Nonequilibrium Interfacial Tension in Simple and Complex Fluids

Interfacial tension between immiscible phases is a well-known phenomenon, which manifests itself in everyday life, from the shape of droplets and foam bubbles to the capillary rise of sap in plants or the locomotion of insects on a water surface. More than a century ago, Korteweg generalized this notion by arguing that stresses at the interface between two miscible fluids act transiently as an effective, nonequilibrium interfacial tension, before homogenization is eventually reached. In spite of its relevance in fields as diverse as geosciences, polymer physics, multiphase flows, and fluid removal, experiments and theoretical works on the interfacial tension of miscible systems are still scarce, and mostly restricted to molecular fluids. This leaves crucial questions unanswered, concerning the very existence of the effective interfacial tension, its stabilizing or destabilizing character, and its dependence on the fluid’s composition and concentration gradients. We present an extensive set of measurements on miscible complex fluids that demonstrate the existence and the stabilizing character of the effective interfacial tension, unveil new regimes beyond Korteweg’s predictions, and quantify its dependence on the nature of the fluids and the composition gradient at the interface. We introduce a simple yet general model that rationalizes nonequilibrium interfacial stresses to arbitrary mixtures, beyond Korteweg’s small gradient regime, and show that the model captures remarkably well both our new measurements and literature data on molecular and polymer fluids. Finally, we briefly discuss the relevance of our model to a variety of interface-driven problems, from phase separation to fracture, which are not adequately captured by current approaches based on the assumption of small gradients.

Popular Summary

Interfacial tension between immiscible fluids plays a role in both the shape of bubbles and the rise of sap in plant capillaries. An open question is whether interfacial tension also exists between miscible fluids. While any interface between miscible fluids must eventually vanish because of mixing, research conducted more than a century ago demonstrated that a nonequilibrium tension does transiently exist between miscible fluids. This result has significant implications in fields as diverse as geoscience, oil recovery, and chemotaxis. However, studies of nonequilibrium interfacial tension have been hampered by a lack of experiments and theoretical modeling beyond molecular fluids, and crucial questions remain unanswered. Here, we address some of these questions both experimentally and theoretically, unveiling new regimes and proposing a simple model that rationalizes both our data and previous measurements.

After injecting water in confined aqueous colloidal and polymer suspensions, we measure the effective interfacial tension by studying the “fingered” morphology of the interface region between the water and the displaced aqueous fluids. We find that such tension has a stronger-than-expected dependence both on the structure of the fluid constituents and the concentration gradient: In some cases, its value spans several orders of magnitude over a narrow composition range. We propose a general analytical form for the interfacial contribution to the system’s free energy.

We expect that our results will shed new light on nonequilibrium interfacial tension and that our simple model will be useful for describing systems with sharp interfaces within other approaches, such as density-functional theory and phase-field methods.

Figure 1

Nonequilibrium interfacial tension ${\mathrm{\Gamma }}_e$ for aqueous solutions of polymers (semifilled symbols), colloidal suspensions (full symbols), and water-glycerol and DA-PDA mixtures (empty symbols, Refs. [13] and [17], respectively). The labels indicate the gap $b$ for the Hele-Shaw experiments; the solvent is water unless specified otherwise. For all LP systems except the shortest polymer, LP-PEG2, the lines are power-law fits with an exponent 2. For all the other samples, the lines are fits of Eq. (9).

Figure 2

Nonequilibrium interface tension (same data as in Fig. 1), plotted using the reduced variables suggested by Eqs. (10), (11), and (13). (a) Data for all polymer systems except LP-PEG2. The line is the law ${\mathrm{\Gamma }}_e\delta /{\kappa }_p(\phi )={\phi }^2$ predicted by the theory. Inset: Interface thickness obtained from the fits as a function of the polymer size. The line is the linear law $\delta =4.4R_g$. (b) Normalized interfacial tension for colloidal suspensions, short polymers (LP-PEG2), and water-glycerol mixtures. The lines are fits of Eq. (10), with ${\mathrm{\Gamma }}^{\mathrm{SG}}=0$ for $CP$ suspensions and ${\mathrm{\Gamma }}^{\mathrm{SG}}$ as in Eqs. (11) and (13) for LP-PEG2 and water-glycerol mixtures, respectively.

Figure 3

Ratio $L/\delta$ obtained by fitting the data shown in Fig. 2 with Eq. (10). Large values of $L/\delta$ are obtained for $CP$ samples where interfaces are probed on a time scale shorter than the diffusion-driven relaxation time of the interface.