Confinement effects in premelting dynamics

We examine the effects of confinement on the dynamics of premelted films driven by thermomolecular pressure gradients. Our approach is to modify a well-studied setting in which the thermomolecular pressure gradient is driven by a temperature gradient parallel to an interfacially premelted elastic wall. The modification treats the increase in viscosity associated with the thinning of films, studied in a wide variety of materials, using a power law and we examine the consequent evolution of the confining elastic wall. We treat (1) a range of interactions that are known to underlie interfacial premelting and (2) a constant temperature gradient wherein the thermomolecular pressure gradient is a constant. The difference between the cases with and without the proximity effect arises in the volume flux of premelted liquid. The proximity effect increases the viscosity as the film thickness decreases thereby requiring the thermomolecular pressure driven flux to be accommodated at higher temperatures where the premelted film thickness is the largest. Implications for experiment and observations of frost heave are discussed.

Figure 1

Schematic of the flow of a premelted film disjoining an elastic capillary tube from the solid phase, from [10].

Figure 2

Least-squares fit (dashed lines) of $\left(\eta -{\eta }_b\right)/\left({\eta }_0-{\eta }_b\right)=\beta {\left(d_0/d\right)}^{\gamma }$ to the experimentally measured viscosity (filled symbols) from [32] (circles; OMCTS), giving $\beta =0.91,\gamma =3.60$ and [31] (squares; squalane), giving $\beta =1.55,\gamma =1.80$.

Figure 3

(a) Similarity solution, Eq. (19), for the short- and long-range electrostatic ($n=3/2$ and $n=2$) and nonretarded van der Waals ($n=3$) interactions. (b) For each type of interaction (i.e., each $n$), a linear least-squares fit gives a power-law relation for the maximum capillary deformation, ${(r_2-r_0)}_{\max }\left(t\right)\propto t^{n/(2n+3)}$, as given by Eq. (19). The $t^{1/4},t^{2/7}$, and $t^{1/3}$ power laws are for $n=3/2$ (lower curve), $n=2$ (middle curve), and $n=3$ (upper curve) respectively. The solid lines correspond to $g_{\max ,n}{t}^{n/(2n+3)}$ and the filled symbols correspond to the parameters for dodecane for $d_0=1\AA$ (black squares) and $d_0=10\AA$ (solid dots). Note, in (b) $t$ denotes dimensional time.

Figure 4

For $n=3,d_0=8\AA ,\mathcal{G}=1$ K/m, ${\eta }_r={10}^7$, and each $\gamma$, a linear least-squares fit is used to obtain a power-law relation, $r_{\max }\propto t^{b(\gamma ,{\eta }_r,d_0,n)}$. Circles represent $b(\gamma ,{\eta }_r,d_0,n)$ from the linear least-squares fit. The lower and upper lines represent the exponents $n/(2n+3+\gamma )$ and $n/(2n+3)$, respectively, corresponding to the similarity solutions (19) and (26).

Figure 5

For ${\eta }_r={10}^7$, the deformation is presented in terms of the similarity variables, viz., Eq. (27): (a) $g_2\left({\zeta }_2\right)$ is self-similar, but (b) $g_{4.75}\left({\zeta }_{4.75}\right)$ is not self-similar. The legend in the inset corresponds to the time in dimensional units. Note the difference in the vertical scale.

Figure 6

Dimensional deformation as a function of position at dimensional time ${10}^2$ h for the parameters shown in the insets, and $\mathcal{G}=1$ K/m, $n=3,d_0=8\AA$. Note the difference in the vertical scales.

Figure 7

Same as Fig. 6, but for $\mathcal{G}={10}^2$ K/m, $n=3,d_0=8\AA$. Note the difference in the scale of the axes relative to Fig. 6.

Figure 8

For $n=3,d_0=8\AA ,\mathcal{G}=1$ K/m, we obtain the relation between ${\gamma }^{★}$ and ${\eta }_r$. Symbols are the data points obtained from the numerical solutions. The dash-dotted line represents the linear least-squares fit: ${\gamma }^{★}=0.8905+0.2358ln{\eta }_r$. The shaded region corresponds to the range of $\gamma$ such that $r_{\max }(T_f;{\eta }_r>1)>r_{\max }(T_f;{\eta }_r=1)$.