Diffusion coefficient of an inclusion in a liquid membrane supported by a solvent of arbitrary thickness

The diffusion coefficient of an inclusion in a liquid membrane is investigated by taking into account the interaction between membranes and bulk solvents of arbitrary thickness. As illustrative examples, the diffusion coefficients of two types of inclusions, a circular domain composed of fluid with the same viscosity as the host membrane and that of a polymer chain embedded in the membrane, are studied. The diffusion coefficients are expressed in terms of the hydrodynamic screening lengths, which vary according to the solvent thickness. When the membrane fluid is dragged by the solvent of finite thickness, via stick boundary conditions, multiple hydrodynamic screening lengths together with the weight factors to the diffusion coefficients are obtained from the characteristic equation. The conditions for which the diffusion coefficients can be approximated by the expression including only a single hydrodynamic screening length are also shown.

Figure 1

Schematic picture showing a planar liquid membrane with 2D viscosity $\eta$ located at $z=0$. It is supported on a solvent with 3D viscosity ${\eta }_s$. A substrate is located at $z=-h$ bounding the solvent.

Figure 2

The smallest positive values for the inverse of the characteristic lengths against the solvent thickness. Both quantities are normalized by $\nu ={\eta }_s/\eta$. The red thick line represents the smallest positive root of the characteristic equation, Eq. (16), calculated numerically. The long dashed line represents $\kappa /\nu =1/\sqrt{\nu h}$. The short dashed line represents the result of Eq. (23). The thin solid line represents the result of Eq. (22).

Figure 3

(a) The pictorial solution of the characteristic equation, Eq. (16); $\cot \left(x\right)$ against $x$ and $x/\left(\nu h\right)$ against $x$ for $\nu h=1$. The cross points of lines are $x_j={\kappa }_{j}h$. The smallest positive value is $x_1$. ${\kappa }_1$ is obtained by ${\kappa }_1=x_1/h$ (b) $C_j$ against ${\kappa }_j/\nu$. $C_j$ represents the weight associated with each hydrodynamic screening length, Eq. (17). Blue squares, red circles, and dots represent $\nu h=0.1$, $\nu h=1.0$, and $\nu h=10.0$, respectively.

Figure 4

The diffusion coefficients of polymer against size for various solvent thicknesses. The solid red lines indicate the complete solution obtained from Eq. (18). The short dashed black lines indicate the approximate solution by assuming a characteristic length scale for hydrodynamic screening given by $1/{\kappa }_1$. The short dashed line for $\nu h=0.1$ overlaps with the exact solution.

Figure 5

Schematic picture showing a liquid domain embedded in a planar liquid membrane located at $z=0$. Both a liquid domain and a membrane have the same 2D viscosity $\eta$. It is supported on a solvent with 3D viscosity ${\eta }_s$. A substrate is located at $z=-h$ bounding the solvent in the lower region.

Figure 6

The diffusion coefficients of a liquid domain against size for various solvent thicknesses. The solid red lines represent the generalized solution of De Koker given by Eq. (37). $\nu h=10$, $\nu h=1$, and $\nu h=0.1$ from right to left. The result in the limit of thin solvent layer, Eq. (44), overlaps with that of $\nu h=0.1$. The red dashed-dotted line represents the asymptotic solution for $\nu h=10$, Eq. (51). The red thin dashed line indicates the Evans-Sackmann expression, Eq. (45). The blue long dashed line represents the original solution of De Koker obtained by taking $h\to \infty$ limit in Eq. (37). The blue dots denote the results of Eq. (40) and the blue dashed-dotted line indicates the asymptotic results, Eq. (41).