Transient anomalous diffusion of tracer particles in soft matter

摘要

This paper is motivated by experiments in which time series of tracer particles in viscoelastic liquids are recorded using advanced microscopy. The experiments seek to infer either viscoelastic properties of the sample [Mason and Weitz, Phys. Rev. Lett. 74, 1250-1253 (1995)] or diffusive properties of the specific tracer particle in the host medium [Suh et al., Adv. Drug Delivery Rev. 57, 63-78 (2005); Matsui et al., Proc. Natl. Acad. Sci. U. S. A. 103, 18131-18136 (2006); Lai et al., Proc. Natl. Acad. Sci. U. S. A. 104, 1482-1487 (2007); Fricks et al., SIAM J. Appl. Math. 69, 1277-1308 (2009)]. Our focus is the latter. Experimentalists often fit data to transient anomalous diffusion: a sub-diffusive power law scaling (t(v), with 0 < v < 1) of mean-squared displacement (MSD) over a finite time interval, with longtime viscous scaling (t(1)). The time scales of sub-diffusion and exponents v are observed to vary with particle size, particle surface chemistry, and viscoelastic properties of the host material. Until now, explicit models for transient sub-diffusive MSD scaling behavior [Doi and Edwards, The Theory of Polymer Physics (Oxford University Press, New York, 1986); Kremer and Grest, J. Chem. Phys. 92, 5057-5086 (1990); Rubinstein and Colby, Polymer Physics (Oxford University Press, New York, 2003)] are limited to precisely three exponents: monomer diffusion in Rouse chain melts (t(1/2)), in Zimm chain solutions (t(2/3)), and in reptating chains (t(1/4)). In this paper, we present an explicit parametrized family of stochastic processes (generalized Langevin equations with prescribed memory kernels) and derive their closed-form solutions which (1) span the complete range of transient sub-diffusive behavior and (2) possess the flexibility to tune both the time window of sub-diffusive scaling and the power law exponent v. These results establish a robust family of sub-diffusive models, for which the inverse problem of parameter inference from experimental data [Fricks et al., SIAM J. Appl. Math. 69, 1277-1308 (2009)] remains to be developed.