Criticality of colloids with distinct interaction patches: The limits of linear chains, hyperbranched polymers, and dimers

We use a simple model of associating fluids which consists of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type $A$ and one is of type $B$; the bonding interactions between each pair of spots have strengths ${ϵ}_{AA}$, ${ϵ}_{BB}$, and ${ϵ}_{AB}$. The theory is applied over the whole range of bonding strengths and the results are interpreted in terms of the equilibrium cluster structures of the phases. In addition to our numerical results, we derive asymptotic expansions for the free energy in the limits for which there is no liquid-vapor critical point: linear chains $\left({ϵ}_{AA}\ne 0,{ϵ}_{AB}={ϵ}_{BB}=0\right)$, hyperbranched polymers $\left({ϵ}_{AB}\ne 0,{ϵ}_{AA}={ϵ}_{BB}=0\right)$, and dimers $\left({ϵ}_{BB}\ne 0,{ϵ}_{AA}={ϵ}_{AB}=0\right)$. These expansions also allow us to calculate the structure of the critical fluid by perturbing around the above limits, yielding three different types of condensation: of linear chains ($AA$ clusters connected by a few $AB$ or $BB$ bonds); of hyperbranched polymers ($AB$ clusters connected by $AA$ bonds); or of dimers ($BB$ clusters connected by $AA$ bonds). Interestingly, there is no critical point when ${ϵ}_{AA}$ vanishes despite the fact that $AA$ bonds alone cannot drive condensation.

Figure 1

Lowest-energy structures (without loops): (a) linear chains $\left({ϵ}_{AB}={ϵ}_{BB}=0,{ϵ}_{AA}\ne 0\right)$ for which $X_A=0$ and $X_B=1$; (b) dimers $\left({ϵ}_{AA}={ϵ}_{AB}=0,{ϵ}_{BB}\ne 0\right)$ for which $X_A=1$ and $X_B=0$; and (c) hyperbranched polymers $\left({ϵ}_{AA}={ϵ}_{BB}=0,{ϵ}_{AB}\ne 0\right)$ for which $X_A=0.5$ and $X_B=0$. The small circles are the interaction sites: $A$ (filled) and $B$ (open).

Figure 2

(a) An $X$ junction; (b) a $Y$ junction.

Figure 3

(Color online) Critical point vs ${ϵ}_{BB}/{ϵ}_{AA}$ for ${ϵ}_{AB}=0$: (a) critical density ${\rho }_c^{\ast }$; (b) reduced critical temperature $T_c/{\left({ϵ}_{AA}{ϵ}_{BB}\right)}^{1/2}$; (c) $X_{Ac}$ and $X_{Bc}$, the fractions of unbonded $A$ and $B$ sites, respectively; and (d) inverse mean size of aggregates, $1/⟨L⟩$. The reduced critical temperature vanishes in the limits of linear chains $\left({ϵ}_{BB}/{ϵ}_{AA}\to 0\right)$ and dimers $\left({ϵ}_{BB}/{ϵ}_{AA}\to \infty \right)$, albeit much more slowly in the latter case. In parts (a) and (b), the solid lines are the numerical solution of Eqs. (7, 8), the dashed lines are the asymptotic solutions, and the insets show the behavior for small ${ϵ}_{BB}/{ϵ}_{AA}$; see the text for details.

Figure 4

(Color online) Critical point vs ${ϵ}_{AB}/{ϵ}_{AA}$ for ${ϵ}_{BB}=0$: (a) critical density ${\rho }_c^{\ast }$; (b) reduced critical temperature $T_c/{\left({ϵ}_{AA}{ϵ}_{AB}\right)}^{1/2}$; (c) $X_{Ac}$ and $X_{Bc}$, the fractions of unbonded $A$ and $B$ sites, respectively; and (d) inverse mean size of aggregates, $1/⟨L⟩$. The reduced critical temperature vanishes in the limits of linear chains $\left({ϵ}_{AB}/{ϵ}_{AA}\to 0\right)$ and hyperbranched polymers $\left({ϵ}_{AB}/{ϵ}_{AA}\to \infty \right)$, albeit much more slowly in the latter case, as in Fig. 3. In parts (a) and (b), the solid lines are the numerical solution of Eqs. (7, 8), the dashed lines are the asymptotic solutions, and the insets show the behavior for small ${ϵ}_{AB}/{ϵ}_{AA}$; see the text for details

Figure 5

(a) A (large) ring of particles connected by $AA$ bonds; its energy is $E=0$. (b) When the ring opens, two ends are created (or an $AA$ bond is broken), so the energy becomes $E=2{ϵ}_e={ϵ}_{AA}$. (c) When a $Y$ junction forms, an end is lost and an $AB$ bond is created; the energy becomes $E={ϵ}_j=-{ϵ}_{AB}+{ϵ}_{AA}/2$.

Figure 6

(a) A cluster composed of two linear strands $\left(n_s=2\right)$, each of size $\ell =7$; the number of unbonded $A$ sites (labeled $\overline{A}$) is $n_s+1=3$, and the number of bonded $B$ sites (labeled $B$) is $n_s-1=1$. (b) When a third linear strand is added, one new bonded $B$ site and one new unbonded $A$ site are created.