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Liquid–crystalline nematic polymers revisited
J. Non-Newtonian Fluid Mech. 120 (2004) 85–92
Liquid–crystalline nematic polymers revisited Daniel Lhuillier a , Alejandro D. Rey b,∗ a
Modélisation en Mécanique, Université P. et M. Curie et CNRS, Case 162, 4 Place Jussieu, 75252 Paris Cedex 05, France b Department of Chemical Engineering, McGill University, Montreal, Que., Canada H3A 2A7 Received 8 September 2003; received in revised form 17 January 2004
This article is part of a Special Volume containing papers from the 3rd International Workshop on Nonequilibrium Thermodynamics and Complex Fluids
Abstract A new model for nematic polymers is proposed, based on the probability ψ(u,∇u,t) for a macromolecule to be oriented along direction u while embedded in a ∇u environment created by its neighbours. The potential of the internal forces is written Φ(u,∇u) accordingly. The free energy contains a contribution ν Φ + kB T ln ψ where the brackets mean an average over the probability distribution, while ν is the (uniform) polymer number density. An equation is derived for the time-evolution of the order parameter S = uu − I/3, together with an expression for the stress tensor. These two results offer a generalization of the Doi Model in so far as they include a distortional energy, analogue to the Frank elastic energy for low molecular mass nematics. Extending the Maier–Saupe variational procedure, we specify the way that the internal potential Φ(u,∇u) must be written for it to favour non-zero values of the order parameter, while giving a penalty to situations with gradients of the order parameter. The result is quite different from the potential proposed a decade ago by Marrucci and Greco (their Φ depends on u only), while it has a clear connection with the so-called Landau-de Gennes (LdG) tensor models, which are based on a free-energy depending on the order parameter and its gradients. © 2004 Elsevier B.V. All rights reserved. Keywords: Nematic polymers; Extended Maier–Saupe potential; Nematic micropolar fluid; Frank elasticity
1. Introduction The wide range of applications of liquid–crystalline materials has created new areas of academic and industrial research. The synthesis of polymer liquid crystals has enlarged the range of applications of these modern materials to areas where mechanical properties are important, as typified by the diverse uses of Kevlar fibres. Liquid crystal biopolymers, such as solutions of DNA, polypeptides, cellulosics, spider and worm silks, are attracting intense interest for biomaterials and health applications. A large number of reviews, textbooks and monographs on the theory, applications and rheology of liquid crystalline materials are available in the literature [1–17]. The flow modeling of liquid–crystalline matter has to take into account the internal structure, as defined by partial positional and orientational order of these phases. A first consequence of this internal structure in the modeling process
∗
Corresponding author. Tel.: +1 514 398 4196; fax: +1 514 398 6678. E-mail address: [email protected] (A.D. Rey).
0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2004.01.016
is the requirement of adding new balance equations to those that govern structureless Newtonian fluids. For example, for a uniaxial nematic liquid crystal an internal momentum balance equation is required to describe the average macroscopic orientation of the liquid. A second consequence is that the constitutive equations must reflect the symmetry properties of the phases. These requirements give rise to a variety of complex macroscopic theories applicable to different phases. This paper is concerned with polymeric nematic liquid crystals. To describe the flow of polymeric nematic liquid crystals three fundamental aspects have to be taken into account: (i) texture, (ii) anisotropy, and (ii) viscoelasticity, as follows. (i) The texture in polymeric nematic liquid crystals is defined by a defect density and defect charge distribution [17]. The charge of a defect reflects the amount of rotation when encircling the defect. Defects break the symmetry of the nematic phase and introduce spatial variations of the tensor order parameter that describe the phase. For uniaxial polymeric nematic liquid crystals the appropriate tensorial order parameter takes into consideration the average molecular orientation, specified by the unit vector director field (n, n·n
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= 1) and by the scalar order parameter S. For biaxial states an orthonormal director triad (n,m,l) and two order parameters are required. The director modes are soft, slow, and of long wave length, while the scalar order parameter modes are hard, fast, and of short wave length. Theories and models for textured nematic polymers must be able to describe defect nucleation and coarsening process, where short and long length scale spatial changes in the tensor order parameter occur. Models that have been successful at describing textural transformation are the Landau-de Gennes Model (LdG) [15,17] and the Edwards-Beris-Grmela (EBG) Model based on Poisson bracket formalism [18]. The anisotropic molecular nature (ii) is reflected in anisotropies in all mechanical properties. This means that the viscosity is now a viscosity tensor. The proliferation of phenomenological parameters in mesoscopic continuum models, such as de Landau-de Gennes Model, has fueled the interest in more molecular theories, such as the Doi Theory based on the orientation distribution function (ODF), first proposed by Hess [19]. The original version of the Doi Theory lacked the ability to describe textures. Recent attempts to correct this shortfall has been made by Marrucci and Greco [9] who incorporated a potential function that depends on gradients of the order parameter, but not on gradients of the molecular orientation. As a consequence, this approach requires the use of independent procedures to find the required stress equations, and leads to expressions that are possibly incompatible with nematostatics. In principle it appears that a rigorous and consistent incorporation of gradient contributions to the nematic potential function would lead to a theory that may share the abilities of the LdG and EBG models while reducing somewhat the number of macroscopic parameters and also providing some molecular information. Since for practical use the Doi Model requires pre-averaging and the use of closure approximations, as discussed at length by Edwards [20], at present it would appear that the main benefits of this approach is to shed light on the molecular origins of the flow-induced processes, as well as a reduction of the parametric space. In this spirit, this paper develops an extension of the Doi Model that is able to describe textures and is consistent with nematostatics. The viscoelasticity (iii) of polymeric nematic arises from couplings of the order parameter with the flow field. One aspect of this coupling is the presence of asymmetric extra (viscous) stress, which must be balanced by elastic restoring torques. The asymmetric stress or viscous torque represents the conversion between internal angular momentum, represented by the rotating microstructure (spin) and the external angular momentum, represented by the vorticity of the flow field. An efficient procedure to take this coupling into effect is to use the polar fluid model adapted to nematic liquid crystals, which makes explicit use of the internal momentum balance. Another benefit of the nematic polar fluid model is that it includes spin diffusion. Thus, when using this model, internal angular momentum and linear momentum are in a
one-to-one correspondence, and double diffusion processes are automatically incorporated. This paper makes use of the nematic polar fluid model in conjunction with the ODF methodology, naturally incorporating spin diffusion. As discussed by Rey and Denn [17], several approaches to model flows of polymeric liquid crystals have been developed. A rigorous comparison between several versions of the Doi Model and the EBG Model has been presented [19]. The comparison between the present model and the LdG and EBG models will only be sketched. On the other hand, a rigorous comparison of the present model and the original Doi Model is presented, as well as a detailed discussion of results obtained when using the present model and the Marrucci-Greco approach. The organization of this paper is a follows: Section 2 presents a brief description of the original Doi Model. Section 3 presents the extended Doi Model developed in this paper. Section 4 develops the equations for the nematic micropolar fluid. Section 5 presents the stress tensor for the nematic micropolar fluid. Section 6 presents limiting approximations that lead to closed form expressions of the stress, the viscous couple stress and the heat flux. Section 7 presents a mean-field expression for the nematic potential. Section 8 presents a discussion of the nematic potential used in this work and compares it with previously proposed expressions. Section 9 presents the main conclusions.
2. A brief presentation of the Doi Model The model proposed by Doi [21,22] (and Edwards [20]) to describe nematic polymers starts from a suspension of long rigid rods with a high and uniform concentration. Each rod has an excluded volume interaction with his neighbours. This interaction is conveniently represented by a mean-field potential Φ(u), depending on the orientation u of the rod. External couples also act on the rods and derive from a potential Φext (u). The probability for observing a given rod with orientation u is noted ψ(u,t). This probability distribution [19] depends on the flow field V (r,t) and is the solution of the conservation equation dψ ∂ ψdu + · =0 (1) dt ∂u dt where d/dt = ∂/∂t + V ·∇ is the convective time-derivative. Doi deduced the time-evolution of u from a balance of the viscous, internal, external and Brownian couples acting on a long and rigid rod: du = (I − uu) · [(u · ∇)V dt ∂ − ζr−1 (Φ + Φext + kB T ln ψ)]. ∂u
(2)
In this evolution equation, I is the unit tensor, δ − uu is the projector on directions orthogonal to u, while ζ r (u) is the
D. Lhuillier, A.D. Rey / J. Non-Newtonian Fluid Mech. 120 (2004) 85–92
rotational friction coefficient. Once the probability distribution is known for a given flow, one can deduce all possible averaged quantities. The most important one is the second moment uu = uuψ(u) du. When the rotational friction coefficient can be given some pre-averaged value ζ(uu), the evolution equation for the second moment becomes dui uj ∂Vj ∂Vi ∂Vk = ui uk + uj uk − 2ui uj uk ul dt ∂x ∂xk ∂xl k 1 ∂ ext − (uj Iik +ui Ijk − 2ui uj uk ) (Φ + Φ ) ζ ∂uk 2kB T (3ui uj − Iij ). (3) − ζ The second important quantity is the stress tensor of the polymer solution, ∂ ext σij = 3νkB T ui uj + ν ui (Ijk − uj uk ) (Φ + Φ ) ∂uk ∂Vk ζ0 , (4) + ν ui uj uk ul 2 ∂xl where ν is the uniform polymer number density, while ζ 0 is the rotational friction coefficient of a dilute suspension of rigid rods (ζ0 ζ). It is clear from Eqs. (3) and (4) that the quantity (I − uu) · ∂/∂u(Φ + Φext ) plays a role of utmost importance. All the external and internal potentials proposed up to now have a quadrupole form. Concerning the internal potential, the most well known is the Doi–Maier–Saupe potential Φ(u) = − 23 kB TU < uu >: uu.
(5)
However, different quadrupole potentials are possible, in which uu is replaced by some non-linear function of uu (see, e.g. Ilg et al. [23]). While all proposed expressions for Φ(u) are intended to favour states in which the order parameter tensor S = uu −
I 3
(6)
is different from zero, they give no penalty to flows exhibiting gradients of this order parameter. This absence of distortion energy was tentatively corrected by Marrucci and Greco [9] who proposed a potential depending on uu and ∇∇ uu and including an octupolar contribution besides the usual quadrupolar one. In what follows, we present a quite different approach in which the internal potential and the probability distribution are supposed to depend not only on u but also on ∇u.
3. The extended Doi Model We introduce ψ(u,∇u,t), the probability for a polymer to be oriented along u when its immediate neighbourhood is characterized by ∇u [24]. The orientation and its gradient are supposed to be independent variables. A first conse-
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quence is that the probability distribution is now the solution of a conservation equation that generalizes Eq. (1). dψ ∂ du ∂ d∇u + · (ψ ) + : (ψ )=0 dt ∂u dt ∂∇u dt
(7)
A second consequence concerns the free energy per unit mass of the nematic polymer. A part of this specific free energy, noted f ψ , depends on the probability distribution and appears in the form ψ mf = [Φ(u, ∇u) + kB T ln ψ(u, ∇u, t)] ×ψ (u, ∇u, t) du d∇u,
(8)
where Φ (u,∇u) is the potential of the internal couples acting on a polymer oriented along u and surrounded by a ∇u environment created by its neighbours, while m is the mass of a polymer. The free energy per unit volume mνf ψ encompasses both the short- and long-range elastic energies. In case of a completely ordered state (with ψ(u,∇u) = δ(u − n)δ(∇u − ∇n), where n is the director) mνf ψ becomes the Frank distortion energy. The time dependence of f ψ is due to the time dependence of the probability distribution and, with the help of Eq. (7) and integration by parts, one obtains ∗ mdf ψ |ν,T ∂Φ du ∂Φ∗ d∇u = · + : dt ∂u dt ∂∇u dt × ψ(u, ∇u, t) du d∇u ∗ ∂Φ du ∂Φ∗ d∇u = · + : , ∂u dt ∂∇u dt where Φ∗ (u,∇u) is the sum of the internal and Brownian potentials Φ∗ = Φ + kB T ln ψ,
(9)
while df ψ |ν,T means the variation of f ψ with constant polymer density and temperature. The polymer orientation u evolves in time according to du =ω×u dt
(10)
where ω is the mean angular velocity of the macromolecules. When writing such a simple evolution equation, we suppose implicitly that the polymer experiences all possible velocities and angular velocities in a very short time, much shorter than the time needed for changing its orientation. From the time evolution for u, it is simple to obtain the time evolution of ∇u and to deduce mdf ψ |ν,T ∂Φ∗ ∂Φ∗ ∂Φ∗ = εipj ωp uj + ∇ k uj − ∇ i uk dt ∂ui ∂∇k ui ∂∇j uk ∂ωi ∂Vi ∂Φ∗ + εipk up − + εijp ωp ∂xj ∂∇j uk ∂xj ∗ ∂Φ × ∇i uk . ∂∇j uk
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Since df ψ /dt must be a Galilean-invariant quantity, it may depend on ∇ω and ∇V + ε·ω, but not on ω. This obliges the potential Φ∗ to obey the rotational identity ∂Φ∗ ∂Φ∗ ∂Φ∗ εipj uj + ∇ k uj = 0. − ∇ i uk ∂ui ∂∇k ui ∂∇j uk To present the above results in a concise form, and to insist on the analogy with the Eriksen–Leslie Model, we define the mesoscopic elastic stress tensor σji∗ = −∇i uk
∂Φ∗ , ∂∇j uk
(11)
the mesoscopic elastic couple stress tensor µ∗ji = εipk up
∂Φ∗ , ∂∇j uk
and the mesoscopic molecular field ∂Φ∗ ∂Φ∗ . + ∇j h∗i = − ∂ui ∂∇j ui
ρ
de = σ T : (∇V + ε · ω) + µT : ∇ω − ∇ · q. dt
In these evolution equations, q is the energy flux, σ the stress tensor, Γ ext stand for the external body force and external body couple per unit volume. Moreover, σ T and µT stand for the transpose of σ and µ. The total free energy per unit mass of polymer nematics is f = f 0 (ρ, T) + f ψ (ν, T, ψ), where f ψ was defined in Eq. (8). The pressure p and specific entropy s of the nematic polymer are defined by the differential form df = − ψ pd(1/ρ) − sdT +df|ν,T . It is then easy to obtain the differential form of the specific internal energy e = f + Ts and to deduce with the help of Eq. (15) that
(12)
ψ df|ν,T p dρ −ρ ρ dt dt ∂Vi ∗ = (σji + pδij − νσji ) + εijk ωk ∂xj ∂qj ∂ωi + (µji − νµ∗ji ) − . ∂xj ∂xj
ds de ρT =ρ − dt dt (13)
The rotational identity can now be rewritten simply as
(14) ∇·µ∗ − ε : σ ∗ − u × h∗ = 0,
and the time-evolution of f ψ reduces to ψ mdf|ν,T ∂Vi ∂ωi + εijk ωk + µ∗ji . = σji∗ ∂xj ∂xj dt
Introducing the viscous stress tensor σ V and the viscous couple stress tensor µV by (15)
These last two results will prove of utmost importance for the dynamics of nematic polymers.
σ = −pI + νσ ∗ + νσ V and,
(18a)
= ν∗ + νV
(18b)
we obtain the ultimate form of the entropy balance of nematic polymers: 4. Nematic polymers and ordered micropolar fluids
Since the translation velocity V and the angular velocity ω are considered as independent kinematic variables, the best suited continuum model to describe the dynamics of nematic polymers is the micropolar fluid [25–27]. Micropolar fluids are endowed with an intrinsic angular momentum such that the total angular momentum and total energy per unit mass are M + r × V and e + 1/2ω · M + 1/2V ·V , respectively. Besides the usual conservation equations for mass and momentum dρ = −ρ∇ · V , (16) dt ρ
dV = ∇ · + F ext , dt
(17)
a micropolar fluid obeys the conservation equations for the total moment of momentum and the total energy. These conservation equations are conveniently expressed as balance equations for the intrinsic angular momentum (spin) and for the internal energy ρ
dM = ∇ · µ − ε : σ + Γ ext , dt
q ρds ∂Vi V T +∇ · = νσji + εijk ωk dt T ∂xj ∂ωi qj ∂T + νµV − . ji ∂xj T ∂xj
(19)
It is noteworthy that with definitions (18) and the rotational identity (14), the angular momentum balance can be rewritten as dM ∂Φext V V ∗ ρ , (20) =ν ∇· +␥ +u× h − dt ∂u where γ V = −ε : σ V is the viscous couple acting on the polymers while the external couple was written in the standard form [21] Γ ext = −νu × ∂Φext /∂u. Nematic polymers are thus described by Eqs. (16), (17), (19) and (20). Since σ ∗ and h∗ were already defined in Eqs. (11) and (13), to close this set of equations, one must propose explicit expressions for the dissipative fluxes σ V , µV and q. In so far as most theoretical models of liquid–crystalline matter have neglected the couple stress and largely ignored the heat flux, we will focus on the stress tensor only.
D. Lhuillier, A.D. Rey / J. Non-Newtonian Fluid Mech. 120 (2004) 85–92
5. The stress tensor of a nematic micropolar fluid
6. Model in the absence of rotary inertia and viscous couple stresses
According to definition (18a), the overall stress tensor is the sum of a pressure stress, an elastic (non-dissipative) stress and a viscous (dissipative) stress. The latter is involved in the entropy production rate and according to Eq. (19), must satisfy the thermodynamic inequality σjiV (∂Vi /∂xj + εijk ωk ) ≥ 0. Taking the definition of the viscous torque into account, one can transform that thermodynamic inequality into σjiVS Aij + γiV (Ωi − ωi ) ≥ 0 where Ω = 1/2 ∇ × V is the mean angular velocity of the fluid, Aij = 1/2(∂Vi /∂xj +∂Vj /∂xi ) is the symmetric part of the velocity gradient, while σ VS is the symmetric part of the viscous stress tensor. Noticing that u × A · u is the only polar vector that can be built from A and containing an even number of vectors u, the most general expression for the viscous couple, which is at a right angle to u, is γ V = ζr (I − uu) · (Ω − ω + λr u × A · u),
(21)
with a positive friction factor ζ r (u) and an arbitrary “tumbling coefficient” λr (u). The most general expression for the symmetric part of the viscous stress tensor, which guarantees a positive entropy production is accordingly νσ
VS
= η : A + νλr [u ⊗ (γ × u)] . V
S
(22)
The fourth-order viscosity tensor η is a function of the orientation, which for an incompressible fluid, appears in the form ηijkl = η2 (Iil δjk + Iik δjl ) + (η3 − η2 )(ui ul Ijk + uj uk Iil + uj ul Iik + ui uk Ijl ) + 2(η1 + η2 − 2η3 )ui uj uk ul . A positive entropy production imposes η1 (u) ≥ 0, η2 (u) ≥ 0 and η3 (u) ≥ 0. Expressions (21) and (22) can be understood as the analogue for nematic polymers of the “Harvard stress tensor” represented by formulas (5.37) and (5.36) of de Gennes and Prost. [3] According to definitions (11) and (9), the elastic part of the stress is ∂Φ νσji∗ = −ν ∇i uk + 2νkB TIij . (23) ∂∇j uk This means that the distortional stress is mainly due to the ∇u dependence of the internal potential Φ, the entropic potential contributing for an (uninteresting) isotropic stress. This distortional stress is absent from Doi Model. After discarding all pressure-like contributions, the expression of the full stress tensor defined in (18a) is finally ν σij = ηijkl Akl + (λr + 1)ui (γ V × u)j 2 ν ∂Φ V + (λr − 1)uj (γ × u)i − ν ∇i uk . 2 ∂∇j uk
89
(24)
Inserting expression (21) of the viscous couple allows one to express the stress tensor in terms of A and Ω − ω.
All the results obtained up to now were a consequence of the dependence on u and ∇u of both the probability distribution and the potential of internal couples. We also considered the evolution Eq. (10) for u but never needed to give any explicit expression similar to Eq. (2) of the Doi Model. This means we have considered the molecular angular velocity ω as a state variable, whose evolution in time is a consequence of the angular momentum balance (20). In general, this implies for ω a complicated evolution equation that can be written formally as dω/dt = F (ψ,Ω − ω,∇∇ω). Solving that equation and introducing the result into Eq. (10) gives an explicit expression for the time evolution of u. Usually complicated, this procedure becomes simple when one is allowed to neglect the rotary inertia and the viscous couple stress. In this case Eq. (20) boils down to a mesoscopic couple balance, which can be presented in the form ∂Φext V . (25) γ × u = (I − uu) · ∂u − h∗ From expression (21) for the viscous couple one deduces a simple expression for ω which, when introduced into Eq. (10), gives du ∂Φext = Ω×u+(δ − uu) · λr A · u + ζr−1 h∗ − , dt ∂u (26) to be compared with Doi’s Eq. (2). The evolution equation of uu is easily deduced from Eq. (26). It simplifies considerably when ζ r (u) and λr (u) can be replaced by some pre-averaged values ζ(uu) and λ(uu). After splitting h∗ into a part h due to Φ and a second part linked to the entropic potential (cf. Eq. (9)), one gets dui uj = uj uk Ωik + uj uk Ωjk + λuj uk Aik dt + λui uk Ajk − 2λui uj uk um Akm 1 ∂Φext (uj Iik + ui δjk − 2ui uj uk ) hk − + ζ ∂uk 2kB T − . (27) ζ(3ui uj − Iij ) Introducing Eq. (25) into the total stress tensor (24) results in σij = ηijkl Akl + 3νkB Tλui uj νλ ∂Φext (uj Iik + ui Ijk − 2ui uj uk ) hk − − 2 ∂uk ext ν ∇i uk ∂Φ ∂Φ + −ν . (uj Iik − ui Ijk ) hk − 2 ∂uk ∂∇j uk (28)
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Results (27) and (28) are the generalizations of Eqs. (3) and (4) respectively. The ubiquitous role of λ, ∂Φ/∂∇u and (I − uu) · h = (I − uu) · [−∂Φ/∂u + ∇ · (∂Φ/∂∇u)] in these two results is noteworthy. These three quantities are the cornerstones of the present approach.
7. A mean-field expression for the internal molecular potential An elegant variational procedure was devised by Maier and Saupe [28] to find the simplest form of intermolecular potential, which would favour a non-isotropic angular distribution. The basis is to define a scalar order parameter S = s(u)ψ(u) du = s(u) where s(u) is a mesoscopic scalar depicting the state of alignment. The only requirement imposed on s(u) is that S vanishes in isotropic states. The free energy per polymer is then written in the form F = F 0 (ρ, T) + kB T ψ(u) ln ψ(u) du + F int (ρ, T, S) The second term represents the entropy associated with the probability distribution. The third term describes the effects of molecular interactions and is a decreasing function of the order parameter so as to favour states with a non-zero value of S. Minimizing F with respect to variations of ψ(u), one obtains ∂F int kB T [1 + ln gψ(u)] δψ(u) du+ S(u) δψ(u) du ∂s = 0, and the equilibrium distribution is thus proportional to exp[−∂/∂S(F int /kB T) s(u)]. The same equilibrium distribution could have been obtained with an internal potential Φ(u) =
∂F int s(u). ∂S
Maier and Saupe considered the scalar order parameter S = 3/2(u·n)2 − 1/3 with a free energy Fint = −1/2kB TUS2 where U is a non-dimensional measure of the molecular 2 interactions and they deduced Φ(u) = −3/2k B TUS (u·n) . For a tensor order parameter Sij = sij (u)ψ(u) du = sij (u) the same variational procedure leads to Φ(u) =
∂F int : s(u). ∂S
Doi [22] chose Sij = ui uj – δij /3 with Fint = −3/4kB TUS:S and deduced his well-known potential (5). Ilg et al. [23] adopted the Doi order parameter but with Fint = kB TU(1 − 3/2S:S)1/2 and they deduced Φ(u) = −3/2kB TU(1 − 3/2S:S)−1/2 S:uu. Finally, the Maier–Saupe variational procedure can be extended to the case of a free energy Fint (S,∇S) depending not only on a tensor order parameter but also on its gradients.
The internal molecular potential associated with Sij = ui uj – Iij /3 is then
∂F int ∂F int ui uj + ∇ k (ui uj ). (29) Φ(u, ∇u) = ∂Sij ∂∇ k Sij There is thus a tight link between the present approach based on the potential of internal forces Φ(u,∇u), and the models based on the dependence of Fint on the order parameter and its gradients [18,15]. Taking Eq. (29) into account and discarding (for simplicity) the role of the external potential, results (27) and (28) now appear in their final forms: dui uj = uj uk Ωik + ui uk Ωjk + λuj uk Aik dt + λui uk Ajk − 2λui uj uk um Akm
2 −∂F int ∂F int + + ∇m ζ ∂Skl ∂(∇m Skl ) × [uj ul Iik + ui ul Ijk − 2ui uj uk ul ] 2kB T − (3ui uj − Iij ). ζ
(30)
and ∂F int σij = ηijkl Akl + 3νkB Tλui uj − ν∇i Skm ∂∇j Skm
int int −∂F ∂F −ν − + ∇m ∂Sk1 ∂∇m Sk1 × (λ − 1)uj ul δik + (λ + 1)ui ul δjk − 2λui uj uk ul
(31)
where the expression of the viscosity tensor ηijkl was given in Section 5 and leads to ηijkl Akl = 2η2 Aij + 2(η3 − η2 )[ui uk Akj + uj uk Aki ] + 2(η1 + η2 − 2η3 )ui uj uk ul Akl with η1 ≥ 0, η2 ≥ 0 and η3 ≥ 0.
8. Some special expressions for the internal potential and their relations to previous works The simplest example of a free-energy which favours a non-zero tensor order parameter S while penalizing its gradient is F int = 43 kB TU[−Sij Sij + L2 ∇k Sij ∇k Sij ], where L is some molecular length. According to Eq. (29), the associated internal potential is Φ(u, ∇u) = 23 kB TU[−Sij ui uj + L2 ∇k Sij ∇k (ui uj )].
(32)
According to Eq. (30), the evolution equation of the configuration tensor is
D. Lhuillier, A.D. Rey / J. Non-Newtonian Fluid Mech. 120 (2004) 85–92
91
dui uj = uj uk Ωik + ui uk Ωjk + λuj uk Aik dt + λui uk Ajk − 2λui uj uk um Akm 3kB TU [Sjk +L2 ∇ 2 Sjk ][uj ul Iik + ui ul Ijk + ζ 2kB T − 2ui uj uk ul ] − (3ui uj − δij ). (33) ζ
cases is not a proof that Eq. (36) is correct (there is a large arbitrariness in the choice of the b3 term) but it means we are not in conflict with thermodynamics when introducing Eq. (36) into Eqs. (30) and (31). Concerning the contribution f(Sij ), the expression kB TU(1 – 3/2S:S)1/2 proposed by Ilg et al. [23] seems appealing if one wants to avoid long polynomial expressions.
It is noteworthy that the same evolution equation could also have been obtained from Eq. (27) with the simple definition h = −∂Φ/∂u and the quadrupolar potential
9. Conclusions
Φ(u) = − 23 kB TU[S + L2 ∇ 2 S] : uu,
(34)
which is nothing but the “one-constant” form of the Marucci–Greco potential [9]. This means that the same physics is contained in Eqs. (32) and (34) as far as the internal molecular field is concerned. However, one cannot get a distortion stress associated with the Marucci–Greco potential, while the distortion stress associated with Eq. (32) is, according to Eq. (31), −ν∇i Skm
∂F 3 = − νkB TUL2 ∇i Skm ∇j Skm . ∂∇j Skm 2
(35)
Feng et al. [29] incorporated the one-constant Marrucci– Greco potential into the Doi Model to formulate a theory for nematic polymers that contains distortion elasticity. Their results are in agreement with Eq. (33) (with λ = 1) but not with Eq. (35). Generalizing Doi’s virtual work procedure, they found −3/4νkB TU L2 [∇i Skm ∇j Skm − ∇i ∇j Skm uk um ] instead of Eq. (35). We believe our approach is much more systematic: the use of Φ(u,∇u) allows one to derive both the molecular field and the distortion stress, while the use of Φ(u) can lead to a physically correct expression for the molecular field but must be completed on an entirely independent calculation of the distortion stress. This double step procedure is likely to be a source of errors. Let us now comment on the dependence of Fint on the gradients of the order parameter. One expects an expression containing three constants to enhance the possibility of a one-to-one correspondence with Frank’s elastic energy. Such an expression was proposed in Eq. (2.15) of Edwards et al. [18] We suspect that their b3 term fails to be positive in any circumstances and we propose instead F int = f(Sij ) +
b2 2 ∇i Sjk ∇j Sjk
+
b2 2 ∇j Sij ∇k Sik
+ 9b23 Sij Sik ∇j Smn ∇k Smn ,
(36)
with b1 > 0, b2 > 0 and b3 > 0. In the uniaxial limit where Sij = S(ni nj − δij /3), this elastic energy boils down to Frank’s with the three elastic constants: K1 = 2S 2 (b1 + b2 + b3 S 2 ),
K2 = 2S 2 (b1 + b3 S 2 ),
K3 = 2S 2 (b1 + b2 + 4b3 S 2 ). As a consequence, K2 < K1 < K3 and 3K2 + K1 > K3 . The fact that these inequalities are fulfilled in all known
With the ultimate objective of describing the textures and defects appearing in the flow of nematic polymers, we have extended the original Doi Model to endow it with orientational elasticity. This was done by considering the molecular orientation and its local gradient as two independent variables. The potential of the internal forces was supposed to be a function of these two variables, which allowed us to define an elastic stress, an elastic couple stress and a molecular field (see Eqs. (11)–(13)) much like in the Ericksen–Leslie Model). The main difference from that well-known model is that those quantities are defined at the mesoscopic level and a probability distribution function (also depending on molecular orientation and its gradient) is necessary to provide the averaged quantities relevant at the macroscopic level. The micropolar fluid model and the thermodynamics of irreversible processes proved to be convenient tools to describe the dynamics of nematic polymers with a consistent picture of orientational elasticity. Among the main simplifying assumptions that were made is the spatial uniformity of the polymer concentration. To remove that assumption is not that simple within the present model, and one must use a more elaborate framework including translational diffusion [30,31] or the mesoscopic theory of Muschik et al. [32]. It is our view that the complexity of the final results shown in Eqs. (30) and (31) is such that we must first test their potentialities prior to disregarding the assumptions which led to them.
Acknowledgements A.D.R. acknowledges support from the Natural Sciences and Engineering Research Council (Canada), the Donors of The Petroleum Research Fund (PRF, American Chemical Society), and the NSF Center for Advanced Fibers and Films (CAEFF/NSF) at Clemson University.
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[3] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd ed., Oxford University Press, Oxford, 1993. [4] M. Kleman, Points, lines and walls, in: Liquid Crystals, Magnetic Systems, and Various Ordered Media, Wiley, New York, 1983. [5] A. Ciferri (Ed.), Liquid Crystallinity in Polymers: Principles and Fundamental Properties, VCH, New York, 1991. [6] C. Noel, P. Navard, Liquid crystal polymers, Prog. Polym. Sci. 16 (1991) 55–110. [7] A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, New York, 1994. [8] M. Srinivasarao, Rheology and rheo-optics of polymer liquid crystals, Int. J. Mod. Phys. B 9 (1995) 2515–2572. [9] G. Marrucci, F. Greco, The elastic constants of Maier–Saupe rod-like molecule nematics, Mol. Cryst. Liq. Cryst. 206 (1991) 17–30. [10] W.R. Burghardt, Molecular orientation and rheology in sheared lyotropic liquid–crystalline polymers, Macromol. Chem. Phys. 199 (1998) 471–488. [11] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, 1999. [12] D. Demus, J. Goodby, G.W. Gray, h-W. Speiss, V. Vill, Physical Properties of Liquid Crystals, Wiley-VCH, Weinheim, 1999. [13] E. Virga, Variational Theories for Liquid Crystals, Chapman & Hall, London, 1994. [14] A.A. Sonin, The Surface Physics of Liquid Crystals, Gordon and Breach Publishers, Luxembourg, 1995. [15] A.D. Rey, T. Tsuji, Recent advances in theoretical liquid crystal rheology, Macromol. Theory Simul. 7 (1998) 623–639. [16] H. Pleiner, H.R. Brand, Hydrodynamics and electrohydrodynamics of liquid crystals, in: A. Buka, L. Kramer (Eds.), Pattern Formation in Liquid Crystals, Springer-Verlag, Vienna, 1996. [17] A.D. Rey, M.M. Denn, Dynamical phenomena in liquid–crystalline materials, Ann. Rev. Fluid Mech. 34 (2002) 233–266. [18] B.J. Edwards, A. N Beris, M. Grmela, Generalized constitutive equations for polymeric liquid crystals. Part I: Model formulation using the Hamiltonian (Poisson bracket) formulation, J. Non-Newtonian Fluid Mech. 35 (1990) 51–72. [19] S. Hess, Fokker-Plank-equation approach to flow alignment in liquid crystals, Z. Naturforschung 31a (1976) 1034–1037. [20] J.E. Edwards, Evaluation of the thermodynamic consistency of closure approximations in several models proposed for the description of
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Liquid–crystalline nematic polymers revisited Daniel Lhuillier a , Alejandro D. Rey b,∗ a
Modélisation en Mécanique, Université P. et M. Curie et CNRS, Case 162, 4 Place Jussieu, 75252 Paris Cedex 05, France b Department of Chemical Engineering, McGill University, Montreal, Que., Canada H3A 2A7 Received 8 September 2003; received in revised form 17 January 2004
This article is part of a Special Volume containing papers from the 3rd International Workshop on Nonequilibrium Thermodynamics and Complex Fluids
Abstract A new model for nematic polymers is proposed, based on the probability ψ(u,∇u,t) for a macromolecule to be oriented along direction u while embedded in a ∇u environment created by its neighbours. The potential of the internal forces is written Φ(u,∇u) accordingly. The free energy contains a contribution ν Φ + kB T ln ψ where the brackets mean an average over the probability distribution, while ν is the (uniform) polymer number density. An equation is derived for the time-evolution of the order parameter S = uu − I/3, together with an expression for the stress tensor. These two results offer a generalization of the Doi Model in so far as they include a distortional energy, analogue to the Frank elastic energy for low molecular mass nematics. Extending the Maier–Saupe variational procedure, we specify the way that the internal potential Φ(u,∇u) must be written for it to favour non-zero values of the order parameter, while giving a penalty to situations with gradients of the order parameter. The result is quite different from the potential proposed a decade ago by Marrucci and Greco (their Φ depends on u only), while it has a clear connection with the so-called Landau-de Gennes (LdG) tensor models, which are based on a free-energy depending on the order parameter and its gradients. © 2004 Elsevier B.V. All rights reserved. Keywords: Nematic polymers; Extended Maier–Saupe potential; Nematic micropolar fluid; Frank elasticity
1. Introduction The wide range of applications of liquid–crystalline materials has created new areas of academic and industrial research. The synthesis of polymer liquid crystals has enlarged the range of applications of these modern materials to areas where mechanical properties are important, as typified by the diverse uses of Kevlar fibres. Liquid crystal biopolymers, such as solutions of DNA, polypeptides, cellulosics, spider and worm silks, are attracting intense interest for biomaterials and health applications. A large number of reviews, textbooks and monographs on the theory, applications and rheology of liquid crystalline materials are available in the literature [1–17]. The flow modeling of liquid–crystalline matter has to take into account the internal structure, as defined by partial positional and orientational order of these phases. A first consequence of this internal structure in the modeling process
∗
Corresponding author. Tel.: +1 514 398 4196; fax: +1 514 398 6678. E-mail address: [email protected] (A.D. Rey).
0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2004.01.016
is the requirement of adding new balance equations to those that govern structureless Newtonian fluids. For example, for a uniaxial nematic liquid crystal an internal momentum balance equation is required to describe the average macroscopic orientation of the liquid. A second consequence is that the constitutive equations must reflect the symmetry properties of the phases. These requirements give rise to a variety of complex macroscopic theories applicable to different phases. This paper is concerned with polymeric nematic liquid crystals. To describe the flow of polymeric nematic liquid crystals three fundamental aspects have to be taken into account: (i) texture, (ii) anisotropy, and (ii) viscoelasticity, as follows. (i) The texture in polymeric nematic liquid crystals is defined by a defect density and defect charge distribution [17]. The charge of a defect reflects the amount of rotation when encircling the defect. Defects break the symmetry of the nematic phase and introduce spatial variations of the tensor order parameter that describe the phase. For uniaxial polymeric nematic liquid crystals the appropriate tensorial order parameter takes into consideration the average molecular orientation, specified by the unit vector director field (n, n·n
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= 1) and by the scalar order parameter S. For biaxial states an orthonormal director triad (n,m,l) and two order parameters are required. The director modes are soft, slow, and of long wave length, while the scalar order parameter modes are hard, fast, and of short wave length. Theories and models for textured nematic polymers must be able to describe defect nucleation and coarsening process, where short and long length scale spatial changes in the tensor order parameter occur. Models that have been successful at describing textural transformation are the Landau-de Gennes Model (LdG) [15,17] and the Edwards-Beris-Grmela (EBG) Model based on Poisson bracket formalism [18]. The anisotropic molecular nature (ii) is reflected in anisotropies in all mechanical properties. This means that the viscosity is now a viscosity tensor. The proliferation of phenomenological parameters in mesoscopic continuum models, such as de Landau-de Gennes Model, has fueled the interest in more molecular theories, such as the Doi Theory based on the orientation distribution function (ODF), first proposed by Hess [19]. The original version of the Doi Theory lacked the ability to describe textures. Recent attempts to correct this shortfall has been made by Marrucci and Greco [9] who incorporated a potential function that depends on gradients of the order parameter, but not on gradients of the molecular orientation. As a consequence, this approach requires the use of independent procedures to find the required stress equations, and leads to expressions that are possibly incompatible with nematostatics. In principle it appears that a rigorous and consistent incorporation of gradient contributions to the nematic potential function would lead to a theory that may share the abilities of the LdG and EBG models while reducing somewhat the number of macroscopic parameters and also providing some molecular information. Since for practical use the Doi Model requires pre-averaging and the use of closure approximations, as discussed at length by Edwards [20], at present it would appear that the main benefits of this approach is to shed light on the molecular origins of the flow-induced processes, as well as a reduction of the parametric space. In this spirit, this paper develops an extension of the Doi Model that is able to describe textures and is consistent with nematostatics. The viscoelasticity (iii) of polymeric nematic arises from couplings of the order parameter with the flow field. One aspect of this coupling is the presence of asymmetric extra (viscous) stress, which must be balanced by elastic restoring torques. The asymmetric stress or viscous torque represents the conversion between internal angular momentum, represented by the rotating microstructure (spin) and the external angular momentum, represented by the vorticity of the flow field. An efficient procedure to take this coupling into effect is to use the polar fluid model adapted to nematic liquid crystals, which makes explicit use of the internal momentum balance. Another benefit of the nematic polar fluid model is that it includes spin diffusion. Thus, when using this model, internal angular momentum and linear momentum are in a
one-to-one correspondence, and double diffusion processes are automatically incorporated. This paper makes use of the nematic polar fluid model in conjunction with the ODF methodology, naturally incorporating spin diffusion. As discussed by Rey and Denn [17], several approaches to model flows of polymeric liquid crystals have been developed. A rigorous comparison between several versions of the Doi Model and the EBG Model has been presented [19]. The comparison between the present model and the LdG and EBG models will only be sketched. On the other hand, a rigorous comparison of the present model and the original Doi Model is presented, as well as a detailed discussion of results obtained when using the present model and the Marrucci-Greco approach. The organization of this paper is a follows: Section 2 presents a brief description of the original Doi Model. Section 3 presents the extended Doi Model developed in this paper. Section 4 develops the equations for the nematic micropolar fluid. Section 5 presents the stress tensor for the nematic micropolar fluid. Section 6 presents limiting approximations that lead to closed form expressions of the stress, the viscous couple stress and the heat flux. Section 7 presents a mean-field expression for the nematic potential. Section 8 presents a discussion of the nematic potential used in this work and compares it with previously proposed expressions. Section 9 presents the main conclusions.
2. A brief presentation of the Doi Model The model proposed by Doi [21,22] (and Edwards [20]) to describe nematic polymers starts from a suspension of long rigid rods with a high and uniform concentration. Each rod has an excluded volume interaction with his neighbours. This interaction is conveniently represented by a mean-field potential Φ(u), depending on the orientation u of the rod. External couples also act on the rods and derive from a potential Φext (u). The probability for observing a given rod with orientation u is noted ψ(u,t). This probability distribution [19] depends on the flow field V (r,t) and is the solution of the conservation equation dψ ∂ ψdu + · =0 (1) dt ∂u dt where d/dt = ∂/∂t + V ·∇ is the convective time-derivative. Doi deduced the time-evolution of u from a balance of the viscous, internal, external and Brownian couples acting on a long and rigid rod: du = (I − uu) · [(u · ∇)V dt ∂ − ζr−1 (Φ + Φext + kB T ln ψ)]. ∂u
(2)
In this evolution equation, I is the unit tensor, δ − uu is the projector on directions orthogonal to u, while ζ r (u) is the
D. Lhuillier, A.D. Rey / J. Non-Newtonian Fluid Mech. 120 (2004) 85–92
rotational friction coefficient. Once the probability distribution is known for a given flow, one can deduce all possible averaged quantities. The most important one is the second moment uu = uuψ(u) du. When the rotational friction coefficient can be given some pre-averaged value ζ(uu), the evolution equation for the second moment becomes dui uj ∂Vj ∂Vi ∂Vk = ui uk + uj uk − 2ui uj uk ul dt ∂x ∂xk ∂xl k 1 ∂ ext − (uj Iik +ui Ijk − 2ui uj uk ) (Φ + Φ ) ζ ∂uk 2kB T (3ui uj − Iij ). (3) − ζ The second important quantity is the stress tensor of the polymer solution, ∂ ext σij = 3νkB T ui uj + ν ui (Ijk − uj uk ) (Φ + Φ ) ∂uk ∂Vk ζ0 , (4) + ν ui uj uk ul 2 ∂xl where ν is the uniform polymer number density, while ζ 0 is the rotational friction coefficient of a dilute suspension of rigid rods (ζ0 ζ). It is clear from Eqs. (3) and (4) that the quantity (I − uu) · ∂/∂u(Φ + Φext ) plays a role of utmost importance. All the external and internal potentials proposed up to now have a quadrupole form. Concerning the internal potential, the most well known is the Doi–Maier–Saupe potential Φ(u) = − 23 kB TU < uu >: uu.
(5)
However, different quadrupole potentials are possible, in which uu is replaced by some non-linear function of uu (see, e.g. Ilg et al. [23]). While all proposed expressions for Φ(u) are intended to favour states in which the order parameter tensor S = uu −
I 3
(6)
is different from zero, they give no penalty to flows exhibiting gradients of this order parameter. This absence of distortion energy was tentatively corrected by Marrucci and Greco [9] who proposed a potential depending on uu and ∇∇ uu and including an octupolar contribution besides the usual quadrupolar one. In what follows, we present a quite different approach in which the internal potential and the probability distribution are supposed to depend not only on u but also on ∇u.
3. The extended Doi Model We introduce ψ(u,∇u,t), the probability for a polymer to be oriented along u when its immediate neighbourhood is characterized by ∇u [24]. The orientation and its gradient are supposed to be independent variables. A first conse-
87
quence is that the probability distribution is now the solution of a conservation equation that generalizes Eq. (1). dψ ∂ du ∂ d∇u + · (ψ ) + : (ψ )=0 dt ∂u dt ∂∇u dt
(7)
A second consequence concerns the free energy per unit mass of the nematic polymer. A part of this specific free energy, noted f ψ , depends on the probability distribution and appears in the form ψ mf = [Φ(u, ∇u) + kB T ln ψ(u, ∇u, t)] ×ψ (u, ∇u, t) du d∇u,
(8)
where Φ (u,∇u) is the potential of the internal couples acting on a polymer oriented along u and surrounded by a ∇u environment created by its neighbours, while m is the mass of a polymer. The free energy per unit volume mνf ψ encompasses both the short- and long-range elastic energies. In case of a completely ordered state (with ψ(u,∇u) = δ(u − n)δ(∇u − ∇n), where n is the director) mνf ψ becomes the Frank distortion energy. The time dependence of f ψ is due to the time dependence of the probability distribution and, with the help of Eq. (7) and integration by parts, one obtains ∗ mdf ψ |ν,T ∂Φ du ∂Φ∗ d∇u = · + : dt ∂u dt ∂∇u dt × ψ(u, ∇u, t) du d∇u ∗ ∂Φ du ∂Φ∗ d∇u = · + : , ∂u dt ∂∇u dt where Φ∗ (u,∇u) is the sum of the internal and Brownian potentials Φ∗ = Φ + kB T ln ψ,
(9)
while df ψ |ν,T means the variation of f ψ with constant polymer density and temperature. The polymer orientation u evolves in time according to du =ω×u dt
(10)
where ω is the mean angular velocity of the macromolecules. When writing such a simple evolution equation, we suppose implicitly that the polymer experiences all possible velocities and angular velocities in a very short time, much shorter than the time needed for changing its orientation. From the time evolution for u, it is simple to obtain the time evolution of ∇u and to deduce mdf ψ |ν,T ∂Φ∗ ∂Φ∗ ∂Φ∗ = εipj ωp uj + ∇ k uj − ∇ i uk dt ∂ui ∂∇k ui ∂∇j uk ∂ωi ∂Vi ∂Φ∗ + εipk up − + εijp ωp ∂xj ∂∇j uk ∂xj ∗ ∂Φ × ∇i uk . ∂∇j uk
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Since df ψ /dt must be a Galilean-invariant quantity, it may depend on ∇ω and ∇V + ε·ω, but not on ω. This obliges the potential Φ∗ to obey the rotational identity ∂Φ∗ ∂Φ∗ ∂Φ∗ εipj uj + ∇ k uj = 0. − ∇ i uk ∂ui ∂∇k ui ∂∇j uk To present the above results in a concise form, and to insist on the analogy with the Eriksen–Leslie Model, we define the mesoscopic elastic stress tensor σji∗ = −∇i uk
∂Φ∗ , ∂∇j uk
(11)
the mesoscopic elastic couple stress tensor µ∗ji = εipk up
∂Φ∗ , ∂∇j uk
and the mesoscopic molecular field ∂Φ∗ ∂Φ∗ . + ∇j h∗i = − ∂ui ∂∇j ui
ρ
de = σ T : (∇V + ε · ω) + µT : ∇ω − ∇ · q. dt
In these evolution equations, q is the energy flux, σ the stress tensor, Γ ext stand for the external body force and external body couple per unit volume. Moreover, σ T and µT stand for the transpose of σ and µ. The total free energy per unit mass of polymer nematics is f = f 0 (ρ, T) + f ψ (ν, T, ψ), where f ψ was defined in Eq. (8). The pressure p and specific entropy s of the nematic polymer are defined by the differential form df = − ψ pd(1/ρ) − sdT +df|ν,T . It is then easy to obtain the differential form of the specific internal energy e = f + Ts and to deduce with the help of Eq. (15) that
(12)
ψ df|ν,T p dρ −ρ ρ dt dt ∂Vi ∗ = (σji + pδij − νσji ) + εijk ωk ∂xj ∂qj ∂ωi + (µji − νµ∗ji ) − . ∂xj ∂xj
ds de ρT =ρ − dt dt (13)
The rotational identity can now be rewritten simply as
(14) ∇·µ∗ − ε : σ ∗ − u × h∗ = 0,
and the time-evolution of f ψ reduces to ψ mdf|ν,T ∂Vi ∂ωi + εijk ωk + µ∗ji . = σji∗ ∂xj ∂xj dt
Introducing the viscous stress tensor σ V and the viscous couple stress tensor µV by (15)
These last two results will prove of utmost importance for the dynamics of nematic polymers.
σ = −pI + νσ ∗ + νσ V and,
(18a)
= ν∗ + νV
(18b)
we obtain the ultimate form of the entropy balance of nematic polymers: 4. Nematic polymers and ordered micropolar fluids
Since the translation velocity V and the angular velocity ω are considered as independent kinematic variables, the best suited continuum model to describe the dynamics of nematic polymers is the micropolar fluid [25–27]. Micropolar fluids are endowed with an intrinsic angular momentum such that the total angular momentum and total energy per unit mass are M + r × V and e + 1/2ω · M + 1/2V ·V , respectively. Besides the usual conservation equations for mass and momentum dρ = −ρ∇ · V , (16) dt ρ
dV = ∇ · + F ext , dt
(17)
a micropolar fluid obeys the conservation equations for the total moment of momentum and the total energy. These conservation equations are conveniently expressed as balance equations for the intrinsic angular momentum (spin) and for the internal energy ρ
dM = ∇ · µ − ε : σ + Γ ext , dt
q ρds ∂Vi V T +∇ · = νσji + εijk ωk dt T ∂xj ∂ωi qj ∂T + νµV − . ji ∂xj T ∂xj
(19)
It is noteworthy that with definitions (18) and the rotational identity (14), the angular momentum balance can be rewritten as dM ∂Φext V V ∗ ρ , (20) =ν ∇· +␥ +u× h − dt ∂u where γ V = −ε : σ V is the viscous couple acting on the polymers while the external couple was written in the standard form [21] Γ ext = −νu × ∂Φext /∂u. Nematic polymers are thus described by Eqs. (16), (17), (19) and (20). Since σ ∗ and h∗ were already defined in Eqs. (11) and (13), to close this set of equations, one must propose explicit expressions for the dissipative fluxes σ V , µV and q. In so far as most theoretical models of liquid–crystalline matter have neglected the couple stress and largely ignored the heat flux, we will focus on the stress tensor only.
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5. The stress tensor of a nematic micropolar fluid
6. Model in the absence of rotary inertia and viscous couple stresses
According to definition (18a), the overall stress tensor is the sum of a pressure stress, an elastic (non-dissipative) stress and a viscous (dissipative) stress. The latter is involved in the entropy production rate and according to Eq. (19), must satisfy the thermodynamic inequality σjiV (∂Vi /∂xj + εijk ωk ) ≥ 0. Taking the definition of the viscous torque into account, one can transform that thermodynamic inequality into σjiVS Aij + γiV (Ωi − ωi ) ≥ 0 where Ω = 1/2 ∇ × V is the mean angular velocity of the fluid, Aij = 1/2(∂Vi /∂xj +∂Vj /∂xi ) is the symmetric part of the velocity gradient, while σ VS is the symmetric part of the viscous stress tensor. Noticing that u × A · u is the only polar vector that can be built from A and containing an even number of vectors u, the most general expression for the viscous couple, which is at a right angle to u, is γ V = ζr (I − uu) · (Ω − ω + λr u × A · u),
(21)
with a positive friction factor ζ r (u) and an arbitrary “tumbling coefficient” λr (u). The most general expression for the symmetric part of the viscous stress tensor, which guarantees a positive entropy production is accordingly νσ
VS
= η : A + νλr [u ⊗ (γ × u)] . V
S
(22)
The fourth-order viscosity tensor η is a function of the orientation, which for an incompressible fluid, appears in the form ηijkl = η2 (Iil δjk + Iik δjl ) + (η3 − η2 )(ui ul Ijk + uj uk Iil + uj ul Iik + ui uk Ijl ) + 2(η1 + η2 − 2η3 )ui uj uk ul . A positive entropy production imposes η1 (u) ≥ 0, η2 (u) ≥ 0 and η3 (u) ≥ 0. Expressions (21) and (22) can be understood as the analogue for nematic polymers of the “Harvard stress tensor” represented by formulas (5.37) and (5.36) of de Gennes and Prost. [3] According to definitions (11) and (9), the elastic part of the stress is ∂Φ νσji∗ = −ν ∇i uk + 2νkB TIij . (23) ∂∇j uk This means that the distortional stress is mainly due to the ∇u dependence of the internal potential Φ, the entropic potential contributing for an (uninteresting) isotropic stress. This distortional stress is absent from Doi Model. After discarding all pressure-like contributions, the expression of the full stress tensor defined in (18a) is finally ν σij = ηijkl Akl + (λr + 1)ui (γ V × u)j 2 ν ∂Φ V + (λr − 1)uj (γ × u)i − ν ∇i uk . 2 ∂∇j uk
89
(24)
Inserting expression (21) of the viscous couple allows one to express the stress tensor in terms of A and Ω − ω.
All the results obtained up to now were a consequence of the dependence on u and ∇u of both the probability distribution and the potential of internal couples. We also considered the evolution Eq. (10) for u but never needed to give any explicit expression similar to Eq. (2) of the Doi Model. This means we have considered the molecular angular velocity ω as a state variable, whose evolution in time is a consequence of the angular momentum balance (20). In general, this implies for ω a complicated evolution equation that can be written formally as dω/dt = F (ψ,Ω − ω,∇∇ω). Solving that equation and introducing the result into Eq. (10) gives an explicit expression for the time evolution of u. Usually complicated, this procedure becomes simple when one is allowed to neglect the rotary inertia and the viscous couple stress. In this case Eq. (20) boils down to a mesoscopic couple balance, which can be presented in the form ∂Φext V . (25) γ × u = (I − uu) · ∂u − h∗ From expression (21) for the viscous couple one deduces a simple expression for ω which, when introduced into Eq. (10), gives du ∂Φext = Ω×u+(δ − uu) · λr A · u + ζr−1 h∗ − , dt ∂u (26) to be compared with Doi’s Eq. (2). The evolution equation of uu is easily deduced from Eq. (26). It simplifies considerably when ζ r (u) and λr (u) can be replaced by some pre-averaged values ζ(uu) and λ(uu). After splitting h∗ into a part h due to Φ and a second part linked to the entropic potential (cf. Eq. (9)), one gets dui uj = uj uk Ωik + uj uk Ωjk + λuj uk Aik dt + λui uk Ajk − 2λui uj uk um Akm 1 ∂Φext (uj Iik + ui δjk − 2ui uj uk ) hk − + ζ ∂uk 2kB T − . (27) ζ(3ui uj − Iij ) Introducing Eq. (25) into the total stress tensor (24) results in σij = ηijkl Akl + 3νkB Tλui uj νλ ∂Φext (uj Iik + ui Ijk − 2ui uj uk ) hk − − 2 ∂uk ext ν ∇i uk ∂Φ ∂Φ + −ν . (uj Iik − ui Ijk ) hk − 2 ∂uk ∂∇j uk (28)
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Results (27) and (28) are the generalizations of Eqs. (3) and (4) respectively. The ubiquitous role of λ, ∂Φ/∂∇u and (I − uu) · h = (I − uu) · [−∂Φ/∂u + ∇ · (∂Φ/∂∇u)] in these two results is noteworthy. These three quantities are the cornerstones of the present approach.
7. A mean-field expression for the internal molecular potential An elegant variational procedure was devised by Maier and Saupe [28] to find the simplest form of intermolecular potential, which would favour a non-isotropic angular distribution. The basis is to define a scalar order parameter S = s(u)ψ(u) du = s(u) where s(u) is a mesoscopic scalar depicting the state of alignment. The only requirement imposed on s(u) is that S vanishes in isotropic states. The free energy per polymer is then written in the form F = F 0 (ρ, T) + kB T ψ(u) ln ψ(u) du + F int (ρ, T, S) The second term represents the entropy associated with the probability distribution. The third term describes the effects of molecular interactions and is a decreasing function of the order parameter so as to favour states with a non-zero value of S. Minimizing F with respect to variations of ψ(u), one obtains ∂F int kB T [1 + ln gψ(u)] δψ(u) du+ S(u) δψ(u) du ∂s = 0, and the equilibrium distribution is thus proportional to exp[−∂/∂S(F int /kB T) s(u)]. The same equilibrium distribution could have been obtained with an internal potential Φ(u) =
∂F int s(u). ∂S
Maier and Saupe considered the scalar order parameter S = 3/2(u·n)2 − 1/3 with a free energy Fint = −1/2kB TUS2 where U is a non-dimensional measure of the molecular 2 interactions and they deduced Φ(u) = −3/2k B TUS (u·n) . For a tensor order parameter Sij = sij (u)ψ(u) du = sij (u) the same variational procedure leads to Φ(u) =
∂F int : s(u). ∂S
Doi [22] chose Sij = ui uj – δij /3 with Fint = −3/4kB TUS:S and deduced his well-known potential (5). Ilg et al. [23] adopted the Doi order parameter but with Fint = kB TU(1 − 3/2S:S)1/2 and they deduced Φ(u) = −3/2kB TU(1 − 3/2S:S)−1/2 S:uu. Finally, the Maier–Saupe variational procedure can be extended to the case of a free energy Fint (S,∇S) depending not only on a tensor order parameter but also on its gradients.
The internal molecular potential associated with Sij = ui uj – Iij /3 is then
∂F int ∂F int ui uj + ∇ k (ui uj ). (29) Φ(u, ∇u) = ∂Sij ∂∇ k Sij There is thus a tight link between the present approach based on the potential of internal forces Φ(u,∇u), and the models based on the dependence of Fint on the order parameter and its gradients [18,15]. Taking Eq. (29) into account and discarding (for simplicity) the role of the external potential, results (27) and (28) now appear in their final forms: dui uj = uj uk Ωik + ui uk Ωjk + λuj uk Aik dt + λui uk Ajk − 2λui uj uk um Akm
2 −∂F int ∂F int + + ∇m ζ ∂Skl ∂(∇m Skl ) × [uj ul Iik + ui ul Ijk − 2ui uj uk ul ] 2kB T − (3ui uj − Iij ). ζ
(30)
and ∂F int σij = ηijkl Akl + 3νkB Tλui uj − ν∇i Skm ∂∇j Skm
int int −∂F ∂F −ν − + ∇m ∂Sk1 ∂∇m Sk1 × (λ − 1)uj ul δik + (λ + 1)ui ul δjk − 2λui uj uk ul
(31)
where the expression of the viscosity tensor ηijkl was given in Section 5 and leads to ηijkl Akl = 2η2 Aij + 2(η3 − η2 )[ui uk Akj + uj uk Aki ] + 2(η1 + η2 − 2η3 )ui uj uk ul Akl with η1 ≥ 0, η2 ≥ 0 and η3 ≥ 0.
8. Some special expressions for the internal potential and their relations to previous works The simplest example of a free-energy which favours a non-zero tensor order parameter S while penalizing its gradient is F int = 43 kB TU[−Sij Sij + L2 ∇k Sij ∇k Sij ], where L is some molecular length. According to Eq. (29), the associated internal potential is Φ(u, ∇u) = 23 kB TU[−Sij ui uj + L2 ∇k Sij ∇k (ui uj )].
(32)
According to Eq. (30), the evolution equation of the configuration tensor is
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91
dui uj = uj uk Ωik + ui uk Ωjk + λuj uk Aik dt + λui uk Ajk − 2λui uj uk um Akm 3kB TU [Sjk +L2 ∇ 2 Sjk ][uj ul Iik + ui ul Ijk + ζ 2kB T − 2ui uj uk ul ] − (3ui uj − δij ). (33) ζ
cases is not a proof that Eq. (36) is correct (there is a large arbitrariness in the choice of the b3 term) but it means we are not in conflict with thermodynamics when introducing Eq. (36) into Eqs. (30) and (31). Concerning the contribution f(Sij ), the expression kB TU(1 – 3/2S:S)1/2 proposed by Ilg et al. [23] seems appealing if one wants to avoid long polynomial expressions.
It is noteworthy that the same evolution equation could also have been obtained from Eq. (27) with the simple definition h = −∂Φ/∂u and the quadrupolar potential
9. Conclusions
Φ(u) = − 23 kB TU[S + L2 ∇ 2 S] : uu,
(34)
which is nothing but the “one-constant” form of the Marucci–Greco potential [9]. This means that the same physics is contained in Eqs. (32) and (34) as far as the internal molecular field is concerned. However, one cannot get a distortion stress associated with the Marucci–Greco potential, while the distortion stress associated with Eq. (32) is, according to Eq. (31), −ν∇i Skm
∂F 3 = − νkB TUL2 ∇i Skm ∇j Skm . ∂∇j Skm 2
(35)
Feng et al. [29] incorporated the one-constant Marrucci– Greco potential into the Doi Model to formulate a theory for nematic polymers that contains distortion elasticity. Their results are in agreement with Eq. (33) (with λ = 1) but not with Eq. (35). Generalizing Doi’s virtual work procedure, they found −3/4νkB TU L2 [∇i Skm ∇j Skm − ∇i ∇j Skm uk um ] instead of Eq. (35). We believe our approach is much more systematic: the use of Φ(u,∇u) allows one to derive both the molecular field and the distortion stress, while the use of Φ(u) can lead to a physically correct expression for the molecular field but must be completed on an entirely independent calculation of the distortion stress. This double step procedure is likely to be a source of errors. Let us now comment on the dependence of Fint on the gradients of the order parameter. One expects an expression containing three constants to enhance the possibility of a one-to-one correspondence with Frank’s elastic energy. Such an expression was proposed in Eq. (2.15) of Edwards et al. [18] We suspect that their b3 term fails to be positive in any circumstances and we propose instead F int = f(Sij ) +
b2 2 ∇i Sjk ∇j Sjk
+
b2 2 ∇j Sij ∇k Sik
+ 9b23 Sij Sik ∇j Smn ∇k Smn ,
(36)
with b1 > 0, b2 > 0 and b3 > 0. In the uniaxial limit where Sij = S(ni nj − δij /3), this elastic energy boils down to Frank’s with the three elastic constants: K1 = 2S 2 (b1 + b2 + b3 S 2 ),
K2 = 2S 2 (b1 + b3 S 2 ),
K3 = 2S 2 (b1 + b2 + 4b3 S 2 ). As a consequence, K2 < K1 < K3 and 3K2 + K1 > K3 . The fact that these inequalities are fulfilled in all known
With the ultimate objective of describing the textures and defects appearing in the flow of nematic polymers, we have extended the original Doi Model to endow it with orientational elasticity. This was done by considering the molecular orientation and its local gradient as two independent variables. The potential of the internal forces was supposed to be a function of these two variables, which allowed us to define an elastic stress, an elastic couple stress and a molecular field (see Eqs. (11)–(13)) much like in the Ericksen–Leslie Model). The main difference from that well-known model is that those quantities are defined at the mesoscopic level and a probability distribution function (also depending on molecular orientation and its gradient) is necessary to provide the averaged quantities relevant at the macroscopic level. The micropolar fluid model and the thermodynamics of irreversible processes proved to be convenient tools to describe the dynamics of nematic polymers with a consistent picture of orientational elasticity. Among the main simplifying assumptions that were made is the spatial uniformity of the polymer concentration. To remove that assumption is not that simple within the present model, and one must use a more elaborate framework including translational diffusion [30,31] or the mesoscopic theory of Muschik et al. [32]. It is our view that the complexity of the final results shown in Eqs. (30) and (31) is such that we must first test their potentialities prior to disregarding the assumptions which led to them.
Acknowledgements A.D.R. acknowledges support from the Natural Sciences and Engineering Research Council (Canada), the Donors of The Petroleum Research Fund (PRF, American Chemical Society), and the NSF Center for Advanced Fibers and Films (CAEFF/NSF) at Clemson University.
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