Simulating Equilibrium Surface Forces in Polymer Solutions Using a Canonical Grid Method Martin Turesson,* ,† Clifford E. Woodward, ‡ Torbjo ¨rn Åkesson, † and Jan Forsman † Theoretical Chemistry, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden, and UniVersity College, ADFA, Canberra ACT 2600, Australia ReceiVed: March 8, 2008; ReVised Manuscript ReceiVed: May 8, 2008 A new simulation method for nonuniform polymer solutions between planar surfaces at full chemical equilibrium is described. The technique uses a grid of points in a two-dimensional thermodynamic space, labeled by surface area and surface separations. Free energy differences between these points are determined via Bennett’s optimized rates method in the canonical ensemble. Subsequently, loci of constant chemical potential are determined within the grid via simple numerical interpolation. In this way, a series of free energy versus separation curves are determined for a number of different chemical potentials. The method is applied to the case of hard sphere polymers between attractive surfaces, and its veracity is confirmed via comparisons with established alternative simulation techniques, namely, the grand canonical ensemble and isotension ensemble methods. The former method is shown to fail when the degree of polymerization is too large. An interesting interplay between repulsive steric interactions and attractive bridging forces occurs as the surface attraction and bulk monomer density are varied. This behavior is further explored using polymer density functional theory, which is shown to be in good agreement with the simulations. Our results are also discussed in light of recent self-consistent field calculations which correct the original deGennes results for infinitely long polymers. In particular, we look at the role of chain ends by investigating the behavior of ring polymers. 1. Introduction An interesting property of polymers is their ability to regulate surface forces and colloid stability. 1,2 The net effect of polymer addition depends on a number of parameters, such as average molecular weight, polydispersity, chain rigidity, solvent quality, and the effective interactions between the monomers and the surface of the colloid particles. There were several early attempts to theoretically rationalize polymer behavior, 3–8 but it has been subsequently shown that, in many cases, the theoretical models, or approximations contained in them, were too simplistic. The treatment by deGennes, 8,9 utilizing the Edwards theory, remains popular, despite its approximate nature. In particular, it relies upon the infinite chain limit. This assumption may lead to predictions that are even qualitatively incorrect, 10 including a theorem stating that 8 polymer-induced surface interactions are always attractive at full equilibrium. The Edwards-deGennes theory has been extended 11–13 to include the effects of finite chain length. One of the predictions of this corrected theory is that dissolved polymers may generate a repulsive surface force even at full equilibrium and that this repulsion is caused by the presence of chain ends. Given that these theories are approximate in nature, it is desirable to investigate polymer-mediated surface forces by more accurate methods. Simulations are, in principle, the best choice for investigating a given model system, as (statistical noise notwithstanding) they provide an exact solution. Unfortunately, simulations are also computationally very demanding, which limits the types of model systems that can be investigated in practice. In this work, we will present a new method for simulating equilibrium surface forces, which is particularly useful for polymer models. It is a general method, that does not require molecule insertions or deletions in order to establish bulk equilibrium. Our method is closely related to techniques utilizing the isotension ensemble 14–17 but offers some advan- tages. For example, it involves many independent simulations and is thus particularly advantageous if computer clusters are used. However, the scope of this work is not just limited to the introduction of a new simulation method. We apply the method to calculate the interaction between adsorbing surfaces in polymer solutions. Our results demonstrate an interesting interplay between surface-monomer adsorption strength and polymer concentration, as they affect these surface interactions. We will also confirm that density functional theory due to Woodward, 18 combined with a generalized Flory-dimer treat- ment of excluded volume interactions, generates remarkably accurate predictions for surface forces in the systems under investigation. In the final part of our work, we will show that the new method is also well-suited for models where the chain archi- tecture is more complex than the simple linear form. In particular, we will investigate ring polymers, which are par- ticularly interesting as they contain no end monomers. Thus, we are able to test the assertion that repulsive corrections to the deGennes approach is entirely due to chain ends. 2. Model We model two interacting colloidal particles 19 via the Derjaguin approximation; 20 that is, they are treated as two infinite, flat, and parallel surfaces. Periodic boundary conditions apply in the lateral (x, y) directions, and the simulation box has * Corresponding author. † Chemical Center. ‡ University College. J. Phys. Chem. B 2008, 112, 9802–9809 9802 10.1021/jp8020529 CCC: $40.75  2008 American Chemical Society Published on Web 07/18/2008