Rheol Acta (2000) 39: 360±370 Ó Springer-Verlag 2000

Valery S. Volkov Valery G. Kulichikhin

Received: 23 March 1999 Accepted: 13 December 1999

V. S. Volkov (&)
V. G. Kulichikhin Institute of Petrochemical Synthesis Russian Academy of Sciences 29 Leninsky Pr., Moscow, 117912 Russia e-mail: [email protected]

ORIGINAL CONTRIBUTION

Non-symmetric viscoelasticity of anisotropic polymer liquids

Abstract The liquid crystalline (LC) polymers are considered as anisotropic viscoelastic liquids with nonsymmetric stresses. A simple constitutive equation for nematic polymers describing the coupled relaxation of symmetric and antisymmetric parts of the stress tensor is formulated. For illustration of non-symmetric anisotropic viscoelasticity, the simplest viscometric

Introduction The theories of viscoelasticity began with the well known papers of Maxwell (1867) and Boltzman (1874). At present it is quite clear that the macroscopic and molecular memory e€ects play an important role in the rheology of polymeric liquids. The appearance and development of LC polymers has aroused interest in the theory of anisotropic liquids with memory (see, for example, De Silva and Kline 1968; Larson and Mead 1989; Edwards et al. 1990; Volkov and Kulichikhin 1990, 1994; Rey 1995; Larson 1996; Singh and Rey 1998; Volkov 1998). In recent years there has been a growing necessity to study models of anisotropic viscoelastic liquids taking into account the features of relaxation anisotropy that these speci®c polymer media exhibit. The principal purpose of investigations in this area is the study of the regularities in rheological behavior of a suciently broad class of these anisotropic viscoelastic liquids, in order to establish the dependence of the rheological properties on the parameters determining viscous and relaxation anisotropy. The simplest phenomenological theory for directed or oriented liquids was developed by Ericksen (1960). He de®ned and investigated a class of liquids with single orientation vector n, called the director. This unit vector

¯ows of polymeric nematics in the magnetic ®eld are considered. The frequency and shear rate dependencies of extended set of Miesowicz viscosities are predicted. Key words Macroscopic theory
Polymer nematics
Non-symmetric viscoelasticity
Relaxation anisotropy
Complex viscosities
Non-linear shear viscosities

characterizes the preferred orientation of liquid particles during ¯ow. Ericksen's concept of transversely isotropic liquid was further used to develop a structural-continuum theory for suspensions of ellipsoidal particles. It is worth mentioning that this theory is widely used to describe blood ¯ow and developed turbulent ¯ows. The concept of a continuum with an inner structure appears to be very promising in order to advance a continuum theory for liquid crystals. Leslie (1968) modi®ed the Ericksen liquid to describe the dynamics of nematic liquid crystals. The popular Leslie±Ericksen theory describes the basic features of the rheological behavior of low molecular weight liquid crystals. However, this theory fails to describe the frequency dependence of dynamic moduli and non-linear e€ects observed in common ¯ows of LC polymer liquids. This fact has stimulated the studies of general principles of anisotropic viscoelastic liquids. In the previous paper (Volkov and Kulichikhin 1990), we presented an anisotropic Maxwell model with Jaumann time derivative for stress tensor. The main feature of the model is anisotropic viscosity and relaxation time represented by fourth-rank material tensors. The anisotropy in elasticity of LC polymers leads to anisotropy of relaxation times. It was assumed that ¯ow destroys initial poly-domains in LC polymers

361

and produces a mono-domain anisotropic structure. As a ®rst approximation in the description of polymer nematics we assumed uniaxial anisotropy in viscosity and relaxation time tensors characterized by a single director. In this way, we obtained a constitutive equation for an incompressible nematic viscoelastic liquid with the symmetric stress tensor. The anisotropic viscoelastic liquid described by this constitutive equation is characterized by two independent (longitudinal and transverse) viscosities and relaxation times, with respect to the director. The director in the constitutive equation is found from an additional equation that describes the change in the orientation of the director under the e€ect of ¯ow. Martins (1994, 1996) developed a similar approach to the nematic viscoelasticity of LC polymers. He proposed a generalization of the Ericksen±Leslie theory that involves an additional time derivative of the stress with a single scalar relaxation time. However, this continuum model cannot describe such important feature of LC polymers as the anisotropy of relaxation processes. In the present work, the simple theory of nonsymmetric viscoelasticity of nematic polymer liquids is formulated. The approach can be considered a generalization of that used in the previous paper (Volkov and Kulichikhin 1990).

General equations of motion From a macroscopic point of view the LC polymers are anisotropic viscoelastic liquids with varying rheological properties in di€erent directions. The motion of a uniaxial anisotropic liquid is described by a velocity ®eld v…r; t† and a director ®eld n…r; t†. Viscoeslastic liquids with one director form the basis for the study of nematic LC polymers. The single preferred direction in equilibrium is the main distinctive feature for anisotropic polymers under consideration. In quasi-linear dynamics of nematic polymers, the in¯uence of ¯ow on axial symmetry can be neglected. We shall hereinafter, limit our consideration to isothermal ¯ow of incompressible LC polymers. In this case, general equations describing the dynamics of nematic polymers include the continuity equation ove ˆ0 ; oxe

…1†

the equation of change for linear momentum q

dvi orji ˆ ‡ fi dt oxj

…2†

and the equation of change for total angular momentum d on Li ˆ ei ‡ Mi : dt oxe

…3†

q is the liquid density, d=dt ˆ o=ot ‡ ve o=oxe is the substantial time derivative and fi is the external body forces. The stress state at any given point in an anisotropic liquid in a general case is described by a non-symmetric stress tensor rij . The total liquid angular momentum L is de®ned as Li ˆ qeijk xj vk ‡ Ieijk nj n_ k ; where I is a constant with dimensions of moment of inertia per unit of volume of liquid, and eijk is the antisymmetric unit tensor. In general, taking into account the body moment mi and surface moment Qi ˆ lei le connected with a couple stress tensor lie , the right part of the equation for angular momentum of continua (3) is de®ned by the relations: nei ˆ eijk xj rek ‡ lei ;

Mi ˆ eijk xj fk ‡ mi :

…4†

Here le is the unit vector in the direction of an external normal to the surface of the volume considered. The equations of motion (1±3) indicate how the mass, momentum, and angular momentum of the nematic liquid change with position and time. According to Eq. (2), the rate of increase of momentum in liquid element is equal to the total force applied to this element. Similarly, according to Eq. (3), the rate of increase of the moment of momentum is equal to the total applied moment. The local notation for the equation of balance of momentum (2) represents Cauchy's ®rst law of motion. In the case of isotropic liquids the balance equation of angular momentum (3) is reduced to a condition of symmetry of a stress tensor rij ˆ rji . This relation de®nes Cauchy's second law of motion, which does not depend on the reference system. Thus, for isotropic liquids the law of conservation of angular momentum does not give any additional di€erential equation of motion. For closing the main equations of motion (Eqs. 1±3) it is necessary to formulate the constitutive equations establishing connection between stress tensors rij ; lij and the kinematic quantities, taking into account the physical properties of the anisotropic polymeric liquids under consideration.

Elementary non-symmetric viscoelastic model We consider an incompressible viscoelastic liquid with a single preferred direction n. The total stress tensor of nematic liquid can be represented as a sum of three parts: rij ˆ

pdij ‡ reij ‡ r0ij

…5†

where p is the pressure, reij is the part of the stress tensor connected with the spatial inhomogeneity of the director ®eld, and r0ij is the dissipative stress tensor. According to

362

Ericksen (1976), the stress tensor reij is related to the Frank elastic energy Fd : oFd ne;j : one;i

reij ˆ

…6†

In an incompressible nematic liquid, for isothermal strains, the free energy density F consists of free energy of the undeformed nematic F0 and the deformation energy. The latter for incompressible medium is Fd ˆ F

F0 ˆ F …ni ; ni;j † ;

where ni;j ˆ oni =oxj is the director gradient. For small strains the energy of nematic medium connected with the distortion of the director ®eld is de®ned as a quadratic form of orientational gradients: Fd ˆ 12 Kijke ni;j nk;e ; where the elastic coecient tensor Kijke is speci®ed from the symmetry conditions. The e€ects caused by the spatial distribution of director orientations are usually analyzed by considering the free energy introduced by Frank (1958) 2Fd ˆ …K1

K2 †ni;i nj;j ‡ K2 ni;j ni;j ‡ …K3

K2 †ni nj nk;i nk;j ; …7†

where K1 ; K2 and K3 are the Frank moduli which have the dimensions of force. The basic task of the present work is designing the constitutive equation for a non-symmetric stress tensor r0ij , taking into account the anisotropic relaxation in polymeric nematics. We shall proceed from the simple assumption of the anisotropic relaxation law of stresses. It is possible to propose a relaxation constitutive equation of the ®rst order, resolved relative to the rate of stress tensor. Thus: D 0 r ˆ fij …r0ke ; cke ; ne ; Ne † : …8† Dt ij The corotational or Jaumann tensor derivative is de®ned by Dr0ij

dr0ij

‡ r0ie xej xie r0ej : Dt dt This is the simplest invariant (frame indi€erent) derivative in its formal mathematical properties. According to experimental data, the stress in liquids induced by ¯ow depends on the rate of strain, instead of strain as in solids. Therefore in Eq. (8), a functional relation between a stress tensor and strain rate tensor cij ˆ …vij ‡ vji †=2 representing a symmetric part of the velocity gradient of a liquid vij ˆ ovi =oxj is considered. The stress state in a moving nematic liquid also depends on its orientation n and the rate of orientation change Ni ˆ Dni =Dt. Here Dni =Dt ˆ dni =dt xie ne is the Jaumann director derivative and xij ˆ …vij vji †=2 is the ˆ

antisymmetric part of velocity gradient in the liquid. The rotation of the director is an additional source of dissipation in the nematic even in the absence of ¯ow. According to expression (8), the general quasi-linear dependence r0ij of cij and Ni is de®ned by the equation Dr0ke ‡ r0ij ˆ gijke cke ‡ bijk Nk ; …9† Dt which includes non-linear terms like x
r0 . The constitutive equation (9) describes the anisotropic viscoelastic liquid in the case of small recoverable strains in comparison with the total strains. Such a case is realized in weakly elastic liquids to which the monodisperse LC polymers should be assigned. The tensors gijke …n† and bijk …n† have dimensions of viscosity and tensor sijke …n† has the dimension of relaxation time. These tensors characterize a nonequivalence of medium properties in various directions. The constitutive equation (9) is a generalization of the known Maxwell equation in the case of anisotropic liquids. The relaxation equation for the deviator stress of an incompressible isotropic Maxwell liquid (1867) is an outcome of the anisotropic relaxation law (9) with special de®nition for viscosity and relaxation time tensors sijke

sijke ˆ sI…ij†…ke† ;

gijke ˆ 2gI…ij†…ke† ;

bijk ˆ 0 ;

…10†

where I…ij†…ke† ˆ …dik dje ‡ die djk †=2 is a unit tensor symmetric with respect to the indices in parentheses. The viscosity and relaxation time tensors of the viscoelastic liquid with di€erent anisotropies can be de®ned on the basis of the general invariant form of connection between the tensorial ®elds characterizing their movement, and their physical and geometrical properties. The di€erent types of liquid crystals that correspond to various symmetries are: nematic, cholesteric and smectic. For nematic viscoelastic liquids, the tensors g, s and b are transversely isotropic relative to the unit vector n. In addition, taking into account the invariance of a stress tensor relative to the transformation ni ! ni (n and n physically are equivalent), we have for incompressible liquids bijk ˆ g5 ni djk ‡ g6 nj dik ; gijke ˆ g1 I…ij†…ke† ‡ g2 nijke ‡ g3 ni dj…k ne† ‡ g4 nj di…k ne† ; sijke ˆ s1 Iijke ‡ s2 nijke ‡ s3 ni djk ne ‡ s4 nj dik ne ‡ s5 ni dje nk ‡ s6 nj die nk ; …11† where Iijke ˆ dik dje is a unit fourth-rank tensor. In these expressions the following notations are introduced: nijke ˆ ni nj nk ne ;

2ni dj…k ne† ˆ ni djk ne ‡ ni dje nk :

Here, the symmetrization is made from the indices given in the parentheses. By virtue of weak magnetization of

363

the considered anisotropic liquids, the in¯uence of an aligning magnetic ®eld H on material functions can be neglected. This considerably simpli®es their constitutive equations. Using tensor relations (11), the constitutive equation for dynamic stress tensor of an incompressible nematic viscoelastic liquids can be derived from Eq. (9): Dr0ij

Dr0ej Dr0ek s1 ‡ s2 nijek ‡ s3 nie Dt Dt Dt 0 0 Dr Drei Dr0ie je ‡ s5 nei ‡ s6 nej ‡ r0ij ‡ s4 nje Dt Dt Dt ˆ g1 cij ‡ g2 nijek cek ‡ g3 nie cej ‡ g4 nje cei ‡ g5 ni Nj ‡ g6 Ni nj :

…12†

This equation (Volkov, 1998) involves six viscosity constants g1 ±g6 and six relaxation times s1 ±s6 . The basic di€erence between the dynamics of liquid crystals and the dynamics of isotropic liquids is that the rotation of the director results in a dissipation of energy. This e€ect is characterized by the viscosity coecients g5 and g6 . The other viscosity coecients g1 ±g4 characterize dissipation due to the presence of a velocity gradient in the moving liquid. The stress tensor r0ij can be decomposed into symmetric r0…ij† and antisymmetric r0‰ijŠ parts: r0ij ˆ r0…ij† ‡ r0‰ijŠ :

…13†

The square brackets about two indices indicate antisymmetrization with respect to those indices r0‰ijŠ ˆ …r0ij r0ji †=2. Proceeding from the Eq. (12), we arrive at a set of coupled di€erential equations. For symmetric stresses, ! Dr0…ij† Dr0…ek† l1 Dr0…ej† Dr0…ei† s1 ‡ s2 nijek ‡ nie ‡ nje Dt Dt 2 Dt Dt ! Dr0‰ejŠ Dr0‰eiŠ l ‡ nje ‡ r0…ij† ‡ 2 nie 2 Dt Dt ˆ g1 cij ‡ g2 nijek cek ‡

m1 …nie cej ‡ nje cei † 2

m2 …ni Nj ‡ Ni nj † ; 2 and for antisymmetric stresses,

…14†

‡

s1

Dr0‰ijŠ Dt

0

‡

Dr‰ejŠ l3 nie 2 Dt 0

Dr…ej† l ‡ 4 nie 2 Dt

nje

nje

Dr0‰eiŠ

Dr0…ei† Dt

!

Dt ! ‡ r0‰ijŠ

m3 m4 …nie cej nje cei † ‡ …ni Nj Ni nj † …15† 2 2 Here the following notations are introduced for relaxation times and viscosities: ˆ

l1 ˆ s3 ‡ s4 ‡ s5 ‡ s6 ;

l2 ˆ s3 ‡ s4

l3 ˆ s 3

l4 ˆ s3

s4

s5 ‡ s6 ;

m1 ˆ g3 ‡ g4 ; m4 ˆ g5

m2 ˆ g5 ‡ g6 ;

s5

s6 ;

s4 ‡ s5

s6 ;

m3 ˆ g3

…16†

g4 ;

g6 :

Let us consider the special case l2 ˆ 0 and l4 ˆ 0 for which the results simplify considerably. According to Eqs. (14) and (15), this choice of values l2 and l4 corresponds to the decoupled relaxation of symmetric and antisymmetric stresses. In this case, we obtain the following constitutive equation from Eq. (12):   Dr0ij Dr0ej Dr0ie Dr0ek s1 ‡ s2 nijek ‡ s3 nie ‡ nej Dt Dt Dt Dt   Dr0ei Dr0je ‡ nei ‡ r0ij ‡ s4 nje Dt Dt ˆ g1 cij ‡ g2 nijek cek ‡ g3 nie cej ‡ g4 nje cei ‡ g 5 n i Nj ‡ g 6 Ni n j ;

…17†

which involves four relaxation times. The director n, which is included in the above rheological equations of nematic polymer liquids, is de®ned using an additional equation describing the change in orientation induced by ¯ow. This equation re¯ects unique properties of anisotropic liquids and has no analogue in the case of isotropic liquids.

Orientation dynamics In conditions of non-stationary external in¯uence the orientation of a nematic liquid n changes in time. Using Eq. (15) for antisymmetric stress, one can derive the motion equation for director in an external magnetic ®eld H directly from the equation of angular momentum (3). The couple stress tensor lie (Aero and Bulygin 1973) is: lei ˆ eijk nj pek ;

…18†

where pij ˆ oFd =onj;i . Using this relation, we obtain the following expression for n from Eq. (3): Ieijk nj

d2 nk ˆ eijk rjk ‡ eijk nj;e pek ‡ eijk nj pek;e ‡ mi : dt2

…19†

The body moment m connected with magnetic ®eld is de®ned as follows: mˆMH where M ˆ v? H ‡ va …n
H†n is the magnetization induced by an applied magnetic ®eld H, va ˆ vk v? is the magnetic anisotropy, and vk ; v? are the magnetic susceptibilities parallel to and perpendicular to the director respectively. The magnetic moment can be considered as a result of the e€ective ®eld hm on the director:

364

mi ˆ eijk nj hm k ;

…20†

where hm i ˆ va ne He Hi has the dimension of body force. The invariance condition of free energy relative to the rigid rotations results in the following relation (Ericksen 1961):   oFd oFd oFd eijk nj ˆ0 : …21† ‡ nj;e ‡ ne;j onk onk;e one;k Taking into account the expression for the Ericksen stress tensor reij , this relation can be written as follows: eijk rejk ˆ eijk nj hek

eijk nj pek;e

eijk nj;e pek :

…22†

According to de Gennes (1974), the molecular ®eld hei has the form   o oFd oFd e : …23† hi ˆ oxj oni;j oni Using the relations (5), (20) and (22), we obtain the following moment equation from Eq. (19): Ieijk nj

2

d nk ˆ eijk nj hk ‡ eijk r0jk : dt2

…24†

According to Eq. (24), the internal elastic forces of a nematic and the external forces acting on the director in a magnetic ®eld can be described by a single molecular ®eld hi ˆ hei ‡ hm i . In order to ®nd the constitutive equation for the director describing orientation of a nematic liquid we vectorially multiply the moment equation (24) by ni : Iie?

d2 ne 0 ˆ h? i ‡ 2ne r‰eiŠ : dt2

…25†

ni ne he is the Here Iie? ˆ I…die ni ne †, and h? i ˆ hi transverse component of the molecular ®eld. The longitudinal component h has no physical sense. Eq. (25) leads to the relation: r0‰ijŠ ˆ n‰i gjŠ ;

…26†

where the following notation gi ˆ I

d2 n i dt2

hi

is introduced. Multiplying both sides of Eq. (15) by 2ni and forming the scalar product 2ne r0‰eiŠ ˆ g? i , we obtain the motion equation for director: l5 Iie

Dg? e ‡ g? i ˆ m3 …ne cei Dt l4 n e

nien cen † ‡ m4 Ni Dr0…ei† Dt

Dr0en nien Dt

! :

…27†

When deriving this equation, we have used the following expression:

2ne

Dn‰e giŠ Dg? e ˆ Iie Dt Dt

In Eq. (27) g? nie ge is the transverse component i ˆ gi of vector ®eld g and Iie ˆ die ni ne . This orientation equation involves two relaxation times l4 , l5 ˆ s1 ‡ l3 =2 and two friction factors m3 ; m4 having dimensions of viscosity. Excluding from the analysis any processes occurring over short time intervals, we can neglect the inertia of microstructural elements of nematics. In this case gi ˆ hi and the director equation (27) for the decoupling approximation is reduced to a simpler form: Dh? e ‡ h? nien cen † ‡ c1 Ni : …28† i ˆ c2 …ne cei Dt The coecients c1 ˆ g6 g5 and c2 ˆ g4 g3 have the dimension of viscosity and are related to the Leslie viscosity coecients. The relaxation time h2 is de®ned as: h2 Iie

h2 ˆ s1 ‡ s3

s4 :

…29†

For a viscous anisotropic model of nematic liquid, when h2 ˆ 0, the orientation equation (28) reduces to the Leslie±Ericksen equation: Dni 1 ˆ …hi c1 Dt

nie he † ‡ k…cie ne

niek cek † :

…30†

On the right-hand side of Eq. (30), the ®rst term describes the relaxation of the director towards equilibrium under the action of a molecular ®eld. The second term leads to a coupling between the orientation of the director and shear ¯ow. The material parameter k is de®ned as k ˆ c2 =c1 and depends essentially on the geometric form of the molecules. According to Eq. (30), two types of nematics can be expected, depending on the material parameter k. In the ®rst case for hi ˆ 0 and jkj < 1, the director under steady-shear ¯ow aligns at a ®xed angle with respect to the ¯ow at all shear rates. Such nematics are called ¯ow aligning. In the second case for hi ˆ 0 and jkj  1, stable orientation does not exist. This type of nematics is called tumbling nematics. The tumbling (rotation) is more probable to occur for polymers than for low molecular weight nematics (Marrucci 1996). It should be noted that the complete orientation equation (30) with hi 6ˆ 0 predicts a shear ¯ow induced transition to tumbling (Pikin 1973; Siebert et al. 1997). The general equations of dynamics (1±3) together with the constitutive equations (5, 6, 12) and the orientation equation (27) de®ne a complete set of equations which allow investigation of the stress state and ¯ow of uniaxial polymer nematics. Using these equations, it is possible to analyze the anisotropy in slow relaxation processes and the rheological anisotropy of nematic polymers under various ¯ow conditions.

365

Asymmetry of shear viscosity We now consider a simple steady-shear ¯ow of anisotropic liquid along the x1 axis. This viscometric ¯ow is given by the velocity ®eld: v1 ˆ c_ x2 ;

v2 ˆ v3 ˆ 0

The shear stress components in (32) are not equal. Thus for anisotropic liquids with a non-symmetric stress tensor, two shear viscosities can be de®ned: r21 r12 g1 …_c† ˆ ; g2 …_c† ˆ ; …33† c_ c_ related to the shear stresses r12 and r21 . The ®rst index of a shear stress shows the direction of the outer normal to the surface element on which it acts. The second index shows the direction of its action. Since the shear viscosities of the liquid crystal (33) depend on the orientation of the director, it is convenient to measure under conditions in which the director is ®xed by an external ®eld (magnetic or electric). By suitably orienting the magnetic ®eld, Miesowicz (1946) was able to determine the three principal shear viscosities ga1 ; gb1 ; gc1 of the low molecular weight liquid crystal when the director was parallel to the direction of ¯ow, perpendicular to the plane of shear stress r21 , and in plane of shear and perpendicular to the ¯ow. The result obtained by Miesowicz shows that all viscosities are shear-rate independent. In a similar manner, the three principal viscosities ga2 ; gb2 ; gc2 associated with another shear stress r12 , which acts in a plane perpendicular to the plane of shear stress, r21 , can be also de®ned. In general, we have ®ve independent components: two shear and three normal stress components. Thus the stresses occurring in simple shear ¯ow of anisotropic polymer systems are characterized in terms of four viscometric functions: r2 …_c† ˆ r12 ;

N1 …_c† ˆ r11

r22 ;

N2 …_c† ˆ r22

ga1 ‡ ha1 ha2 …ga1 ga2 †…_c2 =4† c_ ; 1 ‡ ha1 ha2 …_c2 =2†

…35a†

ra2 …_c† ˆ

ga2 ‡ ha1 ha2 …ga2 ga1 †…_c2 =4† c_ ; 1 ‡ ha1 ha2 …_c2 =2†

…35b†

N1a …_c† ˆ

…ga1 ‡ ga2 †ha2 c_ 2 ; 1 ‡ ha1 ha2 …_c2 =2† 2

…31†

where c_ is the constant shear rate. The stress tensor rij completely characterizes the stress state at any point in the anisotropic liquid. Generally it has nine components. The situation is considerably simpli®ed in the case of simple shear. One of the physical principles requires that the stress tensor has to be invariant with respect to the coordinate system. As a result, the stress state of a liquid subjected to simple shear ¯ow is described by a simple matrix: r11 r12 0 r21 r22 0 : …32† 0 0 r33

r1 …_c† ˆ r21 ;

ra1 …_c† ˆ

r33 :

…34†

These material functions can be obtained directly from rheological equation (17):

N2a …_c† ˆ

ha3 a N …_c† ha2 1

…35c†

for three constant orientations of director, a ˆ a; b; c. Here a is the orientation parallel to the direction of ¯ow, b is the orientation parallel to the velocity gradient, and c is the orientation perpendicular to both the ¯ow and velocity gradient. The relaxation times characterizing the relaxation of shear …r21 ; r12 † and normal stresses …r11 ; r22 † for these basic orientations are de®ned as: a†

ha1 ˆ s1 ‡ s3 ‡ s4 ;

b†

ha3 hb1

ˆ ha1 ;

c†

hc1

hc3

ha2 ˆ s2 ‡ 2…s1 ‡ s3 ‡ s4 †;

ˆ s1 ;

ˆ

…36a† hb2 ˆ ha2 ;

ˆ s1 ;

hc2

hb3 ˆ s1 ‡ s2 ‡ 2…s3 ‡ s4 †;

ˆ 2s1 :

…36b† …36c†

The consetitutive equation (12) leads to the six zero shear viscosities connected with a shear stress r21 ga1 ˆ 12…g1 ‡ g4 ‡ g6 †;

gb1 ˆ 12…g1 ‡ g3 gc1 ˆ 12 g1 ;

g5 †;

…37†

and transverse (with respect to the direction to ¯ow) shear stress r12 ga2 ˆ 12…g1 ‡ g3 ‡ g5 †;

gb2 ˆ 12…g1 ‡ g4

gc2

ˆ

1 2 g1

g6 †;

…38†

:

When deriving the relations (35±37), we have supposed that an orienting (magnetic) ®eld is suciently strong to suppress boundary and shear ¯ow aligning e€ects. The stresses in liquids are restricted by the condition that the entropy production is non-negative. For isothermal ¯ow, this known thermodynamic restriction (De Groot and Mazur 1962) takes the form: T S_ ˆ r0ij …vji

eijm Xm †  0

…39†

and is equal to zero only in rigid motions. Here Xm is the angular velocity of director, and eijk is the completely antisymmetric tensor. Equation (39) shows that shear stress r021 occurring in simple shear ¯ow (31) must be positive in the absence of director rotation. This condition imposes an additional restriction on nonlinear Miesowicz viscosities associated with shear stress r21 :

366

_ ˆ ga1 …c†

_ ra1 …c† c_

…40†

where a ˆ a; b; c (always positive). According to Eq. (35a), the asymptotic behavior of shear stress ra21 for a ˆ a; b at high shear rates is described by linear dependence: ra21 ˆ 12…ga1

ga2 †c_ :

…41†

From Eq. (41) at condition ra21  0, we obtain the inequality ga1  ga2 ; a ˆ a; b. This implies that: Aˆ

ga2  1; ga1

Bˆ

gb2 1 gb1

…42†

where A and B are the asymmetric viscosity parameters. For liquids with a non-symmetric stress tensor, the rotational viscosity can be de®ned in the form: r21 r12 gr ˆ : …43† 2c_

Using these experimental values and Eqs. (37) and (38), we ®nd the following asymmetric viscosity parameters. A ˆ 0:5;

B ˆ 0:003

…48†

for nematic PBG solutions of the tumbling type. Figure 1 shows the reduced non-linear Miesowicz a _ a _ ˆ viscosities ga1 …c† pg 1 …c†=g1 as a function of the reduced shear rate ka ˆ ha1 ha2 =2c_ calculated according to Eq. (35a) for the three special orientations of the director: a ˆ a; b; c. The values of parameters A and B are given _ and gb1 …c† _ show two low by Eq. (48). The curves ga1 …c† and high shear rates plateaus and the intermediate _ decreases as c_ 2 power law region. The viscosity gc1 …c† _ for hign c. _ ˆ Figure 2 shows the reduced viscosities ga2 …c† a _ g2 …c†=ga2 associated with the transverse shear stress r12 . _ and gb2 …c† _ are negative at high shear The viscosities ga2 …c† rates.

Equations (35a, 35b) lead to the two rotational viscosities for simple shear: gar ˆ

ga1

ga2

gbr ˆ

;

gb1

gb2

: …44† 2 2 The third rotational viscosity gcr vanishes for orientation c. Thus the present theory, when specialized to the case of the decoupled relaxation of the symmetric and antisymmetric stresses, yields that the rotational viscosity is independent of shear rate for the considered (special) director orientations. From Eqs. (37), (38) and (44), we have: gar ˆ 14…c1 ‡ c2 †;

gbr ˆ 14…c1

c2 † ;

…45†

and using the Parodi relation g5 ‡ g6 ˆ g4

g3 ;

…46†

we ®nally arrive at: gar ˆ 12 g6 ;

gbr ˆ

1 2 g5

:

Fig. 1 Dimensionless viscosities ga1 …ka †=ga1 …0† as a function of the dimensionless shear rate ka for three special director orientations a ˆ a; b; c. The constant viscosity curve refers to that of the Leslie± Ericksen model

…47†

These relations re¯ect the physical meanings of viscosity coecients g5 and g6 . They control the orientational behavior of the nematic liquid. The coecient g5 is negative for rod-like nematics. The possibility of a positive g5 was pointed out by Carlsson (1982) for the disk-like nematics. Side-chain LC polymers may form prolate or oblate coils, thus negative and positive values of g5 are possible for such LC polymers. The measurement of a complete set of Leslie viscosities was carried out for the lyotropic nematic polymer PBG (poly-c-benzyl-l-glutamate) (Srajer et al. 1989; Han and Rey 1993). The six experimentally measured constants gi for PBG (in units of poise) used in this paper are: g1 ˆ 3:48; g2 ˆ 36:6; g3 ˆ 66:1; g4 ˆ g5 ˆ 69:2; g6 ˆ 0:18 :

2:92;

Fig. 2 Dimensionless viscosities ga2 …ka †=ga2 …0† versus dimensionless shear rate ka for the same director orientations as in Fig. 1

367

Frequency dependencies of Miesowicz viscosities The constitutive equation (17) also allows us to estimate the anisotropy of linear viscoelastic properties for polymeric nematics in the slow relaxation region. The essential feature of anisotropic polymeric liquids is that of complex shear viscosities g1 …x† ˆ r21 …x†=_c…x†;

g2 …x† ˆ r12 …x†=_c…x†

…49†

which depend on the relative orientation of the director and the ¯ow. By aligning the director using an external ®eld parallel to the direction of oscillating ¯ow, parallel to the velocity gradient and perpendicular to both these directions, it is possible to determine the three principal complex viscosities ga1 …x†, gb1 …x†, gc1 …x† associated with the shear stress r21 …x† and the three principal complex viscosities ga2 …x†, gb2 …x†, gc2 …x† associated with r12 …x†. Expression (49) re¯ect the e€ects of shear stress asymmetry for the anisotropic liquid under consideration. During oscillatory ¯ow, the shear stresses arising on platforms perpendicular to axes x1 and x2 are not equal, r12 …x† 6ˆ r21 …x†. We can use the modi®ed Miesowicz method for experimental studies of complex viscosities (49) which then allows measurement of both oscillating shear stresses r21 …x† and r12 …x† for the three basic directions of liquid orientation. Using the constitutive equation (17) and taking into account that for small amplitude motion the Ericksen stress tensor (6) can be neglected (quadratic on amplitude), we can calculate the shear viscosities (49) theoretically. In this case, the following expressions for the complex viscosities connected with a shear stress r21 …x† were obtained:   1 ga1 ‡ ga2 ga ga2 ‡ 1 ; ga1 …x† ˆ 2 1 ixh1 1 ixh2   1 gb1 ‡ gb2 gb gb2 ‡ 1 ; …50† gb1 …x† ˆ 2 1 ixh1 1 ixh2 gc1 …x† ˆ

gc1 : 1 ixs1

The complex viscosities associated with a transverse shear stress r12 …x† are   1 ga1 ‡ ga2 ga1 ga2 a ; g2 …x† ˆ 2 1 ixh1 1 ixh2   1 gb1 ‡ gb2 gb1 gb2 b g2 …x† ˆ ; …51† 2 1 ixh1 1 ixh2 gc2 …x† ˆ

gc2 : 1 ixs1

where h1 ˆ s1 ‡ s3 ‡ s4 and h2 is de®ned by Eq. (29). The relaxation times h1 and h2 characterize the relax-

ation of the symmetric and antisymmetric parts of shear stress. The characteristic relaxation time h2 de®nes the frequency dependence of rotational viscosity (43), which depends on the director orientation. For the anisotropic viscoelastic model (17), we obtain: gar …x† ˆ

gar ; 1 ixh2

gbr …x† ˆ

gbr 1 ixh2

…52†

where gar and gbr are the rotational viscosities in the low frequency limit. They are de®ned by Eqs. (44) and (47). The rotational viscosity gcr …x† vanishes for orientation c. Figure 3a and 3b show the dimensionless real and imaginary components of the complex viscosities ga1 …x† and gb1 …x† as a function of the dimensionless frequency x ˆ xh1 : A and B were accepted to equal to 0.5 and 0.003, respectively. The ®gures show typical behavior of a polymer liquid with two relaxation times. Figure 3c shows the dimensionless real and imaginary parts of the third complex viscosity gc1 …x† as a function of the dimensionless frequency x ˆ xs1 . This ®gure shows the behavior of a Maxwell-type liquid. The dimensionless real and imaginary components of the complex viscosities ga2 …x† and gb2 …x† associated with the transverse shear stress r12 …x† are presented in Fig. 4a and 4b. These frequency dependencies show behavior unknown for isotropic polymer liquids. The real components exhibit an initial growth followed by a decrease. The imaginary components are negative at low frequencies.

Conclusions The subject of this work is anisotropic viscoelasticity of monodisperse LC polymers characterized by the tensor relaxation processes and non-symmetric stress tensor. The important feature of these anisotropic liquids is that they combine the properties of liquid crystals and polymers. The relaxation processes depend on the speci®c structure of these polymers and are anisotropic. The key problem is the experimental and theoretical study of the tensor relaxation processes in anisotropic polymers under various mechanical actions. It is well known that there are signi®cant di€erences between the monodisperse and polydisperse ¯exiblechain polymers. A characteristic feature of monodisperse polymers is that the transition from the terminal zone to the plateau zone of the storage modulus is abrupt. This demonstrates that the viscoelasticity of linear monodisperse polymers of high molecular weight in the slow relaxation region corresponds almost to a single terminal relaxation time. Thus, the simplest hypothesis is to assume that the terminal region of the anisotropic relaxation spectrum of monodisperse LC

368

Fig. 4 Dimensionless real and imaginary parts of the complex viscosities ga2 …x†=ga2 …0†; a ˆ a; b versus dimensionless frequency x. The line types correspond to the same conditions as in Fig. 3

Fig. 3 Dimensionless real (Re) and imaginary (Im) parts of the complex viscosities ga1 …x†=ga1 …0†; a ˆ a; b; c versus dimensionless frequency x ˆ xh1 …a; b† and xs1 …c† for values of the parameter w ˆ h2 =h1 : 10 (dashes) and 103 (full lines)

polymers is de®ned by a single tensor relaxation time. This is the main characteristic feature of our continuum model. The relaxation constitutive equation (12) presented here describes anisotropy of viscosity as well as the

anisotropy of slow relaxation processes in monodisperse polymer nematics. A major prediction of the theory is that the shear thinning characteristics are anisotropic. There is ``more'' shear thinning in one direction than another. For the correct experimental testing model, it is necessary to carry out investigations of non-linear viscoelasticity for monodisperse LC polymers in the slow relaxation region. Unfortunately, such data are still absent. The signi®cance of these observations is not understandable at present. The main result of available experimental works (Zimmermann and Wendor€ 1988; Kulichikhin et al. 1995) is the existence of anisotropy of relaxation times in LC polymers. This fact supports the major supposition of the considered continuum theory. It should be added in this context that existing experimental data for the ®rst normal stresses in LC polymers require better understanding. For instance, using the same polymer, Guskey and Winter (1991) observed the negative values of ®rst normal stress di€erence, whereas Cocchini et al. (1991) observed the positive ones.

369

Our choice of the simplest invariant derivative for stress tensor in anisotropic constitutive equation is based on the research experience in the quasi-linear dynamics of monodisperse ¯exible-chain polymers. According to papers by Vinogradov (1977) and Yakobson and Faitelson (1985), a simple Maxwell model with Jaumann time derivative can describe the viscoelastic properties of practically monodisperse ¯exible-chain polymers in the slow relaxation region. It is of interest to mention that this result is consistent with the predictions of the nonreptation molecular model of non-linear polymer viscoelasticity (Volkov and Vinogradov 1988). It represents the entangled macromolecule as moving in a viscoelastic medium. The principal purpose of future investigations in anisotropic viscoelasticity of LC polymers is the study of the various invariant derivatives for non-symmetric stress tensor.

In this study we have investigated the e€ect of ``asymmetry'' on the shear viscosity of nematics (33). Starting from general arguments, we have pointed out that a transverse shear stress (with respect to the direction of ¯ow) in LC polymers can be negative. The measurement of six principal shear viscosities (50, 51) or shear and rotational viscosities (50, 52) by non-traditional viscometry is of great interest for investigating the anisotropy of relaxation processes in anisotropic polymer liquids. There are still many open problems and this approach would be interesting to pursue. Acknowledgment During this research V. S. Volkov was supported, in part, by the U.S. National Science Foundation through the Department of Polymer Engineering, the University of Akron via grant DMR-9700928 from the Polymers Program.

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