Phase coexistence and line tension in ternary lipid systems

The ternary system consisting of cholesterol, a saturated lipid, and an unsaturated one exhibits a rich phase behavior with multiple phase coexistence regimes. Remarkably, phase separation even occurs when each of the three binary systems consisting of two of these components is a uniform mixture. We use a Flory-Huggins like model in which the phase separation of the ternary system is a consequence of an interaction between all three components to describe the system. From the associated Gibbs free energy we calculate phase diagrams, spinodals, and critical points. Moreover, we use a Van der Waals/Cahn-Hilliard like construction to derive an expression for the line tension between coexisting phases. We show how the line tension depends on the position in the phase diagram, and give an explicit expression for the concentration profile at the phase boundary.

Figure 1

(Color online) Gibbs phase triangle showing phase separation in the ternary system when there is none in any of the binary ones. The thick black line is the binodal, which marks the boundary of the immiscibility region. Any composition corresponding to a point inside the immiscibility region will result in demixing into two states, which are at the ends of the corresponding tie lines (thin black lines). The blue (gray) line inside the immiscibility region is the linear instability line (sometimes called the spinodal): points inside the region bordered by the blue line correspond to compositions that will demix by spinodal decomposition, points outside it will demix by nucleation and growth. The red (gray) dots indicate the critical points. Parameters used: ${\chi }_{xy}=1.5$, ${\chi }_{yz}=1.25$, ${\chi }_{xz}=0.75$, $\overline{\chi }=5.0$.

Figure 2

(Color online) Gibbs phase triangle showing phase separation in the ternary system when there is none in any of the binary ones, but one of the binary interactions is attractive. The thick black line is the binodal, which marks the boundary of the immiscibility region. Any composition corresponding to a point inside the immiscibility region will result in demixing into two states, which are at the ends of the corresponding tie lines (thin black lines). The red (gray) dots indicate the critical points. In this case, we find numerically that the coexistence region vanishes if the value of the ternary interaction parameter $\overline{\chi }$ is set to 0. Parameters used: ${\chi }_{xy}=1.5$, ${\chi }_{yz}=1.0$, ${\chi }_{xz}=-0.5$, $\overline{\chi }=5.0$.

Figure 3

(Color online) Gibbs phase triangle showing phase separation in the ternary system, when one of the underlying binary systems also exhibits phase separation. The thick black line is the binodal, which marks the boundary of the immiscibility region. Any composition corresponding to a point inside the immiscibility region will result in demixing into two states, which are at the ends of the corresponding tie lines (thin black lines). The red (gray) dot indicates the critical point. Parameters used: ${\chi }_{xy}=2.05$, ${\chi }_{yz}=1.25$, ${\chi }_{xz}=0.75$, $\overline{\chi }=5.0$.

Figure 4

(Color online) Gibbs phase triangle showing separation into two phases [the regions with the thin black and red (gray) lines, which represent tie lines] and three phases [inside the blue (gray) triangle; the compositions of the three phases correspond to the vertices of the triangle]. The regions with black tie lines correspond to the coexistence of a gel and a liquid phase; the region with the red (gray) tie lines corresponds to liquid-liquid coexistence, with a critical point indicated by the red (gray) dot. The system is in a homogeneous gel phase in the lower right-hand region and in a homogeneous liquid phase in the left-hand region. Parameters used: ${\chi }_{xy}=2.2$, ${\chi }_{xz}=1.95$, ${\chi }_{yz}=2.15$, $\overline{\chi }=4.0$.

Figure 5

(Color online) Line tension estimates using an optimized quadratic profile for $y\left(x\right)$ in Eq. (41). (a) Gibbs phase triangle showing the binodal (thick black line), tie lines (thin black lines), spinodal (blue/gray line), and critical points (red/gray dots). Some optimal quadratic paths connecting coexisting phases are shown (in red/dark gray and green/light gray), as well as the directions of the eigenvectors of the zero eigenvalues of $\left(g_{ij}\right)$ at the spinodal (in green/light gray). (b) Estimated values of $\tau /\sqrt{2A}$ determined using the optimal quadratic profiles shown in the left figure, as a function of “position” between the critical points (the $z$ coordinate of the center of the corresponding tie line). The figure shows both the estimates determined using the optimal quadratic profiles shown in the left figure (big gray dots), as well as those determined using the optimal fourth-order profile as given in the appendix (small black dots); the positions of the points are indistinguishable in the plot. The line tension vanishes at both critical points and has a maximum when the optimal quadratic profile is a straight line, connecting the points on the binodal with the largest separation (green/light gray line in left figure). Parameters used: ${\chi }_{xy}=1.5$, ${\chi }_{yz}=1.25$, ${\chi }_{xz}=0.75$, $\overline{\chi }=5.0$.