Non-equilibrium steady states of entangled polymer mixtures under shear flow

Advances in Mechanical Engineering 2015, Vol. 7(6) 1–10

Ó_{The Author(s) 2015} DOI: 10.1177/1687814015591923 aime.sagepub.com

Non-equilibrium steady states of

entangled polymer mixtures under

shear flow

Xiao-Wei Guo, Wen-Jing Yang, Xin-Hai Xu, Yu Cao and Xue-Jun Yang

Abstract

By solving the full equations of an extended two-fluid model in two dimensions, we give the first numerical study reveal-ing non-equilibrium steady states in sheared entangled polymer mixtures. This research provides answers for some fun-damental questions in sheared binary mixtures of entangled polymers. Our results reveal that non-equilibrium steady states with finite domain size do exist, and apparent scaling exponents L_k;g_1_:05

andL_?;g_1

are found over six decades of shear rate. Since the wall effects get involved in our simulations, the dependence of average domain size on system size cannot be strictly eliminated. In addition, as an obvious influence of viscoelasticity, the polymer viscosityh_p

appears to induce linear translation of the fitted lines. Through two-dimensional numerical simulations, we show the detailed dynamic evolution of microstructure in binary polymer mixtures with asymmetric composition under shear flow. It is found that the phase patterns are significantly different from symmetric fluids studied previously. Finally, we also identify the importance of wall effects and confirm the irreplaceable role of inertia for a non-equilibrium steady state.

Keywords

Non-equilibrium steady states, Flory–Huggins–Rolie–Poly model, polymer mixtures, phase separation

Date received: 27 March 2015; accepted: 20 May 2015

Academic Editor: Cheng-Xian Lin

Introduction

The non-equilibrium dynamics play an essential role in the applications of numerous areas ranging from soap manufacture to biological systems;1 in fact, most sys-tems found in nature are not in thermal equilibrium. One of the important problems in such systems is to understand the morphological and rheological proper-ties of the fluids undergoing a steady shear flow after a temperature quench. Early experimental2–6and numeri-cal7–12 research revealed the basic features of non-equilibrium phase separation in binary fluids, especially for the spinodal decomposition dynamics. A number of intriguing effects induced by shear were observed; typi-cal results included the highly elongated domains in very weak shear and a string phase in steady state under strong shear. Different from the power law of

the form L(t)_;ta _{in zero-shear systems, the domains}

become elongated in the flow direction under shear. The relation Lx;g_tLy was predicted by theories in which the velocity field did not fluctuate.13–15 It is found that the velocity strongly fluctuates in real fluids and the hydrodynamic effects may balance either inter-facial and viscous or interinter-facial and inertial forces. Nevertheless, it remained unclear whether the shear

State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, China

Corresponding author:

Wen-Jing Yang, State Key Laboratory of High Performance Computing, National University of Defense Technology, No. 109 Deiya Street, Changsha, Hunan 410073, China.

Email: wenjing.yang@nudt.edu.cn

effects interrupted domain coarsening and restored a non-equilibrium steady state independent of the system size, until Stansell and colleagues16,17 answered this question by numerical study. Using lattice Boltzmann approach, they provided convincing evidences for the non-equilibrium steady states with a finite domain size and predicted a power law dependence of characteristic length L_k and L? on shear rate as Lk;g_2=3 and L_?;g_3₌4

. After that, Fielding18presented an in-depth numerical study on the role of inertia in equilibrium steady states and confirmed that the non-equilibrium steady states free of finite-size effects only existed in the systems with inertia.

All the previous studies on non-equilibrium steady states focused on the binary viscous fluids with sym-metric composition and ignored the role of viscoelasti-city. Since all the materials exhibit some viscoelastic response in nature and many complex fluids such as polymer mixtures display significant viscoelastic effects, it is still uncertain whether these conclusions will be maintained in viscoelastic fluids. Deep-quench experi-ments of Tanaka19 have shown that the dynamics of polymer phase transitions can be very different from those of classical binary fluid mixtures. Besides, the asymmetry between the components of polymer mix-tures can also strongly change the morphology of phase separation.19,20 Typically, in the zero-shear system, an asymmetric quench can lead to a droplet pattern, other than the bicontinuous pattern for a nearly symmetric quench.21

Up to now, some fundamental issues, such as the existence of the non-equilibrium steady states free of system size in binary viscoelastic fluid, the rationality for fitting the domain size versus the inverse shear rate to simple power low and the dynamic morphology for an asymmetric quench under shear flow, remain open questions. This article aims to answer these questions through a numerical study considering all the effects of

thermodynamics, hydrodynamics and the

viscoelasticity.

By combining the thermodynamics, hydrodynamics and viscoelasticity in a thermodynamically consistent way, the two-fluid model22–25was widely used to inves-tigate the phase separation in viscoelastic fluids. Such models could be used to replace the previous simple the-oretical frameworks for studying the non-equilibrium viscoelastic phase separation.

Different extended versions of the two-fluid model have been proposed for studying the flow-induced phase separation. Jupp and Yuan’s model (Flory–

Huggins–Johnson–Segalman (FH–JS) model)20,26,27

coupled the FH mixing free energy function and the JS constitutive model in a unified way. They numerically solved the full equations of the two-fluid model without simplifications used widely in early studies. Fielding and Olmsted28,29extended the two-fluid model using a

different approach, which still retained a diffusive stress term in the JS constitutive equation. As the two-fluid model has already accounted for fluid diffusion across streamline in a thermodynamically consistent way, there are no sound physical reasons for an ad hoc mechanical diffusive term to produce numerical results

more compatible with experimental observations.

Cromer et al.30coupled the concentration and the stress through a two-fluid approach to predict the steady-state shear-banding for a monodisperse polymer solution. As the JS constitutive model was known to provide unrealistic rheological predictions for entangled polymeric solutions, they replaced the constitutive equation with a tube theory–based model: the Rolie– Poly (RP) model.31Nevertheless, only one-dimensional calculations had been presented. Recently, we presented a unified model32(FH–RP model) based on the FH–JS framework to study the rheological instabilities in shear-banding flow. The simulation results showed that this model could capture the essential dynamic features of viscoelastic phase separation and the rheochaos reported in the literature.

In this article, we focus on studying the non-equilibrium steady states of entangled polymer mix-tures using the FH–RP model and start by introducing the governing equations and the choice of the para-meter sets, the length and time scales in sections ‘The FH–RP fluid model’ and ‘Length scale, time scale and parameter sets’, followed by the simulation results and discussions in section ‘Numerical results and discus-sions’. Section ‘Conclusion’ contains our conclusions and the future work.

The FH–RP fluid model

After Helfand and Fredrickson33 showed the crucial stress and concentration coupling by analysing the

microscopic Rouse model, Doi and Onuki22 and

Milner23derived the fundamental equations of the two-fluid model for describing the polymer solutions through a more general approach. Yuan’s and col-leagues27solved a full two-fluid model coupling the FH free energy function and the JS constitutive model. Based on the FH–JS framework, we replaced the phe-nomenological JS model by the molecular theory–based constitutive equation. Previous numerical results32have shown that this FH–RP model can capture the essential dynamic features of viscoelastic phase separation and reproduce the rheochaos without introducing extra parameters. We summarize the governing equations of the FH–RP model below.

Considering the polymer mixtures with two compo-nents, A and B. Letf_A(~r,t) and f_B(~r,t) =1_f

A(~r,t)

the components, we can get the continuity equation and the momentum balance equation of an incompressible, isothermal viscoelastic binary fluid as

~

Ñ_~v=0 _ð1Þ

and

rD~v

Dt =h0Ñ

2

~v~Ñ_p(f

AfB)~Ñm+~Ñsve(~r,t) ð2Þ

Here, the volume average velocity~vis calculated by the velocities of components as ~v(~r,t) = f_A~vA(~r,t) + f_B~vB(~r,t).pis the pressure field, the material derivative

(D~v=Dt) = (∂_~v=∂_{t) +~v}~Ñ_{~vandh0is the viscosity of the} Newtonian stress.

In equation (2), except for the viscoelastic stress term ~

Ñs_ve, the osmotic stress is another significant

contri-bution originated from the thermodynamic effects. The chemical potential differencem=m_Am_B is defined as the functional derivative of the mixing free energy with respect to local volume fraction, that is

m=dFmix½fA(~r) dfA(~r)

=dFmix½fB(~r) dfB(~r) ð

3_Þ

A first-order approximation of the mixing free energy functional may be given by the FH–de Gennes form as

Fmix½fA(~r)=

ð

d~r fmix+ G 2

½Ñ_f

A

2

ð4_Þ

and

fmix

kBT

= 1

MA

fA lnfA+ 1

MB

fB lnfB+xfAfB ð5Þ

whereG is the interfacial tension coefficient, Mi is the molecular weight of each component polymer andx is the FH interaction parameter.

Bringing the osmotic stress and the viscoelastic stress term together, we can get the following evolution equa-tion for the volume fracequa-tion

DfA(~r,t) Dt =~Ñ

f2_Af2_B

§ (~Ñma~Ñsve)

ð6_Þ

where§is a frictional coefficient andais a dimension-less coefficient given bya= (1_G0_Z0_)=(f

A+G0Z0fB).

The chemical potential difference can be calculated from equation (3), and the total viscoelastic stresss_ve

should be obtained by solving constitutive equations. Jupp and Yuan20argued that the tube velocity should be used in the viscoelastic constitutive equation. The tube velocity may be expressed in terms of the volume average velocity by

~vT=~v+f_Af_Ba(~vA~vB) ð7Þ

and

~vA~vB=fAfB

§ ½~Ñm+a~Ñsve ð8Þ

As equation (8) shows, the velocity difference between

components A and B depends on the thermodynamic

and viscoelastic forces.

To calculate the viscoelastic forces of the mixed fluid, we split the viscoelastic stress into three parts as

s_ve= X

i=AA,BB,AB

GiWi ð9Þ

where the quantityW_{iis the polymer conformation}

ten-sor, whose stress GiW_{i is parameterized by the elastic}

modulusGi. Nows_veincludes the contribution of

com-ponent A, comcom-ponent B and an interacting part. By applying the linear rule to the Rouse process and the ‘double reputation’ rule to the disentanglement process, the elastic modulus for the three parts can be expressed as

GAA=fAGAR+f

2

AGdA ð10Þ

GBB=fBGBR+f

2

BGdB ð11Þ

and

GAB=2_f

AfB

ffiffiffiffiffiffiffiffiffiffiffiffi

GA dGBd

q

ð12Þ

Corresponding rules for the relaxation times will be tAA

d =tAd,tdBB=tBd andtABd =2tAdtBd=(tAd+tBd). For the

Rouse time, there is no interacting part, thus we have tAA

R =tAR,tBBR =tBRandtABR =0.

The RP constitutive equation for the polymer con-formation tensorW

imay be written as34

DW_i

Dt =2D+ (Ñ~vT) T_W

i+Wi(Ñ~vT)

1

ti d

W_if1i_(I+W_i+f2iW_i)

ð13Þ

whereD=1₌2₍Ñ_{~vT+ (}Ñ_~vT)T_{) and the material} deriva-tive,DW_{i=Dt, is defined in terms of the tube velocity as}

(DW_{i)=(Dt) = ((}∂W_i)=(∂_{t)) +~vT}ÑW_{i; the coefficients}

involving the trace of the strain tensor are defined by

f1i= 2

ti R

1 1₊tr(

W

i)

3

12!

fori=AA,BB _ð14_Þ

and

f2i=b 1+

tr(W

i)

3

d

forW_{AB, ast}AB

R !0leading to a non-stretching limit,

31

we obtain

f1AB= 2

3 (Ñ~vT)tr(

W

i+I)

ð Þ ð16Þ

and

f2AB=b ð17Þ

In the FH–RP fluid model,brepresents the convective constraint release (CCR) magnitude coefficient anddis a negative power specifying the exponent for the relaxation due to the CCR. For simplicity, but without much loss of generality, we assume the same value forbanddfor the three parts of s_ve. By selecting parameter space for the

FH–RP model in terms ofh0,tA_R,tA_d,tB_R,tB_d,G_RA,GB_R,GA_d,

GB

d (Gi=GiR+Gid) and the shear rate, we can obtain

appropriate rheology behaviour in shear flow.

Length scale, time scale and parameter

sets

To study the phase behaviour under shear, we consider a system with the size of Lx3Ly, where Lx=2Ly

defines a rectangular simulation box. Conceptually, a shear flow of rateg_ can be applied by moving the upper wall boundary to the right direction at a constant speed

_

gLy. Since eliminating the finite-size effects requires a

large system size, we must ensure the typical domain size LLx,y. Most of the recent publications15,16,18

about the non-equilibrium phase separation tune the system size between 512 and 1024, and it is found that the finite-size effects are under control with such a system size. In this study, all simulations are done for asymmetric quenches on a Lx3Ly=10243512

mesh.

There exist many possible measures of the character-istic domain size. The structure factor based on Fourier transforms is usually used to define a characteristic domain size in zero-shear systems. While under shear the system is no longer periodic and the morphology will in general be anisotropic, we define the length scales using a gradient statistic for fA across the

simu-lation box. We define a symmetric matrix16,18,35

Dab=

lÐdxÐdy∂_af A∂bfA

Ð

dxÐ

dyf2

A

ð18Þ

the reciprocal eigenvalues of which give us two ortho-gonal length scales L_k and L_?, characterizing the long and short principal axes of the domain morphology. Here, l is the interface width, and it can be calculated by Helfand and Tagami’s self-consistent field theory.36–38 Taking into account the finite chain length, Broseta et al.39,40 predicted a broader interface and smaller

interfacial tension. According to our simulation obser-vations, Broseta’s formulation gave a more accurate description of the interfaces as

l= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b 6 _x2_ln2 1

MA + 1

MB

h i

r ð19Þ

where x is the FH interaction parameter in equation (4) and the statistical segment lengthbmay be given in terms of the interfacial tension coefficient G as

b= ffiffiffiffiffiffiffiffi 18_G

p

.41 Similarly, the interfacial tension s0 may be calculated by

s0=br0kBT ffiffiffi x 6 r

1 p 2

12_x 1

MA +

1

MB

ð20_Þ

where r0 represents the uniform number density of monomers. The physics of binary fluid demixing can be non-dimensionalized via length scaleL0 and time scale

T0,16both of which depend on the fluid densityr, the viscosityhand the interfacial tensions0. Here, consid-ering the existence of viscoelastic stresss_ve, we replace hby the total viscosity

h_t=h0+h_p ð21Þ

thus the characteristic length and time can be given by

L0= h2

t

rs0 ð

22Þ

and

T0= h3

t

rs2 0

ð23Þ

As described in equation (2), h0 is the Newtonian viscosity contributed from the solvent or fast dynamics. The polymer zero-shear-rate viscosity hp, governed by

the viscoelastic stress of the polymer mixtures, can be expressed in terms of the elastic modulusGiand the dis-entanglement relaxation timesti

das

h_p= X

i=AA,BB,AB

Giti_d ð24Þ

Taking into account the mean effects of the composi-tion field, here we assume that the polymer zero-shear-rate viscosity only depends on the initial (mean) value off_A, thus the initial values of the composition filedf0_A can be substituted into equations (10)–(12) to calculate h_pvia equation (24).

In our research on non-equilibrium steady states, the shear rate and domain size are non-dimensionalized as

_

sizedx=dy=1, then in any given run, our aim is to

ensure a separation of the length scales

dx,ylLk,?Lx,y ð25Þ

yet we realize it is difficult to restrict theL_? to signifi-cantly larger than mesh spacing with asymmetric com-positions, and we will discuss this situation later in the next section.

For simplicity but without much loss of generality, we choose the following parameters: MA=MB=1, kBT=1:3, G=1, §=0:1, r0=1, b=0:2 and d=0_:5_{. Thus, the equilibrium phase diagram is} always symmetric with respect to f0_A=0_:5_{. Figure 1} shows the equilibrium phase diagram in thex=f0_A para-meter space of an FH fluid. ∂_fmix=∂_f

A=0 and ∂2_fmix=∂_f2

A=0 define the binodal and the spinodal

curves, respectively. Below the binodal curve shown as a broken line in the figure, the system favours a homo-geneous state in equilibrium. For (f0_A,x) above the spi-nodal (solid) line, homogeneous states are unstable and systems will spontaneously be decomposed into two coexisting phases. The critical point is at x=2_:0_and f0_A=0_:5 _{for this parameter set. Focusing on the} non-equilibrium phase separation of binary polymer mix-tures with asymmetric composition, we choose the (f0_A,x) in the two-phase region as f0_A=0_:3 _and x=3_:0_.

As the bold cross marked in the figure shows, this point is far away from the homogeneous region at equi-librium, thus the spontaneous decomposition will take place with infinitesimal amplitude non-local fluctua-tions of composition. A Gaussian noise with an inten-sity of 103 _{is superimposed on the initial uniform} composition field f0

A in all simulations used for this

article.

All the data sets used in our simulation runs are listed in Table 1. These runs are ordered by the values of T0. Ignoring the inertia (r=0), R000 is a special case which will be used to check the role of inertia in binary viscoelastic fluids. The others could be classified into three collections with significantly different poly-mer zero-shear-rate viscosityhp. A large value of

poly-mer viscosity h_p=22_:25 _{is set for the cases named}

R0_{xx. R}3_{xx and R}1_{xx are allocated a smaller value,} h_p=5_:5887 _{and h}

p=1:2425, respectively. Since high

shear rates give inaccuracies as described in previous publications,16and low values of shear rates give unac-ceptably long run times, we tune the shear rates in an acceptable range as1_:95₃103

g_2_:93₃102

Numerical results and discussions

We used a pressure implicit with splitting of operator (PISO)-based iterative solution algorithm to solve the two-fluid model. This algorithm has been well tested in a numerical study for the dynamics of polymer solu-tions in contraction flow.42 Recently, it is adopted to study the shear-banding flows with a macroscopic two-fluid model.32 To discretize the governing equations, we implemented the algorithms based on an Open Source computational fluid dynamics (CFD) toolbox released by the OpenCFD Ltd, named OpenFOAM. The equations are discretized through finite volume method, which locally satisfied the physical conserva-tion laws through computing each term of the govern-ing equations by integral over a control volume. For spatial discretization terms, the second-order Gauss MINMOD and Gauss Linear scheme are applied and the temporal terms are discretized using a simple Euler scheme. Finally, these equations will reduce to linear systems, thus using the iterative solvers predefined in OpenFOAM we can get the solutions of the equations at every time step. Typical solvers in the toolbox include the preconditioned conjugate gradient (PCG) and preconditioned biconjugate gradient (PBiCG) methods. For details, please refer to the OpenFOAM manual.

First, we will turn to the question that whether or not a non-equilibrium steady state with finite domain size exists in binary viscoelastic polymer mixtures. For all the parameter sets in Table 1 from R001 to R103, we find the domain lengths to saturate at long times with temporal fluctuations around constant mean val-ues. Some typical runs are shown in Figure 2; both time series (except for the zero-inertia case) forL_k and L?

with g_=1_:953₃102

reach a length-scale saturation in a regime that seems free of any finite-size effects. A number of additional tests are performed by changing the system size in the range of Lk=5122048 and L?=2561024. Other tests with different shear rates 0 0.2 0.4 0.6 0.8 1

0 1 2 3 4 5 6

φ A 0

χ

two−phase

homogeneous binodal spinodal

Figure 1. Equilibrium phase diagram of Flory–Huggins free energy with parametersMA=MB=1:0 andkBT=1:3. The homogeneous and two-phase fluid areas are bordered by binodal and spinodal lines, and between the two is a metastable region. The bold cross marks the parameters used in this article:

between1_:95₃103

and2_:93₃102

also do not change the conclusions we report in this article.

With L_k,?Lx,y=6, we may conclude from these

tests that the finite-size effects are fully under control. We tested several different mesh spacings with this model, which should provide us some confidences. Regarding that no any previous numerical research about the non-equilibrium steady states sets up an asymmetric composition field, we hope to give an impression with asymmetric composition field in non-equilibrium steady states.

Two typical snapshots of the steady-state composi-tion field are presented in Figure 3 for parameter set R005. It is found that the morphologies are signifi-cantly different from the 50:50 symmetric quenches.16,18 Although the finite-size effects appeal to be under con-trol, the wall effects apparently get involved in our

simulations. As shown in Figure 3, there are notable differences between the moving (top) wall and the static (bottom) wall. The domains near the moving wall are elongated into some short and smooth band structures, while the system favours a much smaller domain length with irregular droplets around the static wall boundary. Since our characteristic domain sizeL_k,?only measures the average lengths across the simulation box, it cannot be concluded that such a steady state is independent of the system size. Additional tests with different system sizes support our supposition: the characteristic domain size does increase as the system size changes from 1024₃512 _to 2048₃1024_{. Thus, we conclude that a} non-equilibrium steady state with finite domain size does exist in binary viscoelastic polymer mixtures, but the dependence of average domain size on system size cannot be strictly eliminated in our simulations.

Table 1. Parameter sets used in simulations, along withL0andT0.

Name h₀ h_p r s0 l GA tAd GB tBd L0 T0

R000 0.1 22.25 0 1.7616 7.2642 2.5 30 2.5 5.0 – –

R001 0.1 22.25 1 1.7616 7.2642 2.5 30 2.5 5.0 283.5618 3597.642 R002 1.0 22.25 10 1.7616 7.2642 2.5 30 2.5 5.0 30.68588 404.9993 R003 0.1 22.25 100 1.7616 7.2642 2.5 30 2.5 5.0 2.835618 5.97642 R006 0.5 22.25 260 1.7616 7.2642 2.5 30 2.5 5.0 1.13001 14.59339 R005 1.0 22.25 1000 1.7616 7.2642 2.5 30 2.5 5.0 0.306859 4.049993 R007 1.0 22.25 2000 1.7616 7.2642 2.5 30 2.5 5.0 0.153429 2.024996 R301 0.05 5.5887 30 1.7616 7.2642 2.5 10 2.5 1.0 0.591013 1.875007 R302 0.05 5.5887 50 1.7616 7.2642 2.5 10 2.5 1.0 0.354608 1.125004 R303 0.05 5.5887 100 1.7616 7.2642 2.5 10 2.5 1.0 0.177304 0.562502 R304 0.05 5.5887 1000 1.7616 7.2642 2.5 10 2.5 1.0 0.01773 0.05625 R102 0.01 1.2415 10 1.7616 7.2642 0.5 10 0.5 1.0 0.087489 0.061656 R105 0.01 1.2415 50 1.7616 7.2642 0.5 10 0.5 1.0 0.017498 0.012331 R103 0.01 1.2415 100 1.7616 7.2642 0.5 10 0.5 1.0 0.008749 0.006166

0 50 100 150 200

0 20 40 60 80 100 120 140 160 180

˙ γt

L

(

t

)

0 50 100 150 200

0 1 2 3 4 5

˙ γt L⊥

(

t

)

(a) (b)

Figure 2. Plots ofL_kandL_?for various runs withg_=1:9533102_{. Data sets for decreasing average values ofL}

kcorrespond to

More detailed results of the phase transition from an initial homogeneous state are presented in Figure 4. Atg_t=4_:85_{, the system has formed a two-phase state.} The droplet patterns are rapidly stretched along the flow direction in the early diffusive regime. From

_

gt=4_:85 _{to g_t=}9_:75_{, the morphology is becoming} coarser and coarser through diffusive mechanism. At late times g_t=48_:82 _{and g_t=}97_:3_{, the structures of} composition fieldf_A are highly elongated, and the con-tinuous stretching and breaking lead to slower domain growth than the diffusive regime. After a long time simulation, the system reaches a dynamical steady state similar to Figure 3. It is seen that the characteristic domain length of R001 is much smaller than R005, and wall effects still play an important role in the phase separation process. Actually, such steady states are observed in all our simulations. Phase transitions com-ing from visual observations for composition fieldf_A suggest that the length saturation stems from hydrody-namic balance between the stretching, breaking and the coalescence of domain.

Another important question of non-equilibrium steady state for binary viscoelastic fluids under shear is about the dependence of domain lengthLon the shear

Figure 3. Snapshots of the steady-state composition fieldf_A

for R005 at strain: (a)gt_ =177:71 and (b)gt_ =234:37.

Figure 4. Snapshots of the composition fieldf_Aat various timestfor an initially homogeneous viscoelastic binary fluid. Results are obtained under applied shear rateg_=1:9923102_{with parameter set R001: (a)gt_ =4:85, (b)gt_ =9:75, (c)gt_ =48:82, (d)}

_

rate. Previous research predicts a simple power law as

L;g_a_{, but the value of exponent a is not yet agreed}

upon. Some early studies2,8suggested a value between 1/4 and 1/3, while, even with some caution, recent numerical research16–18 proposed the value a=3₌4_. Considering the role of viscoelasticity, the dimension-less scaling plots of steady-state length scales L_k,?

against shear rate are presented in Figure 5, for a series of runs in Table 1 under shear rate g_=1_:992₃102

.

L_k,? are obtained as the temporal means of the time

series L_k,?(t), after discarding data for g_t\100. It is found that (a) the data can be fitted by simple power law and apparent scaling exponents are estimated as

Lk;g_1:05

andL?;g_1

over six decades of shear rate, with correlation coefficient R.0_:99_{; (b) the polymer} zero-shear-rate viscosity h_p induces linear translation: the lines translate to the right ashpdecreases. This

phe-nomenon has not been observed in pure viscous fluids and may be attributed to the presence of viscoelastic stress.

The main distinction between our work and the pre-vious research is that the viscoelastic effects introduced by the constitutive equations have been considered. As shown in Table 1, the viscoelastic stress is governed by the modulus Gi and the disentanglement relaxation timesti

d. While the extra parameterhpcan also be

com-puted byGi andti

d according to equation (24),

chang-ing the modulus and the relaxation times will tune both the elastic forces and the total viscosity of the fluid. Focusing on the viscoelastic effects, here we tested

different parameters to generate the fluid systems with

hp ranging from low (hp=1:2415) to high

(h_p=22_:25_).

For the scaling exponents, the main difference between our result and previous studies of Newtonian fluids is the larger value a. One possible explanation may be that the total viscosity of the viscoelastic fluid in the simulations is much higher than the pure Newtonian viscosity. Since the numerical studies of zero-shear system using Boltzmann methods predict different dependencies of the characteristic domain size upon time, especially, in the viscous dominated regime

LV;pt, which is larger than the diffusive dominated regime LD;t1=3

and the inertia regimeLI;t2=3

. By sub-stitutingt=g_1

in the shear-free coarsening plot, our results for binary polymer mixtures are consistent with the scaling exponents of viscous regime asa_’1_.

Finally, we also test the zero-inertia systems by set-tingr=0_{. Typical parameters are included in Table 1,} named R000. From the plot of L_k(t) shown in Figure 2(a), the domain length appears to increase indefinitely as in zero-shear systems. Domain morphologies of R000 underg_=1_:992₃102_{are presented in Figure 6.} Initially, the scattered droplets are rapidly elongated along the flow direction; afterg_t=100_{, the domain has} formed into extremely long string-like patterns, which is similar to the experimental observations under strong shear.2,4 We also test some other parameter sets with r=0_{and no evidence of non-equilibrium steady states} with finite domain size is found. Our results confirm the irreplaceable role of inertia for a non-equilibrium steady state.

Conclusion

Non-equilibrium steady states in pure viscous binary fluid have been extensively studied, and many impor-tant conclusions are made through numerical studies with simple models ignoring the viscoelastic stress. We have numerically studied the phase separation in binary polymer mixtures with an extended two-fluid model: the FH–RP fluid model, which directly couples the thermodynamics, hydrodynamics and viscoelasticity. By solving the full equations in two dimensions, we give the first numerical study revealing non-equilibrium steady states in sheared viscoelastic fluids.

This research provides answers for some fundamen-tal questions. First, in binary polymer mixtures, our results reveal that non-equilibrium steady states with finite domain size do exist, but the dependence of aver-age domain size on system size cannot be strictly elimi-nated in our simulations. Second, we find apparent scaling exponents Lk;g_1:05

and L?;g_1

over six decades of shear rate. In addition, the polymer viscosity hp appears to induce linear translation of the fitted 10−2 100 102 104

10−2 100 102 104

1/( ˙γT0)

L

,

⊥

/L

0

α= 1.0052

α= 1.0427

α= 1.0449

α= 0.9909

α= 1.0721

α= 1.0021

Figure 5. Dimensionless scaling plots of lengths versus shear rate. Error bars show the standard deviation. Solid lines: power law fits to the crosses, suggestingL_k;g_1:05. Dashed (red) lines: power law fits to the circles, suggestingL_?;g_1_(L

kandL?

represent two orthogonal length scales characterizing the long and short principal axes of the domain morphology,

lines. It is an obvious evidence of the important role of viscoelasticity. Third, our two-dimensional (2D) simu-lation results show the dynamic evolution of micro-structure in binary polymer mixtures with asymmetric composition under shear flow. It is found that the phase patterns are significantly different from sym-metric fluids studied previously. Finally, we also iden-tify the importance of wall effects and confirm the irreplaceable role of inertia for a non-equilibrium steady state. Apparently different domain characteristic lengths are observed near the moving wall and the sta-tic wall, which also increase the dependence of average domain size on system size in steady states. The inertia is still playing an essential role in the non-equilibrium steady states in viscoelastic polymer mixtures, since no such steady state with finite domain size is found in inertialess systems.

In future work, we aim to investigate the non-equilibrium phase separation in large-scale three-dimensional (3D) simulations. Given a more realistic free energy function, our model can accurately predict the dynamic behaviours of the systems that are not in thermal equilibrium. We will further extend this approach with the latest equations which would pro-vide a much more realistic description of polymer free energy. We hope the realistic computational model pre-sented here could provide valuable guidance to new experimental work and could also be used to quantita-tively study some outstanding problems by appropri-ately selecting the parameters to mimic the real physical systems.

Acknowledgements

The authors would like to thank Prof. Xue-Feng Yuan for constructive comments and helpful discussions.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos 61221491, 61303071 and 61120106005) and Open fund from State Key Laboratory of High Performance Computing (Nos 201303-01, 201503-01 and 201503-02).

References

1. Cates ME and Evans MR.Soft and fragile matter:

none-quilibrium dynamics, metastability and flow (PBK). Lon-don: CRC Press, 2010.

2. Hashimoto T, Matsuzaka K, Moses E, et al. String phase

in phase-separating fluids under shear-flow. Phys Rev

Lett1995; 74: 126–129.

3. Lauger J, Laubner C and Gronski W. Correlation between shear viscosity and anisotropic domain growth

during spinodal decomposition under shear-flow. Phys

Rev Lett1995; 75: 3576–3579.

4. Hobbie EK, Kim SH and Han CC. Stringlike patterns in

critical polymer mixtures under steady shear flow.Phys

Rev E1996; 54: R5909–R5912.

5. Matsuzaka K, Koga T and Hashimoto T. Rheological response from phase-separated domains as studied by

shear microscopy.Phys Rev Lett1998; 80: 5441–5444.

Figure 6. Snapshots of the composition fieldf_Aat various timestfor an initially homogeneous viscoelastic binary fluid. Results are obtained under applied shear rateg_=1:9923102_{with parameter set R000: (a)}

_

6. Qiu F, Ding JD and Yang YL. Real-time observation on deformation of bicontinuous phase under simple shear

flow.Phys Rev E1998; 58: R1230–R1233.

7. Corberi F, Gonnella G and Lamura A. Two-scale

com-petition in phase separation with shear. Phys Rev Lett

1999; 83: 4057–4060.

8. Shou ZY and Chakrabarti A. Ordering of viscous liquid

mixtures under a steady shear flow.Phys Rev E2000; 61:

R2200–R2203.

9. Corberi F, Gonnella G and Lamura A. Phase separation

of binary mixtures in shear flow: a numerical study.Phys

Rev E2000; 62: 8064–8070.

10. Lamura A and Gonnella G. Lattice Boltzmann simula-tions of segregating binary fluid mixtures in shear flow.

Physica A2001; 294: 295–312.

11. Lamura A, Gonnella G and Corberi F. The segregation

of sheared binary fluids in the Bray-Humayun model.Eur

Phys J B2001; 24: 251–259.

12. Berthier L. Phase separation in a homogeneous shear flow: morphology, growth laws, and dynamic scaling.

Phys Rev E2001; 63: 051503.

13. Cavagna A, Bray AJ and Travasso RDM. Ohta-Jasnow-Kawasaki approximation for nonconserved coarsening

under shear.Phys Rev E2000; 62: 4702–4719.

14. Bray AJ. Coarsening dynamics of phase-separating

sys-tems. Philos Trans A Math Phys Eng Sci 2003; 361:

781–791.

15. Gonnella G and Lamura A. Long-time behavior and

dif-ferent shear regimes in quenched binary mixtures. Phys

Rev E2007; 75: 011501.

16. Stansell P, Stratford K, Desplat JC, et al. Nonequilibrium

steady states in sheared binary fluids.Phys Rev Lett2006;

96: 085701.

17. Stratford K, Desplat JC, Stansell P, et al. Binary fluids

under steady shear in three dimensions.Phys Rev E2007;

76: 030501.

18. Fielding SM. Role of inertia in nonequilibrium steady

states of sheared binary fluids. Phys Rev E 2008; 77:

021504.

19. Tanaka H. Critical-dynamics and phase-separation

kinetics in dynamically asymmetric binary fluids – new dynamic universality class for polymer mixtures or

dynamic crossover.J Chem Phys1994; 100: 5323–5337.

20. Jupp L and Yuan XF. Dynamic phase separation of a binary polymer liquid with asymmetric composition

under rheometric flow. J Non-Newton Fluid 2004; 124:

93–101.

21. Tanaka H. Coarsening mechanisms of droplet spinodal

decomposition in binary fluid mixtures. J Chem Phys

1996; 105: 10099–10114.

22. Doi M and Onuki A. Dynamic coupling between stress

and composition in polymer-solutions and blends.J Phys

II1992; 2: 1631–1656.

23. Milner ST. Dynamical theory of concentration

fluctua-tions in polymer-solufluctua-tions under shear.Phys Rev E1993;

48: 3674–3691.

24. Ji H and Helfand E. Concentration fluctuations in

sheared polymer-solutions. Macromolecules 1995; 28:

3869–3880.

25. Clarke N and McLeish TCB. Shear flow effects on phase

separation of entangled polymer blends. Phys Rev E

1998; 57: R3731–R3734.

26. Yuan XF and Jupp L. Interplay of flow-induced phase

separations and rheological behavior. Europhys Lett

2002; 60: 691–697.

27. Jupp L, Kawakatsu T and Yuan XF. Modeling

shear-induced phase transitions of binary polymer mixtures.J

Chem Phys2003; 119: 6361–6372.

28. Fielding SM and Olmsted PD. Early stage kinetics in a unified model of shear-induced demixing and mechanical

shear banding instabilities. Phys Rev Lett 2003; 90:

224501.

29. Fielding SM and Olmsted PD. Nonlinear dynamics of an

interface between shear bands. Phys Rev Lett2006; 96:

104502.

30. Cromer M, Villet MC, Fredrickson GH, et al. Shear

banding in polymer solutions. Phys Fluid 2013; 25:

051703.

31. Likhtman AE and Graham RS. Simple constitutive equa-tion for linear polymer melts derived from molecular

the-ory: Rolie–Poly equation.J Non-Newton Fluid2003; 114:

1–12.

32. Guo XW, Zou S, Yang X, et al. Interface instabilities and chaotic rheological responses in binary polymer mixtures

under shear flow.RSC Adv2014; 4: 61167–61177.

33. Helfand E and Fredrickson GH. Large fluctuations in

polymer solutions under shear.Phys Rev Lett 1989; 62:

2468.

34. Adams JM, Fielding SM and Olmsted PD. Transient shear banding in entangled polymers: a study using the

Rolie-poly model.J Rheol2011; 55: 1007–1032.

35. Wagner AJ and Yeomans JM. Phase separation under

shear in two-dimensional binary fluids.Phys Rev E1999;

59: 4366–4373.

36. Helfand E and Tagami Y. Theory of the interface

between immiscible polymers. II.J Chem Phys1972; 56:

3592–3601.

37. Clarke CJ, Eisenberg A, LaScala J, et al. Measurements of the Flory–Huggins interaction parameter for

polystyr-ene–poly(4-vinylpyridine) blends. Macromolecules 1997;

30: 4184–4188.

38. Sferrazza M, Xiao C, Jones RAL, et al. Evidence for capillary waves at immiscible polymer/polymer

inter-faces.Phys Rev Lett1997; 78: 3693–3696.

39. Broseta D, Fredrickson GH, Helfand E, et al. Molecular-weight and polydispersity effects at polymer-polymer

interfaces.Macromolecules1990; 23: 132–139.

40. Schubert DW and Stamm M. Influence of chain length on the interface width of an incompatible polymer blend.

Europhys Lett1996; 35: 419–424.

41. Forrest BM and Toral R. The phase-diagram of the Flory-Huggins-de Gennes model of a binary polymer

blend.J Stat Phys1994; 77: 473–489.

42. Omowunmi SC and Yuan XF. Time-dependent non-linear dynamics of polymer solutions in microfluidic contraction flow – a numerical study on the role of