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Fractional Brownian Motion Run with a Multi-Scaling Clock Mimics Diffusion of Spherical Colloids in Microstructural Fluids Moongyu Park, John Howard Cushman, and Dan O'Malley Langmuir, Just Accepted Manuscript • Publication Date (Web): 11 Sep 2014 Downloaded from http://pubs.acs.org on September 12, 2014
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Fractional Brownian Motion Run with a Multi-Scaling Clock Mimics Diffusion of Spherical Colloids in Microstructural Fluids Moongyu Park1,2, John Howard Cushman1,2*, Dan O’Malley3 1
Department of Earth, Atmospheric and Planetary Sciences, 550 Stadium Mall Drive, Purdue University, West Lafayette, IN. 47907, USA. 2
Department of Mathematics, Math Science Building, Purdue University, West Lafayette, IN. 47907, USA. 3
Computational Earth Science, Los Alamos National Laboratory, Los Alamos, NM, USA
*
Corresponding Author. E-mail:
[email protected] ABSTRACT The collective molecular reorientations within a nematic liquid crystal fluid bathing a spherical colloid cause the colloid to diffuse anomalously on a short time scale (i.e. as a non-Brownian particle). The deformations and fluctuations of long-range orientational order in the liquid crystal profoundly influence the transient diffusive regimes. Here we show that an anisotropic fractional Brownian process run with a non-linear multi-scaling clock effectively mimics this collective and transient phenomenon. This novel process has memory, Gaussian increments and a multi-scale mean square displacement that can be chosen independently from the fractal dimension of a particle trajectory. The process is capable of modeling multiscale sub, super, or classical diffusion. The finite-size Lyapunov exponents for this multiscaling process are defined for future analysis of related mixing processes.
KEY WORDS Transient diffusion, liquid crystal fluid, nonlinear multi-scaling clock, fractional Brown motion, finite size Lyapunov exponent.
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Introduction The collective molecular reorientations within a nematic liquid crystal fluid bathing a spherical colloid may cause the colloid to diffuse anomalously (1) on a short time scale (i.e. as a nonBrownian particle). The deformations and fluctuations of long-range orientational order in the liquid crystal profoundly influence the transient diffusive regimes. Polar fluids, fluids with microstructure (see (2) for a review) such as liquid crystals, have many technological applications; the most common of which is the liquid crystal display (LCD). Less well known are their appearances in nature; e.g., moist smetitic clay (3) (such as a montmorillonite gel) and the character of fluids in slit nano-pores when the fluid molecule is commensurate with the planner confining pore walls (4). When one deals with the physical continuum of a polar fluid many additional degrees of freedom arise beyond those in more classical fluids. To each point in the fluid a 3x3 matrix is assigned. These additional 9 degrees of freedom correspond to micro shear, micro stretch and micro rotation. In the simplest case, that of a uniaxial nematic liquid crystal, there is no micro shear or micro stretch. For these liquid crystals it suffices to define, at every point in space, an average orientation of the long axis of the molecule called the director (these fluids have stiff rod-like or cylindrical molecules that are very lengthy in the axial direction relative to their diameter). The classical theory of such fluids is called the LeslieErickson director theory (2) (see (5) for its application to colloid diffusion in a LC when there is no time dependent viscosity or diffusivity). Unfortunately, this theory has not been successfully applied to study the dynamic (possibly memory dependent) relaxation of a LC fluid surrounding a spherical colloid (1). The main finding in (1) was that if you imbed a microsphere in an oriented uniaxial nematic liquid crystal (the diameter of the sphere is much greater than the axial dimension of the liquid crystal molecule) then it both displays diffusional anisotropy (faster in the direction of the director field) over all scales and an anomalous (super or sub) diffusional character (often called transient diffusion) over short time scales. This is consistent with earlier observations in polymers (6,7,8). Anisotropic diffusion of colloids in LC fluids has been observed elsewhere (9), including diffusive changes associated with modified surface functionality. The purpose of this report is to show that both multi-scale anomalous diffusion and anisotropic diffusive behavior are mimicked by a very special stochastic process called fractional Brownian motion run with multi-scaling non-linear clock. This process provides a very general model for transient diffusion in sub, classical or super diffusive regimes. Different, but related avenues of thought on transient sub-diffusion are provided in (10,11). Stochastic models have the advantage that they can often represent very complex physics in a very simple fashion. This is especially advantageous when memory associated with finite timescale relaxation is apparent, such as the non-Markovian (memory dependent) short time-scale diffusion of colloids in liquid crystal fluids where no comprehensive theory of the physics has succeeded in modeling sub-diffusive behavior. 2 ACS Paragon Plus Environment
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Theory An arbitrary stochastic processes (12), X(t), with mean-square displacement (MSD) not proportional to time, t, over some range of scales has historically been called anomalous. For given t, X(t) is a random variable and for a given realization, X(t) is a classical function. By definition, the MSD is where denotes expected value (ensemble average). If X(t) is Brownian motion, which we henceforth label as B(t), then ∝ t which is why historically a process was said to be anomalous if ∝ tβ , β ≠ 1 . A standard Brownian motion (13) is simply a continuous random walk that starts at 0 with increments, B(t)B(s), t>s, that are independent and Gaussian with mean zero and variance t-s. Historically a process was said to be superdiffusive if β>1, subdiffusive if βs is (t-s)2H where the Hurst exponent, H, lies between 0 and 1. This variance implies that the MSD is proportional to t2H, so except when H=1/2 the process is anomalous with 0 < β < 2. A simple extension (25) of fractional Brownian motion, XH(t), called fractional Brownian motion run with a non-linear clock (fBm-nlc), involves replacing the usual linear clock, t, with a nonnegative, non-decreasing, non-linear function, F(t), i.e. XH(t)=BH(F(t)). If we take F(t)=tα , α>0, then the MSD of XH(t) is proportional to t2Hα and thus XH(t) can be used as a model of anomalous diffusion for any β > 0. It is well known that the fractal dimension, a measure of the space filling character, for a trajectory generated by BH(t) is 2-H and it has recently been proven that the fractal dimension for a trajectory generated by XH(t), with power-law F(t), is also 2-H, and from this we see that, unlike fractional Brownian motion, the fractal dimension and MSD for XH(t) can be chosen independently. That is, H fixes the fractal dimension and Hα fixes the MSD. Another very interesting point about the MSD of XH(t) is that if you take α=1/2H then the MSD is t and yet except when H is near ½, the process is anything but classical (it has memory and a fractal dimension different from Brownian motion). Thus XH(t) provides a method of generating an uncountable number of anomalous processes with linear MSD, which suggests that the classification based on a power-law argument alone does not make much sense. That is, though it is clear a process whose MSD is not linear in time is anomalous, it cannot be said that a process with linear MSD is Brownian unless substantially more information is provided about 3 ACS Paragon Plus Environment
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the structure of the increments. A way around the classification problem, using renormalization groups and their relation to p-self-similarity, is provided elsewhere (26) and will not be discussed further in this article. A good measure of mixing that displays scale dependence is the finite-size Lyapunov exponent, FSLE (27). The FSLE, λa (r ) , is the average exponential rate that two particles, initially separated by a distance r, separate to a distance ar. For XH(t) defined above, up to a good numerical approximation this can be shown to be α λ (r ) = a
2
−1/ 2 H
ln( a ) < Tα ( −1,1) > r1/α H ( a 2 − 1)1/2α H
(1)
Here we have superscripted with α to remind the reader that a power-law clock has been employed in defining XH(t). Also, < Tα (−1,1) > is the average time for a particle to first exit the interval (-1,1). The reader should note that the distribution of XH(t) does not change as a function of particle size, and hence r can be taken as the distance between the surface of large particles, rather than their center of mass. Though not obvious in Eq. 1, λ αa is a two-particle statistic as compared to the MSD which is a one-particle statistic. Often the MSD displays a multiscaling character, that is, the MSD takes on different power-law exponents over successive increments of time. This is illustrated in (1) where it was found that “The dynamic complexity of the director field around the Brownian particles and finite span (in terms of time lags) make each of the anomalous regimes unlikely to follow a single power law…” (see Figs. S13D, S14 and S15 of (1)). Let 0=t0
