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42281-AC7
Efficient Sampling Techniques for Field-Based Computer Simulations of Complex Fluids

Glenn H. Fredrickson, University of California (Santa Barbara)

In the past year, graduate student Erin Lennon has worked to develop improved algorithms for the numerical integration of the stochastic partial differential equations that form the basis of the complex Langevin (CL) simulation technique for simulating field theory models of polymeric fluids.  Erin Lennon has been co-advised by Profs. Hector Ceniceros and Carlos Garcia-Cervera of UCSB's Mathematics Department, and by myself.

The CL technique involves writing stochastic differential equations whose steady state solution yields a probability distribution of field configurations that can be used to compute the structure and thermodynamics of polymer solutions and melts. Explicit time stepping of these equations is virtually useless because of a very small stability threshold caused by insufficient damping of the highest wavevector modes. Erin developed a second-order accurate (weak sense), semi-implicit scheme that has far superior stability than any explicit time integration scheme that we have tested to date. The second order accuracy of the method is also superior to previous first-order explicit algorithms that had been proposed for solving the CL equations.

Erin has tested her algorithm extensively on a model of diblock copolymer melts and has shown more than an order of magnitude speedup relative to first-order schemes. For example, the figure at left shows how the average energy <H_R> converges with the timestep of the integration Δt. Erin's second-order scheme 2S provides accurate values of the energy using timesteps more than a factor of 10 larger than those required for the first-order schemes EM (explicit, Euler-Maruyama) and 1S (semi-implicit, first-order). Erin has recently submitted a paper on the performance of this algorithm, where she also discusses the application of Fourier acceleration techniques [1].

Postdocs Kirill Katsov, Yuri Popov, and Jonghoon Lee also worked with Erin on adapting the new CL algorithm to field theory models of polyelectrolyte solutions. Using this algorithm, Yuri and Jonghoon completed a first report on a field-theoretic simulation study of complexation of oppositely charged polyelectrolytes [2]. A second more detailed paper is in preparation.  Such complexation produces a concentrated liquid polymer phase known as a complex coacervate, which has fascinating physical properties, but because of strong charge correlations and high concentration, has defied attack by conventional computer simulation methods. Coacervates have a wide range of uses including as drug delivery vehicles, adhesives and cements, printing media, and food and pharmaceutical coatings, so we are excited about the possibilities of our new simulation method in the design of polyelectrolytes for these types of applications.

Another continuing focus has been on the development of methods for computing free energies in field-theoretic simulations. In particle-based computer simulations, particle insertion methods are commonly used to compute free energies of fluid phases. These methods are known to become inefficient at high density and for long polymer chains. Curiously, we have found that the mathematical structure of polymer field theories lend themselves better to a method proposed by Frenkel and Ladd for solids, even when it is a fluid phase that is of interest. We are currently adapting this technique to polymer field theory models and testing it for the accurate determination of order-disorder transitions of block copolymer melts. Once the technique is optimized, we expect to be able to tackle a wide variety of difficult problems in polymer science – situations where strong fluctuations dominate the fluid phase behavior, such as the polyelectrolyte coacervation problem, yet traditional particle simulations are difficult due to the long chain lengths and high densities involved.

Finally, I mentioned last year that my monograph, The Equilibrium Theory of Inhomogeneous Polymers, G. H. Fredrickson (Oxford Press, 2006) is now in press. This book is the first to provide a comprehensive and consistent approach to the theory of inhomogeneous polymeric fluids, and an introduction to the new “field-theoretic computer simulation” technique that is the subject of the present grant. Without the consistent support of my research over the past decade by the PRF, this book would not have been possible. I am extremely grateful to the foundation for this support, which has enabled me to develop this new field of scientific research.

[1] “Numerical Solutions of the Complex Langevin Equations in Polymer Field Theory,” E. M. Lennon, G. O. Mohler, H. D. Ceniceros, C. J. Garcia-Cervera, and G. H. Fredrickson, SIAM Multiscale Modeling and Simulation, submitted.

 [2] “Field-Theoretic Simulation of Polyelectrolyte Complexation,” Y. O. Popov, J. Lee, and G. H. Fredrickson, J. Polym. Sci. B: Polym. Phys., in press (invited Viewpoint article).

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