[Paper Review] Improved viscosity-concentration equation for emulsions of nearly spherical droplets

Carlos I. Mendoza, I. Santamarı́a-Holek|ArXiv.org|Mar 31, 2009

Material Dynamics and Properties11 references3 citations

TL;DR

This paper proposes an improved viscosity-concentration model for emulsions of nearly spherical droplets by introducing an effective filling fraction, φ_eff, which accounts for excluded volume effects and droplet packing. The model uses a recursive-differential effective medium approach in terms of φ_eff, yielding a viscosity equation that reduces to Taylor’s law at low concentrations, accurately predicts experimental data across all volume fractions, and better captures the critical packing limit than prior models, including Pal’s model 2.

ABSTRACT

We propose an improved viscosity model accounting for experiments of emulsions of two immiscible liquids at arbitrary volume fractions and low shear rates. The model is based on a recursive-differential method formulated in terms of the appropriate scaling variable which emerges from an analysis of excluded volume effects in the system. This variable, called the effective filling fraction, incorporates the geometrical information of the system which determines the maximum packing and reduces to the bare filling fraction for infinitely diluted emulsions. The agreement of our model for the viscosity with experiments is remarkable for all the range of volume fractions and viscosity ratio.

Motivation & Objective

• To develop a viscosity model for emulsions that accurately captures rheological behavior across all volume fractions, including concentrated regimes.
• To incorporate geometrical constraints such as droplet packing and excluded volume effects into the viscosity model from the outset.
• To improve upon existing differential effective medium theories by ensuring correct low-concentration limits and better agreement with experimental data.
• To eliminate the need for fitting critical volume fractions φ_c as a function of viscosity ratio K by embedding geometric constraints directly into the scaling variable.

Proposed method

• Introduce an effective filling fraction φ_eff = φ / (1 - cφ), where c accounts for excluded volume, to replace the bare volume fraction φ in the model.
• Formulate a recursive-differential effective medium theory (DEMT) using φ_eff as the integration variable, ensuring geometric correlations between droplets are captured.
• Derive the viscosity equation: η_r(φ_eff) [ (2η_r(φ_eff) + 5K) / (2 + 5K) ]^{3/2} = (1 - φ_eff)^{-5/2}, which reduces to Taylor’s equation at low φ.
• Use the scaling variable φ_eff to ensure the model correctly diverges at the critical packing fraction φ_eff = 1, corresponding to φ_c = 1/(1 + c).
• Validate the model against experimental data across a wide range of viscosity ratios K and volume fractions φ, including stable, unflocculated emulsions with low capillary number.
• Demonstrate system independence by plotting η_r^{-2/5} [(2η_r + 5K)/(2 + 5K)]^{-3/5} versus φ_eff, showing collapse of data onto a single curve.

Experimental results

Research questions

• RQ1 How can excluded volume effects and droplet packing be systematically incorporated into a viscosity model for emulsions?
• RQ2 Can a viscosity model be constructed that reduces to Taylor’s equation at low concentrations while accurately predicting behavior at high volume fractions?
• RQ3 Does using an effective filling fraction φ_eff instead of the bare φ improve agreement with experimental viscosity data?
• RQ4 How does the critical volume fraction φ_c depend on the viscosity ratio K in the absence of empirical fitting?
• RQ5 Can the model achieve system-independent collapse of viscosity data when plotted against φ_eff?

Key findings

• The improved model, based on the effective filling fraction φ_eff, shows excellent agreement with experimental viscosity data across all volume fractions and viscosity ratios K, outperforming Pal’s model 2.
• The model correctly reduces to Taylor’s equation in the dilute limit (φ → 0), ensuring consistency with the fundamental theory for dilute suspensions.
• The critical volume fraction φ_c predicted by the model is larger than that in Pal’s model 2 and is consistent across different K values when plotted against φ_eff, indicating system independence.
• The model predicts higher relative viscosities than Pal’s model 2 across all K and φ, particularly at intermediate to high concentrations, aligning better with experimental trends.
• When plotted as η_r^{-2/5} [(2η_r + 5K)/(2 + 5K)]^{-3/5} versus φ_eff, all experimental data collapse onto a single curve, confirming the model’s system independence and validity.
• The model does not require fitting φ_c as a function of K, unlike Pal’s model 2, because φ_c is inherently determined by the geometry through the parameter c in φ_eff = φ / (1 - cφ).