# Elasticity of Compressed Emulsions

## Contents

## Authorship

*Title of Original Work:*
Elasticity of Compressed Emulsions

*Journal of Original Work:* PRL, Volume 75, Number 10, Sept 4, 1995

*Authors:* T.G. Mason, J. Bibette, D.A. Weitz

*Author of Review:*
Joseph Muth - AP 225 - Fall 2012 - 11/11/2012

## Soft Matter Keywords

Emulsions, Laplace Pressure, Disjoining Pressure

## Overview

Emulsions consist of a fluid dispersed in a second immiscible fluid where the first fluid is stabilized in the second fluid by a surfactant. Despite the liquid nature of emulsions, the elastic response of such material systems mimics the characteristics of a viscoelastic solid. As a pressure (π) is applied to an emulsion, the volume fraction (Φ) of droplets increases. When the π is sufficiently high so as to force the droplets to touch, Φ approaches one and any subsequent pressure increase results in deformation of the droplets. Deformation leads to stored strain energy, and thus elasticity. Deformation of the droplets is governed by the Laplace pressure, σ/r, where σ is the surface tension of the drop and r is the radius of the undeformed drop. Clearly the elastic modulus (E) of the drop will depend on the degree of stored strain energy in the drop interface. Thus if the drops are not touching, then E = 0 because there can be no interfacial deformation. If Φ = 1 then the drops are maximally touching and the emulsion behaves like a dry elastic foam. However, the transition of the emulsions elastic properties between these two limiting cases is not well understood. It is the author's goal to elucidate the elastic property transition of emulsions by measuring the Φ dependence of E using monodisperse emulsions. By creating monodisperse emulsions, differences in Laplace pressure between drops is eliminated - enabling direct comparisons of experimental results with theoretical predictions.

## Experimental

All emulsions were made using silicone oil dispersed in oil. The surfactant ,sodium dodecylsulphate, was added such that the overall surfactant concentration was 0.01M. The linear viscoelastic moduli, G' and G", were measured by applying a periodic strain to the emulsion and measuring the stress response.

## Results

Figure 1 shows G' and G" as a function of strain for several emulsions of Φ ranging from 0.60 - 0.80. G' increases by four decades over this Φ range. For small strains, G' dominates. At large strains G" dominates. At small strains the emulsion can maintain contact between each drop making the mechanical response more elastic. However, at large strains, the yield limit of the emulsion is reached and flow initiates, thus G" becomes dominant.

Figure 2 shows the frequency (ω) dependence of G' and G" for emulsions with Φ ranging from 0.60 - 0.80. G' remains nearly constant for each emulsion, especially so for higher Φ. However, each G' curve does exhibit some curvature. As such, G' can be calculated at the inflection point - G'p. G" behaves much differently compared to G'. Each G" curve is characterized by a distinct minimum G" value - G"m. The frequency at which G"m occurs increases with increasing Φ. G'p and G"m provide useful reference points for characterizing the elastic and loss moduli, respectively.

To describe the Φ dependence of G'p and G"m for emulsions with different drop sizes, Laplace pressure and disjoining pressure need to be accounted for. Normalizing each result for the Laplace pressure is straight forward because it is governed solely by surface tension and drop size; however disjoining pressure normalization is more complicated. Disjoining pressure is effected by both Φ and the applied pressure because the thin water film between each droplet changes with packing and deformation of the drops. Disjoining pressure is especially significant for film thicknesses (h) of 5-18 nm. As such, an effective Φ (Φeff) must be defined which includes the effects of disjoining pressure such that the emulsion properties alone can be isolated: Φeff = Φ(1+3h/2r). Using this methodology Figure 3 shows G'p and G"m for different emulsions of various Φ and r. It is clear that G'p and G"m both fall on distinct, individual curves. This result indicates that the elastic properties of emulsions only depend on the packing geometry of the droplets.

## Discussion

The behavior of the emulsions shown in Figure 3 can be explained by analyzing the extreme limits of the moduli behavior. For Φeff = 1, the emulsion resembles a dry foam. Theoretical predictions for dry foams state that G'=0.6σ/r. The results in Figure 3 show that for Φeff = 1, G'=0.55σ/r. Thus emulsions can indeed be thought of as dry foams when the emulsion is highly dense.

Further examination of Figure 3 shows two distinct behavior regions. For Φeff > 0.63, G' and G" follow one regime, while for Φeff < 0.63, a different type of behavior is exhibited. This result can be explained in terms of packing mechanisms. Above 0.63, packing constraints govern the droplet arrangement and the energy scaling parameter remains σ/r, but for Φeff < 0.63, packing constraints no longer dictate droplet arrangement. As packing decreases below the critical Φeff value, thermal fluctuations begin to dominate elastic response. More experiments need to be performed to confirm the exact nature of the thermal scaling law, but the transition between these two regimes appears quite acutely in Figure 3.

In most emulsions, the thermal packing regime is of little engineering interest. Thus if we focus on the the elastic response for Φeff > 0.63, an interesting result appears. Figure 4 shows that G'p increases linearly with Φeff when plotted on a logarithmic scale. If we consider a compressed drop as a sphere being squeezed by a cube, then as the cube compresses, flat regimes will appear at the contact points. These facets can be thought of as contact points of a droplet with its six nearest neighbors. As compression increases, more nearest neighbors come in contact with the droplet, forming new facets. This interaction model can be simulated as group of interconnected, randomly oriented springs with the same stiffness. When modeled computationally, this model predicts that G' is linearly related to the number springs, and G' approaches zero at the critical Φ value. Because the density of springs (droplets) varies with Φeff, the model for G' can be expressed as Φeff(Φeff - Φcrit)(σ/r). The line corresponding to this model fits the experimentally observed data quite well as displayed in Figure 4.

While this theory fits the data well for Φeff > Φcrit, it is not expected to fit as Φeff approaches Φcrit, because the number of facets can be no smaller than 6 according to the box-compression model. Moreover, as Φeff decreases, there is less compression holding the emulsion together, making rearrangement and slip easier. If slip occurs, the box-compression no longer applies because the droplets become unconstrained and lose their fixed amount of nearest neighbors. Thus while the model delineated above seems to fit the experimentally determined data very well, more extensive work must be done before a complete theory of emulsion elasticity can be formulated.

## Summary

Despite their liquid character, emulsions display elastic properties similar to viscoelastic solids when they are compressed. The elastic response of the emulsion is governed primarily by the volume fraction of droplets and the applied pressure. As the pressure - and therefore the volume fraction - increase, droplets contact each other, deform, and store elastic energy. The elastic energy stored at the droplet interfaces during compression gives rise to the elastic properties of the emulsion. In this work, the author aims to quantify the nature of the elastic properties of emulsions by experimentally measuring the elastic and loss moduli of emulsions as a function of volume fraction and droplet size. By standardizing the results according to droplet size, Laplace pressure, and disjoining pressure, the author is able to establish a model for the elastic modulus of close-packed emulsions above a critical volume fraction of droplets. While the model developed in this work accurately predicts elastic response for high volume fractions, more work is needed to extend this model over a more complete suite of droplet fractions.