Models of surface viscosities ofparticle-laden fluid interfaces

Sergey Lishchuk

Materials & Engineering Research Institute, Sheffield Hallam University, UK

Motivation

● Many foods and beverages are soft interface-dominated materials.

● Many of them contain food-grade particles adsorbed at fluid interfaces.

● Surface rheology plays an important role in:➢ formation, stability and dynamics of food emulsions and foams;

➢ sensory physics.

● There is a need in powerful constitutive laws for particle-laden fluid interfaces capable to predict their rheological properties in a range of conditions relevant to food manufacturing.

● A good model should be able to➢ agree with existing experimental data;

➢ successfully predict future experimental results

➢ guide engineering of foods and beverages with desired rheological properties.

Pickering emulsions

Walter Ramsden Proceedings of the Royal Society of London

72 (1903) 156-164

Spencer PickeringJournal of the Chemical Society, Transactions

91 (1907) 2001-2021

● any fine particles are able to stabilise oil-in-water emulsions

● enhanced stability compared to surfactant-stabilised emulsions

Pickering emulsions

Walter Ramsden Proceedings of the Royal Society of London

72 (1903) 156-164

Spencer PickeringJournal of the Chemical Society, Transactions

91 (1907) 2001-2021

The Arnold Principle (formulated by Michael Berry):

“If a notion bears a personal name, then this name is not the name of the discoverer.”

Particle at a fluid interface

water

oil

particle water

oil

particleθ

E1=γ pw⋅4 π R2 E2=γ po A po+γ pw A pw−γow A ow

Δ E=π R2γow (1−cosθ)2 where cosθ=

γpw−γ poγow

For R ~ 1 μ : ΔE ~ 107 kBT

Work of detachment

Work of detachment: W=ℱ detachment−ℱ equilibrium

Small-particle approximation, 90-degree contact angle: W=−π2γ a2 [ ln (√2+1)ν

8−√2−

12 ]

Ettelaie and Lishchuk, Soft Matter (2015)

Particle-laden fluid interface

● Continuous on large scale

● Static properties, eg:

➢ surface tension

➢ bending modulus *

➢ saddle-splay modulus **

➢ …

● Dynamic properties, eg:

➢ surface shear viscosity

➢ surface dilatational viscosity

➢ ...

* Aveyard et al., Prog. Coll. Pol. Sci. (2002)* Kralchevsky et al., Langmuir (2005)

** Lishchuk, EPL (2009)

Models of surface viscosity

● We focus on the viscous part of viscoelastic rheological response.

● Principal mechanisms of viscous dissipation of energy:

➢ bulk fluid flow

➢ interfacial fluid flow

➢ motion / deformation of particles

Saffman-Delbrück length

Small particles – bulk not important

Large particles – bulk is important

(a≪ l0)

l 0≡ s

(a≫ l0)

Saffman-Delbrück length:

(sets length scale over which momentum transfers from the surface to the bulk)

Saffman and Delbrück, PNAS (1975)

Einstein formula

Albert Einstein doctoral thesis (1905):

● Suspension of a large number of identical rigid spherical particles in the incompressible fluid

● Suspension is dilute: The motion of any particle is not affected by other particles

● The inertia of the particles and their rotational motion can be neglected

● The velocity on the surface of the particles is zero

Einstein formula

A. Einstein, Ann. Phys. 19 (1906) 289:

Einstein formula

A. Einstein, Ann. Phys. 34 (1911) 591:

A. Einstein, Ann. Phys. 19 (1906) 289:

Corrected by Ludwig Hopf

N. Straumann, arXiv:physics/0504201A. Pais, "Subtle is the Lord”, Oxford, 2005

Surface viscosities

σαβ=(γ+ζs g

γδ Sγ δ)gαβ+ηsE

αβγ δS γδ

Sαβ=12( gαβ+ vα ,β+ vβ ,α)

Eαβ γ δ≡g αγ gβδ+ gα δgβ γ−gαβ g γ δ ,

Newtonian isotropic surface fluid — Boussinesq-Scriven constitutive equation

Surface stress tensor:

ηs

ζs

gαβ

– surface rate-of-deformation tensor

– surface metric tensor

– surface tension

– surface shear viscosity

– dilatational viscosity

vα – surface velocity

Boussinesq, Ann. Chim. Phys (1913)Scriven, Chem. Eng. Sci. (1960)

γ

Effective surface viscositiesof a particle-laden fluid interface

Surface shear viscosity:

Surface dilatational viscosity:

Area fraction of particles:

(N particles of radius R per area A)

s=5312R

s=512R

= R2NA

Lishchuk and Halliday, PRE (2009)

Conditions of applicability

Elasticity is small: α ≫u ' ' (r)ηRϕ

Reynolds number is small: α ≪η

ρR2

Interface is flat: α ≪γ

|η1−η2|R

Particles are well separated: ϕ ≪ 1

Surface viscosity is small: ϕ ≫ηs

ηR

Arbitrary contact angle

Lishchuk, PRE (2016)

Incompressible fluid interface

Streamlines of incompressible surface flow:● The presence of even a small amount of

surfactant at the particle-laden fluid interface subjected to shear makes surface flow incompressible if the shear rate is small enough.

● Surface continuity equation for incompressible interfacial flow:

● Hydrodynamic equations are solved in vector spherical harmonic representation.

● The effective surface shear viscosity is found to be 7.6% larger than the value obtained without account for surface incompressibility.

∇ s⋅vs=0

Fischer et al., J. Fluid Mech. (2006)Lishchuk, PRE (2014)

Larger concentrations

Lishchuk and Halliday, PRE (2009)

ηs=53(η1+ η2)Rϕ

Lattice Boltzmann simulation results:

Very large concentrations

Assumptions:

● highly concentrated particle arrays in a plane form a hexagonal structure

● the dominant contribution to the viscous dissipation rate arises in the thin gaps between the neighbouring particles [Frankel and Acrrivos, Chem. Eng. Sci. (1967)]

Result:

A: Lishchuk and Halliday, PRE (2009)B: Lishchuk, PRE (2014)

ζ s=3√3π(η1+η2)R

16(√ϕm /ϕ−1), ϕm=

π

2√3

Summary

● The Einstein's approach to calculating the effective shear viscosity of suspensions has been generalized to the case of a flat interface separating two immiscible fluids laden with monodisperse rigid spherical non-Brownian particles.

● This approach has been used to calculate the following effective viscous properties of this system:

a) dilatational viscosity in the limit of small and large concentration of the adsorbed particles

b) dependence of the dilatational viscosity upon the contact angle between the fluid interface and the particle surface

c) surface shear viscosity at small concentration of particles

d) the modification of the surface shear viscosity due to the surface flow incompressibility

● Some other approaches and models to bereviewed in the upcoming book chapter: