from a technological point of view because of the good stabilizing properties of diblock copolymer chains in colloidal dispersions. However, the same approach can be used to model the interaction between (polymeric) micelles or other interacting systems such as small particles covered by physisorbed (co)polymers.
Self-consistent-FieldTheory We describe the polymer chains by a walk of N - 1 steps linking the points r1...rN. Within a self-consistentfield approach we can use the composition law1z2 to calculate the volume fraction of the sth segment a t r
Equation 1 is most easily solved using lattice approximations. Scheutjens and Fleer3 have done this for inhomogeneous systems where 4 is a function of one coordinate only (e.g. polymer adsorption on an infinitely large surface), so that r can be replaced by the distance coordinate z of a Cartesian coordinate system, while simultaneously applying a mean-field approximation in the x and y directions (1D theory). Huang and Balazs4 generalized the Scheutjens-Fleer model for adsorption onto a laterally heterogeneous surface by allowing for concentration gradients in two perpendicular directions in a Cartesian coordinate system and applying a meanfield approximation in the remaining one (2D theory). We adopt a same sort of approach but use a cylindrical coordinate system, as done previously by Leermakers et aL5 for the modeling of inhomogeneous lipid membranes. The axial coordinate z denotes the position parallel to the axis between both particle centers, and the radial coordinate R gives the distance to this central axis. In the lateral direction the lattice is built up of parallel layers. Each layer, z = 1, 2, ..., is divided into concentric rings, numbered R = 1, 2 , ..., starting a t the axis between the particle centers. The spacing between two layers is the same as the spacing between two rings and equals the polymer segment diameter. A ring ( 2 3 ) consists of L(R) = n(2R- 1)equally sized lattice sites. We define aprzori step probabilities A(z’,R’lz,R) to connect a segment a t ( 2 3 ) to one at (z’,R’). These quantities define the lattice geometry. We use a simple cubic lattice with coordination number 6 and allow only steps where 12’ - z / IR‘ - RI is 0 (for steps within one ring, which can go in two directions or 1(steps to an adjacent ring in the same layer or to the same ring in an adjacent layer):
+
where u(r)is the potential energy of a polymer segment with ,d = ( k ~ 2 7 - l .The function G plays the role of Green’s function in the diffusion equation and obeys the following relation: e-8u(r)
G(r;slrl)= -j
4~1’
dr’ &Z-lr-r’I)G(r’;s-llr,) (2) r‘
2R-1
(3)
where I defines the polymer step length. Physically, G(r,slrl)is the statistical weight of all possible conformations of the chain fragment s’ = l, ..., s, with segment number 1 at coordinate r1 and segment s a t r. Present address: Physical Chemistry 1, Chemical Center, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden. Abstract published inAduance ACSAbstructs, October 1,1994. (1) Edwards, S.F.Proc. Phys. SOC.1965, 85, 613. (2) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY,1979; p 249.
The system is incompressible (we do not consider free
+
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(3) Scheutjens, J. M. H. M.; Fleer, G . J. J.Phys. Chem. 1979, 83, 1619. (4) Huang, K.; Balazs, A. C. Phys. Reu. Lett. 1991, 66,620. ( 5 ) Leermakers, F. A. M.; Scheutjens, J. M. H. M.; Lyklema, J. Biochim. Biophys. Acta 1990, 1024, 139.
Q743-7463/94/241Q-4514$Q4.5Q/Q 0 1994 American Chemical Society
WGmans et al.
4516 Langmuir, Vol. 10, No. 12, 1994 40
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0
Figure 2. Free energy ofinteraction between two spheres with various radii of curvature. The free energy Aintdivided by the particle radius Re is given as a functionofthe particle separation for Re = 2 , 5 , and 10. Athermal system; u = 0.1; N = 50. The solid curve is the Derjaguin approximation.
Q 0.2
0.1
0 0
5
10
15
20
Figure 3. Polymer volume fractions in the direction perpendicular to the axis between the particle centers. The profiles are shown for various particle separations M ,and illustrate that the chains are pushed away from the region between the two particles.
responding free energy per unit areaAflat(M)between two equivalent planar surfaces:
For small particles the interaction is far less repulsive. The particles only interact at shorter distances and the interaction curve is less steep than predicted by eq 9. When R, is increased the repulsive interaction does indeed increase. Due to computational limitations, no 2D results are shown for R, > 10 for this system. The fact that for small particles the repulsion is less steep than predicted by Derjaguin's approximation can be explained by the possibility of the polymer molecules to move away out of the gap between the particles when these approach each other. This is illustrated in Figure 3, where volume fraction profiles of the polymer are drawn as a function of the distance to the particle surface, in the direction perpendicular to the axis between the particle
centers. When the distance M between the particles is decreased, the volume fractions in this direction increase. Polymer segments that were originally in the gap between both particles redistribute themselves in order to lower the free energy of this system. This redistribution is not accounted for in the Derjaguin approximation. The failure of the Derjaguin approximation to predict the onset of the interaction curve for small particle radius can be explained by the fact that the brush height H strongly depends on the particle radius R,. In ref 7 it was shown that in the high curvature regime, where H and R, are of the same order of magnitude, H = R,2I5. This scaling correlates with the onset of the interaction curves for R, I10. In conclusion, we have shown that the interaction between two small spherical particles bearing adsorbed polymer layers is far less repulsive than would be expected from the interaction between two equivalent flat surfaces. For larger particles than considered here, this should still be the case when the polymer chain length is increased correspondingly. We have restricted our analysis to the case that the grafting point of the chains remain homogeneously distributed over the surface, even when the particles were highly interacting. In the case of, for example, interacting polymeric micelles, this constraint can be removed to allow for the movement of the grafting point over the surface. The force between two mesoscopic particles has never been directly measured. Direct measurements of the interaction between adsorbed polymer layers have only been made for layers adsorbed onto macroscopic surfaces. However, particle interaction potentials can, in principle, be calculated from osmotic pressure measurements of colloidal dispersion^,^ from the high-frequency limit of the shear modulus of dispersions of monodisperse spherical particles,l0 or from the observation of the Brownian motion of particles by a suitable microscope.ll An alternative method to experimentally investigate our predictions would be the use of an atomic force microscope with a single colloidal particle attached to its tip. Then the interaction could be measured between this tip (onto which a polymer layer would first have to be adsorbed) and a flat surface with an adsorbed polymer layer. The interaction between a sphere and a plane should be in between the interaction between two planes and the interaction between two spheres. Our lattice model can easily be extended to model the interaction between a sphere and a plate, using again cylindrical coordinates.
Acknowledgment. In the early stages of this research we benefited greatly from discussions with the late Dr. J a n Scheutjens, who initiated the work. (9) Evans, R.; Napper, D. H. J . Colloid Interface Sci. 1978, 63, 43. (10)Zwanzig, R.; Mountain, R. D. J . Chem. Phys. 1965, 43, 4464.
(11)Vondermassen, K.; Bongers, J.; Mueller, A,; Versmold, H. Langmuir 1994, IO, 1351.
