Rheology of Emulsions and Suspensions

A suspension is a heterogeneous mixture of dispersed solid particles in a liquid, for example, pastes and clays. An emulsion, on the other hand, is a mixture of two or more liquids where one liquid is dispersed in the other such as oil in water. The presence of particles suspended in a Newtonian liquid is known to lead to non-Newtonian behavior, as the mixture viscosity becomes increasingly dependent on the volume fraction of the dispersed phase.

Krieger and Dougherty [491] developed relative (mixture) viscosity models to describe the non-Newtonian behaviour of suspensions of particles in a liquid. For these models, the relative viscosity depends only upon the volume fraction of the particles. As the concentration of particles increases, an exponential increase in viscosity occurs as the particles move towards contact causing moments of hydrodynamic lubrication forces to become large. Eventually jamming occurs at the critical volume fraction or maximum packing volume fraction. The maximum packing volume fraction is defined as the physically largest volume the particles can occupy when they are packed (usually assumed as close random packing). Maximum packing volume fraction is an empirical parameter. Experiments show that the most compact way to pack hard perfect spheres randomly gives a maximum packing volume fraction of approximately 0.64.

Similar relative viscosity models were also created for emulsions by Pal and Rhodes [524].

In suspension rheology, the dimensionless relative viscosity is used to describe the mixture viscosity. The relative mixture viscosity prescribes how the mixture viscosity increases as more particles interact with each other, along with hydrodynamic forces that are present in the fluid between the particles. As the maximum packing of the particles approaches and the mixture jams, the relative viscosity tends to infinity, essentially encapsulating a yield stress behaviour.

Generally, the relative viscosity is defined as:

Figure 1. EQUATION_DISPLAY

η_{r} = \frac{η}{μ_{c}}

(2422)

where:

$η$ is the mixture viscosity
$μ_{c}$ is the continuous phase (Newtonian) viscosity.

The non-Newtonian behaviour of suspensions and emulsions in terms of the relative viscosity influences the flow through the stress tensor $T_{i}$ in the momentum equation Eqn. (1905).

Constitutive Equations

For a Newtonian fluid, the stress tensor for a generic phase $i$ is given by (see also Eqn. (696)):

Figure 2. EQUATION_DISPLAY

T_{i, Newtonian} = 2 α_{i} μ_{i} (D_{i} - \frac{1}{3} (\nabla \cdot v_{i}) I)

(2423)

where:

$α_{i}$ is the volume fraction
$μ_{i}$ is the (Newtonian) viscosity of phase $i$
$D_{i}$ is the rate-of-deformation tensor.

For a suspension, the continuous liquid stress tensor ( $i = c$ ) takes into account an extra stress tensor $T_{c, extra}$ in addition to the Newtonian stress of the phase. Therefore, the total stress tensor for the continuous phase is:

Figure 3. EQUATION_DISPLAY

T_{c, total} = T_{c, Newtonian} + T_{c, extra}

(2424)

The extra stress tensor for the continuous phase is defined as:

Figure 4. EQUATION_DISPLAY

T_{c, extra} = 2 α_{c} μ_{c} (η_{r} (ϕ) - 1) D_{c}

(2425)

where $ϕ$ is the volume fraction of the dispersed phase, that is, of the suspended particles. Note that in suspension rheology, the volume fraction of the dispersed phase is generally denoted by $ϕ$ instead of $α_{d}$ . Therefore, $ϕ$ is used in all following equations of this chapter. Eqn. (2425) contains a shear stress contribution from the relative viscosity from which one is subtracted to give only the extra-stress due to the presence of the particles in the mixture.

The dispersed particle phase stress tensor ( $i = d$ ) $T_{d, extra}$ has a split contribution from shear along with the normal stress contribution and the osmotic pressure $Π$ :.

Figure 5. EQUATION_DISPLAY

T_{d, extra} = 2 ϕ μ_{c} (η_{r} (ϕ) - 1) D_{d} - μ_{c} η_{n} (ϕ) {\dot{γ}}_{d} Q - Π I

(2426)

where:

$η_{n}$ is the normal relative viscosity
$Q$ is the anisotropy tensor
${\dot{γ}}_{d}$ is the shear-rate of the dispersed phase
$Π$ is the osmotic pressure.

Generally, emulsions use the same stress equation as suspensions. Since for emulsions, the dispersed phase does not consist of hard spheres as for suspensions, maximum packing can be higher before jamming occurs. Jamming occurs at a critical packing fraction, also called the maximum packing fraction. For emulsions, the relative viscosity inverts at volume fractions above the inversion volume fraction:

Figure 6. EQUATION_DISPLAY

η_{r} (ϕ) \to η_{r} (1 - ϕ) for ϕ \geq ϕ_{i n v}

(2427)

where:

$ϕ$ is the packing volume fraction
$ϕ_{inv}$ is the inversion volume fraction.

Simcenter STAR-CCM+ provides several methods to model the relative viscosity $η_{r}$ , the normal relative viscosity $η_{n}$ , and the osmotic pressure $Π$ in Eqn. (2426). One of the earliest from the study of suspensions is the Krieger-Dougherty model [491].

Krieger-Dougherty Model

The Krieger-Dougherty model was based on the assumption that crowding plays a vital role in the origin of non-Newtonian flow behavior in rigid-sphere suspensions. Measurements on latex systems did not detect non-Newtonian behavior at concentrations below 20% by volume of suspended polymer. The Krieger-Dougherty model takes into account the interactions between neighboring spherical particles. The resulting flow equation compares well with experimental viscometric data gathered from tests using synthetic latexes and solutions of polymer particles.

In Simcenter STAR-CCM+, the Krieger-Dougherty model has the form:

Figure 7. EQUATION_DISPLAY

η_{r} = {(1 - \frac{ϕ}{ϕ_{m}})}^{- [η] ϕ_{m}}

(2428)

where:

$[η]$ is the intrinsic viscosity. For spherical particles, $[η] = 2.5$ .
$ϕ$ is the volume fraction.
$ϕ_{m}$ is the maximum critical packing fraction. For hard spheres, $ϕ_{m} ≃ 0.645$ .

For the Krieger-Dougherty model, the normal relative viscosity is zero. Therefore, the second term in Eqn. (2426) vanishes for this model.

Morris and Boulay Model

The Morris and Boulay model was based on an examination of the role of normal stresses in causing particle migration and macroscopic spatial variation of the particle volume fraction $ϕ$ in a mixture of rigid, neutrally buoyant spherical particles that are suspended in a Newtonian fluid.

The entire $ϕ$ dependence of the compressive shear-induced normal stresses is captured by a normal relative viscosity $η_{n} (ϕ)$ . This quantity vanishes (as $ϕ^{2}$ ) at $ϕ = 0$ and diverges at maximum packing, in the same fashion as does the shear viscosity $η_{s} (ϕ)$ . Anisotropy of the normal stresses arising from the presence of the particles is modeled as independent of $ϕ$ .

In Simcenter STAR-CCM+, the relative shear viscosity of the Morris and Boulay model [521] is defined as:

Figure 8. EQUATION_DISPLAY

η_{r} (ϕ) = 1 + 2.5 ϕ {(1 - \frac{ϕ}{ϕ_{m}})}^{- 1} + K_{s} {(\frac{ϕ}{ϕ_{m}})}^{2} {(1 - \frac{ϕ}{ϕ_{m}})}^{- 2}

(2429)

where:

$K_{s}$ is the contact contribution
$ϕ_{m}$ is the maximum packing fraction.

The Morris and Boulay model was derived from experiments in pipes and Couette devices where the flow becomes aligned in a particular direction that depends on the geometry. The normal stress was shown to be anisotropic. The normal relative viscosity leads to migration of the particles in different directions.

For shear flow in the x direction, the anisotropy tensor reads:

Figure 9. EQUATION_DISPLAY

Q = [\begin{matrix} λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{2} \end{matrix}]

(2430)

where the anisotropy parameters are:

$λ_{1}$ = 1.0
$λ_{2}$ = 0.8
$λ_{3}$ = 0.5

An isotropic version gives equal weighting to all directions by setting the anisotropy tensor equal to the identity tensor, [520].

Figure 10. EQUATION_DISPLAY

Q = I

(2431)

This form is valid for 3D simulations, but the contributions are no longer anisotropic.

The normal relative viscosity of the Morris and Boulay model is defined as:

Figure 11. EQUATION_DISPLAY

η_{n} (ϕ) = K_{n} {(\frac{ϕ}{ϕ_{m}})}^{2} {(1 - \frac{ϕ}{ϕ_{m}})}^{- 2}

(2432)

where:

$K_{n}$ is the normal contact contribution
$ϕ_{m}$ is the maximum packing fraction.

This is equivalent to a particle pressure. It leads to migration of particles towards regions of low shear-rate.

Shear Thinning Model

The shear thinning model blends a zero-shear and infinite-shear Morris and Boulay relative viscosity model [521] with a Carreau Generalized Newtonian model. The shear thinning model is not experimentally verified. You are advised to use it with caution and only if the physics is known to deviate from the Morris and Boulay model.

The zero-shear relative viscosity is:

Figure 12. EQUATION_DISPLAY

η_{r 0} (ϕ) = 1 + 2.5 ϕ {(1 - \frac{ϕ}{ϕ_{m 0}})}^{- 1} + K_{s 0} {(\frac{ϕ}{ϕ_{m 0}})}^{2} {(1 - \frac{ϕ}{ϕ_{m 0}})}^{- 2}

(2433)

where

$K_{s 0}$ is the contact parameter at zero shear rate
$ϕ_{m 0}$ is the maximum packing at zero shear rate.

The infinite-shear relative viscosity is:

Figure 13. EQUATION_DISPLAY

η_{r \infty} (ϕ) = 1 + 2.5 ϕ {(1 - \frac{ϕ}{ϕ_{m \infty}})}^{- 1} + K_{s \infty} {(\frac{ϕ}{ϕ_{m \infty}})}^{2} {(1 - \frac{ϕ}{ϕ_{m \infty}})}^{- 2}

(2434)

where:

$ϕ$ is the dispersed phase volume fraction
$K_{s∞}$ is the contact parameter at infinite shear rate
$ϕ_{m∞}$ is the maximum packing at infinite shear rate.

Blended with a Carreau model:

Figure 14. EQUATION_DISPLAY

η_{r} (\dot{γ}, ϕ) = \frac{τ_{y}}{\dot{γ}} + η_{r \infty} (ϕ) + (η_{r 0} (ϕ) - η_{r \infty} (ϕ)) {(1 + {(λ \dot{γ})}^{a})}^{\frac{(n - 1)}{a}}

(2435)

where:

$τ_{y}$ is the yield stress
$λ$ is the relaxation time
$a = 2$ is a parameter to control shear thinning
$n$ is the shear-thinning index

The default value for $n$ is 0.5. To describe $n$ as a function of $ϕ$ , Simcenter STAR-CCM+ provides the following linear relation:

Figure 15. EQUATION_DISPLAY

n = - n_{1} ϕ + n_{2}

(2436)

where $n_{1}$ and $n_{2}$ are constant values.

To prevent shear-thickening, $n$ is limited to a miximum value of 1.

Following the same approach as for the shear relative viscosity, the Morris and Boulay normal relative viscosity equation is applied to the zero and infinite shear-rate limits:

Figure 16. EQUATION_DISPLAY

η_{n 0} (ϕ) = K_{n 0} {(\frac{ϕ}{ϕ_{m 0}})}^{2} {(1 - \frac{ϕ}{ϕ_{m 0}})}^{- 2}

(2437)

where $K_{n 0}$ is the normal contact parameter at zero shear rate.

and:

Figure 17. EQUATION_DISPLAY

η_{n \infty} (ϕ) = K_{n \infty} {(\frac{ϕ}{ϕ_{m \infty}})}^{2} {(1 - \frac{ϕ}{ϕ_{m \infty}})}^{- 2}

(2438)

where $K_{n∞}$ is the normal contact parameter at infinite shear rate.

Again, the normal viscosity limits are blended to give the normal relative viscosity:

Figure 18. EQUATION_DISPLAY

η_{n} (\dot{γ}, ϕ) = \frac{τ_{y}}{\dot{γ}} + η_{n \infty} (ϕ) + (η_{n 0} (ϕ) - η_{n \infty} (ϕ)) {(1 + {(λ \dot{γ})}^{a})}^{\frac{(n - 1)}{a}}

(2439)

Yield Stress

For mono-disperse emulsions, the yield stress in Eqn. (2435) and Eqn. (2439) is [516]:

Figure 19. EQUATION_DISPLAY

τ_{y} = τ_{y 0} (\frac{σ}{a}) ϕ_{e f f} {(ϕ_{e f f} - ϕ_{m})}^{2}

(2440)

where:

$τ_{y 0}$ is a constant prefactor. The default value is 0.5, as used in [516].
$σ$ is the inter-facial tension on the surface of the droplets.
$a$ is the radius of the droplets.
$ϕ_{e f f}$ is the effective volume fraction, accounting for the presence of the thin film between droplets.
$ϕ_{m}$ is the critical packing fraction (for which $ϕ_{c} = 0.645$ for randomly close-packed hard spheres).

Emulsions can have a much larger maximum packing fraction than particle suspensions due to the deformability of liquid droplets.

For suspensions of solid particles, the yield stress develops below maximum packing, due to the formation of a network of particles in contact with one another. Several studies [[446]], [500] have revealed a power-law behavior relating yield stress to volume fraction with a variable power index (from 2 to 4). The following model also fits much of the data available for suspension yield stress:

Figure 20. EQUATION_DISPLAY

τ_{y} = τ_{y 0} {(\frac{ϕ}{ϕ - ϕ_{m y}})}^{2}

(2441)

where:

$τ_{y 0}$ is a constant prefactor of order unity
$ϕ_{m y}$ is the yield maximum packing.

Osmotic Pressure

Osmotic pressure in Eqn. (2426) is due to the particle-excluded volume and can be derived from the stress due to interparticle forces. $Π_{o}$ is the osmotic pressure of the particles given by Mewis and Wagner [519]:

Figure 21. EQUATION_DISPLAY

Π_{o} = n k_{B} T (\frac{1 + ϕ + ϕ^{2} - ϕ^{3}}{{(1 - ϕ)}^{3}})

(2442)

where:

$n$ is the particle density
$k_{B}$ is the Boltzmann constant
$T$ is temperature.

Simcenter STAR-CCM+ employs the hard sphere osmotic pressure, which defines a step function when the suspended particles come into contact:

Figure 22. EQUATION_DISPLAY

Π_{hs} = {\begin{array}{l} Π_{o} (ϕ), & if ϕ < ϕ_{m} \\ Π_{\max}, & if ϕ \geq ϕ_{m} \end{array}

(2443)

where $Π_{o} (ϕ)$ is given by Eqn. (2442) and $Π_{\max}$ is the user-specified maximum osmotic pressure.

When the dispersed phase volume fraction is above maximum packing, the osmotic pressure is given a large maximum osmotic pressure value to represent a hard sphere interaction that repels the particles from packing beyond the maximum packing limit.

For an emulsion the hard sphere repulsion reduces to the inverted osmotic pressure above the inversion volume fraction:

Figure 23. EQUATION_DISPLAY

Π_{hs} = {\begin{array}{l} Π_{o} (ϕ), & if ϕ < ϕ_{m} \\ Π_{\max}, & if ϕ_{m} \leq ϕ \leq ϕ_{i} \\ Π_{o} (1 - ϕ), & if ϕ > ϕ_{i} \end{array}

(2444)

Suspension and Emulsion Drag

Drag models for particles in a liquid are functions of the single particle Reynolds number. In turn, the particle Reynolds number is a function of the continuous liquid dynamic viscosity. Numerous experimental studies [440] correlate drag models with parameters of single phase Generalized Newtonian models. Simcenter STAR-CCM+ employs a modified Schiller-Naumann model where the continuous phase dynamic viscosity $μ_{c}$ is replaced by $μ_{c} \cdot η_{r}$ . This combined viscosity reflects the viscosity of the mixture rather than the viscosity of the dilute continous phase.

The inversion volume fraction $ϕ_{i}$ is defined within the emulsion drag model.