I Center for Composite Materials and Department of Chemical Engineering,. University of ... 'AT&T Bell Laboratories, Murray Hill, New Jersey 07974 (U.S.A.).
243
Journal of Non-Newtonian Fluid Mechanics, 36 (1990) 243-254 Elsevier Science Publishers B.V., Amsterdam
GENERALIZED CONSTITUTIVE LIQUID CRYSTALS PART 2. NON-HOMOGENEOUS
BRIAN J. EDWARDS ‘, ANTONY and RONALD G. LARSON 3
EQUATION
FOR POLYMERIC
SYSTEMS
N. BERIS *, MIROSLAV
GRMELA
*
I Center for Composite Materials and Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 (U.S.A.) * Ecole Polytechnique de Montreal, Mont&al, Qu&bec H3C 3A7 (Canada) ‘AT&T Bell Laboratories, Murray Hill, New Jersey 07974 (U.S.A.) (Received
November
27, 1989)
Abstract The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets was used in Ref. 1 to develop a constitutive equation for the stress and the order parameter tensor for a polymeric liquid crystal. These equations were shown to reduce to the homogeneous Doi equations as well as to the Leslie-Ericksen-Parodi (LEP) constitutive equations under small deformations [l]. In this paper, these equations are fitted against the non-homogeneous Doi equations through the simulation of the spinodal decomposition of the isotropic state when it is suddenly brought into a parameter region in which it is thermodynamically unstable. Linear stability analysis reveals the wavelength of the most unstable fluctuation as well as its initial growth rate. Results predicted from this theory compare well with the predictions of Doi for the spinodal decomposition using an extended molecular rigid-rod theory in terms of the distribution function. . This completes the development of a generalized constitutive equation for polymeric liquid crystals initiated in Part 1. Keywords: Hamiltonian fonnulation; Poisson bracket; polymeric liquid crystals; constitutive equation; tensorial order parameter
1. Introduction Recently, the Hamiltonian (Poisson bracket) formulation was used to produce a more general theory for polymeric liquid crystals [l]. The gener-
244 alized constitutive equation which is generated in Ref. 1 gives predictions very close to the Doi theory for homogeneous flows and reduces to the equivalent expressions obtained from Leslie-Ericksen-Parodi (LEP) theory for small deformations. In Part 1, the translational diffusion parameter, B, was unspecified so that the initial model formulation remained as simple as possible. Now, however, the issue of non-homogeneous systems must be addressed. To do this, we must first specify the dissipation parameter, B, which will be accomplished by a direct comparison with the extended model of Doi [2,3] involving the distribution function of polymer molecules. In so doing, we are able to rewrite his equations in a manner which is consistent with the order parameter representation. A linear stability analysis is then performed to test the model predictions in the simple case of spinodal decomposition of the isotropic state (in a thermodynamically unstable concentration regime) to the nematic state. Results of this test are analogous to the much more involved calculations of Doi [3], involving the distribution function, and allow specification of the parameters b,. Thus, all parameters needed in the new formulation are specified here and tested so that they agree with the presently available molecular theories of liquid crystalline structure. However, previous models have suffered from limited regions of applicability or from the complexity introduced into the formulation by using a distribution-function approach. Our new formulation in terms of the order parameter tensor remains simple enough, yet faithful enough to the physics of the liquid crystalline systems, to use as a basis for future stability analyses. This paper is organized as follows. In Section 2, the parameters in the constitutive equation for the orientation order parameter are determined by a direct comparison with Doi’s extended rigid-rod theory. In Section 3, spinodal decomposition is simulated through a linear stability analysis of the constitutive equation for the order parameter tensor. Section 4 briefly presents our conclusions from this work. 2. Generalized constitutive
equation for polymeric liquid crystals
Here, use is made of a second-order
orientational
parameter,
S, definbd
as
(2-l) where m is the second-moment
of the distribution (2.2)
245 where $( r, it, t) is the distribution function of the molecular orientation, and r, n and t are the position vector, the unit vector of orientation and the time, respectively. From the definition (2.2), both S and m are symmetric and = 0,
(2.3a)
tr(m) = 1.
(2.3b)
tr(S)
The portion of the free energy (Hamiltonian) density dependent upon the order parameter tensor for the generalized model is H(S)
= -&J&(S)
dV-
&J&(S)
dV,
(2.4)
where a,, is the homogeneous contribution to the entropy, WS)
= - L%YSYol + $GG&3SP~
- &J,,S,pSP,SC~
- $a;( S&J*, (2.5)
and Qe is the inhomogeneous (Frank elastic) contribution, @e(S)
= - &%/G&u
- %%,s,nSvp,u - &%,S,+%,,.
(2.6)
Following Ref. 1, the evolution equation for the order parameter tensor is
(2.7) where 6H/6Sa,, the functional derivative of H with respect to Sny, is given by Dl, 6H = %J( mx,
a2Syu- a3( Sy&3, - t~yaS2k3)
+ a4(Sy&3S& - ~s,&?&S~/3) + d%,S~aSa~ - WauJ3.P - b2(+Spu,p,a+ :Spu,p,y- %Spc.s.c) +b&+&,~
- S&%y,c,~- %dav.r
- &xuS~&3~.s)).
(2.8)
246 Here and in the following, we use Einstein’s summation convention (i.e. repeated indices imply summation) and a comma to denote spatial differentiation. Subsequently, the term proportional to b, will be neglected since it is higher-order than the other non-homogeneous terms and does not add qualitatively to the physics of the situation. The stress constitutive equation is
(2.9) With the exclusion of the B and bi parameters, which control the effects of spatial inhomogeneities in the orientation, all the parameters involved in the above three expressions have already been determined in Ref. 1 in terms of the Doi theory for homogeneous systems of rigid rods. This was done by comparing the molecular equations for the time evolution of the order parameter tensor and the free energy with the equivalent expressions provided by eqns. (2.4), (2.5) and (2.7). The two expressions are identical for spatially homogeneous distributions if (2.10a) and al=-
9 l-2 i
u , 3 1
a3=$U,
ai=
%U,
(2.10b)
provided that the Doi closure relationship for the fourth-order average is used [l]. The term proportional to a4 has been dropped, since tr(S . S - S. S) = (1/2)tr(S. S)2 [4], and since it gives identical contributions with the aA term to the free energy. (For a much simpler proof than [4], see Appendix A.) As shown in [l], the viscous stress of the Doi model can also be recovered as well, simply by letting Mayfir = ~{~~~in,,,rn~~, and the solvent stress is obtained when r = 7,. Thus, the parameters of the present theory are based on a consistent averaging of the molecular Doi theory for rigid rods. As such, some of the molecular details (but hopefully not the important ones) are lost in the final form of the equations, (eqns. 2.7-2.10); however, a major advantage is realized. Namely, the molecular distribution function I,L is eliminated from the equations while the thermodynamic consistency of the equations is
247 preserved because of the adherence to the Poisson bracket formalism. Elimination of 4 = I/.J( r, n, t)-which is, in general, a six-dimensional function-makes the numerical solution of the resulting equations practical for interesting inhomogeneous problems, such as spinodal decomposition in its early, intermediate and late stages and flow-induced texture evolution in liquid crystals. Furthermore, the thermodynamic consistency guaranteed from the Poisson bracket formalism lowers significantly the risk of observing aphysical behavior introduced from the averaging of the more detailed molecular description. The remaining parameters, B and bj, are determined from the more recent, inhomogeneous Doi theory for concentrated solutions of rigid rods [2,3] as follows. The parameter B is determined from Doi’s translational diffusivity term in the following paragraphs. The parameters bi are obtained from the inhomogeneous excluded volume (molecular interaction) effects by comparing the linear stability analyses of spinodal decomposition from the isotropic to the nematic phase, as described in the next section. The translational term in the extended Doi diffusion equation [2,3] (neglecting the effects of the flow field and the rotational terms, which were addressed in Ref. 1) can be rewritten as (2.11) which also can be identified as (2.12) where pA denotes the free energy density, defined from the free energy expression in terms of the distribution function [2,3,5] A=
1
p,G(r,
= ck,T
/
n, t) dn dV
(In #( r, n, t) + W(r,
n, t))$(r,
n, t) dn dV,
(2.13)
where w is the potential of the molecular field expressed in terms of the interaction potential W( r - r’, n, n’) [2,3] as @=
1
W(r-
r’, n, n’)#(r’,
n’, t) dn’ dV’.
(2.14)
In eqns. (2.4), (2.5) and (2.6), the Landau-de Gennes expression for the free energy is used in lieu of the alternative free energy expression provided by eqn. (2.13). Furthermore, the parameters a, in eqn. (2.59, listed in eqn. (2.10), were obtained by fitting the Doi free energy expression [5]. Therefore,
248
in order to compare the translational term of eqn. (2.7) with eqn. (2.11), we need to obtain an alternative expression to eqn. (2.11) for the portion of the free energy depending on the order parameter tensor, H(S), which will involve eqn. (2.4) in terms of an integral over the entire space r 8 n. One way to obtain this is to assume an approximation for the free energy of the form 1 dH
H=Ho+-2-dS:S.
(2.15)
This approximation becomes exact in the limit of small S when the free energy reduces to a quadratic functional of S. Thus, the free energy A’ is now written as
A’=&+
:~[n,n,-',S,,]~#(r,
n,
t) dn dV/,
Ya
(2.16)
where A, involves the portion of the free energy which does not depend S (kinetic energy). Then, pA can be approximated as
on
(2.17) As a consequence, our formalism as
eqn. (2.11) can be rewritten
in a form compatible
with
.
(2.18) If now eqn. (2.18) is multiplied by n,np and integrated over n, then an equivalent expression for the evolution of the order parameter tensor is obtained:
where the decoupling approximation (n,npnin,n,n,) = (n,n,) ( npng)(n,n,) is used in order to preserve the symmetry of B, and 6,,( ~H/&l$,,) = 0 since the functional derivative is traceless. Equation (2.19) does not, in general, guarantee the symmetry and unit trace characteristics of m. However, the corresponding equations for mpa and tr(m) can be formulated and used together, as in eqn. (2.7), for the definition of am,,/&. (We have tried to make the comparison as simple as possible.) Then, comparison of eqn. (2.7) with a modified form of eqn. (2.19) leads to the following expression for B B ally& = mq [(h,,-A,)m,,+h,6,,]mgj,
(2.20)
249 where A,,=---
Dll 2ck,T
D,
’
(2.21)
___ 2ck,T’
AL=
Equation (2.20) is positive-definite, provided that A,, 2 A _L, as is always the case for polymeric liquid crystals. Thus, this definition is acceptable from a thermodynamic point of view (see Refs. 1 and 6). Although the qualitative character of the sixth-order tensor B is well represented by the above theoretical description, the exact numerical values of it are not, due to the assumptions involved in the derivation. An alternative approach can be constructed as follows. In the small concentration limit, 0, and at equilibrium, the distribution function is constant, #=$,=1/47r. F or cases which are perturbed slightly from equilibrium, the distribution function can be approximated by a truncated series in terms of the order parameter tensor S, which as a first-order approximation is
w=
#=A exp(PS:nn)=&(l+jB:nn), with the parameter S = $(nn
- +a)#
p determined dn = &S:
or, /3 = H/2. Therefore, eqn. (2.13) reduces to A = ck,T = ck,T = ck,T
s / /
(2.22) from the consistency
/nnnn
I
dn = %S,
in this limit, the free energy expression
(2.23) provided
by
In+ 4 dn dV nn)-&l+
(-ln(4r)
+ FS:
( - ln(4r)
+ YS : S) dV;
where use was made of the identities &
requirement
n,np dn = +Saa,
F-S:
nn) dn dV
(2.24) [7] (2.25a)
and (2.25b)
SH 6s = lSck,TS.
(2.26)
250
On the other reduces to
hand,
in the same small concentration
limit,
eqn. (2.11)
(2.27) and gives, through multiplication by n,np - (1/3)~?,~ n, the evolution equation for Sup
and integration
over
which leads, through pre-averaging of the translational diffusivity, to an expression similar to eqn. (2.7) with the sixth-rank tensor B identified as
BWY&r= &4&,,-
b)m,,+ Ls,c]s,p
(2.30)
Note that the expression for the sixth-rank tensor B obtained through eqn. (2.30) is very similar to that of eqn. (2.20) in the limit of an isotropic (S -+ 0) dilute solution of rigid rods, the only difference being that the factor 2/15 is replaced by l/9. This small difference is attributed to the closure approximation necessary for the development of eqn. (2.20) even in the limit of no intermolecular interactions. Either eqn. (2.20) or eqn. (2.30) can, in principle, be used to describe translational effects for concentrated solutions, each one involving specific simplifying assumptions. Which one is more appropriate can be seen only after the completion of more detailed (numerical) calculations in the non-linear regime. As seen in the following section, for the linear stability analysis of the spinodal decomposition. both formulae for B give practically identical results. 3. Kinetics of spinodal decomposition As an example of the applicability of the present model to the study of non-homogeneous systems, the initial stages of the spinodal decomposition are calculated as a liquid crystal in the isotropic state (S = 0) is brought into a parameter region where the isotropic state is unstable. Recently, Doi [3] has investigated the same problem by examining the initial behavior of the fluctuation of the concentration and orientation in the isotropic phase. A linear stability analysis of a Fourier component of the fluctuation led to a system of coupled differential equations for the components of the order parameter tensor which revealed three different fluctuation modes; the growth rates and the most unstable wavelength were subsequently determined. In this section we show that the same results can be produced using the present model with much less effort.
251 In the initial stages of the disturbance, it (u = O), and that the parallel translational mechanism for the molecules (A = 0, A I = and retaining only terms linear with respect parameter tensor, eqn. (2.7) becomes
VDll
= - 27 ( a2Sap -
is reasonable to assume no flow diffusion is the only relaxation 0) [3]. Under these assumptions, to the components of the order
blKQ3,p.p - b&&,p,s
+ :Spp,r,,
- f%~%+,~));~,~; (3 -2)
where we have used (3.3) which is the limiting (S = 0) expression for B arising from either of eqns. (2.20) or (2.30) with the numerical (order one) constant v assuming the values of l/2 or 3/5 respectively. To investigate the time evolution of various modes, let Sk be the kth Fourier component of the order parameter tensor Sk = Real[ Ak( t) eikx3],
(3.4
in general complex, where Ak(t) is a traceless, symmetric tensor. Substitution of eqn. (3.4) into eqn. (3.2) leads to a system of five independent, ordinary differential equations coupling the five independent components of Ak( t). As already observed by Doi [3], these equations can be separated into five independent sets of equations, each one governing the (initial) evolution in time of five orientational modes. These are equations involving Ak12, A kll - Ak22, Ak13, Ak23 and Ak33In particular, the equations have exactly the same form discovered by Doi [3] and separate the fluctuation modes into three types. (1) The “twist” A k22, is &A
2vDll
k12=
-
-
3L2
mode, with similar equations
8bl
-K4 9L2
1
followed
Ak12,
by A,,,
and A,,, -
(3.5)
length of the molecule. where K = kL/2 with L being a characteristic equation (eqn. (3.2) in Ref. Equation (3.5) is the same as the corresponding
252 3) with the only (minor) difference being that the numerical factor 2v/3 appears instead of 4/7, provided that the coefficient b, is defined as h&.
(3.6)
to a For U < 3 the coefficient of Akr2 in eqn. (3.5) is negative, corresponding negative eigenvalue, which implies a decaying fluctuation for every wavelength k. For UP 3 however, the fluctuation will grow for small enough wavelengths with the maximum growth rate A, h
= 6vL+(1 - u/3)* n2 UL2
attained
for the “most
(3.7)
) unstable”
wavenumber
k,,,
k,, = ;
(3.8)
Thus U = 3 corresponds to the critical concentration beyond which the isotropic state becomes unstable to infinitesimal perturbations, in agreement with the free energy analysis of the static system. (2) The “bend” $Ah13
mode, with similar equations
= - -2vDll
K*+
3L2
followed
by A,,,
and A,,,,
is
(3.9)
4(269'L:b2)K4]Ak,3.
Equation (3.9) is the same as the corresponding equation (3.3) in Ref. 3 with the only (minor) difference that the numerical factor 2v/3 appears instead of 12/7 provided that the coefficient h, is defined as (3.10) (3) The “splay” and Ak22, is $A,;;=
--
mode, with the following
2vDll
1
3L2 [i
coupled
equations
for AA33, A,,,
1h33? K4A f%1 w 8bI
8(3h, + 2&) 27L2 -K4 9L2
A,,,
K4
- -
A,
3L2
(3.11)
27L2
x33>
(3.12)
2vDll 1 $A,,,= - __ 3L2 [i
1
K4 Am
2vD,, - 3~2
81’,, ~K”A,33-
(3.13)
253 This mode has two independent equations (if the traceless condition is satisfied at all times). There are two eigenvalues, both of them remaining negative as long as U is less than the critical value of 3. If U > 3, however, these eigenvalues can become positive for small wavenumbers. These characteristics are also exhibited by the corresponding equations for the “splay” mode of Doi [3]. Finally, all the growth rates of the above modes have the same dependence on the wavenumber as that discovered by Doi [3] (3.14) In conclusion, this similarity of the predictions of the present model for the spinodal decomposition with the results from the molecular theory of Doi provides additional evidence that the essential physics is correctly incorporated in the present formulation. 4. Conclusion In this part of the paper, we have introduced the dissipation tensor B by a first-order comparison with the more complex, non-homogeneous theory of Doi [3]. Using the spinodal decomposition results of Doi [3] as a reference, it was possible to calculate the parameters b, and b, in terms of molecular variables. Then the Frank elasticity parameters can be estimated from Doi’s molecular theory through their interrelationship with the bi shown in Refs. 1 and 8. Although the non-homogeneous Doi theory, based upon the distribution function, 4, is too complicated for general stability analyses, the new formulation in terms of the order parameter tensor is simple enough for application to complex flow studies. This represents a unique opportunity to study the phenomenon of spinodal decomposition, as well as the physics of flow-induced texture evolution. Future work will address this subject. Acknowledgement The authors (BJE, ANB) would like to acknowledge financial support provided by NSF through an ERC grant to the Center for Composite Materials. References 1 B.J. Edwards, A.N. Beris and M. Grmela, J. Non-Newtonian 51-72. 2 M. Doi, J. Chem. Phys., 88 (1988) 4070-4075.
Fluid Mech., 35 (1990)
254 3 M. Doi, J. Chem. Phys., 88 (1988) 7181-7186. 4 D.C. Wright and N.D. Mermin, Crystalline Liquids: The Blue Phases, Rev. Mod. Phys., 61 (1989) 385-432. 5 M. Doi, J. Polym. Sci., Polym. Phys. Ed., 19 (1981) 229-243. 6 A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55-78. 7 R.B. Bird, C.F. Curtiss, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory, 2nd Ed., John Wiley, New York, NY, 1987. 8 B.J. Edwards and A.N. Beris, J. Rheol., 33 (1989) 1189-1193.
Appendix A For any matrix
c, the Cayley-Hamilton
c *c - l,c + I$ - I$-l
theorem
states that
(~4.1)
= 0,
where
64.2)
I, = trc, I,=
(1/2)[(trc)*-
tr(c.c)],
(A.3)
and I3 = det(c). Multiplying tr(c.c.c.c) Substitution tr(S.S.S.S)
(A.4) eqn. (A.l) by c*, and taking the trace yields -I$r(c.c.c)
+I,tr(c.c)
of S for c, knowing = (1/2)[tr(S.S)]*.
-I,trc=O.
that trS = 0, immediately
(A-5) yields (A@
