Thermostability analysis of line-tension-associated nucleation at a gas-liquid interface

The influence of line tension on the thermostability of a droplet nucleated from an oversaturated vapor at the interface of the vapor and another immiscible liquid is investigated. Along with the condition of mechanical equilibrium, the notion of extremization of the reversible work of formation is considered to obtain the critical parameters related to heterogeneous nucleation. From the energetic formulation, the critical reversible work of formation is found to be greater than that of homogeneous nucleation for high value of the positive line tension. On the other hand, for high value of the negative line tension, the critical reversible work of formation becomes negative. Therefore, these thermodynamic instabilities under certain substrate wettability situations necessitate a free-energetics-based stability of the nucleated droplet, because the system energy is not minimized under these conditions. This thermostability is analogous to the transition-based stability proposed by Widom [B. Widom, J. Phys. Chem. 99, 2803 (1995)] in the case of partial wetting phenomena along with the positive line tension. The thermostability analysis limits the domain of the solution space of the present critical-value problem as the thermodynamic transformation in connection with homogeneous and workless nucleation is considered. Within the stability range of the geometry-based wetting parameters, three limiting modes of nucleation, i.e., total-dewetting-related homogeneous nucleation, and total-wetting-associated and total-submergence-associated workless nucleation scenarios, are identified. Either of the two related limiting wetting scenarios of workless nucleation, namely, total wetting and total submergence, is found to be favorable depending on the geometry-based wetting conditions. The line-tension-associated nucleation on a liquid surface can be differentiated from that on a rigid substrate, as in the former, the stability based on mechanical equilibrium and a typical case of workless nucleation with complete submergence are observed.

Figure 1

Schematic of the geometric model for liquid-substrate-mediated heterogeneous nucleation. A stable cluster (b), in the shape of a lenticular droplet of lateral radius $R$, forms from its oversaturated vapor (a) on an immiscible liquid substrate (c). $\alpha$ and $\beta$ are the pseudowetting angle and the submergence angle, respectively. The associated radii of curvature, $r_{ab}=R/\phantom{Rsin\alpha }sin\alpha$ and $r_{bc}=R/\phantom{Rsin\beta }sin\beta$, describe the lens-shaped droplet.

Figure 2

Effect of the sign of line tension on the geometric angles of a lenticular droplet. It can be found from Eq. (12) that positive (a) and negative (b) line tension act as compressive and tensile force, respectively. Positive line tension tries to constrict the triple line and the true geometric angles $(\alpha ,\beta )$ are greater than the apparent geometric angles $({\alpha }_0,{\beta }_0)$. On the other hand, negative line tension tries to enlarge the length of the triple line and the true geometric angles $(\alpha ,\beta )$ are smaller than the apparent ones $({\alpha }_0,{\beta }_0)$.

Figure 3

Three limiting nucleation scenarios obtained in the present thermostability analysis. Two modes of wetting in connection with workless nucleation ($\alpha =0^{\circ }$) are identified: (a) total wetting ($\beta =0^{\circ }$), associated with the formation of a thin film over a substrate acting like a rigid and ideal wet substrate, and (b) total submergence ($\beta ={180}^{\circ }$) in which the spherical droplet is completely surrounded by the immiscible liquid only with a point contact with the free surface. The only mode of wetting related to homogeneous nucleation ($\alpha ={180}^{\circ }$ and $\beta =0^{\circ }$) can be considered as (c) total dewetting in which the liquid substrate behaves like a rigid and ideal dry substrate.

Figure 4

Variation of the potency factor, $z$ (a) and the critical dimensionless line tension, $m$ (b) with respect to $\alpha$ and $\theta$ for $\beta =0^{\circ }$ (i.e., rigid substrate). The present thermostability analysis constrains the domain of the solution space. The line segments AB, BC, and CD represent workless, heterogeneous, and homogeneous nucleation, respectively. B and C are the points where the transformations related to workless and homogeneous nucleation occur, respectively. The potency factor ($z$) varies from 0 to 1 when line tension is absent ($m=0$ curve) in the analysis.

Figure 5

Variation of the potency factor $z$ with respect to $\alpha$ and $\theta$ for different values of $\beta$: (a) $\beta ={45}^{\circ }$, (b) $\beta ={90}^{\circ }$, and (c) $\beta ={135}^{\circ }$. The variation in $z$ is shown using Eq. (24). At the limiting energetic situations ($z=0$ and $z=1$), the wetting conditions jump discontinuously. To satisfy the condition $\psi <{180}^{\circ }$, the maximum value of $\alpha$ is decreasing continuously with the increase in $\beta$ as can be seen from (a–c).

Figure 6

Variation of the critical dimensionless line tension $m$ with respect to $\alpha$ and $\theta$ for different values of $\beta$: (a) $\beta ={45}^{\circ }$, (b) $\beta ={90}^{\circ }$, and (c) $\beta ={135}^{\circ }$. The variation in $m$ is shown using Eq. (22). The limiting stability conditions ($m_{\mathrm{less}}$ and $m_{\mathrm{homo}}$) are expressed with the help of Eqs. (27) and (28). The wetting conditions jump discontinuously at the limiting nucleation scenarios (workless and homogeneous nucleation).

Figure 7

Variation of the characteristic wetting angle ${\theta }_c$ with respect to $\alpha$ for different values of $\beta$. For $\beta =0^{\circ }$, i.e.; when the liquid substrate acts as a rigid substrate, ${\theta }_c=\alpha$ as can be found from Eq. (30). In general, ${\theta }_c>\alpha$ when $\beta \ne 0^{\circ }$. Line tension acts as a tensile force (negative) in the upper regime of any curve. On the other hand, line tension acts as a compressive force (positive) in the lower regime of any curve.

Figure 8

Typical regime map based on the thermodynamic transformation for $\alpha ={30}^{\circ }$ and $\beta ={45}^{\circ }\left(\psi ={75}^{\circ }\right)$. There are four regimes separated by two vertical lines and one sigmoidal curve. The regimes above the green and the red solid curve are associated with the total-wetting-related workless nucleation and the total-dewetting-related homogeneous nucleation, respectively. The regime of heterogeneous nucleation is further divided into two regimes corresponding to the tensile and the compressive line tension.

Figure 9

Typical regime map based on the thermodynamic transformation for $\alpha ={90}^{\circ }$ and $\beta ={60}^{\circ }\left(\psi ={150}^{\circ }\right)$. This map also illustrates the four regimes. However, the total-submergence-associated workless nucleation is viable in this case.