Numerical study of chain conformation on shear banding using diffusive Rolie-Poly model

Abstract

Shear-banding phenomenon in the entangled polymer systems was investigated in a planar Couette cell with the diffusive Rolie-Poly (ROuse LInear Entangled POLYmers) model, a single-mode constitutive model derived from a tube-based molecular theory. The steady-state shear stress σ _s was constant in the shear gradient direction while the local shear rate changed abruptly, i.e., split into the bands. We focused on the molecular conformation (also calculated from the Rolie-Poly model) around the band boundary. A band was found also for the conformation, but its boundary was much broader than that for the shear rate. Correspondingly, the first normal stress difference (N ₁) gradually changed in this diffuse boundary of the conformational bands (this change of N ₁ was compensated by a change of the local pressure). For both shear rate and conformation, the boundary widths were quite insensitive to the macroscopic shear rate but changed with various parameters such as the diffusion constant and the relaxation times (the reptation and the Rouse times). The broadness of the conformational banding, associated by the gradual change of N ₁, was attributed to competition between the molecular diffusion (in the shear gradient direction) and the conformational relaxation under a constraint of constant σ _s.

Appendix: Calculation details in simulation

The finite element method was employed to discretize the governing Eqs. 1–6 with stabilizing schemes for viscoelastic fluids such as DEVSS-G (Liu et al. 1998) and SUPG (Brooks and Hughes 1982). We reformulate the continuity Eq. 1 and the momentum Eq. 2 with DEVSS-G scheme (Liu et al. 1998) into the following weak form:

$$\left\langle {\phi \,;\nabla \cdot {\rm {\bf u}}} \right\rangle =0, $$

(A1)

$$\begin{array}{rll} &&{\kern-8pt} -\left\langle {\nabla \psi \,;p{\rm {\bf I}}} \right\rangle +\left\langle {\nabla \psi \,;\eta (\nabla {\rm {\bf u}}+ \nabla{\rm{\bf u}}^T)-\eta \phi_p ({\rm {\bf G}}^T+{\rm {\bf G}})} \right\rangle \\ &&{\kern6pt} +\, \left\langle{\nabla \psi \,;{\mathbf{\sigma}}_p } \right\rangle =0, \end{array} $$

(A2)

$$ \left\langle {\phi \,;{\rm {\bf G}}-\nabla {\rm {\bf u}}^T} \right\rangle =0, $$

(A3)

where ϕ and ψ are linear and quadratic shape functions, respectively, and $\langle$;$\rangle$ denotes integral along the finite elements. Variables such as p, G (the velocity gradient tensor), σ _p are approximated in terms of linear shape functions, while u is discretized with quadratic shape function.

We also employed the matrix logarithm (Hulsen et al. 2005) to enhance the numerical stability of calculation. The conformation tensor C can be diagonalized with the relationship C = R · c · R ^T, where R is a matrix composed of the eigenvectors of C and the diagonal tensor c have the corresponding eigenvalues c _i as its components. We can replace the C-based constitutive model with the logarithm tensor based formulation. Thus, we dealt with the evolution equation of ${\rm {\bf s}}=\log {\rm {\bf c}}=\sum\limits_{i=1}^2 {\log (c_i){\rm {\bf n}}_i {\rm {\bf n}}_i =\sum\limits_{i=1}^2 {s_i {\rm {\bf n}}_i {\rm {\bf n}}_i } }$, with s, s _i , and n _i being the logarithm tensor in the principal frame, the eigenvalue of the logarithm tensor, and the principal direction conjugated with the eigenvalues c _i of C. The time derivative of s for the Rolie-Poly model can be written as

$$\begin{array}{rll} \dot{\mbox{\bf s}}&=&\sum\limits_{i=1}^2 \left( 2G_{ii} -\frac{1}{c_i }\left( {\frac{1}{\tau_d }(c_i -1)} \right)\right.\\ &&-\frac{2\left( {1-\sqrt {2/(c_1 +c_2 )} } \right)}{\tau_R }\\ &&\left.\times\left( {c_i +\beta_{\rm{CCR}} \left( {\frac{c_1 +c_2 }{2}} \right)^\delta (c_i -1)} \right)\frac{1}{c_i } \right){\rm {\bf n}}_i {\rm {\bf n}}_i \\ &&+\mathop{\sum\limits_{i=1}^2 {\sum\limits_{j=1}^2}}\limits_{i \neq j} {\frac{s_i -s_j }{c_i -c_j }\left({c_j G_{ij} +c_i G_{ji}} \right){\rm {\bf n}}_i {\rm {\bf n}}_j \,.} \end{array} $$

(A4)

Here, G _ij is the components of the velocity gradient tensor in the principal frame. Consequently, the constitutive model with the diffusive term is described by the logarithm tensor S in the global frame as

$$ \frac{\partial {\rm {\bf S}}}{\partial t}+{\rm {\bf u}}\cdot \nabla {\rm {\bf S}}=\dot{\mbox{\bf S}}+D\nabla^2{\rm {\bf S}}, $$

(A5)

where ${\dot{\mbox{\bf S}}}$ is the tensor transformed from $\dot{\mbox{\bf s}}$ through the matrix diagonalization; $\dot{\mbox{\bf S}}={\rm {\bf R}}\cdot \dot{\mbox{\bf s}}\cdot {\rm {\bf R}}^T$. The discrete form of Eq. A5 with SUPG scheme (Brooks and Hughes 1982) can be written as

$$\begin{array}{rll} &&{\kern-6pt} \left\langle {\phi +\phi^s;\frac{{\rm {\bf S}}^{n+1}-{\rm {\bf S}}^n}{\Delta t}+{\rm {\bf u}}^{n+1}\cdot \nabla {\rm {\bf S}}^{n+1}} \right\rangle \\ &&{\kern6pt} =\left\langle {\phi +\phi^s;\dot{\mbox{\bf S}}^n+D\nabla^2{\rm {\bf S}}^n} \right\rangle . \end{array} $$

(A6)

Here, ϕ ^s is the element-wise upwinding shape function, $\phi ^s=\alpha \left({\left| {{\rm{\bf u}}_c \cdot {\rm{\bf h}}} \right|} \right)/\left({2{\rm {\bf u}}_c \cdot {\rm {\bf u}}_c }\right)$, u _c is the velocity vector at center node of an element, and h is a characteristic size of the element. Following previous literatures (Baaijens 1998; Chung et al. 2008; Kim et al. 2004; Ramirez and Laso 2005), we utilized the streamline upwinding coefficient α = 2 in Eq. A6. The superscripts n and n + 1 appearing in Eq. A6 denote the present and the next time steps, respectively.

The numerical solution of Eq. A6 was transformed into the principal frame through a relationship s = R ^T·S · R to obtain the conformation tensor C( = R · c · R ^T = R · e ^s · R ^T). Finally, the stress tensor for the polymeric component, σ _p , was calculated by Eq. 5. Then, the set of the desired variables, G, u, and p, was obtained after solving the coupled Eqs. A1–A3 at every time step.