Analytical theory of effective interactions in binary colloidal systems of soft particles

While density functional theory with integral equations techniques are very efficient tools in the numerical analysis of complex fluids, analytical insight into the phenomenon of effective interactions is still limited. In this paper, we propose a theory of binary systems that results in a relatively simple analytical expression combining arbitrary microscopic potentials into effective interaction. The derivation is based on translating a many-particle Hamiltonian including particle-depletant and depletant-depletant interactions into the occupation field language, which turns the partition function into multiple Gaussian integrals, regardless of what microscopic potentials are chosen. As a result, we calculate the effective Hamiltonian and discuss when our formula is a dominant contribution to the effective interactions. Our theory allows us to analytically reproduce several important characteristics of systems under scrutiny. In particular, we analyze the following: the effective attraction as a demixing factor in the binary systems of Gaussian particles, the screening of charged spheres by ions, which proves equivalent to Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, effective interactions in the binary mixtures of Yukawa particles, and the system of particles consisting of both a repulsive core and an attractive/repulsive Yukawa interaction tail. For this last case, we reproduce the “attraction-through-repulsion” and “repulsion-through-attraction” effects previously observed in simulations.

Figure 1

Effective interaction between Gaussian particles, according to formula (65). ${\sigma }_1$ is the unit length, and the scaling reads $\beta {\epsilon }_1^2/{\epsilon }_2=1$. $U_{\text{eff}}$ is a negative Gaussian function for every ${\sigma }_2$.

Figure 2

Effective attraction in binary mixtures of Gaussian particles as a driving force behind phase separation. This plot visualizes inequality (69) for $D=3$, where $\tilde{\epsilon }={\epsilon }_1/\sqrt{{\epsilon }_0{\epsilon }_2}$ and $c={\sigma }_2/{\sigma }_0$. Shaded region—total interaction is purely repulsive, no driving force for de-mixing. Plain region—total interaction has an attractive tail stimulating demixing. The dashed line illustrates mean-field condition (70) for spinodal decomposition in the Gaussian mixture.

Figure 3

Effective potential for a binary mixture of Yukawa particles, according to formula (87), for which $\beta {\epsilon }_1^2/{\epsilon }_2=1$, and ${\sigma }_2$ and ${\kappa }_i$ are given in units $\left[{\sigma }_1\right]$ and $\left[{\sigma }_1^{-1}\right]$, respectively. Curves 1–3: growing depletion attraction for decreasing size of depletant particles, ${\kappa }_i={\sigma }_i^{-1}$, to match the size of the particle. Curves 4–6: for higher values of ${\kappa }_1$, a ${\kappa }_2$-dependent energy barrier appears.

Figure 4

Model (91) (dashed black line) fitted to simulation data (red solid line) from [25]. In all cases, core exponents read ${\lambda }_1=3{\kappa }_1$ and ${\lambda }_2=2.4{\kappa }_2$; $c_i$ are determined from the fitting procedure. Insets: a comparison between potential wells resulting from the fitting procedure and the hard-core potentials applied in [25]; gray, small-small interaction; black, big-small interaction; dashed lines, soft-core potential generated with ${\lambda }_i$ and $c_i$; solid lines, referential hard-core potential from [25].

Figure 5

Effective interaction in the binary mixture of particles consisting of a repulsive core and a Yukawa interaction tail. Left: effective interaction generated from formula (91) for soft-core particles. Right: effective interaction measured in the simulations of hard-core particles, reprinted from [25]. ${\sigma }_0$ is the radius of bigger particles, ${\sigma }_1=0.6{\sigma }_0$, ${\sigma }_2=0.2{\sigma }_0$, ${\kappa }_1=6/{\sigma }_0$, and ${\kappa }_2=15/{\sigma }_0$. Core parameters for all curves: ${\lambda }_1=\frac{5}{3}{\kappa }_1$, ${\lambda }_2=\frac{32}{15}{\kappa }_2$, $\beta c_1=1$, and $\beta c_2=2.2$. Curves 1–3: for ${\epsilon }_1=0$, the behavior of $U_{\text{eff}}$ depends on the sign of ${\epsilon }_2$. Curves 4–6: for ${\epsilon }_1>0$, $U_{\text{eff}}$ is attractive. Curves 7–9: for ${\epsilon }_1<0$, $U_{\text{eff}}$ is repulsive.