Generalized potentials for a mean-field density functional theory of a three-phase contact line

We investigate generalized potentials for a mean-field density functional theory of a three-phase contact line. Compared to the symmetrical potential introduced in our previous article [Phys. Rev. E 85, 011120 (2012)], the three minima of these potentials form a small triangle located arbitrarily within the Gibbs triangle, which is more realistic for ternary fluid systems. We multiply linear functions that vanish at edges and vertices of the small triangle, yielding potentials in the form of quartic polynomials. We find that a subset of such potentials has simple analytic far-field solutions and is a linear transformation of our original potential. By scaling, we can relate their solutions to those of our original potential. For special cases, the lengths of the sides of the small triangle are proportional to the corresponding interfacial tensions. For the case of equal interfacial tensions, we calculate a line tension that is proportional to the area of the small triangle.

Figure 1

Schematic diagram for the mapping between physical space and the Gibbs triangle (briefly, the Gibbs space). (a) Sketch of our system consisting of three bulk phases, $\alpha$, $\beta$, and $\gamma$, divided by three interfaces, $\alpha \beta$, $\beta \gamma$, and $\gamma \alpha$. The interfaces, if extrapolated, are each perpendicular to sides of the dashed triangle (illustrative computational domain) and meet at a three-phase contact line. The corresponding dihedral angles are ${\theta }_{\alpha }$, ${\theta }_{\beta }$, and ${\theta }_{\gamma }$ and $(x,y)$ are Cartesian coordinates in physical space. The region within three solid curves represents approximately the diffuse region of the interfaces and the contact line. (b) The Gibbs space, in which the three bulk phases are located at three points and $(u,v)$ are Cartesian coordinates in units of mole fractions. The three curved lines that connect pairs of points for bulk phases are trajectories of two-phase transitions. (c) Slice of an interface in a cuboid far from the contact line. $s$ is a coordinate perpendicular to the $\alpha \beta$ interface and $w$ is the width of the cuboid in a direction perpendicular to the contact line.

Figure 2

(a) Geometry of the three minima of the potential $\tilde{f}$ given by (7) [or $f^o$ in (4) when $a=1$]. $\tilde{f}$ is a function of scaled mole fractions ${\left\{Y_i\right\}}_{i=1,2,3}$, where ${\sum }_{i=1}^3{Y}_i=1$. $Y_i$ can be expressed in terms of two independent Cartesian coordinates $(u,v)$. $\alpha =(u_1^0,v_1^0)=(0,0)$, $\beta =(u_2^0,v_2^0)=(2/\sqrt{3},0)$, and $\gamma =(u_3^0,v_3^0)=(1/\sqrt{3},1)$ are the three minima of $\tilde{f}$ located at the three corners of the Gibbs triangle. $L_i=0$ represents a line that passes through two points $(u_j^0,v_j^0)$ and $(u_k^0,v_k^0)$ $(j\ne k\ne i)$, and $I_i=0$ is a line parallel to $L_i=0$ but passing through only one point $(u_i^0,v_i^0)$. (b) Contour plot of $\tilde{f}$. The value of $\tilde{f}$ decreases from its value at the center of the inner area as one proceeds toward the three vertices where it is zero.

Figure 3

(a) Geometry of the three minima of the quartic potential $f_g$ given by (10). $\alpha =(u_1,v_1)$, $\beta =(u_2,v_2)$, and $\gamma =(u_3,v_3)$ are the three minima of this potential arbitrarily located at any three points within the Gibbs triangle. $L_i=0$ represents a line that passes through $(u_j,v_j)$ and $(u_k,v_k)$ for $j\ne k\ne i$, and $I_i=0$ is a line parallel to $L_i=0$ but passing through the vertex $(u_i,v_i)$. (b) Contour plot of potential $f_g$ for $(u_1,v_1)=(0.3087,0.1667)$, $(u_2,v_2)=(0.9060,0.1167)$, $(u_3,v_3)=(0.4974,0.6667)$, and $(d_1,d_2,d_3)=(1.0,1.1,1.2)$.

Figure 4

(a) Geometry of the three minima of the quartic potential $f_G$ given by (17). $\alpha =(u_1,v_1)$, $\beta =(u_2,v_2)$, and $\gamma =(u_3,v_3)$ are the three minima of this potential arbitrarily located at any three points within the Gibbs triangle. $L_i=0$ represents a line passing through $(u_j,v_j)$ and $(u_k,v_k)$, and $K_i=0$ is a point located at $(u_i,v_i)$. (b) Contour plot of potential $f_G$ for $(u_1,v_1)=(0.3087,0.1667)$, $(u_2,v_2)=(0.9060,0.1167)$, $(u_3,v_3)=(0.4974,0.6667)$, and $(a_i,b_i,c_i)=(1,1,1)$, $(i=1,2,3)$.

Figure 5

Schematic representation of the generalization from a potential $\tilde{f}$ in (7) to the potential $f_g$ in (10) of similar form and to the more general potential $f_G$ in (17). At the left-hand side of this figure, there are three pairs of two parallel lines. The three minima of $\tilde{f}$ are located at the three nodes where lines intersect and form a small equilateral triangle. In the middle of this figure, we generalize $\tilde{f}$ to $f_g$ by allowing the three minima of $f_g$ to be at arbitrary locations. In this case, the three pairs of parallel lines determine the three minima. The right-hand side of this figure shows a further generalization in which each minimum of $f_G$ is located by two lines and a point.

Figure 6

Schematic linear transformation from the three corners of the Gibbs triangle to three arbitrary internal points $(u_1,v_1)$, $(u_2,v_2)$, and $(u_3,v_3)$ within it.

Figure 7

Analytic far-field solutions for the mole fraction $Y_i$ for a special case of the potential $f_g$ in (10) at the $\alpha \beta$ interface. In this special case, $d_i=1$ and ${\ell }_i=1$ (${\epsilon }_u={\epsilon }_v=3/4$ and ${\epsilon }_{uv}=0$). The three minima of $f_g$ are $(u_1,v_1)=(0.3087,0.1667)$, $(u_2,v_2)=(0.9060,0.1167)$, and $(u_3,v_3)=(0.4974,0.6667)$.

Figure 8

Geometry of the three interfacial tensions in the Gibbs space for a special case of the potential $f_g$ given by (10), in which the weighting coefficients are equal to 1, $d_i=1$, and the gradient energy is isotropic, ${\ell }_i=1$. The three vertices 1, 2, and 3 correspond to the three minima $\alpha$, $\beta$, and $\gamma$ of $f_g$. The length of the side of the small triangle opposite to the vertex $i$ is labeled by $S_i$, and the area of the small triangle is $A$. The interfacial tension of the $\alpha \beta$ interface ${\widehat{\sigma }}_{\alpha \beta }$ is proportional to the length of the side of the small triangle that connects $\alpha$ and $\beta$, i.e., $S_3$. Similarly, ${\widehat{\sigma }}_{\beta \gamma }\propto S_1$ and ${\widehat{\sigma }}_{\gamma \alpha }\propto S_2$. ${\theta }_{\alpha }$, ${\theta }_{\beta }$, and ${\theta }_{\gamma }$ are the three dihedral angles corresponding to the three-phase contact line.

Figure 9

Geometry of the line tension in the Gibbs triangle for a special case of $f_g$ with equal weighting coefficients $d_i=1$, isotropic gradient energy ${\ell }_i=\ell$, and the small triangle formed by the three minima is equilateral. The line tension and the interfacial tensions are proportional to the area $A$ and the lengths of the sides of the small triangle, respectively.

Figure 10

(a) Geometry of the three minima of the quartic potential $f_1$ given by (A1). $\alpha =(u_1,v_1)$, $\beta =(u_2,v_2)$, and $\gamma =(u_3,v_3)$ are the three minima of this potential arbitrarily located at any three points in Gibbs space. $L_i=0$ represents a line that passes through $(u_j,v_j)$ and $(u_k,v_k)$ for $j\ne k\ne i$. (b) Contour plot of potential $f_1$ for $(u_1,v_1)$ $=$ $(0.3087,0.1667)$, $(u_1,v_1)$ $=$ $(0.3087,0.1667)$, $(u_2,v_2)$ $=$ $(0.9060,0.1167)$, $(u_3,v_3)$ $=$ $(0.4974,0.6667)$, and $(a_1,b_1)=(1,0.1)$, $(a_2,b_2)=(-1,0.1)$, and $(a_3,b_3)=(-0.1,1)$.

Figure 11

(a) Geometry of the three minima of the quartic potential $f_2$ given by (A2). $\alpha =(u_1,v_1)$, $\beta =(u_2,v_2)$, and $\gamma =(u_3,v_3)$ are the three minima of this potential arbitrarily located at any three points in Gibbs space. $L_i=0$ represents a line that passes through $(u_j,v_j)$ and $(u_k,v_k)$ for $j\ne k\ne i$. (b) Contour plot of potential $f_2$ for $(u_1,v_1)$ $=$ $(0.3087,0.1667)$, $(u_1,v_1)$ $=$ $(0.3087,0.1667)$, $(u_2,v_2)$ $=$ $(0.9060,0.1167)$, $(u_3,v_3)$ $=$ $(0.4974,0.6667)$, and $d_{12}=d_{23}=d_{31}=1$.