From Random Walk to Single-File Diffusion

We report an experimental study of diffusion in a quasi-one-dimensional (q1D) colloid suspension which behaves like a Tonks gas. The mean squared displacement as a function of time is described well with an ansatz encompassing a time regime that is both shorter and longer than the mean time between collisions. The ansatz asserts that the inverse mean squared displacement is the sum of the inverse mean squared displacement for short time normal diffusion (random walk) and the inverse mean squared displacement for asymptotic single-file diffusion (SFD). The dependence of the 1D mobility in the SFD on the concentration of the colloids agrees quantitatively with that derived for a hard rod model, which confirms for the first time the validity of the hard rod SFD theory. We also show that a recent SFD theory by Kollmann [Phys. Rev. Lett. 90, 180602 (2003)] leads to the hard rod SFD theory for a Tonks gas.

Figure 1

Typical probability density of particle displacement evolving with time (for $\eta =0.57$). The solid lines are fits of the data to Eq. (6).

Figure 2

Mean squared displacement as a function of $t$ at different concentrations. Note that $⟨x(t{)}^2⟩$ for large spheres is scaled by the factor ${\sigma }_2/{\sigma }_1$. The data (symbols) are shifted downward a factor of 3 from one another for clarity. The error bars are smaller than the symbols used. For $t\le 1\mathrm{s}$ the movies were grabbed at $30\mathrm{\text{frames}}/\mathrm{s}$, and for $t>1\mathrm{s}$ the images were grabbed at $4$ and $5\mathrm{\text{frames}}/\mathrm{s}$ for small and large spheres, respectively (only a subset of the data are plotted for clarity). The solid lines are fits of the data to Eq. (8).

Figure 3

Quasi-1D mobility (solid circles) determined with the empirical expression [Eq. (8)] as a function of concentration [$F$ for the large colloids is scaled by the factor $({\sigma }_1/{\sigma }_2)\sqrt{D_{01}/D_{02}}$]. The solid line represents $F^{\mathrm{HR}}$. Other symbols represent $F^q$ determined from Eq. (4). The inset zooms into the data at higher $\eta$.

Figure 4

The experimentally determined static structure factor $S(q)$ for small $q$ as a function of concentration, compared with that derived from the equation of state for a Tonks gas.