Vector Imitation Model of Semiflexible Polymers: Application to Polymer Adsorbed on a Spherical Particle Iliya Kusner and Simcha Srebnik* Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel 32000 ReceiVed NoVember 27, 2006; ReVised Manuscript ReceiVed April 10, 2007 ABSTRACT: We introduce a new type of model that imitates polymer behavior of semiflexible polymers. In this model, the energy and entropy contributions to the free energy are treated as forces that act on the individual monomers, and thus guide the “movement” or “growth” of the chain. The model falls at the interface between theory and simulation, allowing statistical prediction of polymer conformational behavior of long polymer chains at a small computational cost. Results are compared with theoretical predictions by Spakowitz and Wang (SW) (Phys. ReV. Lett. 2003, 91, 166102) of the behavior of semiflexbile chain confined to the surface of a sphere and our own Monte Carlo simulations. The SW and Monte Carlo results are reproduced with the imitation model with surprising accuracy. The simple nature of the imitation model allows us to carry out calculations for very long chains, with which we show that the surface coverage of the sphere is a non-monotonic function of persistence length for a given fixed chain length, distinguishing between random, wrapping, and ring conformations. 1. Introduction The large number of degrees of freedom of even a single polymer chain presents a formidable problem for all but the simplest case of the ideal Gaussian chain. Though it is straightforward to write an expression for the partition function and associated thermodynamic averages for a given model of a polymer chain, including nonbonded interactions between distant segments, other chains, or an external potential, it is in general not feasible to carry out exact calculations analytically. For the semiflexible chain, the Krakty-Porod 2 wormlike chain model provided the first rigorous analysis, with an analytic expression for the end-to-end distance as a function of chain persistence length. However, the inability to calculate the end- to-end distance distribution from the Kratky-Porod model meant that direct comparison with experiments was not pos- sible. 3,4 Kholodenko 5 introduced the Dirac chain model, which provides an analytical expression for the chain propagator. However, while the Dirac model accurately predicts chain behavior at the flexible and rigid limits, it is not accurate in the semiflexible regime. 6 Wilhelm and Frey arrived at an ap- proximate analytical expression for the end-to-end distance distribution of semiflexible chains, which showed good agree- ment with MC simulations for the stiffer chains but poorly predicted the end-to-end distance distribution at the flexible regime. 7 Polymer theory appears specifically underdeveloped for problems involving semiflexible chains in confined environ- ments because of finite dimensionality. Moreover, there is no consistent theory at the present time for the behavior of confined semiflexible macromolecules, such as that developed for flexible chains. 8-10 Using the Kratky-Porod wormlike chain model, 2 Yamakawa and Stockmayer 11 were the first to address the way that degrees of freedom may be lost when a stiff chain is coiled tightly into a circle of radius less than its persistence length, λ p . Similarly, in narrow pores, a macromolecule is heavily deformed, and its behavior is no longer universal and becomes dependent both on the conformational properties of the mac- romolecule (its topology, rigidity, excluded volume effects) and on the pore form, i.e., on the nature of boundary constraints. More recently, Kholodenko et al. 12 extended the Dirac chain model to treat the problem of semiflexible chains confined between adsorbing flat parallel plates and similarly showed that chain dimensions of stiff chains are determined by the confining geometry. However, the complexity of these theories renders their extension to other geometries and physically relevant problems difficult. In this communication, we introduce an imitation model of semiflexible chains that is based on thermodynamic consider- ations, and which provides a simple means for qualitatively describing polymer behavior. In the model, the competition between energy and entropy is interpreted as separate “ener- getic” and “entropic” forces that determine the location of the next monomer in a growing chain. In its basic form, the model reduces to simulating ideal random walk behavior. Our method can be modified to include various complications in the dilute- solution regime, such as irregular confining geometries or self- avoidance, 13 which are difficult to incorporate into existing theories. We apply the concept to a semiflexible chain and then treat the problem of a semiflexible chain confined to the surface of a sphere. We compare the results with the theoretical predictions by Spakowitz and Wang 1 (SW) and Monte Carlo (MC) simulations, which provide us with an instrument to check the SW theory. We find that the SW model is very accurate in predicting the behavior of semiflexible chains confined to the surface of a sphere, and agreement with the MC results is excellent. In addition, we show that the imitation method confirms the results of ref 1 and sheds additional insight into the problem of very long chains. The remainder of the manuscript is organized as follows. In section 2, we present the vector imitation model (VIM), detailing its thermodynamic basis and its basic construction for a semiflexible polymer chain. In section 3, we briefly describe the Monte Carlo simulation used to test the model. In section 4, we present and discuss our results, and compare them with the SW model. We offer concluding remarks in section 5. * Corresponding author. E-mail: simchas@technion.ac.il. 6432 Macromolecules 2007, 40, 6432-6438 10.1021/ma062721+ CCC: $37.00 © 2007 American Chemical Society Published on Web 07/18/2007