Interactions between droplets in immiscible liquid suspensions and the influence of surfactants

TL;DR

This work develops a general, lattice-DFT–based framework to compute the effective interaction potential between immiscible-liquid droplets, $\\Delta\\Omega_2(L)$, and extends it to ternary oil–water–alcohol mixtures to capture ouzo-like surfactant effects. By constraining droplet centers with weak Gaussian fields, the authors map the free-energy landscape as a function of center separation and reveal how alcohol reduces interfacial tension and attraction, while stronger surfactant-like terms can create a repulsive barrier that stabilizes droplets. The study also demonstrates non-additive three-body interactions, and uses dynamical density functional theory to show that stronger adsorption slows coarsening and increases droplet discreteness, with implications for long-lived emulsions. Overall, the method provides a broadly applicable route to quantify and control droplet stability in immiscible mixtures, with potential extensions to continuum DFT and charged systems.

Abstract

We develop a general method for determining the effective interaction potential between two or more droplets suspended within a fluid phase. Our approach is based on classical density functional theory. Here, we apply the method to determine the interaction potential between oil droplets suspended in water and also consider the influence of adding a third species, alcohol. This ternary mixture is that found in the ouzo beverage. The ouzo system exhibits spontaneous emulsification when the neat spirit is mixed with water. The oil emulsion that forms has been observed to be surprisingly long-lived. Here we show that the alcohol in the system does indeed play a role in making the droplets more stable, by decreasing the oil-water interfacial tension and therefore also the strength of the attractive interactions between droplets. Within our theory, the surfactant nature of the alcohol can be enhanced without changing the bulk fluid thermodynamics. In fact, our theory can be used to model surfactant mixtures. In this model, the effective interaction between pairs of oil droplets can become repulsive, with a free-energy barrier to droplets merging, thus making them stable.

Figures (10)

• Figure 1: The bulk phase diagram of the ternary oil-water-alcohol (ouzo) system, using the pair interaction parameters given in Eq. \ref{['eq:epsilons']}. Each of the corners correspond to the respective (as labelled) pure liquids, with the concentration of each species decreasing with distance from each respective corner. Below the binodal, the system exhibits two-phase coexistence. Note that in this representation the tie-lines between coexisting state points on the binodals are not horizontal archer2024experimental. The critical point is located at the unique point where the binodal and spinodal curves meet tangentially. Note that this phase diagram is for the incompressible mixture, where we assume the total number density $(n_{\rm a}+n_{\rm o}+n_{\rm w})=1$. However, the phase diagram hardly changes if re-calculated for fixed oil chemical potential $\mu_{\rm o}$, in the range $-3.5\lesssim\beta\mu_{\rm o}\lesssim1$. The points A--C correspond to bulk state points where results in Figs. \ref{['fig:pot1']}--\ref{['fig:three_body']} are obtained.
• Figure 2: The effective interaction potential $\Delta\Omega_2(L)$ between a pair of oil droplets of diameter $2R\approx20\sigma$ plotted as a function of the distance between the droplet centres $L$, for $\beta\mu_{\rm a}=-10$ (i.e. effectively no alcohol in the system) and $\beta\mu_{\rm w}=-3.5$ (the bulk water-rich phase surrounding the droplets corresponds to point A in Fig. \ref{['fig:phase_diag']}). The potential $\Delta\Omega_2(L)$ has two branches, one corresponding to the droplets advancing forward 'F' towards each other and the other corresponding to a single droplet of diameter $28\sigma$ being pulled apart into two droplets and reversing 'R' away from each other. Examples of five typical configurations are indicated (these do not show the whole computational domain). In all cases, the total number of oil molecules in the system of area $80\sigma\times80\sigma$ is fixed to be $N_{\rm o}=800$.
• Figure 3: The effective interaction potential $\Delta\Omega_2(L)$ between pairs of oil droplets, plotted as a function of the distance between the droplet centres $L$. These are calculated for the three alcohol chemical potential $\mu_{\rm a}$ values given in the key. The corresponding bulk water-rich phases surrounding the droplets are indicated as points A--C in Fig. \ref{['fig:phase_diag']}. The chemical potential of the water $\beta\mu_{\rm w}=-3.5$. The total number of oil molecules $N_{\rm o}=800$ is fixed, in a domain of size $80\sigma\times80\sigma$. The potential $\Delta\Omega_2(L)$ has two branches, one corresponding to the droplets advancing forward 'F' towards each other and the other corresponding to the droplets being pulled apart and reversing 'R' away from each other. Examples of four typical configurations are indicated.
• Figure 4: Density profiles for a fixed amount of oil $N_{\rm o}=800$ and $L=30\sigma$, corresponding to the plots of $\Delta\Omega_2(L)$ displayed in Fig. \ref{['fig:pot2']}. These profiles all correspond to the solution branch where the pair of droplets are joined. The chemical potential of the water is $\beta\mu_{\rm w}=-3.5$, while that of the oil increases in each row from top to bottom, as indicated (corresponding to points A--C in Fig. \ref{['fig:phase_diag']}, respectively). In each row, the left hand profile is that of the oil, the middle that of the water and the right that of the alcohol.
• Figure 5: Density profiles of the alcohol/surfactant, for varying $\epsilon_3$ and for $N_{\rm o}=800$ and $L=30\sigma$.