Dynamics and rheology of immiscible polymer-liquid

THE JOURNAL OF CHEMICAL PHYSICS 123, 014906 共2005兲

Dynamics and rheology of immiscible polymer-liquid-crystal systems Wei Yua兲 and Chixing Zhou Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

共Received 1 March 2005; accepted 29 April 2005; published online 12 July 2005兲 The morphology evolution in immiscible polymer-liquid-crystal systems is quite different from flexible polymer–polymer mixtures due to the anisotropic properties of liquid crystals. The dynamics and rheology of such system are discussed. A theoretical model is proposed to describe the dynamics of liquid-crystal droplets in a flow field and the rheological properties of immiscible liquid-crystal/polymer mixtures. The deformation of liquid-crystal droplet is found to be greatly dependent on the interfacial properties of polymer-liquid crystal. The scaling relationships of interfacial contribution to the stresses are found to be quite different from the isotropic mixtures with droplet morphology. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1940049兴 I. INTRODUCTION

Polymer-liquid-crystal 共LC兲 mixtures have received much attention recently.1 For example, polymer matrix is used to stabilize liquid-crystal displays; addition of liquidcrystalline polymers into isotropic polymers can lead to enhanced processability and mechanical performance of the bulk polymers.2 In order to better control the properties of such materials, it is crucial to understand the rheology and the dynamics of morphology evolution in such system. The evolution of phase morphology of a mixture of two immiscible flexible-chain polymers has been investigated extensively by experiments,3 theoretical models,4,5 and numerical simulations.6 However, most theoretical works focus on the isotropic mixtures with droplet4,5 or complex7 morphology. Less extensive works have been done to understand the effect of the director orientation and anchoring on the morphology evolution in flow field in a polymer-liquid-crystal system. Some cases considering the anisotropic properties of LC include the phase separation process8 and linear viscoelasticity9 of phase-separated polymer/LC blends. Most of our knowledge on the morphology and rheology of a mixture of two immiscible components is limited to isotropic, flexible-chain molecules. Two crucial factors that influence the shape of droplets are the viscosity ratio p = ␩d / ␩m 共␩d and ␩m are viscosity of droplet and matrix, respectively兲 and the capillary number Ca= ␩mR␥˙ / ⌫ 共R is the radius of droplet, ␥˙ is the strain rate, and ⌫ is the interfacial tension兲. The capillary number can be regarded as a ratio between the viscous shear stress 共␩m␥˙ 兲 and the interfacial tension 共⌫ / R兲. When the components become viscoelastic, a counterpart of Ca starts to play a role,5 which is defined as qd = Ded / Ca for droplet and qm = Dem / Ca for matrix, where De= ␭␥˙ is the Deborah number and ␭ = ␭d or ␭m is the characteristic relaxation time of droplet or matrix. When the dispersed droplet becomes a nematic liquid crystal, parameters such as the orientation of director, tumbling parameter of director, and the interfacial anchoring energy are going to a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

0021-9606/2005/123共1兲/014906/10/$22.50

affect the dynamics of the shape evolution of droplet. In this paper, we want to propose a model to describe the dynamics of morphology evolution of LC droplets in a flow field and rheology of immiscible polymer-liquid-crystal mixtures. We use here the recently developed Hamiltonian formalism of dissipative flow processes in media with internal microstructure10–12 in order to systematically investigate the coupled effects of the polymer viscoelasticity, anisotropy of LC, and the imposed flow field on the morphology of droplets, conformation of polymer molecules, and rheology of mixtures. The Hamiltonian approach has enabled the description of the dynamics of many polymer flowing systems.5,11–13 II. MODEL

The two phases of the system are assumed to be completely immiscible. The models for immiscible blends have been developed by many authors.4,5 All these models and present model are established on a continuum level for a unit volume of fluid, which is sufficiently small as compared to the characteristic length of flow field, but sufficiently large to contain enough droplet whose concentration is a constant 共␸兲. Such system can be characterized by the following independent variables: the mass density ␳, taken to be a constant for the incompressible system considered here; the momentum density M = ␳u , u is the velocity field; a secondorder tensor G to represent the ellipsoidal morphology, which satisfies for every point x on the droplet interface G : xx = 1;4,5 two second-order tensors C1 and C2 to represent two-mode relaxation of the flexible-chain molecules of matrix,5 which can be regarded as the second moment of the distribution function for the end-to-end distance vector of the polymer chains; a second-order tensor m to represent the conformation of long rodlike molecules of liquid crystal; volumetric disclination density ␳V inside the liquid-crystal droplet. The conformation tensors Ci are assumed to be free of constraint. Although more reasonable finite extension conformation tensors can be used, the unconstraint conformation tensors Ci are adopted due to its simplicity.5 Usually one conformation tensor is used for the flexible polymer, such as the single mode Oldroyd B model. To describe a polymer

123, 014906-1

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J. Chem. Phys. 123, 014906 共2005兲

W. Yu and C. Zhou

with a spectrum of relaxation times, it is convenient to use multiple conformation tensors with different relaxation times and moduli. The reason that we use two conformation tensors for flexible polymer comes from a previous study on droplet dynamics in viscoelastic matrix by a rigorous analysis based on fluid mechanics,5 where two relaxation modes with different relaxation times have been found for small droplet deformation. The morphology tensor G has a constraint of det G = 1, which represents the volume preservation of a droplet. The constraint on the conformation tensor m is tr m = 1, which means that only the rotation of the molecules of liquid crystal is allowed. Usually the nematics are described by a traceless symmetric tensor Q␣␤.14 The relationship between Q␣␤ and the conformation tensor m␣␤ used in this paper is Q␣␤ = m␣␤ − ␦␣␤ / 3. These two tensors are actually equivalent. It will be shown below that the dynamic equation of m␣␤ 关Eq. 共39兲兴 can be transformed to the mesoscopic theory of Larson–Doi when the influence of interface is ignored and taking use of the relation Q␣␤ = m␣␤ − ␦␣␤ / 3. The system is assumed to be characterized by the following free-energy functional H:

H=

冕

⍀

共⌽u + ⌽m + ⌽C + ⌽G兲dV,

共1兲

where ⌽u = M iM i / 2␳ is the kinetic-energy density, ⌽m, ⌽C, and ⌽G are the free-energy density of liquid-crystal phase, flexible polymer matrix, and interface, respectively. The following forms of the free-energy density are taken here: 2

⌽C = 共1 − ␸兲兺 i=1

冋

冋

册

1 1 nCiKCiI1共Ci兲 − nCikBT ln I3共Ci兲 , 2 2

册

1 1 nmKmI1共m兲 − nmkBT ln I3共m兲 , 2 2

⌽G = KG ln I1共G兲 +

␸⌫␪ 关2m:G − 共m:G兲2兴I2共G兲, 2R6

dF = 兵F,H其 + 关F,H兴 dt

共3兲

共4兲

where I1共a兲 = tr共a兲, I2共a兲 = 21 关共tr a兲2 − tr共a · a兲兴, and I3共a兲 = det共a兲 are the invariants of tensor a. The free-energy density ⌽C in Eq. 共2兲 is a classical one from which a series of dumbbell models with linear elastic spring can be obtained.10 nCi is the elasticity density, KCi is the character elastic constant, kB is the Boltzmann constant, and T is the absolute temperature. A similar free-energy density, Eq. 共3兲, is adopted for liquid crystal, where Km is the character elastic constant and nm is the elasticity density. Although we can expect certain form of free energy of liquid crystal other than Eq. 共3兲, it is shown below that the mesoscopic theory of Larson–Doi14 can be obtained from such free-energy density ⌽m. The free-energy density of interface ⌽G, Eq. 共4兲, is composed of two parts. The first part is the contribution from

共5兲

once the Poisson bracket 兵·,·其 and the dissipative bracket 关·,·兴 are fully determined for arbitrary functionals F and H. The Poisson bracket and dissipative bracket will give the conservative and dissipative parts of the dynamic equations. Hence, the time evolution equations can be written as

冏冏冏冏冏冏冏冏冏冏冏冏

⳵Ci,␣␤ ⳵Ci,␣␤ = ⳵t ⳵t ⳵m␣␤ ⳵m␣␤ = ⳵t ⳵t ⳵G␣␤ ⳵G␣␤ = ⳵t ⳵t

共2兲

⌽m = ␸

the isotropic interfacial tension ⌫, which has been used for Newtonian system15 and viscoelastic system.5 The second part of ⌽G represents the contribution from the anisotropic part of interfacial tension, which is consistent with the average anisotropic interfacial energy of Rapini–Papoular16 theory for spherical droplet. ␪ is the ratio of the anchoring energy due to the nematic orientation at the interface to the isotropic interfacial tension. If ␪ = 0, the interfacial anisotropy vanishes; for ␪ ⬎ 0, the interface easy axis is parallel to the interface 共planar anchoring兲; for ␪ ⬍ 0, the interface easy axis is perpendicular to the interface 共homeotropic anchoring兲. According to the Hamiltonian approach followed in this work, the governing equations are obtained from the dynamic equations developed for an arbitrary functional F:

+

cons

+

cons

+

cons

⳵Ci,␣␤ ⳵t

,

i = 1,2,

共6兲

diss

⳵m␣␤ ⳵t

diss

⳵G␣␤ ⳵t

diss

,

共7兲

.

共8兲

The Poisson bracket 兵F , H其 is written as a sum of 兵F , H其C, 兵F , H其m, and 兵F , H其G with the subscripts representing the matrix polymer, liquid crystal, and interface, respectively. The Poisson bracket 兵F , H其C of unstrained tensor Ci can be found in the monograph of Beris and Edwards.10 The Poisson brackets 兵F , H其m and 兵F , H其G of constrained tensors m and G have been suggested by Edwards et al.17 Therefore, the conservative part of dynamic equations can be written as

冏冏 ⳵Ci,␣␤ ⳵t

冏冏 ⳵m␣␤ ⳵t

冏冏 ⳵G␣␤ ⳵t

= − u␥ⵜ␥Ci,␣␤ + Ci,␥␣ⵜ␥u␤ cons

+ Ci,␤␥ⵜ␥u␣,

i = 1,2,

共9兲

= − u␥ⵜ␥m␣␤ + m␥␣ⵜ␥u␤ + m␤␥ⵜ␥u␣ cons

− 2m␣␤m␥␧ⵜ␥u␧ ,

共10兲

= − u␥ⵜ␥G␣␤ − G␣␥ⵜ␤u␥ − G␤␥ⵜ␣u␥ cons

2 − G␣␤ⵜ␥u␥ . 3

共11兲

In fact, Eqs. 共9兲 and 共10兲 are upper convected derivatives, and Eq. 共11兲 is a lower convected derivative if the incompressible condition is adopted.


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Immiscible polymer-liquid crystal systems

Next, the dissipative bracket 关F , H兴 can involve five terms: three terms for the dissipative contributions from the matrix polymer 关F , H兴C, the liquid-crystal droplet 关F , H兴m, and the interface 关F , H兴G, two terms for coupling contributions from the matrix polymer-interface interaction 关F , H兴CG and the liquid-crystal-droplet-interface interaction 关F , H兴mG:

关F,H兴mG = − −

共12兲

−

冕

关F,H兴C = − 兺

⍀

i=1

冕

Ci ⌳␣␤␥ ␧

␦H

␦Ci,␣␤ ␦Ci,␥␧

冉冊冉冊

␦F ␦H ⵜ␥ dV, ␦u␤ ␦u␧

关F,H兴CiG = −

where the first term accounts for the viscoelasticity of the Ci matrix polymer with a relaxation coefficient ⌳␣␤␥ ␧, and the second term represents for the viscous effect with a viscous C dissipative coefficient Q␣␤␥ ␧. The coupling effect between the conformation tensors C1 and C2 is not considered in this work. The dissipative bracket related with the constraint tensors m and G can be written as follows by using the transformation rule suggested by Edwards et al.:17

+

⍀

关F,H兴m = −

冕冉 m ⌳␣␤␥ ␧

− ␦␣␤ − −

+ −

冊

␦H ␦F m dV ␦ m ␥␧ ␦ m ␩␮ ␩␮

冕冉冊冉冊冕冋冉冊冉 m L␣␤␥ ␧ ⵜ␣

− ⵜ␣

关F,H兴G = −

␦F ␦H ␦m␣␤ ␦m␥␧

m Q␣␤␥ ␧ⵜ ␣

冉冊冉 ␦H ␦u␤

␦F ␦H ⵜ␥ dV ␦u␤ ␦u␧ ␦F ␦u␤

␦H ␦H − ␦ ␥␧ m ␦ m ␩␮ ␩␮ ␦ m ␥␧

␦F ␦F − ␦ ␥␧ m ␦ m ␩␮ ␩␮ ␦ m ␥␧

冕冕冕冋冉冊 G ⌳␣␤␥ ␧

冊册

冊

dV, 共14兲

␦F ␦H dV ␦G␣␤ ␦G␥␧

1 G ␦F ␦H G−1 G␩␯ dV ⌳ 3 ␣␤␥␧ ␣␤ ␦ G ␩␯ ␦ G ␥␧

G L␣␤␥ ␧ ⵜ␣

冉冊冉冊

␦F ␦H ␦ u ␤ ␦ G ␥␧

册

冊

冕冉 mG ⌳␣␤␥ ␧

␦H ␦F ␦H ␦F − ␦ m ␦G␥␧ ␦ma␤ ␦G␥␧ ␣␤ ␦m␩␮ ␩␮

冊

− +

冕冕冕冕

CiG ⌳␣␤␥ ␧

共16兲

␦F

␦H dV ␦Ci,␣␤ ␦G␥␧

1 CiG ␦F ␦H −1 ⌳ G G␩␮dV 3 ␣␤␥␧ ␦C␣␤ ␦G␩␮ ␥␧ CiG ⌳␣␤␥ ␧

␦H

␦F dV ␦Ci,␣␤ ␦G␥␧

1 CiG ␦H ␦F −1 ⌳ G G␩␮dV, 3 ␣␤␥␧ ␦C␣␤ ␦G␩␮ ␥␧

共17兲

m G where ⌳␣␤␥ ␧ and ⌳␣␤␥␧ denote the relaxation of liquid crysm tal and droplet shape, respectively, Q␣␤␥ ␧ represents the vism G cous dissipation in liquid crystal, L␣␤␥␧ and L␣␤␥ ␧ represent the coupling effects between the microstructures 共texture of mG liquid crystal and droplet兲 and the flow field, ⌳␣␤␥ ␧ and CiG ⌳␣␤␥␧ represent the coupling effects of the liquid-crystaldroplet shape and the matrix polymer-droplet shape. The dissipative coefficients in Eqs. 共13兲–共17兲 can be functions of microstructural variables Ci, m, and G Ci ⌳␣␤␥ ␧=

1 共Ci,␣␥␦␤␧ + Ci,␣␧␦␤␥ + Ci,␤␥␦␣␧ 2␭CinCiKCi共1 − ␸兲 + Ci,␤␧␦␣␥兲,

m ⌳␣␤␥ ␧=

i = 1,2,

⌳m 0 共m␣␥␦␤␧ + m␣␧␦␤␥ + m␤␧␦␣␥ + m␤␥␦␣␧兲, ␸

共18兲共19兲

G G ⌳␣␤␥ ␧ = ⌳0 共G␣␥G␤␧ + G␣␧G␤␥兲,

共20兲

C Q␣␤␥ ␧ = ␩ms共␦␣␥␦␤␧ + ␦␣␧␦␤␥兲共1 − ␸兲,

共21兲

m Q␣␤␥ ␧ = ␸关␮共␦␣␥␦␤␧ + ␦␣␧␦␤␥兲 + 2␮1m␣␤m␥␧ + ␮2共m␣␥␦␤␧

冉冊

1 ␦F ␦H ␦H ␦F − ⵜ␣ G G−1 − ⵜ␣ 3 ␦ u ␤ ␦ G ␩␯ ␩␯ ␥␧ ␦ u ␤ ␦ G ␥␧ 1 ␦H ␦F + ⵜ␣ G G−1 dV, 3 ␦ u ␤ ␦ G ␩␯ ␩␯ ␥␧

1 ␦F −1 ␦H G G ␩␮ 3 ␦m␣␤ ␥␧ ␦G␩␮

1 ␦F ␦H G␨␯ m dV, + G␥−1␧␦␣␤ 3 ␦G␨␯ ␦ m ␩␮ ␩␮

dV 共13兲

−

C Q␣␤␥ ␧ⵜ ␣

␦F

␦H ␦F ␦F ␦H − ␦␣␤ m ␦ G ␥␧ ␦ m ␩␮ ␩␮ ␦m␣␤ ␦G␥␧

1 ␦H ␦F − G␥−1␧ G 3 ␦m␣␤ ␦G␩␮ ␩␮

The dissipative bracket for the unconstrained tensor Ci is5 2

mG ⌳␣␤␥ ␧

1 ␦H ␦F + ␦␣␤G␥−1␧ m G dV 3 ␦m␨␯ ␨␯ ␦G␩␮ ␩␮

关F,H兴 = 关F,H兴C + 关F,H兴m + 关F,H兴G + 关F,H兴CG + 关F,H兴mG .

冕冉

共15兲

+ m␣␧␦␤␥ + m␤␧␦␣␥ + m␤␥␦␣␧兲兴,

共22兲

m L␣␤␥ ␧=

␭−1 共m␣␥␦␤␧ + m␣␧␦␤␥ + m␤␥␦␣␧ + m␤␧␦␣␥兲, 2

共23兲

G L␣␤␥ ␧=

1 − f2 共G␣␥␦␤␧ + G␣␧␦␤␥ + G␤␥␦␣␧ + G␤␧␦␣␥兲, 2

共24兲


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W. Yu and C. Zhou

CiG CiG ⌳␣␤␥ ␧ = ⌳0 共Ci,␣␥G␤␧ + Ci,␣␧G␤␥ + Ci,␤␥G␣␧

+ Ci,␤␧G␣␥兲,

i = 1,2,

共25兲

mG mG ⌳␣␤␥ ␧ = ⌳0 共m␣␥G␤␧ + m␣␧G␤␥ + m␤␥G␣␧ + m␤␧G␣␥兲,

共26兲

where ␭Ci is the relaxation time of conformation Ci, ␩ms is the Newtonian contribution to the viscosity of polymer ma-

冏冏 ⳵Ci,␣␤ ⳵t

=− diss

trix, ␮, ␮1, and ␮2 are viscosity parameters which have the same meaning as those in Ericksen’s transversely isotropic fluid model,14 ␭ is the tumbling parameter of liquid crystal, and f 2 is a coupling parameter which depends on the viscosCiG mG are the scalar ity ratios of the system. ⌳m 0 , ⌳0 , and ⌳0 relaxation parameters. The dissipative parts of dynamic equations can be obtained from the dissipative brackets of Eqs. 共13兲–共17兲 together with the dissipative coefficients defined in Eqs. 共18兲–共26兲:

冋

册

冉

1 2共Ci,␣␥G␤␥ + Ci,␤␥G␣␥兲 4 ⌫␪ 共1 − m:G兲2I2共G兲 Ci,␣␥m␥␧G␤␧ − Ci,␣␤ − ⌳C0 iG␸ 共Ci,␣␤ − ␦␣␤兲 − ⌳C0 iGKG 3 R6 ␭ Ci I1共G兲

冊

冋

2 ⌫␪ „2m:G − 共m:G兲2… I1共G兲共Ci,␣␥G␤␥ + Ci,␤␥G␣␥兲 − 共Ci,␣␥G␥␧G␤␧ + G␣␧m␥␧Ci,␤␥ − m:G Ci,␣␤ − ⌳C0 iG␸ 3 R6

册

4 + Ci,␤␥G␣␧G␥␧兲 − I2共G兲 Ci,␣␤ , 3

冏冏 ⳵m␣␤ ⳵t

共27兲

m = − ⌳m 0 2nmkBT共3m␣␤ − ␦␣␤兲 + 共␭ − 1兲共m␣␥D␥␤ + D␣␤m␤␥兲 − 2共␭ − 1兲m␥␧ⵜ␥u␧m␣␤ − ⌳0 diss

⫻共m␣␥G␥␤ + m␤␥G␣␥ − 2m␣␤m:G兲 − ⌳mG 0 − 2m␣␤m:G兲 − ⌳mG 0 ␸

冏冏 ⳵G␣␤ ⳵t

冋

册

2KG ⌫␪ 共2m:G − 共m:G兲2兲I1共G兲共m␣␥G␤␥ + m␤␥G␣␥ +␸ R6 I1共G兲

⌫␪ ⌫␪ 共1 − m:G兲2I2共G兲共m␣␥m␥␧G␤␧ + G␣␧m␥␧m␤␥ − 2m␣␤m␨␥m␥␧G␨␧兲 + ⌳mG „2m:G 0 ␸ R6 R6

− 共M:G兲2…共m␣␥G␥␧G␤␧ + G␣␧G␥␧m␤␥ − 2m␣␤m␨␥G␥␧G␨␧兲,

冉

⌫␪ 2共1 − m:G兲I2共G兲 R6

冊

冋

共28兲

冉

2 3G␣␥G␤␥ ⌫␪ − G␣␤ + 共1 − f 2兲共G␣␥D␥␤ + D␣␥G␤␥兲 − ⌳G = − ⌳G 共1 − m:G兲I2共G兲2 G␣␥m␥␧G␤␧ 0 KG 0␸ 3 R6 I1共G兲 diss

冊

冉

1 2 − m:GG␣␤ + „2m:G − 共m:G兲2… I1共G兲G␣␤G␤␥ − G␣␥G␥␧G␤␧ − I2共G兲 G␣␤ 3 3

冊

冊册冋

冉

− ⌳mG ␸nmkBT3 m␣␥G␤␥ 0

2 ⌫␪ 共1 − m:G兲I2共G兲2„m␣␥G␥␧G␤␧ + G␣␧G␥␧m␤␥ − m:G共m␣␥G␤␥ + G␣␥m␤␥兲… + G␣␥m␤␥ − G␣␤ + ␸ 3 R6 2

冉

冊

2 − 兺 ⌳C0 iG共1 − ␸兲nCikBT Ci,␣␥G␤␥ + G␣␥Ci,␤␥ − I1共Ci兲 G␣␤ . 3 i=1

In Eqs. 共27兲 and 共29兲, the conformation tensor Ci are made ˜ dimensionless by C i,␣␤ = Ci,␣␤KCi / kBT, and the tilde over the tensors are omitted for simplicity. The tensor D␣␤ is defined as the symmetric part of velocity gradient tensor, D␣␤ = 1 / 2共ⵜ␣u␤ + ⵜ␤u␣兲. It can be seen from Eqs. 共27兲–共29兲 that the evolution of certain microstructure, such as the conformations of matrix polymer, the orientation of director of liquid crystal, and the shape of droplet, depends on the other microstructures, which represents the full coupling effects in the system. The evolution of droplet shape depends not only on its previous

册

共29兲

shape, but also on the conformation of matrix polymer and the orientation of director inside the droplet. Also, the evolution of the orientation of director depends on its previous state of orientation and the droplet shape. More important thing is the evolution of the conformation of matrix polymer depends on its previous state, droplet shape, and the conformation of liquid-crystal molecules. This means that the orientation of director of liquid crystal inside the droplet can have certain influences on the conformation of matrix polymer molecules, which has been verified by experiments.18 The complete dynamic equation can be obtained by substituting Eqs. 共27兲–共29兲 and Eqs. 共9兲–共11兲 into Eqs. 共6兲–共8兲.


014906-5

J. Chem. Phys. 123, 014906 共2005兲


The dynamic evolution of tensor m, if the coupling terms are neglected, is equivalent to the Larson–Doi mesoscopic model of liquid-crystalline polymer, when we take ⌳m 0 =−

␭ , 共 ␣ 2 + ␣ 3兲

n mk BT = −

共31兲

5pms , 2p + 3pms

⌳G 0 KG =

共35兲

⌳C0 2G =

1 ␧共␣2 + ␣3兲␳V , 6␭

共32兲

60共p + pms兲 1 , ␭G 共2p + 3pms兲共19p + 16pms兲

共34兲

192共1 − pms兲 1 1 , 2 nC2kBT 5共19p + 16pms兲 ␭G 1 − ␸ 共36兲

19p + 16pms ␭m , ␭ C2 = 19p + 16 where p is the viscosity ratio between the droplet viscosity ␩d and matrix viscosity ␩m, pms = ␩ms / ␩m denotes the Newtonian contribution in flexible polymer matrix, R is the radius of a droplet, ␸ is the volume fraction of droplets, and ␭G = ␩mR / ⌫ is the characteristic relaxation time of a droplet. The scalar coefficient of the coupling effect between the droplet shape and the liquid crystal, however, is hard to determine. We suggest the following expression for this coefficient, which includes both contributions from liquid crystal 共␳V兲 and interface 共⌫ / R兲:

共33兲

6⌫ 共p + 1兲共2p + 3兲␸ KG = , 5R 5共p + 1兲 − 共5p + 2兲␸

12共1 − pms兲 1 1 , 2 nC1kBT 5共2p + 3pms兲 ␭G 1 − ␸

2p + 3pms ␭m , ␭ C1 = 2p + 3

共30兲

where ␣2 and ␣3 are the Leslie viscosity coefficients and ␧ is a dimensionless constant. The relaxation parameters for droplet shape and the parameters of the coupling effects between the droplet and the flexible polymer matrix have been determined in the model for viscoelastic blends,5 and can be written as

f2 =

⌳C0 1G =

⌳mG 0 =

a␳V , ␸⌫/R

共37兲

where a is a dimensionless constant. The final expressions can be obtained by substituting the coefficients 共30兲–共37兲 into the dynamic equations 共27兲–共29兲.

冋

⳵G␣␤ g1 3G␣␥G␤␥ + u␥ⵜ␥G␣␤ = ␻␣␥G␤␥ − G␣␥␻␥␤ − f 2共G␣␥D␥␤ + D␣␥G␤␥兲 − − G␣␤ ⳵t ␭G I1共G兲 −

冋

冉

冊

册

1 ␪ 1 g2 共1 − m:G兲I2共G兲2 G␣␥m␥␧G␤␧ − m:GG␣␤ + „2m:G − 共m:G兲2… 3 ␭G 6

冉

2 ⫻ I1共G兲G␣␥G␤␥ − G␣␥G␥␧G␤␧ − I2共G兲 G␣␤ 3

冊册

+

冉

a␳V2 ␭G␧共␣2 + ␣3兲 2 m␣␥G␤␥ + G␣␥m␤␥ − G␣␤ 3 2␭␩m

␪ − 2a␳V 共1 − m:G兲I2共G兲关m␣␥G␥␧G␤␧ + G␣␧G␥␧m␤␥ − m:G共m␣␥G␤␥ + G␣␥m␤␥兲兴 6 −

冋冉

冊冉

2 2 1 g3 C1,␣␥G␤␥ + G␣␥C1,␤␥ − I1共C1兲 G␣␤ + g4 C2,␣␥G␤␥ + G␣␥C2,␤␥ − I1共C2兲 G␣␤ 3 3 ␭G

冉

冊册

,

冊共38兲

冊

1 1 2␭␩m ␪ ⳵m␣␤ + u␥ⵜ␥m␣␤ = ␻␣␥m␥␤ − m␣␥␻␥␤ + ␭共D␣␥m␥␤ + m␣␥D␥␤兲 − 2␭m␣␤m:D − ␧␳V m␣␤ − ␦␣␤ + 3 ⳵t ␭ G 共 ␣ 2 + ␣ 3兲 6 ⫻共1 − m:G兲I2共G兲共m␣␥G␥␤ + m␤␥G␣␥ − 2m␣␤m:G兲 − ␣␳V共m␣␥G␤␥ + m␤␥G␣␥ − 2m␣␤m:G兲 ⫻

冋

册

12共p + 1兲共2p + 3兲 1 ␪ ␪ + „2m:G − 共m:G兲2…I1共G兲 − ␣␳V 共1 − m:G兲2I2共G兲共m␣␥m␥␧G␤␧ 6 I1共G兲 5关5共p + 1兲 − 共5p + 2兲␸兴 2

␪ + G␣␧m␥␧m␤␥ − 2m␣␤m␨␥m␥␧G␨␧兲 + a␳V „2m:G − 共m:G兲2…共m␣␥G␥␧G␤␧ + G␣␧G␥␧m␤␥ 6 − 2m␣␤m␨␥G␥␧G␨␧兲,

共39兲


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W. Yu and C. Zhou

冋

⳵C1,␣␤ g5 g6␭m 共C2,␣␥G␥␤ + G␣␥C2,␥␤兲 2 + u␥ⵜ␥C1,␣␤ = C1,␥␣ⵜ␥u␤ + C1,␤␥ⵜ␥u␣ − 共C1,␣␤ − ␦␣␤兲 − 2 − C1,␣␤ 3 ⳵t ␭m I1共G兲 ␭G −

冉

冊

册

2 ␭m ␪ ␭m ␪ 2 2 g7共1 − m:G兲I2共G兲 C1,␣␥m␥␧G␤␧ + G␣␧m␥␧C1,␤␥ − m:G C1,␣␤ − 2 g8„2m:G − 共m:G兲 … 3 6 ␭G ␭G 6

冋

册

4 ⫻ I1共G兲共C1,␣␥G␤␥ + C1,␤␥G␣␥兲 − 共C1,␣␥G␥␧G␤␧ + C1,␤␥G␣␧G␥␧兲 − I2共G兲 C1,␣␤ , 3

冋

共40兲

⳵C2,␣␤ g9 g10␭m 共C2,␣␥G␥␤ + G␣␥C2,␥␤兲⬘ 2 + u␥ⵜ␥C2,␣␤ = C2,␥␣ⵜ␥u␤ + C2,␤␥ⵜ␥u␣ − 共C2,␣␤ − ␦␣␤兲 − 2 − C2,␣␤ 3 ⳵t ␭m I1共G兲 ␭G −

冉

冊

册

2 ␭m ␪ ␭m ␪ 2 2 g11共1 − m:G兲I2共G兲 C2,␣␥m␥␧G␤␧ + G␣␧m␥␧C2,␤␥ − m:G C2,␣␤ − 2 g12„2m:G − 共m:G兲 … 3 6 ␭G ␭G 6

冋

册

4 ⫻ I1共G兲共C2,␣␥G␤␥ + C2,␤␥G␣␥兲 − 共C2,␣␥G␥␧G␤␧ + C2,␤␥G␣␧G␥␧兲 − I2共G兲 C2,␣␤ . 3

共41兲

The tensor ␻␣␤ is defined as the asymmetric part of velocity gradient tensor, ␻␣␤ = 21 共ⵜ␣u␤ − ⵜ␤u␣兲. The coefficients gi 共i = 1 – 12兲 are functions of viscosity ratio p, pms, and volume fraction ␸, and listed in the Appendix. The volumetric disclination density ␳V inside the liquid-crystal droplet will change during the deformation or relaxation of droplet in a flow field. A simple scaling dynamic equation of ␳V 共Ref. 14兲 is adopted here to close the dynamic equations: d␳V ¯ = ␥˙ ␳V − ␳V2 , dt

共42兲

where ¯␥˙ denotes the mean amplitude of strain rate inside the droplet. The extra stress of the mixture can be obtained simultaneously as

␴␣␤ = − 2f 2KG

冉

冊

冉

冊

2 ⌫␪ ⌫␪ G␣␤ 1 „2m:G − 共m:G兲2… I1共G兲G␣␤ − G␣␥G␥␤ − I2共G兲 ␦␣␤ − ␸ 共1 − m:G兲I2共G兲 − ␦␣␤ − f 2␸ 3 R6 R6 I1共G兲 3

冋冉

冊

册

1 ⫻ 2m:G ␭m␣␤ − f 2 ␦␣␤ + 共f 2 − ␭兲共G␣␥m␥␤ + m␣␥G␥␤兲 + 共1 − ␸兲␩m 3

冋兺 2

i=1

p Ci ␭ Ci

共Ci,␣␤ − ␦␣␤兲 + pms共ⵜ␣u␤ + ⵜ␤u␣兲

+ ␸关␮2D␣␤ + 2␮1m␣␤m␥␧ⵜ␥u␧ + 2␮2共m␣␥D␥␤ + D␣␥m␤␥兲 + ␭nmkBT共3m␣␤ − ␦␣␤兲兴.

The last two terms in Eq. 共42兲 account for the stress contribution from the matrix polymer and the liquid crystal, and represent a linear mixing rule which comes from the linear mixing rule taken in the system free energy, Eqs. 共1兲–共3兲. It should be stressed that the contribution from the matrix polymer is also influenced by the director orientation of liquid crystal, as shown in the coupling effect in the dynamic equation of conformation tensors Ci, Eqs. 共40兲 and 共41兲. The others terms are due to the interfacial contribution, with the first term from the isotropic interfacial tension, and the sec-

册

共43兲

ond and third terms from the anisotropic interfacial tension. III. ASYMPTOTIC ANALYSIS

It is crucial to know how the properties of liquid-crystal influence the droplet shape under the flow field. We will consider a liquid-crystal droplet immersed in an infinite Newtonian matrix, so the viscoelasticity of matrix can be neglected. The dynamic equation for the droplet shape can be written in a dimensionless form as

冉

冊

冋

冉

3G␣␥G␤␥ dG␣␤ ␪ − G␣␤ − g2s 共1 − m:G兲I2共G兲2 G␣␥m␥␧G␤␧ = Ca共G␣␤␻␥␤ − ␻␣␥G␤␥兲 − f 2sCa共G␣␥D␥␤ + D␣␥G␤␥兲 − g1s dt* I1共G兲 6

冊

冉

1 2 − m:GG␣␤ + „2m:G − 共m:G兲2… I1共G兲G␣␥G␤␥ − G␣␥G␥␧G␤␧ − I2共G兲 G␣␤ 3 3

冊

冊册

+

冉

2 a␳V2 ␭G ␧共␣2 + ␣3兲 m␣␥G␤␥ 2␭␩m

2 ␪ + G␣␥m␤␥ − G␣␤ − 2a␳V␭G 共1 − m:G兲I2共G兲关m␣␥G␥␧G␤␧ + G␣␧G␥␧m␤␥ − m:G共m␣␥G␤␥ + G␣␥m␤␥兲兴, 3 6

共44兲


014906-7

J. Chem. Phys. 123, 014906 共2005兲


FIG. 1. Three possible states of the orientation of droplet and director of liquid crystal.

where the time is made dimensionless by t* = t / ␭G, the rates of the deformation tensor and the vorticity tensor are normalized by the shear rate of the flow, and f 2s = 5 / 共2p + 3兲, g1s = 40共p + 1兲 / 关共2p + 3兲共19p + 16兲兴, and g2s = 250共p + 1兲 / 关共2p + 3兲2共19p + 16兲兴. The flow field is assumed to be a steady simple shear with the normalized velocity gradient given by

冢冣

m12 =

共45兲

It is assumed that the flow field is sufficiently weak, such that the droplet shape deviates a little from a sphere. Therefore, the droplet shape can be expanded to the first order of Ca as G␣␤ = ␦␣␤ + CaS␣␤ ,

共46兲

where S␣␤ is a traceless second-order tensor. In steady flow, the steady value of ␳V equals to the mean amplitude of strain rate ¯␥˙ inside the droplet, which is proportional to the applied shear rate. Substituting Eq. 共46兲 and ␳V into 共44兲 and keeping the terms to the first order of Ca, we obtain Ca

␪ dS␣␤ * = − f 2s2CaD␣␤ − g1sCaS␣␤ − g2sCa关S␣␤ dt 6 − 2m:S共3m␣␤ − ␦␣␤兲兴.

共47兲

It is noticed that the terms containing a in Eq. 共44兲 disappear in Eq. 共47兲, which means that it will affect the droplet shape to the higher order of deformation. The steady values of m␣␤ are needed to solve Eq. 共47兲 for the droplet shape. This can be readily obtained from Eq. 共39兲 to the zero order of droplet deformation. For ␭ ⬍ 1, we have m12 =

␭␧ + O共␧2兲, 3共1 − ␭2兲

m11 =

1+␭ + O共␧2兲, 3 − ␭2

1−␭ 1 − ␭2 2 + O共␧ 兲, m = + O共␧2兲 m22 = 33 3 − ␭2 3 − ␭2 and for ␭ ⬎ 1

共48兲

2␭

+

共3 − 2␭2兲␧ + O共␧2兲, 6␭共␭2 − 1兲

m11 =

␭+1 共␭ + 3兲␧ + O共␧2兲, − 2␭ 6␭冑␭2 − 1

m22 =

共␭ − 3兲␧ ␭−1 + O共␧2兲, − 2␭ 6␭冑␭2 − 1

0 1 0

ⵜu = 0 0 0 . 0 0 0

冑␭2 − 1

共49兲

Ⲑ

m33 = ␧ 3冑␭2 − 1 + O共␧2兲.The steady solution of 共47兲 can be obtained for ␭ ⬍ 1 to the first order of ␧ S12 = −

f 2s + O共␧兲 g1s + g2s␪/6

共50兲

with all other components to be zero. The steady deformation parameter D can be calculated as15 D=

3f 2s L−B = Ca + O共␧,Ca2兲, L + B 兩6g1s + ␪g2s兩

共51兲

where L and B are the lengths of the semiaxes of droplet in shear plane. When ␪ = 0 which corresponds to the isotropic interface, the deformation parameter reduces to DN = Ca共19p + 16兲 / 共16p + 16兲, which is the steady deformation of Newtonian droplet.19 When ␪ ⬍ 0, the deformation of liquid crystal is larger than that of Newtonian system with same viscosity ratio, which means that the homeotropic anchoring tends to increase the steady deformation of liquid-crystal droplet. When ␪ ⬎ 0, D / DN ⬍ 1, which means that the planar anchoring tends to decrease steady deformation of liquidcrystal droplet. The orientation angle of droplet, which is defined as the angle between the long axis of droplet and the flow direction, equals to ␲ / 4, which is same as the Newtonian system. Although the tumbling parameter ␭ does not appear in Eq. 共51兲, the deformation parameter of droplet is influenced by the tumbling parameter through the viscosity of liquid crystal. For flow aligning liquid crystal 共␭ ⬎ 1兲, the steady solutions of Eq. 共47兲 are written as S11 = −

2␪共␭ + 3兲冑␭2 − 1f 2sg2s + O共␧兲, ␭2共2g1s − ␪g2s兲共6g1s + ␪g2s兲

共52兲


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J. Chem. Phys. 123, 014906 共2005兲

W. Yu and C. Zhou

S22 = −

S33 =

2␪共␭ − 3兲冑␭2 − 1f 2sg2s + O共␧兲, ␭ 共2g1s − ␪g2s兲共6g1s + ␪g2s兲 2

4␪冑␭2 − 1f 2sg2s + O共␧兲, ␭共2g1s − ␪g2s兲共6g1s + ␪g2s兲

共53兲

The deformation parameter of droplet can be calculated as

D= 共54兲

=

2 2 − 4␪g1sg2s + ␪2g2s 3f 2s冑4␭2g1s Ca + O共␧,Ca2兲 ␭兩共2g1s − ␪g2s兲共6g1s + ␪g2s兲兩

冋

册

f 2sg2s共2␭2 − 3兲 f 2s + ␪ + o共␪2兲 Ca + O共␧,Ca2兲, 2 2g1s 12␭2g1s 共56兲

S12 = −

6f 2s共2␭2g1s − ␪g2s兲 + O共␧兲. ␭2共2g1s − ␪g2s兲共6g1s + ␪g2s兲

␣ = tan−1 =

冋冑

共55兲

and the orientation angle is

共2␭2g1s − ␪g2s兲

2 2 ␭2 − 1␪2s + ␭冑4␭2g1s − 4␪g1sg2s + ␪2g2s sgn共2g1s − ␪g2s兲sgn共6g1s + ␪g2s兲

␲ 冑␭2 − 1g2s − ␪ + O共␧, ␪2,Ca兲. 4 4g1s␭2

␣m = tan−1

冉冊

1/2

= tan−1

冉冊 ␭−1 ␭+1

1/2

.

共58兲

Therefore, three possible states of the orientation of droplet and the orientation of director can exist. The orientation of droplet is larger than the orientation of director for ␪ ⬍ 0, as shown in Fig. 1共a兲. The orientation of droplet can be slightly smaller than the orientation of director for ␪ ⬎ 0 and ␭ Ⰷ 1, as shown in Fig. 1共c兲. If the tumbling parameter ␭ is not so large, the orientation of droplet is always larger than that of director. The orientation angle of droplet and director can equal to each other only when ␪ = 0 and ␭ Ⰷ 1, as shown in Fig. 1共b兲. This analysis is valid only when the droplet shape is not deviated too much from a sphere. The rheological properties can be readily calculated from the above expansion. The interfacial contribution of the stress is 2 ⌫␪ s CaS␣␤ ␴␣␤ = − f 2KGCaS␣␤ − f 2␸ 3 R6 +␸

冉

+ O共␧,Ca兲共57兲

When absolute value of ␪ is small, the steady deformation of liquid-crystal droplet can be larger than that of Newtonian droplet if ␪ ⬍ 0 and 1 ⬍ ␭ ⬍ 冑3 / 2 ⬇ 1.225, or ␪ ⬎ 0 and ␭ ⬎ 1.225. The orientation angle is larger than ␲ / 4 when ␪ ⬍ 0 共homeotropic anchoring兲, and smaller than ␲ / 4 when ␪ ⬎ 0 共planar anchoring兲. An interesting phenomenon here is that the orientation of director and the orientation of droplet may be different. The orientation of director can be readily calculated as m22 m11

册

冊

1 ⌫␪ Cam:S6f 2 m␣␤ − ␦␣␤ . 3 R6

共59兲

The interfacial stress for ␭ ⬍ 1 can be obtained by substituting Eqs. 共48兲 and 共50兲 into Eq. 共59兲,

冊

共60兲

Ns1 = Ns2 = O共␧,Ca2兲.

共61兲

s = ␴12

冉

2 f 2s 2 ⌫␪ KG + ␸ Ca + O共␧,Ca2兲, 3 R 6 g1s + g2s␪/6

This represents a constant viscosity at low shear rate, and vanishing normal stress differences. For ␭ ⬎ 1, the interfacial stresses are s ␴12 =

冋

2 ⌫␪ 共− 4␭2g1s − ␭2␪g2s KG共2␭2g1s − ␪g2s兲 + ␸ 3 R6

+ 6g1s兲

册

2 6f 2s 共2␭2g1s − ␪g2s兲Ca + O共␧,Ca2兲, ␭2共2g1s − ␪g2s兲共6g1s + ␪g2s兲

共62兲 Ns1 =

冉

冊

2 12␪冑␭2 − 1f 2s 2 ⌫ KGg2s − g1s␸ Ca 3 R ␭2共2g1s − ␪g2s兲共6g1s + ␪g2s兲

+ O共Ca2兲, Ns2 =

冉

共62a兲

冊

2 6␪共␭ − 1兲冑␭2 − 1f 2s 2 ⌫ Ca KGg2s − g1s␸ 2 3 R ␭ 共2g1s − ␪g2s兲共6g1s + ␪g2s兲

+ O共Ca2兲.

共63兲

We obtain the following scaling relationship at low shear s rate, i.e., ␴12 ⬃ ␥˙ , Ns1 ⬃ ␥˙ , and Ns2 ⬃ ␥˙ . The ratio between the second normal stress difference and the first normal stress difference is Ns2 / Ns1 = 共␭ − 1兲 / 2. Therefore, Ns2 can be positive and larger than Ns1 with a sufficient large tumbling parameter ␭. The scaling relationships of normal stress differences are quite different from those of other mixtures. The interfacial normal stress difference for isotropic mixture with droplet


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J. Chem. Phys. 123, 014906 共2005兲


morphology scales with ␥˙ 2, and Ns2 / Ns1 = −1 / 2.15 For isotropic mixture with cocontinuous morphology, Ns1 ⬃ ␥˙ and Ns2 / Ns1 = −1.7 The scaling relationship of Ns1 is similar to the prediction of Doi–Ohta model.7 However, it should be noticed that a great difference in morphology exists between the present model and the Doi–Ohta model. Such scaling relationship for Ns1 in polymer-liquid-crystal system has been experimentally verified by Riise and Denn.20

ACKNOWLEDGMENT

This work was financially supported by Natural Science Foundation of China Grant Nos. 20204007, 20174024, and 20490220. APPENDIX: SCALAR COEFFICIENTS gi „i = 1 − 12… IN EQS. „38…–„41…

IV. CONCLUSIONS

A model of the microstructural evolutions and rheology in polymer/liquid-crystal systems is proposed by a Hamiltonian approach. The coupling effects between the LC droplet shape and the director and the LC droplet shape and the conformation of polymer chain are well integrated in the model. The dynamic equations of the evolution of droplet shape tensor G, order parameter tensor m, and conformation tensors Ci are obtained. The asymptotic behavior of a LC droplet under sufficient weak shear flow was analyzed. It is found that for the tumbling nematics, the deformation of a LC droplet could be larger 共smaller兲 than a Newtonian droplet with same viscosity ratio if the homeotropic 共planar兲 anchoring on the interface is favored. For the shear aligning liquid crystal, the deformation of a LC droplet depends not only on the interfacial anchoring state ␪, but also on the tumbling parameter ␭. The orientation of a LC droplet is found to be different from the orientation of the director. The orientation of the director is more aligned with the flow direction under most cases. The dependence of interfacial stresses on shear rate is also obtained. The interfacial shear stresses for tumbling liquid-crystal scales with shear rate and the normal stress difference are zero to the second order of s , Ns1, and Ca and first order of ␧. The interfacial stresses 共␴12 s N2兲 for flow aligning liquid crystal scale linearly with the shear rate. The linear scaling relationships of normal stress difference are quite different from the quadratic scaling relationship for isotropic mixtures with droplet morphology.

g10 =

2

40共p + pms兲 , 共2p + 3pms兲共19p + 16pms兲

共A1兲

g2 =

50共p + pms兲关5共p + 1兲 − 共5p + 2兲␸兴 , 共p + 1兲共2p + 3兲共2p + 3pms兲共19p + 16pm兲

共A2兲

g3 =

12共1 − pms兲 , 5共2p + 3pms兲2

共A3兲

g4 =

192共1 − pms兲 , 5共19p + 16pms兲2

共A4兲

g5 =

2p + 3 , 2p + 3pms

共A5兲

g6 =

144共1 − pms兲共p + 1兲␸ , 25pC1共2p + 3pms兲关5共p + 1兲 − 共5p + 2兲␸兴共1 − ␸兲共A6兲

g7 =

24共1 − pms兲␸ , 5pC1共2p + 3兲共2p + 3pms兲共1 − ␸兲

1 g8 = g7 , 2 g9 =

2304共1 − pms兲共p + 1兲共2p + 3兲␸ , 25pC2共19p + 16兲共19p + 16pms兲关5共p + 1兲 − 共5p + 2兲␸兴共1 − ␸兲

19p + 16 , 19p + 16pms

共A7兲

共A8兲

共A9兲

共A10兲

6

384共1 − pms兲␸ , g11 = 5pC2共19p + 16兲共19p + 16pms兲共1 − ␸兲

共A11兲

1 g12 = g11 . 2

共A12兲

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g1 =

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014906-10

E. G. Virga, Variational Theories for Liquid Crystals 共Chapman Hall, London, 1994兲. 17 B. J. Edwards, M. Dressler, M. Grmela, and A. Ait-Kadi, Rheol. Acta 42, 64 共2003兲. 16

J. Chem. Phys. 123, 014906 共2005兲

W. Yu and C. Zhou

F. Roussel, C. Canlet, and B. M. Fung, Phys. Rev. E 65, 021701 共2002兲. G. I. Taylor, Proc. R. Soc. London, Ser. A 138, 41 共1932兲. 20 B. L. Riise and M. M. Denn, J. Non-Newtonian Fluid Mech. 86, 3 共1999兲. 18 19
