Diffusion of a Rouse chain in porous media: A mode-coupling-theory study

We use a kinetic mode-coupling theory (MCT) combining with generalized Langevin equation (GLE) to study the diffusion and conformational dynamics of a bead-spring Rouse chain (RC) dissolved in porous media. The media contains fluid particles and immobile matrix ones wherein the latter leads to the lack of translational invariance. The friction kernel $\zeta \left(t\right)$ used in the GLE can be obtained directly by adopting a simple density-functional approach in which the density correlators calculated by MCT equations of porous media serve as inputs. Due to cage effects generated by surrounding particles, $\zeta \left(t\right)$ shows a very long tail memory in the high volume fraction of fluid and matrix. It is found that the long-time center-of-mass diffusion constant $D_{\text{CM}}$ of the RC decreases with the increment of volume fraction, influencing more strongly by the matrix particles than by the fluid ones. The auto-correlation function (ACF) of the end-to-end distance fluctuation can also be calculated theoretically based on GLE. Of particular interest is that the power-law region of ACF has a nearly fixed length in logarithmic scale when it shifts to longer time range, with increasing the volume fraction of media particles. Moreover, the effect of lack of translational invariance has been investigated by comparing the results between fluid-matrix and pure fluid cases under identical total volume fraction.

Figure 1

Sketch of porous model. The black (blue) spheres represent for matrix (fluid) particles which are immobile (mobile). The red spheres represent for the Rouse chain.

Figure 2

Short-time diffusion coefficient $D_0$ vs. ${\phi }_f$ and ${\phi }_m$ in logarithm scales. The black and red lines are corresponding to the left and right axis and fixing ${\phi }_m=0.0524$ and ${\phi }_f=0.419$, respectively. $d_f$ and $d_m$ denote the diameters of fluid and matrix particles, respectively.

Figure 3

The scattering functions ${\mathrm{\Phi }}_k\left(t\right)$ and ${\varphi }_k^s\left(t\right)$. The lines of same color in (a) and (b) are corresponding to the same volume fraction systems. $k=6.44$ is the frequency space position of the first peak of $S_k^{ff\left(c\right)}$. The dash-dot line is the system with ${\phi }_f=0.209$ and ${\phi }_m=0.157$. In this situation, system enters localized state due to the influence of matrix, which is reflected by the plateau of ${\varphi }_k^s\left(t\right)$.

Figure 4

Friction kernel $\zeta$ vs. $t$ in logarithmic space. The volume fraction of matrix ${\phi }_m$ is fixed at 0.0524. The multiexponential fitting is shown in the inset. The open circle are the data obtained from calculating. And the red line is the fitting curve ${\sum }_{i=1}^6{\eta }_{i}exp\left(-v_{i}t\right)$, the fitting is very well in almost 11 order of magnitudes.

Figure 5

MSD of rouse chain CM as a function of $t$ with different ${\phi }_f$ and ${\phi }_m$. Insets: The long-time diffusion coefficient $D_{\text{CM}}$ as functions of ${\phi }_f$ and ${\phi }_m$. The black lines are drawn for guiding eyes. The small insets show the dependencies of the subdiffusion scaling exponent on ${\phi }_f$ or ${\phi }_m$.

Figure 6

$D_{\text{CM}}$ as functions of bead number $M$ with different ${\phi }_f$ and ${\phi }_m$. The values of $M$ from left to right are 8, 10, 12, 15, 20, 25, 30, 35, 40. The straight lines are the linear fitting of these data.

Figure 7

The distance ACF $C_{\text{DF}}\left(t\right)/C_{\text{DF}}\left(0\right)$ as function of $t$.

Figure 8

(a) The black line: fluid-matrix (FM) case. The red line: pure fluid case. (b) $D_{\text{CM}}$ as functions of ${\phi }_{\text{total}}$ in different cases, where ${\phi }_{\text{total}}$ is the total volume fraction of particles including fluid and matrix. The lines are drawn for guiding eyes.