Brownian Motion in Viscoelastic Media

   Recently there has been strong interest in determining the local and bulk viscoelasticity of soft materials by monitoring the thermal or Brownian motion of spherical probe particles dispersed within the soft material of interest with various optical techniques.  The possibility of utilizing such measurements has been proposed off and on for more than eighty years or so.  However, until recently only relatively low frequencies could be probed owing to a reliance on single scattering dynamic light scattering techniques to probe particle motion.  Phillies and his coworkers have been pioneers in measuring long-time spherical probe diffusion coefficients in polymer solutions.  It was in early 1995 that Mason and Weitz first reported the use of diffusing wave spectroscopy (DWS) to monitor the short-time motion of dispersed spherical probes and, therefore, greatly increased the accessible frequency range.  In the five years since, many others have also utilized diffusing wave spectroscopy and a whole host of other particle tracking methods to carry out similar measurements on a whole host of other complex fluid systems with a particular focus being on actin solutions and other biologically derived soft materials.

    Brownian motion in simple viscous liquids is well understood and the connection between this thermal motion and hydrodynamic response is readily apparent.  In a similar manner, bulk moduli should be recoverable if the thermal motion of spherical probes dispersed within a viscoelastic medium can be measured.  The so-called local viscoelasticity should be representative of small-scale structure and dynamics as sampled by small spherical probes, while larger spherical probes should probe bulk moduli.  The ability to characterize small samples would prove to be invaluable to the biomedical community where the available quantities of materials can be exceedingly minute.  In addition, these optical techniques can access much higher frequencies than conventional mechanical rheometry, are noninvasive in that they probe quiescent or unperturbed dynamics in theory and provide a potential means for testing theoretical models over large frequency ranges.

    Since viscoelastic fluids store energy, a certain 'memory' of the particle's past motion must exist.  For this reason, a memory function is oftentimes utilized to account for the frictional resistance experienced by a diffusing particle in a Langevin description of Brownian motion in such systems. This frictional resistance is non-local in time and the frictional force experienced by a diffusing particle at time t is influenced by its velocity at some earlier time t'.  One of the first descriptions of the connection between correlation functions and memory kernels was that of Zwanzig.  Berne and coworkers derived similar results and applied them to molecular velocity autocorrelation functions where they proposed a two-parameter exponentially decaying function as a memory kernel ansatz.  Later, Zwanzig and Bixon established the connection between a frequency dependent friction coefficient in the Stokes-Einstein formalism and the memory function in a Langevin description of Brownian motion via the hydrodynamic theory.  A Langevin description of Brownian motion in more complicated media would entail a proper model and development of an appropriate memory kernel pertinent to the suspending medium which, for example, has recently been done for the case of concentrated colloidal dispersions where interparticle interactions are present and by this research group for the case of Brownian motion in a single relaxation-time Maxwell fluid.

     Several groups have investigated the connection between a colloidal particle's Brownian motion and the bulk rheological properties of the medium in which the particle is suspended.  There have been two different, yet related approaches.  In the first approach, originally pioneered by Mason and Weitz, elasticity is built into a framework which is exact for a purely viscous fluid.  The foundation of this approach is the assumption that the no-slip Stokes-Einstein relationship can be generalized to all frequencies.  Here a mean field assumption is made wherein macroscopic stress relaxations are directly connected to microscopic stress relaxations or, more simply put, there is no delineation between local and bulk viscoelasticity.  This assumption establishes a direct relationship between the suspending medium's shear modulus and the mean square displacement of a Brownian particle.  The second approach, which originated with the Michigan group led by MacKintosh and Schmidt, makes a more direct accounting of the elastic component of the suspending medium.  The equation of elastic equilibrium is solved exactly for a rigid spherical surface exhibiting the no-slip boundary conditions to yield an effective compliance for sphere displacement.  At sufficiently high frequencies this effective compliance is directly proportional to the inverse of the shear modulus when it is assumed that the suspending medium is incompressible owing to viscous coupling between the solvent and the matrix material.  However, at lower frequencies this coupling does not exist and the suspending medium's osmotic compressibility may also influence the Brownian motion of any suspended probes.  Therefore, both the suspending medium's longitudinal and transverse moduli may influence a probe's Brownian motion below some critical frequency.

    Our group recently utilized a model for Brownian motion in a single relaxation-time Maxwell fluid to analyze DWS measurements of probe motion in a series of cetyltrimethylammonium bromide/potassium bromide (CTAB/KBr) solutions.  At certain conditions this surfactant-salt system forms wormlike micelle solutions.  According to mechanical rheometry measurements these CTAB/KBr wormlike micelle solutions exhibit the dynamic signature of a single relaxation-time Maxwell fluid.  An example of our group's measurements of Brownian motion in these systems is shown in the figure below.

Current and Past Group Members

    S. Amin, C.J. Kloxin, S.J. Dees ('99-'01), K.P. Rufener ('96-'98), D.C. Miao ('97-'98), K.P. Rufener ('96-'98), R.M. van Zanten ('99-'00), M.P. Weinmann ('97-'99)
 

Current and Past Collaborators

    K. Ganesan ('97) , S.C. Kuo ('97), T.G. Mason ('97), D. Wirtz ('96-'98)

Publications to Date

• J.H. van Zanten & K.P. Rufener "Brownian Motion in a Single Relaxation Time Maxwell Fluid", Physical Review E 62, 5389-5396 (2000).
• T.G. Mason, K. Ganesan, J.H. van Zanten, D. Wirtz & S.C. Kuo, "Particle-Tracking Microrheology of Complex Fluids", Phys. Rev. Lett.79, 3282-3285 (1997).