Perspectives on shear banding in complex fluids

Abstract

In this review, I present an idiosyncratic view of the current state of shear banding in complex fluids. Particular attention is paid to some of the outstanding issues and questions facing the field, including the applicability of models that have “traditionally” been used to model experiments; future directions and challenges for experimentalists; and some of the issues surrounding vorticity banding, which has been discussed theoretically and whose experiments are fewer in number yet, in many ways, more varied in character.

Appendix

Models that exhibit shear banding

JS model

In the JS model, the total stress is assumed to be divided as in Eq. 3b, and the viscoelastic stress $\boldsymbol{\Sigma}$ is assumed to obey (Johnson and Segalman 1977; Olmsted et al. 2000)

$$\stackrel{\blacklozenge}{\boldsymbol{\Sigma}}+\frac{1}{\tau} \boldsymbol{\Sigma} = 2 \frac{\mu}{\tau}\boldsymbol{D} +{\mathcal D} \nabla^2 \boldsymbol{\Sigma},$$

(A1)

where

$$\stackrel{\blacklozenge}{\boldsymbol{\Sigma}} = (\partial_t + \mathbf{v}\cdot \nabla) \boldsymbol{\Sigma}+(\boldsymbol{\Omega}\boldsymbol{\Sigma} - \boldsymbol{\Sigma} \boldsymbol{\Omega})-a(\boldsymbol{D}\boldsymbol{\Sigma} + \boldsymbol{\Sigma}\boldsymbol{D})$$

(A2)

is the Gordon–Schowalter derivative (Larson 1988), τ is a relaxation time, the “polymer” viscosity μ determines a modulus G = μ/τ, and $\boldsymbol{\Omega} =\tfrac12\left[ \nabla \mathbf{v} - (\nabla \mathbf{v})^T \right]$. The total stress comprises the viscoelastic stress of the DJS model and a Newtonian contribution, according to Eq. 3b, and the viscosity ratio ε ≡ η/μ controls the balance between the two stresses. The “slip parameter” a, which describes nonaffine stretch of the dumbbell with respect to the extension of the flow, allows for a nonmonotonic constitutive curve for 0 < |a| < 1 and ε < 1/8.

The “diffusion” term $\mathcal D\nabla^2\boldsymbol{\Sigma}$ describes nonlocal relaxation of the viscoelastic stress and is necessary to describe strongly inhomogeneous flow profiles (Olmsted et al. 2000). Because of this term, the steady shear banding state obeys a spatial differential equation, which must be solved subject to boundary conditions specified at the walls of the flow cell. The solvability condition for a stationary interface leads to a unique total shear stress plateau for imposed average shear rates in the nonmonotonic portion of the constitutive curve (Lu et al. 2000). The characteristic width ℓ of the interface between shear bands is given by $\ell=\sqrt{{\mathcal D}\tau}$.

Yuan has proposed an alternative form for a diffusion term, which is similar to the “curvature viscosity” term proposed by Dhont (1999). In this version, spatial gradients in the local shear rate contribute to stress relaxation (Yuan 1999)

$$\stackrel{\blacklozenge}{\boldsymbol{\Sigma}}+\frac{1}{\tau} \boldsymbol{\Sigma} = 2 \frac{\mu}{\tau}\boldsymbol{D} -{\mathcal D} \nabla^2 \boldsymbol{D}.$$

(A3)

The DJS model has been coupled to concentration degrees of freedom in several different guises (Cook and Rossi 2004; Fielding and Olmsted 2003a, b, c; Rossi 2006; Yuan and Jupp 2002).

Liquid crystal hydrodynamics

A suspension of long rigid rods undergoes an isotropic-to-nematic phase transition at a sufficiently high concentration. Flow can induce the nematic phase, and the dynamical equations of motion support shear banding. In this case, the microstructural stress is a function of the nematic order parameter tensor $\boldsymbol{Q}=\langle\boldsymbol{\hat{u}}\boldsymbol{\hat{u}}\rangle-\tfrac13\boldsymbol{I}$, where the unit vector $\boldsymbol{\hat{u}}$ denotes the orientation of a rigid rod. There have been many essentially equivalent independent derivations of the hydrodynamic equations of motion, both phenomenologically and from a molecular point of view. Perhaps the most precise derivation is the recent one of Stark and Lubensky (2003), based on microscopic Poisson Brackets. The general form is as follows (Doi 1981; Edwards et al. 1990; Hess 1975; Kuzuu and Doi 1983; Olmsted and Goldbart 1990):

$$\stackrel{\bullet}{\boldsymbol{Q}} = \frac{1}{\zeta_{\scriptscriptstyle Q}}\,\stackrel{\circ}{\boldsymbol{H}} (\boldsymbol{Q}) + \boldsymbol{\Lambda}\boldsymbol{:}\boldsymbol{D},$$

(A4)

where

$$\stackrel{\bullet}{\boldsymbol{Q}} = (\partial_t + \mathbf{v}\cdot \nabla) \boldsymbol{Q}+(\boldsymbol{\Omega} \boldsymbol{Q} - \boldsymbol{Q}\boldsymbol{\Omega})$$

(A5)

is the corotational derivative and $\tau_{\scriptscriptstyle Q}$ is a relaxation coefficient. The reactive fourth rank tensor $\boldsymbol{\Lambda}$ describes nondissipative evolution of the order parameter tensor. The molecular field that drives relaxation to equilibrium is

$$\boldsymbol{H} = -\frac{\delta{\cal F\/}}{\delta \boldsymbol{Q}},$$

(A6)

where the free energy functional $\cal F$ includes homogeneous and inhomogeneous (Frank elastic) terms. The stress has a viscous part $\boldsymbol{T}^v$, a symmetric elastic part $\boldsymbol{T}^{el}$, and an antisymmetric part $\boldsymbol{T}^{a}$, as follows:

$$\boldsymbol{T} = \boldsymbol{T}^v + \boldsymbol{T}^{el} + \boldsymbol{T}^{a} - p\boldsymbol{I}$$

(A7)

$$\boldsymbol{T}^v =\boldsymbol{\eta}\boldsymbol{:}\boldsymbol{D}$$

(A8)

$$\boldsymbol{T}^a =\boldsymbol{HQ} -\boldsymbol{QH}$$

(A9)

$$\boldsymbol{T}^{el} = -\boldsymbol{\Lambda}\boldsymbol{:}\,\stackrel{\circ}{\boldsymbol{H}} - \boldsymbol{\nabla}Q_{\alpha\beta}\cdot {\delta{\cal F\/}\over\delta\boldsymbol{\nabla}Q_{\alpha\beta}},$$

(A10)

where $\boldsymbol{\eta}$ is a fourth rank viscosity tensor that generally depends on $\boldsymbol{Q}$, and the term $-\boldsymbol{\Lambda}\boldsymbol{:}\,\,\stackrel{\circ}{\boldsymbol{H}}$ ensures Onsager reciprocity. Recall that $\,\stackrel{\circ}{\boldsymbol{H}}$ is the traceless-symmetric version of $\boldsymbol{H}$. The elastic stress is equivalent to the elastic stress due to Frank elasticity (de Gennes and Prost 1993), generalized to a description in terms of the nematic order parameter $\boldsymbol{Q}$ rather than the nematic director. The free energy ${\cal F}(\boldsymbol{Q})$ describes the isotropic–nematic phase transition, and the coupling to flow $\boldsymbol{\Lambda}\boldsymbol{:}\boldsymbol{D}$ in Eq. A4 perturbs the isotropic state and induces a phase transition. The choice of $\boldsymbol{\Lambda}$ is somewhat a matter of taste and has been derived from a microscopic picture by Doi et al. and by Stark and Lubensky (Doi 1981; Edwards et al. 1990; Hess 1975; Kuzuu and Doi 1983; Olmsted and Goldbart 1990; Stark and Lubensky 2003).

The simple choice $\boldsymbol{\Lambda}\boldsymbol{:}\boldsymbol{D} =\nu\boldsymbol{D}$ was chosen in an early study of the effects of flow on the isotropic–nematic phase transition (Olmsted and Goldbart 1990, 1992). From a microscopic model, Stark and Lubensky derived the following:

$$\boldsymbol{\Lambda}\boldsymbol{:}\boldsymbol{D} = (\boldsymbol{D}\boldsymbol{Q} + \boldsymbol{Q}\boldsymbol{D}) + \frac{I}{3\Delta I}\boldsymbol{D}$$

(A11)

$$\;\;\;- 2\left[\frac{1}{3}\boldsymbol{I} + \left(1 + \frac{I}{\Delta I}\right)\boldsymbol{Q} \right]\,\hbox{\rm Tr}(\boldsymbol{Q}\boldsymbol{D}),$$

(A12)

where I is the principal moment of inertia and ΔI the difference between the two moments of inertia of uniaxial rods. In the Doi model $\boldsymbol{\Lambda}$ is defined by

$$\boldsymbol{\Lambda}\boldsymbol{:}\boldsymbol{D} = (\boldsymbol{D}\boldsymbol{Q} + \boldsymbol{Q}\boldsymbol{D}) + \frac23\boldsymbol{D} - 2\left(\boldsymbol{Q}+\frac13\boldsymbol{I}\right)\,\hbox{\rm Tr}(\boldsymbol{Q}\boldsymbol{D}).$$

(A13)

This model was coupled to concentration to explore the effects of shear flow on the isotropic–nematic phase transition in suspensions (Olmsted and Lu 1997, 1999a). In the Beris–Edwards model, the same choice is made with an additional prefactor ξ that can be related to the aspect ratio of the rigid rods (Edwards et al. 1990; Kuzuu and Doi 1983) and is essentially the same as the slip parameter a that appears in the Gordon–Schowalter derivative; this was used by Denniston and Yeomans in a lattice Boltzmann study (Denniston et al. 2001). Hess and Kröger used the choice (Rienacker et al. 2002a)

$$\boldsymbol{\Lambda}\boldsymbol{:}\boldsymbol{D} = a \left[\boldsymbol{D}\boldsymbol{Q} + \boldsymbol{Q}\boldsymbol{D} - \frac{2}{3}\boldsymbol{I}{\rm Tr}\left({\boldsymbol{QD}}\right)\right] + \alpha\boldsymbol{D}$$

(A14)

in their study of chaotic dynamics in nematic liquid crystals, where α was related to a ratio of Leslie–Erickson coefficients. A similar choice was made in another study of chaotic dynamics (Das et al. 2005). Other microscopically derived constitutive equations include those of Dhont and Briels (2003a, b) and of Calderer et al. (2004); the former is notable for its incorporation of inhomogeneous flows.

Rolie–Poly model

The original DE theory for entangled polymer dynamics predicted a nonmonotonic constitutive curve and an instability that could lead to shear banding (Doi and Edwards 1989; McLeish 1987; McLeish and Ball 1986). This early pioneering model neglected an important physical source of stress under flow, which has been recently understood and modeled: the enhanced release of polymer entanglements due to convection (Marrucci 1996). This increases the polymer stress and, for sufficiently strong convected constraint release (CCR), can "cure" the DE instability. CCR and tube stretching were incorporated in Milner et al. (2001) to describe polymer melts and wormlike micelles and later simplified to the following differential version, called the Rolie–Poly model (Likhtman and Graham 2003):

$$\stackrel{\blacktriangledown}{\boldsymbol{\Sigma}} + \frac{1}{\tau}\boldsymbol{\Sigma} = 2 G \boldsymbol{D} - \frac{2}{3} \boldsymbol{\Sigma}:\nabla\mathbf{v} \left[ \boldsymbol {I} + (1+\beta) \frac{\boldsymbol{\Sigma}}{G} \right] + {\mathcal D} \nabla^2 \boldsymbol{\Sigma}.$$

(A15)

The CCR parameter β (0 ≤ β ≤ 1) is proportional to the frequency of the release of polymer entanglement constraints due to convection by the flow. Likhtman and Graham (2003) used β = 1 to model a well-entangled polymer melt without a constitutive instability. For small enough β, a constitutive instability results, akin to the original DE instability (Adams et al. 2007). The stress diffusion included above did not appear in the original formulation. In the version of the Rolie–Poly model used in Eq. A15, the tube length is assumed to be relaxed (“nonstretching”).

Interestingly, the limit β = 0 corresponds to a differential version of the Cates model (Cates 1990) of wormlike micellar solutions. This constitutive equation can be obtained by rewriting the original integral equation as a differential equation and then using a decoupling approximation to remove fourth-order moments (Adams et al. 2007; McLeish 1997).