# Effective viscosity of a dilute emulsion of spherical drops containing soluble surfactant

Effective viscosity of a dilute emulsion of spherical drops containing soluble surfactant The effective viscosity of a dilute emulsion of spherical drops containing a soluble surfactant is calculated under a linear creeping flow. It is assumed that convection of surfactant is small relative to diffusion, and thus the Peclet number, P e, is small. We calculate the effective viscosity of the emulsion to O ( P e ϕ μ ) $\mathcal {O}(Pe\phi \mu )$ , where ϕ is the small volume fraction of the dispersed drops and μ is the viscosity of the surfactant-free suspending fluid. This O ( P e ϕ μ ) $\mathcal {O}(Pe\phi \mu )$ contribution is a sensitive function of the bulk and interfacial surfactant transport. Specifically, soluble surfactant molecules diffuse from the bulk to the interface and then adsorb to the interface. The ratio of the time scale for bulk diffusion to the time scale for adsorption to the interface is quantified by a Damkohler number, D a. The adsorption of surfactant to the interface may cause a significant decrease in the bulk concentration, which is known as depletion. The impact of depletion is characterized by two parameters: h, which is a dimensionless depletion depth; and k, which is the ratio of the desorption time scale to the adsorption time scale. We analytically determine how the O ( P e ϕ μ ) $\mathcal {O}(Pe\phi \mu )$ contribution to the effective viscosity depends on h, k, and D a. Surprisingly, for certain regimes in the h − k − D a parameter space, we predict the effective viscosity of the emulsion to be greater than Einstein’s result for the viscosity of a suspension of rigid spheres. Large Marangoni stresses driven by convective transport of soluble surfactant molecules are responsible for this result. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Rheologica Acta Springer Journals

# Effective viscosity of a dilute emulsion of spherical drops containing soluble surfactant

, Volume 57 (7) – May 29, 2018
11 pages

/lp/springer_journal/effective-viscosity-of-a-dilute-emulsion-of-spherical-drops-containing-xEVvG0U0du
Publisher
Springer Journals
Subject
Materials Science; Characterization and Evaluation of Materials; Polymer Sciences; Soft and Granular Matter, Complex Fluids and Microfluidics; Mechanical Engineering; Food Science
ISSN
0035-4511
eISSN
1435-1528
D.O.I.
10.1007/s00397-018-1092-x
Publisher site
See Article on Publisher Site

### Abstract

The effective viscosity of a dilute emulsion of spherical drops containing a soluble surfactant is calculated under a linear creeping flow. It is assumed that convection of surfactant is small relative to diffusion, and thus the Peclet number, P e, is small. We calculate the effective viscosity of the emulsion to O ( P e ϕ μ ) $\mathcal {O}(Pe\phi \mu )$ , where ϕ is the small volume fraction of the dispersed drops and μ is the viscosity of the surfactant-free suspending fluid. This O ( P e ϕ μ ) $\mathcal {O}(Pe\phi \mu )$ contribution is a sensitive function of the bulk and interfacial surfactant transport. Specifically, soluble surfactant molecules diffuse from the bulk to the interface and then adsorb to the interface. The ratio of the time scale for bulk diffusion to the time scale for adsorption to the interface is quantified by a Damkohler number, D a. The adsorption of surfactant to the interface may cause a significant decrease in the bulk concentration, which is known as depletion. The impact of depletion is characterized by two parameters: h, which is a dimensionless depletion depth; and k, which is the ratio of the desorption time scale to the adsorption time scale. We analytically determine how the O ( P e ϕ μ ) $\mathcal {O}(Pe\phi \mu )$ contribution to the effective viscosity depends on h, k, and D a. Surprisingly, for certain regimes in the h − k − D a parameter space, we predict the effective viscosity of the emulsion to be greater than Einstein’s result for the viscosity of a suspension of rigid spheres. Large Marangoni stresses driven by convective transport of soluble surfactant molecules are responsible for this result.

### Journal

Rheologica ActaSpringer Journals

Published: May 29, 2018

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