Interfacial tension and a three-phase generalized self-consistent theory of non-dilute soft composite solids

Abstract

In the dilute limit Eshelby's inclusion theory captures the behavior of a wide range of systems and properties. However, because Eshelby's approach neglects interfacial stress, it breaks down in soft materials as the inclusion size approaches the elastocapillarity length L≡γ/E. Here, we use a three-phase generalized self-consistent method to calculate the elastic moduli of composites comprised of an isotropic, linear-elastic compliant solid hosting a spatially random monodisperse distribution of spherical liquid droplets. As opposed to similar approaches, we explicitly capture the liquid-solid interfacial stress when it is treated as an isotropic, strain-independent surface tension. Within this framework, the composite stiffness depends solely on the ratio of the elastocapillarity length L to the inclusion radius R. Independent of inclusion volume fraction, we find that the composite is stiffened by the inclusions whenever R < 3L/2. Over the same range of parameters, we compare our results with alternative approaches (dilute and Mori-Tanaka theories that include surface tension). Our framework can be easily extended to calculate the composite properties of more general soft materials where surface tension plays a role.

Finite-element modeling of soft solids with liquid inclusions

“…We obtain excellent agreement between our simulation results and the experiments of Style et al (2015a) and Ducloué et al (2014) for both isolated droplets and composite materials. The simulation results also agree well with theoretical predictions of the ground-state shear modulus of the composite material based on both the Mori-Tanaka theory (Ducloué et al, 2014;Mancarella et al, 2016a) and the three-phase GSC theory (Mancarella et al, 2016b). Since our simulation capability makes no linearizing assumptions, it may be used to model the response of the composite material under large deformations.…”

“…This solution has then been used to analytically determine the infinitesimal shear modulus of a composite material made up of a soft solid matrix with a given volume fraction of fluid-filled droplets using a Mori-Tanaka approach (Mori and Tanaka, 1973;Ducloué et al, 2014;Mancarella et al, 2016a) and a three-phase, self-consistent approach (Christensen and Lo, 1979;Mancarella et al, 2016b), and good agreement with experiments has been attained. To move beyond the small-deformation, linearized-kinematics response and determine the full stress-strain response of these composite materials, nonlinear, numerical methods are required.…”

“…4 From these values of the elasto-capillary numbers along with the reported droplet size and matrix shear moduli, we may estimate the interfacial surface tension as γ = 18.3 mJ/m 2 , which is close to the value estimated by Style et al (2015a) of γ = 14 mJ/m 2 . Figure 3b proach, derived by both Ducloué et al (2014) and Mancarella et al (2016a), and a three-phase generalized self-consistent (GSC) theory, derived by Mancarella et al (2016b). First, for the Mori-Tanaka approach (see Mancarella et al (2016a) for details of the derivation), the ground-state composite shear modulus is given through the droplet volume fraction and elasto-capillary number as…”

“…We also consider the three-phase, generalized self-consistent approach of Mancarella et al (2016b), which extended the work of Christensen and Lo (1979) to include the effect of surface tension. This approach begins by considering a single droplet of radius R inside of a thick spherical shell of matrix material with shear modulus G and outer radius chosen to achieve the desired droplet volume fraction φ, which is then embedded inside of an infinite matrix of the homogenized composite material with unknown properties.…”

“…This approach begins by considering a single droplet of radius R inside of a thick spherical shell of matrix material with shear modulus G and outer radius chosen to achieve the desired droplet volume fraction φ, which is then embedded inside of an infinite matrix of the homogenized composite material with unknown properties. Through their analysis -explicitly accounting for the interfacial surface tension γ -they determine an implicit equation for the shear modulus of the homogenized composite as a function of φ and γ/GR (see Equation (15) of Mancarella et al (2016b)), which is quite lengthy and for the sake of brevity is not reproduced here. The two analytical models for the normalized composite shear modulus G c /G are plotted as a function of the elasto-capillary number for droplet volume fractions of φ = 0.1, 0.2, and 0.3 along with our simulation results in Figure 4.…”