# Difference between revisions of "Many-Body Force and Mobility Measurements in Colloidal Systems"

Jason W. Merrill, Sunil K. Sainis, Jerzy Bławzdziewicz and Eric R. Dufresne

Soft Matter 6 (2010) p.2187-2192

wiki entry by Emily Russell, Fall 2010

The article can be found here.

## Overview

This paper introduces a technique whereby the mobility tensor of a system of particles can be determined from measurements of trajectories. Particles in close proximity can affect one another's mobility via hydrodynamic interactions through the medium, so that a force exerted on one particle indirectly causes motion of another. Thus the scalar mobility used in elementary fluid dynamics is not sufficient to describe the system; instead a mobility tensor is needed to take into account interactions between particles. The authors describe the calculation of this tensor in systems of three and seven particles, and find that it is well described by theoretical predictions.

## Experiments

The experiments performed use the same system as in another recent paper by three of the same authors, Many-Body Electrostatic Forces Between Colloidal Particles at Vanishing Ionic Strength: colloidal PMMA particles of 600 nm radius, in hexadecane with here 500 $\mu$M NaAOT surfactant to introduce charging of the particles. The particles were arranged using optical tweezers, and then released and trajectories recorded using a fast camera. This paper considers both an equilateral triangle of three particles, and a hexagonal arrangement of seven particles. Note that the experiments and analysis are all done in two dimensions; since the particles and forces are all in the same plane, the analysis remains valid.

## Results Mean displacement and displacement covariance. (a) Mean displacement and (b) displacement covariance versus time for each coordinate of three particles arranged in an equilateral triangle as shown with side length s = 4.4a = 2.6 $\mu$m. Lines through the data are best fits of eqn (7) and (8) to the data. Light gray lines are drawn as an aid to the eye in locating zero.

From the trajectories, the authors extract the mean displacement of each particle in each dimension as a function of delay time, and compute the displacement covariance of each pair of particle coordinates. The results are given in Figure 1.

From the mean displacement data are calculated the drift velocities. (The authors note that a contribution to the mean displacements can also come from the gradients of the diffusion constants; however these effects would be significant only at small particle separations, and the authors verify a posteriori that the gradients are much smaller than the drift velocities.) Note that the data in Fig. 1a are well-fit by the lines giving the velocity.

A linear fit of the displacement covariance data gives the elements of the diffusion tensor: $cov_{\tau}(x_i(t+\tau) - x_i(\tau), x_j(t+\tau)-x_j(\tau)) = 2D_{ij}(t) + \epsilon_{ij}$. In Fig. 1b, the diagonal elements, $cov(x_i,x_i)$, are all linear with time with roughly the same slope, giving the scalar diffusion constant of the particles. The interesting physics, however, is in those off-diagonal elements which are non-zero, indicating that the fluctuations of one particle are correlated with the fluctuations of another; this is a deeper statement than that the particle exert forces on each other, and is caused by the hydrodynamic interactions through the viscous medium.

The authors measure the drift velocities and diffusion tensors for arrangements of various side-lengths, investigating the effect of distance on these interactions (Fig. 2; for brevity, I have only included the diffusion tensor, which I think is the more interesting). Note that the diagonal elements are all high, constant, and have roughly the same value, while the off-diagonal elements, indicating the hydrodynamic influence of one particle on another, decay to zero with increasing distance between the particles, as expected. Velocity and diffusion. (b) diffusion/mobility tensor for particles arranged in an equilateral triangle as a function of side length, s. Diffusion (mobility) values are normalized to $D_0 = k_BT/6\pi\eta a = 117 nm^2 ms^{-1} (b_0 = 1/6\pi\eta a = 29.5 mm s^{-1} pN^{-1}$). Lines through the data on the diffusion/mobility tensor plot are predictions based on eqn (11) and (12).

The experimental results are compared to theoretical predictions under the Stokeslet Superposition Approximation, and agree well with the predictions. The authors note that the hydrodynamic interactions seem to be pairwise (that is, the hydrodynamic effect of one particle on another is not so great as to change the interaction with a third), unlike the electrostatic forces as found in Many-Body Electrostatic Forces Between Colloidal Particles at Vanishing Ionic Strength.

In their previous paper, the authors considered only the force on the breathing mode of each configuration; here, they determine the forces on each particle in each direction, from which they can also extract the forces on the normal modes of the system. The present technique is of course more general.

## Discussion

The paper is well-written, with a good introduction and by and large clear explanations of the meaning of, for example, the mobility tensor. The results are a nice demonstration that complex fluids - even relatively simple complex fluids with only a few particles - do indeed have complex interactions, such that even so basic a concept as a diffusion constant is no longer scalar and constant, but depends on nearby particles. The paper also clearly states assumptions, and argues well that their technique is quite general and makes few assumptions about the particle interactions.

I find the closing sentence intriguing: "It should be possible to use the same technique to measure torques on anisotropic particles." This is an even cooler idea, and I look forward to seeing if anything comes of it.