Rheological response of soft Solid/Liquid Composites

TL;DR

This study addresses the dissipative rheology of soft solid–liquid composites formed by liquid PEG droplets in a PDMS matrix. It demonstrates that viscous dissipation increases linearly with the droplet volume fraction and saturates beyond Φ ≈ 0.4, while the elastic response remains largely unchanged; Palierne’s model fails to describe the full spectrum, prompting a two-branch fractional Kelvin–Voigt formulation. The authors establish a time–volume fraction master curve by rescaling high-frequency data and show that the continuous phase dominates at high frequencies whereas droplets govern the low-frequency regime, with dissipation adding approximately linearly with Φ. These insights advance understanding of how liquid inclusions modify dissipation in soft solids and inform design strategies for soft adhesives and impact-resistant materials. Future work points to creep/relaxation experiments and simulations to deepen the theoretical framework and explore different liquid inclusions.

Abstract

Understanding a material's dissipative response is important for their use in many applications, such as adhesion or fracture resistance. In dispersions, the interplay between matrix and inclusions complicates any description. Fractional rheology is conveniently used to fit the storage and loss moduli of complex materials. In conjugation with superposition methods, they allow to better capture the behavior of materials of complex rheology. We study the rheology of soft solid/liquid composites of liquid poly(ethylene glycol) (PEG) droplets in a soft poly(dimethylsiloxane) (PDMS) matrix. We analyze the influence of the droplets through fractional rheology and a time-concentration superposition in the continuous-phase-dominated region. Viscous dissipation increases proportionally with volume fraction, independently of the frequency, whereas the elastic response is almost unchanged.

Figures (6)

• Figure 1: Photograph of a solid emulsion on the rheometer in plate-plate configuration.
• Figure 2: (a) Comparison of the frequency dependency of the storage ($\CIRCLE$) and loss ($\blacklozenge$) moduli of the pure continuous phase (pink, $\Phi=0$), and a solid emulsion of volume fraction $\Phi=0.15$ (red). (b) Evolution of the storage ($\Circle$) and loss ($\lozenge$) moduli of solid emulsions with the angular frequency. The colors correspond to the volume fraction of each sample. (c) Evolution of the loss factor of solid emulsions with the angular frequency. (d) Evolution of the loss factor with the volume fraction of solid emulsions at two angular frequencies, $20\pi~\radian.\second^{-1}$ (red) and $0.2\pi~\radian.\second^{-1}$ (blue). The lighter colors represent the range of behaviors between these frequencies.
• Figure 3: Fit by Palierne's model palierne_linear_1990 of the storage ($\CIRCLE$) and loss ($\blacklozenge$) moduli of three solid emulsions of volume fraction $\Phi=0.05$, $0.15$ and $0.35$.
• Figure 4: (a) Fractional rheology models used to represent (Left) the PDMS (FKV0) and (Right) the solid emulsions (FKV1). The changing branch corresponds to the different power-laws visible at low frequencies and on $G^\prime$. $\mathds{V}_0$ is set at $2.06~\pascal.\second^{\alpha_0}$ and $\alpha_0$ at $0.4$. (b) Left Evolution of the quasi-property $\mathds{V}_1$ with the volume fraction. The grey circles correspond to samples for which $R^2<0.9$. Right Evolution of the fractional power-law exponent $\alpha_1$ with volume fraction.
• Figure 5: Left Storage ($\CIRCLE$) and loss ($\blacklozenge$) moduli evolution with angular frequency for three samples of volume fraction $\Phi=0$ (top), $0.1$ (middle), and $0.25$ (bottom). The high frequency evolution has been fitted with a FKV0 model. Right Rescaled moduli for all samples. The superposition was obtained from the FKV0 fits, with $\tilde{G^\prime}=G^\prime/G_0$, $\tilde{G^{\prime\prime}}=G^{\prime\prime}/G_0$, and $\tilde{\omega}=\omega/\omega_0$ where $\omega_0=(G_0/(\mathds{V}_0\sin(2\alpha_0/\pi)))^{1/\alpha_0}$. The black curves represent the normalized FKV0 model $\tilde{G}^*=1+\mathds{V}_0/G_0(\mathrm{i}\tilde{\omega})^\alpha$, with $\mathds{V}_0/G_0=1.2$ and $\alpha=0.37$. $G_0$ and $\omega_0$ are represented in the inset as functions of the volume fraction. The grayed-out area points a change in behavior, especially visible on $\omega_0$, for $\Phi>0.4$.