Translational diffusion across a free-standing smectic film above the bulk smectic-$A$–isotropic transition temperature

Calculations of translational self-diffusion coefficient in free-standing smectic films during a series of layer-thinning transitions as the temperature is raised above the bulk smectic-$A$–isotropic transition have been carried out. A molecular model based upon the random walk theory is applied for calculating the translational diffusion coefficient (TDC) $D_{\parallel }$ across the smectic film both in the bulk of the film, as well as in the vicinity of the bounding surfaces. Calculations of $D_{\parallel }$ require the set of the orientational and translational order parameters (OPs) which have been obtained by using the extended McMillan approach with anisotropic forces. The effect of $\mathbf{\text{E}}$ on the orientational and translational OPs, as well as on the TDC of smectic films has been investigated. A reasonable agreement between the theoretically predicted and the experimentally obtained data on the TDC in the bulk of the partially fluorinated H10F5MOPP film has been obtained. We also found, in agreement with the experimentally observed behavior of $D_{\parallel }\left(\mathrm{N}\right)(N=25,13,11,10)$, that the translational diffusion coefficient in the bulk of the film gradually increases as the film thickness $N$ is decreased.

Figure 1

Orientational $q_i\left(\mathrm{\Delta }\right)$ ($i=1,...,25$) (a) and translational ${\sigma }_i\left(\mathrm{\Delta }\right)$ ($i=1,...,25$) (b) OPs as functions of layer number $i$, and for different values of electric field: $\mathrm{\Delta }=0.0$ (curves 1) and 0.08 (curves 2), respectively. In all these cases the temperature $\theta =0.67, \alpha =1.05$, and $W_0=5V_0$, respectively.

Figure 2

Dimensionless translational diffusion coefficient $D_{i,i+1}\left(\theta ,\mathrm{\Delta }=0\right)/D_{N/2,N/2+1}\left(\theta ,\mathrm{\Delta }=0\right)$ as the function of layer number $i$, corresponding to $N=25$ film thickness, in the absence of the external electric field ($\mathbf{\text{E}}=0$), and for three values of the dimensionless temperature $\theta =0.67$ (squares), 0.675 (up triangles), and 0.677 (down triangles), respectively.

Figure 3

The distribution of the number of $D_{i,i+1}\left(\theta ,\mathrm{\Delta }=0\right)/D_{N/2,N/2+1}\left(\theta ,\mathrm{\Delta }=0\right)$ profiles across the smectic films, during the sequence of the layer-thinning transitions $25\to 13\to 11\to 10$, in the case of absence ($\mathrm{\Delta }=0.0$) of the electric field $\mathbf{\text{E}}$.

Figure 4

Dimensionless translational diffusion coefficient $D_{i,i+1}\left(\theta ,\mathrm{\Delta }\right)/D_{N/2,N/2+1}\left(\theta ,\mathrm{\Delta }\right)$ as the function of layer number $i$, corresponding to $N=25$ film thickness, both in the absence ($\mathrm{\Delta }=0.0$) and with accounting for ($\mathrm{\Delta }=0.08$) the external electric field $\mathbf{\text{E}}$. In all these cases $\theta =0.67, \alpha =1.05$, and $W_0=5V_0$, respectively.