Phase behavior near and beyond the thermodynamic stability threshold

The phase behavior of stabilized dispersions of macromolecules is most easily described in terms of the effective interaction between the centers of mass of solute particles. For molecules such as polymer chains, dendrimers, etc., the effective pair potential is finite at the origin, allowing “particles” to freely interpenetrate each other. Using a double-Gaussian model (DGM) for demonstration, we study the behavior of the system as a function of the attraction strength $\eta$. Above a critical strength ${\eta }_{\mathrm{c}}$, the infinite-size system is Ruelle unstable, in that it collapses to a cluster of finite volume. As ${\eta }_{\mathrm{c}}$ is approached from below, the liquid-vapor region exhibits an anomalous widening at low temperature, and the liquid density apparently diverges at the stability threshold. Above ${\eta }_{\mathrm{c}}$, the thermodynamic plane is divided in two regions, differing in the value of the average waiting time for collapse, being finite and small on one side of the boundary line, while large or even infinite in the other region. Upon adding a small hard core to the DGM potential, stability is fully recovered and the boundary line is converted to the spinodal line of a transition between fluid phases. We argue that the destabilization of a colloidal dispersion, as induced by the addition of salt or other flocculant, finds a suggestive analogy in the process by which a strengthening of the attraction pushes a stabilized-DGM system inside the fluid-fluid spinodal region.

Figure 1

DGM below ${\eta }_{\mathrm{c}}$: Liquid-vapor coexistence densities from GEMC simulations performed for a number of $\eta$ values (0.020, red; 0.025, magenta; 0.0263, blue; and 0.02631, black). The various symbols correspond to different initial $N$ values in the two simulation boxes ($864+864$, triangles; $4000+108$, squares; $8192+128$, open dots; and $16000+128$, solid dots). In each run, we generated a trajectory of $4\times {10}^5$ MC cycles, of which only the second half was used to compute the averages. Also reported is the BL for the same $\eta$ values (dotted lines). Inset: BL for a few $\eta$ values close to ${\eta }_{\mathrm{c}}$ (0.026 315, red; 0.026 315 7, magenta; 0.026 315 79, blue; and 0.026 315 80, black, further magnified in the top-right corner).

Figure 2

DGM above ${\eta }_{\mathrm{c}}$: Liquid-vapor coexistence densities from GEMC simulations performed for a number of $\eta$ values (0.026 32, red; 0.026 35, magenta; 0.0265, green; 0.027, blue; and 0.028, black—same symbols as in Fig. 1). For $\eta =0.028$ no coexistence occurred above $T=0.194$. Inset: BL for a few $\eta$ values above ${\eta }_{\mathrm{c}}$ (from bottom to top: 0.026 35, red; 0.0265, magenta; 0.027, green; 0.03, blue; and 0.05, black).

Figure 3

DGM above ${\eta }_{\mathrm{c}}$: MC loci of the metastable-to-unstable crossover. Each color refers to a different $N$ (500, red; 1372, magenta; 4000, blue; and 8788, black crosses). In these simulations, $\rho$ was increased in steps of 0.01 at constant $T$ (for $\eta =0.03,T$ was reduced in steps of 0.01 at constant $\rho$). Any state point where collapse occurred abruptly was drawn as a large symbol, whereas a small symbol (dot or cross) marks the approximate location of a more diffuse (in density or in temperature) energy drop. For each $\eta$, the BL is plotted for comparison (green lines).

Figure 4

DGM for $\eta =0.2$ and $T=6$: Collapse times (red crosses) for a large number of MD runs carried out in the metastable-to-unstable crossover region ($N=864$, right; $N=1372$, center; and $N=2048$, left). Some of the crosses fall outside the $t$ range shown. The black dots are arithmetic averages over each set of 31 instances (51, for every of the lowest five densities in each group). Also shown are the median times (open blue dots).

Figure 5

SDGM for $\eta =0.0265$: BL for low densities ($\delta =0.1$, solid dots; $\delta =0$, open dots). Inset: High-density behavior of the BL for $\eta =0.03$ and a number of $\delta$ values (0, black; 0.08, red; 0.1, green; and 0.15, blue).