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Orientational dynamics of nematic liquid crystals under shear flow
Physica A 267 (1999) 294–321
www.elsevier.com/locate/physa
Orientational dynamics of nematic liquid crystals under shear ow G. Rienacker ∗ , S. Hess Institut fur Theoretische Physik, TU Berlin, Hardenbergstr. 36, 10623 Berlin, Germany Received 9 December 1998
Abstract The orientational dynamics of low molecular weight and polymeric nematic liquid crystals in a ow eld is investigated, based on a nonlinear relaxation equation for the second rank alignment tensor. Various approximations are discussed: Assuming uniaxial alignment with a constant order parameter, the results of the Ericksen–Leslie theory are recovered. The detailed analysis to be presented here for plane Couette ow concerns (i) uniaxial alignment with a variable degree of order and (ii) the tensorial analysis involving the three symmetry-adapted components of the ve components of the alignment tensor. The transitions between tumbling, wagging and aligning behavior observed in polymeric liquid crystals and described by the Doi theory of rod-like nematic polymers are recovered. Consequences for the rheological behavior c 1999 Elsevier Science B.V. All rights reserved. are indicated. PACS: 61.30.Gd; 05.70.Ln; 62.15.+i; 05.60.+w Keywords: Nematic liquid crystals; Nematic polymers; Order parameter; Orientational dynamics; Shear ow
1. Introduction The nonsteady, oscillating response of a physical system to a steady driving “force” is a fascinating phenomenon. In nematic liquid crystals undergoing a shear ow, such behavior can occur when the anisotropy of the viscosity does not allow steady ow alignment of the director. This e ect is referred to as director tumbling [1,2]. In low molecular weight liquid crystals, the transition from ow aligned to tumbling behavior can occur with decreasing temperature. In polymeric liquid crystals, such as PBLG (poly- -benzyl L-glutamate) and HPC (hydroxypropylcellulose), still richer behavior of ∗
Corresponding author. E-mail address: [email protected] (G. Rienacker)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 6 9 - 4
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the molecular orientation has been observed via the resulting changes of the rheological properties with increasing shear rate [3–7]. Solutions of the Doi equation for rod-like liquid-crystalline polymers [8] showed that shear induced transitions between tumbling, wagging and ow aligned regimes are responsible for the unusual behavior of the rst and second normal stress di erences [9,10]. It is the purpose of this article to present theoretical results for the dynamics of the molecular orientation based on a nonlinear relaxation equation for the second rank alignment tensor [11–13] which characterizes the molecular orientation in a ow eld, which is treated here as an external variable. Various approximations are discussed. Assuming uniaxial alignment with a constant order parameter, the results of the well-known Ericksen–Leslie theory [14,15] are recovered. The detailed analysis to be presented here for plane Couette ow concerns (i) uniaxial alignment with variable degree of order and (ii) the tensorial analysis involving the three symmetry-adapted components of the ve components of the alignment tensor. Consequences for the rheological behavior are indicated. Compared to those of previous theoretical studies [16,17], the constitutive equations given here are relatively simple and contain a smaller set of parameters. The steady states can be calculated analytically and some conclusions concerning stability and, in case (ii), biaxial distortions of the steady states can be drawn. We believe that the qualitative dynamical behavior for low and medium shear rates can be captured by this model. Because terms higher than third order in the second-rank alignment tensor are neglected, deviations are to be expected for high shear rates. Further restrictions are given by the in-plane assumption and the spatially homogeneous ansatz for the alignment tensor, and the disregarded back ow e ects in the solution of the equations. Spatially inhomogeneous ow phenomena of liquid-crystalline polymers, such as ow instabilities, banded and striped textures, are beyond the scope of this model. This article is divided into four sections. Section 2 introduces the second-rank alignment tensor, discusses uniaxial and biaxial alignment, and the constitutive equations for the alignment tensor and pressure tensor. In Section 3, stationary and transient solutions of the alignment tensor equation are presented (i) for the in-plane uniaxial and (ii) symmetry-adapted tensorial ansatz mentioned above. The stability of the stationary solutions is investigated, the tumbling, wagging and aligning regimes are discussed, and the critical shear rates for the transition between these regimes are calculated. Using ansatz (ii), also the biaxiality of the alignment is discussed and consequences for the normal stress di erences are indicated. Concluding remarks are given in Section 4.
2. Theory 2.1. Alignment tensor properties (1) Orientational distribution function and alignment tensor. The orientation of a nematic liquid crystal, consisting of e ectively uniaxial molecules with the molecular
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direction given by the unit vector u, can be described by the one-particle orientational distribution function (u): 1 (1 + 2 a : + higher-order terms) : (1) = 4 The second rank alignment tensor a = 2 h
i;
(2)
sometimes also denoted by Q or S, is directly connected with mechanical and electromagnetic properties of the liquid crystal and, p time. R in general, depends on space and The average h:R : :i is de ned as h i:= S 2 d 2 u, and the constant 2 = 15=2 is chosen so that S 2 d 2 u = 1. The expression := 12 (b + bT ) − 31 (tr b) denotes the symmetric-traceless (irreducible) part of the tensor b where is the unit tensor. When possible, the compact notation (bold italic C for vectors and bold upright t for second rank tensors) is used where s · t and s : t are de ned by s t and s t respectively. By convention, summation over equal Greek indices is assumed. (2) Uniaxial ansatz, biaxiality. If in the nematic phase the orientational distribution is uniaxial, a has two equal eigenvalues 1 = 2 6= 3 and can be written as p (3) a = 3=2a with the nematic director n being the direction corresponding to the third eigenvalue 3 and the symmetry axis of the orientational distribution. The scalar order parameter a with a2 = a : a is related √to the commonly used Maier-Saupe order parameter S2 = h 32 (n · u)2 − 12 i by S2 = a= 5. In general, however, the alignment tensor has three di erent eigenvalues and leads to a biaxial orientational distribution. A measure of biaxiality is given via the parameter b de ned by [18] b2 = (I23 − I32 )=I23
(4)
with the scalar invariants I2 = tr(a · a) = a : a; 2
I3 =
√
6tr(a · a · a) =
√
6(a · a) : a :
(5)
2
The cases b = 0 and b = 1 correspond to uniaxial and planar biaxial alignment, respectively. 2.2. Constitutive equations for alignment and pressure (1) Alignment tensor. Constitutive equations for the dynamics of alignment tensor a and friction pressure tensor p in a ow eld C have been derived within the framework of irreversible thermodynamics [11–13] and describe the isotropic and nematic phase of a liquid crystal. The equation for the alignment tensor can be written for the spatially homogeneous case as √ ap da − 2 + −1 : (6) −2 a =− 2 dt a
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The tensors = 12 ((∇C)T +∇C) and = 12 ((∇C)T −∇C) are the symmetric traceless and asymmetric parts of the velocity gradient ∇C. In the case of the shear ow considered, C = ˙ ye x (plane Couette ow), with the shear rate ˙ , they are given by = 12 ˙ (e x ey + ey e x )
and
1
= ˙ (e x ey − ey e x ): 2
(7)
The constants a and ap are phenomenological relaxation times with a ¿ 0 and ap having either sign, and is a coecient explicitly describing the change of alignment caused by the deformation rate . It is needed to obtain the full anisotropy of the viscosity in the nematic phase [13]. In Eq. (6), = @=@a is the derivative of the Landau–de Gennes potential (a) = 12 Aa : a −
1 3
√
6B(a · a) : a + 14 C(a : a)2
(8)
which is proportional to the free energy of the alignment. The coecients A = A0 (1 − T ∗ =T ); B and C are constrained by the conditions A0 ¿ 0; B ¿ 0; C ¿ 0 and B2 ¿ 92 A0 C. The pseudo-critical temperature T ∗ is somewhat below the nematic-isotropic transition temperature Tk . Eq. (6) can also be obtained from a Fokker–Planck type equation for ellipsoidal particles [19]. From the latter equation, coupled nonlinear equations for the second and fourth rank alignment tensors a = a(2) and a(4) were derived. The coupling to alignment tensors of rank six and higher was ignored, and a(4) was expressed in terms of a(2) , based upon the assumption that the relaxation of a(4) is considerably faster and the in uence of the velocity gradient on it is considerably weaker than on a(2) . This molecular approach yields the result that and ap = a are related to a “nonsphericity” parameter R which is positive for prolate, p zero for spherical and negative for oblate particles by = 37 R and ap = a = − 3=5R. The ane transformation model for perfectly aligned uniaxial ellipsoidal particles with the axis ratio Q suggests R = (Q2 − 1)=(Q2 + 2) [20]. The Doi theory [8–10] is based on the same Fokker–Planck equation, but uses a di erent decoupling scheme (closure) for the elimination of the higher rank tensors. Recently, a variety of other closure approximations have been discussed in [21]. It should be mentioned that a relaxation equation with the same mathematical structure as (6) was suggested by Hess and Hess [18] for the friction pressure tensor, which describes the non-Newtonian behavior of the viscosity and the normal pressure di erences of uids showing shear thickening. (2) Pressure tensor. The full pressure tensor p consists of a hydrostatic pressure p = 13 trp, an asymmetric part pa = 12 (p − pT ) and the symmetric traceless part referred to as friction pressure tensor. The latter can be split into an “isotropic part” and a part explicitly depending on the alignment tensor: = −2iso
+
(9)
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with iso = kB T p m =
kB T m
2ap 1− a p
! ;
√ ap 2 − 2 a
(10) :
(11)
The constant p is another phenomenological relaxation time, and there are two more restrictions p ¿ 0 and a p ¿ 2ap for the relaxation times. In general, there is also an antisymmetric part of the pressure tensor (see [13]). However, it is not needed for the calculation of the normal stress di erences and is therefore omitted. (3) Dimensionless form. With the scalings t = t ∗ a =Ak , C = C∗ Ak = a and a = a∗ ak , Eq. (6) can be transformed into a dimensionless form r √ 3 da∗ ∗ ∗ ∗ ∗ k ∗ ; −2 − 2 + (#a − 3 6 + 2(a : a )a ) = ∗ dt 2 (12) where Ak = A0 (1 − T ∗ =Tk ) = 2B2 =9C, ak = aeq (Tk ) = 2B=3C is the (nonzero) equilibrium value of the scalar order parameter a at the transition temperature Tk , 1 2 √ ap k = − 3 ; (13) 3 a a k ∗
and ∗ are the parts of the dimensionless velocity gradient and T∗ T∗ 1− #= 1− T Tk
(14)
is a reduced temperature variable chosen so that # = 0 for T = T ∗ and # = 1 for T = Tk . For the sake of simplicity, the asterisks for the rescaled variables are omitted in the following.
3. Orientational dynamics in a plane Couette ow 3.1. Dynamics of director and order parameter Inserting (3) into (12), applying the operations · = 13 + 29 and n · n=1, d(n · n)=dt =0, for a and n da = (a) : − a (a) ; dt 1
: and n ×n· and using the relations : = 23 leads to coupled equations (15)
Assuming the critical order parameter Sk = S(Tk ) ≈ 0:4, one has ak ≈ 0:9. Measurements of S yielded, e.g., Sk = 0:4 for 4:40 -di-methoxy-azoxybenzol [22], Sk = 0:28 for 5CB and Sk = 0:32 for 8CB [23].
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dn = · n + (a)( · n − : nnn) : (16) dt with the abbreviations k 3 (17) (a) = a + k and (a) = + 2 3 a and the derivative @ = (#a − 3a2 + 2a3 ) (18) a (a) ≡ @a of the dimensionless Landau–de Gennes potential (a) for the uniaxial case. If no ow is present, the order parameter a reaches an equilibrium value aeq , which in the nematic phase is given by √ (19) a (aeq ) = 0; aeq = 34 + 14 9 − 8# : (1) Relation to the Ericksen–Leslie theory. In the low shear rate limit ˙ 1, the in uence of the ow on the order parameter vanishes and one can identify a with aeq . Eq. (16) then reduces to the spatially homogeneous case of the Ericksen–Leslie equation (see, e.g., [24]) for the director, usually written as dn − · n + 2 · n = 0 ; (20) n × 1 dt which says that the viscous torque on the director has to vanish. The Leslie coecients
1 and 2 are related to the coecients of (6) by 2 √
1 = 3 kB T (ak aeq )2 a and 2 = kB T (2 3(ak aeq ) ap − (ak aeq )2 a ): (21) m m The tumbling parameter = − 2 = 1 of the Ericksen–Leslie theory is then given by k : (22) = (aeq ) = + 3 aeq It is positive (negative) for nematics built from rod-like (disk-like) molecules. In a steady shear ow with the shear rate ˙ , for ||¿1, the torque at the director vanishes when it aligns in the shear plane at a speci c angle ’ = 12 cos−1 (1=)
(23) 3
to the ow direction ( ow-aligning). In the case || ¡ 1, no stationary solution with zero torque is possible and the director, when it initially lies in the shear plane, rotates continuously in this plane (tumbling). 4 The tumbling period is given by 2 p 1 − 2 : (24) T=
˙ For an out-of-plane director, the movement is more complicated, but the projection of n to the shear plane still rotates with the period T . 2
Note that ak appears here explicitly because aeq is given in units of ak . Low molecular ow-aligning substances are, e.g., p0 -methoxybenzylidene-p-n-butylaniline (MBBA) with ◦ ◦ = 1:02 for T = 22 C [1] and 4-n-pentyl-40 -cyanobiphenyl (5CB) with = 1:16 for T = 32:5 C [25]. 4 The substance 4-n-octyl-40 -cyanobiphenyl (8CB) with = 0:27 for T = 34:0◦ C [26] is an instance of a low molecular tumbling nematic. 3
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The ow-aligning or tumbling behavior described by (20) is completely determined by the tumbling parameter , i.e., independent of the shear rate. This is practically true for low molecular weight nematics, where the relaxation times a , ap are very small, and therefore the dimensionless shear rate ˙ ∗ = ( a =Ak )˙ is much smaller than one and the order parameter is not de ected much from its equilibrium value. For nematic polymers, the situation is di erent. Because of their high relaxation times, 5 even small experimental shear rates lead to a strong coupling of (15) and (16). The e ective tumbling parameter (a) now becomes time dependent. (2) In-plane ansatz. To investigate the dynamics of scalar order parameter and director in a plane Couette ow C = ˙ ye x with the dimensionless shear rate ˙ , the most simple ansatz for the director is the restriction to the shear plane n = cos ’ e x + sin ’ ey
(25)
with the director angle ’. Insertion of (25) into (15) and (16) leads to the dynamical system a˙ = 12 ˙ (a)sin 2’ − a ;
(26)
’˙ = 12 ˙ ((a)cos 2’ − 1) :
(27)
In the following, we only consider liquid crystals composed of elongated particles (k ; ¿ 0) and positive shear rates. 3.1.1. Steady states As a rst step, the stationary solutions a(˙ ), ’(˙ ) are computed by setting a˙ and ’˙ to zero. It is more convenient to solve ˙ as a function of a in (26), where sin 2’ can be expressed through (a) with the help of (27): cos 2’ = (a)−1 ;
˙ (a) =
2a 2(#a − 3a2 + 2a3 ) p : =± (a)sin 2’ (a) 1 − (a)−2
(28)
The solutions are plotted in Fig. 1 for the temperature variable # = 0, the coecient = 0 and characteristic values of k . For # ¡ 0 and a¿0, the numerator a of ˙ (a) has two zeros 0 and aeq and is negative for 0 ¡ a ¡ aeq . The denominator is de ned for a ¡ amax = k =(1 − =3) and tends to zero for a → amax . Depending on the ratio of amax and aeq there are three cases: (i) amax ¡ aeq , (ii) amax =aeq and (iii) amax ¿ aeq . They correspond to eq ≡ (aeq ) ¡ 1, =1 and ¿ 1, the tumbling and aligning cases of the Ericksen-Leslie theory. In the rst case, one has a ¡ aeq and ’ ¡ 0. This means the alignment angle is negative here. In the second case, one has solutions with a ¡ aeq and ’ ¡ 0 for low shear rates and an additional solution a = aeq and ’ = 0 for arbitrary ˙ . The third 5
The viscosities of liquid-crystalline polymers are often some orders of magnitude larger than those of low molecular weight nematics.
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Fig. 1. Stationary values of a and ’ in system (26), (27) for # = 0 and = 0 (amax = k ). Top: ’(a=k ) (positive branch). Bottom: ˙(a) for (a) k = 1:4 (eq = 0:93), (b) k = 1:5 (eq = 1) and (c) k = 1:6 (eq = 1:07). For a ¡ aeq = 3=2, the alignment angle ’ is negative.
case again permits a ¡ aeq and ’ ¡ 0 for low shear rates, but also has solutions with amax ¿ a ¿ aeq and positive ’ for arbitrary ˙ . In cases (i) and (iii), the order parameter a tends to amax and the alignment angle ’, to zero for high shear rates. Although the parameters and k are not accessible by experiment, the tumbling parameter eq ≡ (aeq ) can be obtained from the Leslie coecients 1 and 2 . When (a) and (a) are expressed by eq and , aeq 3 ; (a) = aeq eq + a − 2 2
(a) =
aeq aeq eq + 1− ; a 3 a
(29)
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one can see that, for a given eq , the in uence of a realistic 6 is relatively small and one can expect that the essential dynamics is already captured in the simpler case = 0, where (a) = k =a, (a) = 32 k and amax = k . Indeed, for, say, = 3=7, the same qualitative behavior of the dynamical systems occurs for slightly di erent ˙ (see Fig. 12). (1) Stability analysis. A linear stability analysis reveals whether the steady states calculated are stable against small perturbations. For the trace and determinant of the Jacobian matrix, 1 0
˙ (a)sin 2’ − aa
˙ (a)cos 2’ ; (30) J (a;’) = 2 1 0 ˙ (a)cos 2’ −˙ (a)sin 2’ 2 one obtains, using the relations ˙ (a) and ’(a), = tr J (a; ’) =
= det J
(a; ’)
a ( − 2(a)) − aa (a)
2a = (a)
(31)
k a (a)a − + (a)aa a2 ((a)2 − 1) (a)
= (a)sin 2’ a ˙ 0 (a);
(32)
where aa ≡ @2 =@a2 and ˙ 0 (a) ≡ @˙ =@a. The xed points of (26), (27) are hyperbolic and asymptotically stable (of type focus or node) if ¿ 0 and ¡ 0. The case ¡ 0 leads to a saddle point. Eq. (32) shows a relation between saddle point behavior and the shape of ˙ (a). Because both sin 2’ and a are positive (negative) for a ¿ aeq (a ¡ aeq ), a saddle point occurs when ˙ 0 (a) is negative. The stability regions are discussed for the reduced temperature # = 0 and coecient = 0. In this case, is independent of k and negative for a ¿ as ≈ 1:154. The real and imaginary parts of the eigenvalues 1 , 2 of J (a; ’) for k = 1:4 (eq = 0:93), 1:5 (eq = 1) and 1:6 (eq = 1:07) are shown in Figs. 2– 4. • For k = 1:4, one has ¿ 0 (no saddle points) throughout. When ˙ ¿ ˙ c2 ≈ 1:55, a is bigger than as and the stationary state with negative alignment angle is stable. Fig. 2 shows that the imaginary part of the eigenvalues is nonzero for approximately a ¡ 1:29 and a ¿ 1:347. This means one has a focus in these cases. In particular, at a = as , the real part crosses zero with a nonvanishing slope indicating a supercritical Hopf bifurcation (birth of a stable limit cycle for a ¡ as or ˙ ¡ ˙ c2 ). • If k = aeq = 1:5, one has ¿ 0 for a ¡ 1:171 (left of the local maximum of ˙ (a)). That means the steady states in the small interval as ¡ a ¡ 1:171 are stable. In the special case a = aeq and ’ = 0, the relation for ˙ (a) is invalid and, therefore, and have to be taken directly from (30): = −aa = −# + 6aeq − 6a2eq ¡ 0
and
= − 12 ˙ 2 (a)0 (a) = 34 ˙ 2 ¿ 0 : (33)
6
The ane transformation model, e.g., admits ||63=7.
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Fig. 2. Real and imaginary part of the eigenvalues 1 , 2 of the Jacobian matrix of system (26), (27) for # = 0, = 0 and k = 1:4 (eq = 0:93).
Fig. 3. Real and imaginary part of the eigenvalues 1 , 2 of the Jacobian matrix of system (26), (27) for # = 0, = 0 and k = 1:5 (eq = 1).
Consequently, this state is asymptotically stable for all ˙ 6= 0. The type of the xed point changes at approximately ˙ ¿ 2:6 where 2 =4 ¡ from node to focus causing a relaxation to the steady state via damped oscillations. • For k = 1:6 (aligning case), is positive when a ¿ aeq = 1:5 or a ¡ 1:122. In this case, one has a stable shear aligning solution with a ¿ aeq and ’ ¿ 0 for all
˙ ¿ 0. For a ¡ 1:122, one has ¿ 0, i.e., an unstable focus or node. The eigenvalue plot, Fig. 4, shows that the type of the xed point changes from node to focus for a ¿ 1:588 (˙ ¿ 3:02) indicating the presence of damped oscillations of a and ’ towards the steady state.
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Fig. 4. Real and imaginary part of the eigenvalues 1 , 2 of the Jacobian matrix of system (26), (27) for # = 0, = 0 and k = 1:6 (eq = 1:07).
No shear aligning state is possible for amax = k ¡ as , because then is positive for all values of a. 3.1.2. Dynamical behavior of the system To investigate the dynamical behavior of (26), (27), the system is numerically integrated for the temperature # = 0 and coecient = 0 using the initial conditions a = aeq = 1:5 and ’ = 0. The results are shown in Figs. 5 –11. For k = 1:4, the system shows three dynamical modes: • When ˙ ¡ ˙ c1 ≈ 1:44 tumbling occurs. The director rotates in the x − y plane and the order parameter is periodically de ected from its equilibrium value. The periodic minima of a become deeper when ˙ is increased (Figs. 5 and 7). • For ˙ in the small interval between ˙ c1 and ˙ c2 ≈ 1:55, the xed point a(˙ ) is still unstable but the tumbling is replaced by wagging [10], an oscillation of the director angle. Increasing ˙ leads to a shift of the center of the oscillation towards a negative value and to a decrease of the amplitude (Figs. 6a–c and 8). • For ˙ ¿˙ c2 , the xed point is stable and the oscillations of a and ’ are damped. The director aligns to a negative alignment angle that tends towards zero with further increase of the shear rate (Figs. 6d and 9). ◦ For k =1:5, the xed point is at a=aeq and ’=0 and, therefore, the initial conditions are chosen di erently. At higher shear rates, a and ’ undergo damped oscillations around their stationary values as indicated (Fig. 10). For k = 1:6, the director aligns at a positive angle which tends towards zero when
˙ is increased. Again, damped oscillations occur at higher shear rates (Fig. 11). In the rst case, k = 1:4 (tumbling regime for low shear rates), the system described
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Fig. 5. Dynamics of system (26), (27) for # = 0, = 0 and k = 1:4. Top: a(˙ t) and bottom: ’(˙ t) for (a)
˙ = 0:1, (b) ˙ = 0:5, (c) ˙ = 1:0, (d) ˙ = 1:2, (e) ˙ = 1:3, (f) ˙ = 1:4.
by (26), (27) shows at low and medium shear rates qualitatively the same dynamics as reported, e.g., by Larson [10] for the (direct) solutions of the Doi equation. As the shear rate increases, the order parameter deviates more and more from its equilibrium value and a transition from tumbling to wagging occurs. Note that the solutions of the Doi equation are actually no longer uniaxial and, therefore, Larson introduces the concept of the birefringence axes as a generalization of the director. With further increase of the shear rate, the oscillations become damped, leading to a steady state with slightly negative alignment angle and diminished uniaxial order parameter. The di erence to the results obtained here is that the alignment angle again becomes positive and the order parameter exceeds its equilibrium value for even higher shear rates.
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Fig. 6. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Top: a(˙ t) and bottom: ’(˙ t) for (a)
˙ = 1:45, (b) ˙ = 1:5, (c) ˙ = 1:55, (d) ˙ = 1:6.
Coupled dynamical equations for the in-plane director and scalar order parameter dynamics based upon a phenomenological continuum theory for the alignment tensor have also been derived by Farhoudi and Rey [16]. They predict, in analogy to the results presented here, a complex orientation mode in simple shear ow at suciently high nematic potentials which exhibits tumbling, wagging and stationary regimes depending on the shear rate. (2) Critical shear rates. The critical shear rates ˙ c1 and ˙ c2 of the tumbling–wagging and wagging–aligning transition are plotted vs. the equilibrium tumbling parameter eq in Fig. 12 for various values of temperature # and coecient . The values decrease with eq . For low eq , rst the aligning and then the wagging state vanishes.
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Fig. 7. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Plot of a(t) vs: ’(t) for ˙ = 1:4 (tumbling regime, just below ˙c1 ).
Fig. 8. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Plot of a(t) vs: ’(t) for ˙ = 1:45 (wagging regime).
Plots a, b for # = 0; = 0:4 and plots c, d for # = 0; = 0 show that has not much in uence on the dynamics. For lower # (plots e, f), ˙ c1 and ˙ c2 are shifted towards higher values. 3.2. Dynamics of the symmetry-adapted alignment tensor After the investigation of the uniaxial in-plane system, now the dynamics of the alignment tensor itself is considered and compared to that of system (26), (27). Because
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Fig. 9. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Plot of a(t) vs: ’(t) for ˙ = 1:56 (aligning regime, just above ˙c2 ).
the driving ow eld ∇C is biaxial, the resulting alignment is also expected to be biaxially distorted in the nematic phase. (1) Basis tensors. The tensorial equation (12) is equivalent to ve equations for ve linearly independent scalars because a is symmetric traceless. For the plane Couette
ow considered, a special geometry-adapted ansatz with three components can be made [27] which includes, e.g., those uniaxial states with the director either lying in the shear plane (x–y) or in the vorticity direction (z): a = a0 T0 + a1 T1 + a2 T2
(34)
with the three orthonormalized symmetric traceless tensors r √ 3 1√ x x 0 ; T1 = 2(e e − ey ey ); T2 = 2 T = 2 2
:
(35)
The following relations are satis ed Ti : Tk = ik
(i; k = 0; 1; 2) ;
(36)
a20 + a21 + a22 = a2 = a : a :
(37)
(2) Equations for three tensor components. Ansatz (34) leads to three equations for the three components a0 ; a1 and a2 √ a˙0 = −#a0 − 3(a21 + a22 − a20 ) − 2a2 a0 − 13 3˙ a2 ; (38) ˜ 1; a˙1 = ˙ a2 − #a ˜ 2+ a˙2 = −˙ a1 − #a
(39) 1 2
√
3k ˙ −
1 3
√
3˙ a0 ;
(40)
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Fig. 10. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:5 (from initial conditions a = 1:6 and ◦ ’ = 45 ). Top: a(˙ t) and bottom: ’(˙ t) for (a) ˙ = 0:1, (b) ˙ = 0:5, (c) ˙ = 1:0, (d) ˙ = 5:0, (e) ˙ = 10.
with #˜ = # + 6a0 + 2a2 :
(41)
(3) Equations for one angle and two order parameters. Using the de nitions a1 = p cos 2’; 2
2
2
a2 = p sin 2’; 2
r =a =p +q ;
p2 = a21 + a22 ;
q = a0 ;
(42)
Eqs. (38) – (40) are equivalent to three equations for one angle ’ and two order parameters p and q [27,18] √ √ (43) p˙ = −(# + 6q + 2r 2 )p + ˙ ( 12 3k − 13 3q) sin 2’ ;
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Fig. 11. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:6. Top: a(˙ t) and bottom: ’(˙ t) for (a) ˙ = 0:1, (b) ˙ = 0:5, (c) ˙ = 1:0, (d) ˙ = 5:0, (e) ˙ = 10.
√ √ ’˙ = 12 ˙ (p−1 ( 12 3k − 13 3q) cos 2’ − 1) ; q˙ = −#q − 3(p2 − q2 ) − 2qr 2 −
1 3
√
3˙ p sin 2’ :
(44) (45)
With the de nitions (42), the alignment tensor has two eigenvectors within the shear plane enclosing the angle ’ √ and =2 + √ ’ with the ow direction, which correspond to the eigenvalues 1; 2 = (± 3p −pq)= 6, and a third eigenvector in the vorticity direction with the eigenvalue 3 = 2=3q. There are two uniaxial cases: (a)√ n = e z for p = 0 and q = r and (b) n = cos’ e x + sin’ ey (in-plane case) for p = 12 3r and q = − 12 r. For the general, biaxial case, three mutually orthogonal directors (Larson: birefringence axes) exist, which are parallel to the principal axes.
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Fig. 12. Critical shear rates ˙c1 ; ˙c2 of system (26), (27) for the tumbling-wagging and wagging-aligning transitions vs. equilibrium tumbling parameter eq for a, b: #=0; =0:4; c, d: #=0; =0; e, f: #=−1; =0.
Inserting the in-plane case (b) into (43) – (45), one does not recover the uniaxial system (26), (27) (Eqs. (43) and (45) are not compatible for, e.g., = 0). Therefore trajectories of (43) – (45) are, in general, not uniaxial for ˙ 6= 0, even when they start from uniaxial initial conditions. 3.2.1. Steady states In the following, = 0 is assumed for the sake of simplicity: √ ˜ + ˙ 1 3k sin 2’ ; p˙ = −#p 2 1 √ k 1 3 cos 2’ − 1 ; ’˙ = ˙ 2 2 p q˙ = −#q − 3(p2 − q2 ) − 2qr 2 :
(46) (47) (48)
Again, as a rst step, the stationary solutions are obtained by setting p; ˙ ’˙ and q˙ to zero. The procedure is similar to the previous case: ˙ is expressed as a function of p; ’ and q in (46). Then ’ and p are eliminated using (47) and (48). As in the uniaxial case, only positive k will √ be considered. With the abbreviation k = 12 3k and the assumption k ¿ 0 one gets p2 (q) = −
#q − 3q2 + 2q3 ; 3 + 2q
(49)
cos 2’(q) = p(q)=k ;
˙ (q) =
(50)
˜ ˜ #(q)p(q) #(q)p(q) ; = ±q k sin 2’(q) 2 − p2 (q) k
(51)
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Fig. 13. Stationary solutions of system (46) – (48). Order parameters p(q) and r(q) for (a) # = −1, (b) # = 0, (c) # = 1.
with #˜ expressed by q 3(# + 6q + 8q2 ) ˜ : #(q) = # + 6q + 2q2 + 2p2 (q) = 3 + 2q
(52)
Relations p(q) and r(q) are plotted in Fig. 13 for various reduced temperatures #, the rst plot is similar to the corresponding gure in [27]. The shear rate dependency ˙ (q) of the steady states is plotted in Fig. 14 for # = 0 and various k . The domain of q is limited by the condition 06p(q)6k , i.e., for # = 0, one has q ¡ 1:5 independent of k and q¿ − 0:719; q¿ − 0:75, and q¿ − 0:779 for k = 1:4; 1:5 and 1:6, respectively.
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Fig. 14. Stationary solutions of system (46) – (48). Shear rate ˙(q) for # = 0 and (a) k = 1:4, (b) k = 1:5, (c) k = 1:6.
For q ¡ 0, Fig. 14 looks similar to the plot of ˙ (a) in Fig. 1 when q is replaced by −q. In the interval 0 ¿ q ¿ qeq = −0:75, the alignment angle is negative. If qmin denotes the minimum value of q, such that p(q) ¡ k , then, for k = 1:4; k = 1:5 and k = 1:6, one has qmin ¿ qeq ; qmin = qeq and qmin ¡ qeq . In the rst case, only steady states with ’ ¡ 0 exist for high shear rates, whereas for k = 1:6 also states with ’ ¿ 0 are possible. If the uniaxiality condition p2 = 3q2 = 34 r 2 (see below) is assumed, one obtains
˙ (r) =
#r − 3r 2 + 2r 3 q ; k2 − r 2
(53)
which is, except for the factor 3=4, the same as ˙ (a) in (28). (1) Uniaxiality and biaxiality. The biaxiality can be generally expressed as a function of p and q. With I2 = r 2 = p2 + q2 and I3 = q3 − 3qp2 one has b2 = 1 −
I32 p2 (p2 − 3q2 )2 = 3 (p2 + q2 )3 I2
and
1 − b2 =
I32 q2 (q2 − 3p2 )2 = : 3 (p2 + q2 )3 I2
(54)
If one excludes the isotropic case r 2 = p2 + q2 = 0, the uniaxiality condition b2 = 0 is equivalent to p(p2 − 3q2 ) = 0 :
(55)
With use of the stationarity relation p2 (q) (49) and under the assumption q ¿ − 32 , condition p = 0 is ful lled for √ (56) q = 34 ± 14 9 − 8# :
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Fig. 15. Biaxiality b(q) of the stationary solutions of system (46) – (48) for (a) # = −1, (b) # = 0, (c) # = 1.
Because of p2 − 3q2 = −q(# + 6q + 8q2 )=(3 + 2q), the second condition p2 − 3q2 = 0 holds for √ (57) q = − 38 ± 18 9 − 8# : Both cases lead to the order parameter √ r = 34 ± 14 9 − 8# :
(58)
Furthermore, using (52), one can see that the uniaxiality condition (55) is equivalent to ˜ = 0. Consequently, all equilibrium states (steady states for ˙ = 0) are uniaxial. The #p reverse is also true: except for p =k (’=0), all uniaxial steady states are equilibrium states. The special case p = k is reached for #˜ = 0. Then the states for any ˙ are uniaxial. The planar biaxial states correspond to b2 = 1, i.e. q(q2 − 3p2 ) = 0 :
(59)
For q = 0, one obtains the isotropic case p = r = 0. Because of q2 − 3p2 = q(3# − 6q + 8q2 )=(3 + 2q), condition q2 = 3p2 leads to √ (60) q = 38 ± 18 9 − 24# : The corresponding order parameter is given by r 1√ 1√ 4 3± q= 3 − 8# : r= 3 4 4
(61)
The biaxiality √ b(q) of the steady states is plotted in Fig. 15. For # = 0, the √ states q = − 34 , p = 34 3 and q = 32 ; p = 0 are uniaxial whereas the state q = 34 ; p = 14 3 is planar biaxial.
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Fig. 16. Real part of the eigenvalues 1 ; 2 ; 3 of the Jacobian matrix of system (38) – (40) for # = 0; = 0 and k = 1:4.
(2) Stability analysis. The steady states are stable against small perturbations when the real parts of all eigenvalues of the Jacobian matrix are negative. The Jacobian matrix is given by −6p2 − 6q − 2q2 − # 2k ˙ cos 2’ −p(6 + 4q) (p; q; ’) : = −(k =2p2 )˙ cos 2’ −(k =p)˙ sin 2’ J 0 2 2 −p(6 + 4q) 0 −2p + 6q − 6q − # (62) q ˜ and k ˙ cos 2’ = |#|p ˜ 2 = 2 − p2 , using the fact Relation (51) yields k ˙ sin 2’ = #p k ˜ that ’ and # are of the same sign for p ¿ 0. Inserting the steady states into (62) yields for the case # = 0 the following intervals of q where the steady states are stable: −0:719 ¡ q ¡ − 0:62 or q ¿ 1:17 for k = 1:4 and −0:779 ¡ q ¡− 0:75 or q ¿ 1:18 for k = 1:6. Consequently, for q ¡ 0, the steady states for k = 1:4 are stable when ˙ is suciently high, whereas for k = 1:6 stable steady states exist for any ˙ as in the uniaxial case. √ In the case k = 1:5, the state p = 34 3, ’ = 0; q = − 34 with p = k is inserted directly into (62), and one obtains stability for all ˙ . The real parts of the eigenvalues of (62) are shown in Figs. 16 –18. Fig. 14 shows that for low shear rates, there are also steady states with positive q. In this case, q must be to the right of the local maximum of ˙ (q) to ensure stability. 3.2.2. Dynamical behavior To investigate the dynamical behavior, system (38) – (40) is integrated numerically and the values p; q; r; b and ’ are extracted from a0 ; a1 , a2 . As initial conditions, uniaxial √ equilibrium states are chosen with the director oriented in the ow direction: p = 12 3aeq ; q = − 12 aeq ; ’ = 0. The results for # = 0; = 0; k = 1:4; 1:6 and various
˙ are shown in Figs. 19 –22.
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Fig. 17. Real part of the eigenvalues 1 ; 2 ; 3 of the Jacobian matrix of system (38) – (40) for # = 0; = 0 and k = 1:5.
Fig. 18. Real part of the eigenvalues 1 ; 2 ; 3 of the Jacobian matrix of system (38) – (40) for # = 0; = 0 and k = 1:6.
Again, for k ¡ aeq , the system is of the tumbling type for low shear rates and switches to wagging and aligning at higher ˙ . The alignment angle is negative and the order parameter r is smaller than aeq . The biaxiality b is nonzero and becomes considerably large when the order parameters p and r are low. If k ¿ aeq , aligning occurs for all ˙ with ’ ¿ 0 and r ¿ aeq . The biaxiality is low but nonzero (see Section 3.2.1). In both cases, the alignment angle tends towards zero for high ˙ . (1) Critical shear rates. For # = 0; = 0 and k = 1:4, the critical shear rates of the tumbling-wagging-aligning transitions are ˙ c1 ≈ 1:35 and ˙ c2 ≈ 1:45. Other parameter values lead to di erent ˙ c1; 2 . The values of ˙ c1 and ˙ c2 for # = 0, = 0 and various k are plotted vs. the equilibrium tumbling parameter eq ≡ k =aeq in Fig. 23.
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Fig. 19. Dynamics of system (38) – (40) in the tumbling regime for k ¡ aeq . Order parameters p; q; r, biaxiality b and angle ’ for ˙ = 1:2; # = 0; = 0; k = 1:4.
Fig. 20. Dynamics of system (38) – (40) in the wagging regime for k ¡ aeq . Order parameters p; q; r, biaxiality b and angle ’ for ˙ = 1:4; # = 0; = 0; k = 1:4.
3.2.3. Consequences for the normal stress di erences of liquid-crystalline polymers The normal stress di erences N1 and N2 are de ned as N1 = xx − yy = pyy − pxx and N2 = yy − zz = pzz − pyy . They can be related to the components of in the tensor basis (35): r 3 1√ 1√ p0 ; 2p1 ; N2 = 2p1 + (63) N1 = − 2 2 2 where is calculated from the alignment tensor using Eq. (11). Because samples of liquid-crystalline polymers have a domain structure, i.e., the alignment tensor is spatially inhomogeneous, measured normal stress di erences are always the result of spatial averaging of the individual domains. As done in previous
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Fig. 21. Dynamics of system (38) – (40) in the aligning regime for k ¡ aeq and higher shear rates. Order parameters p; q; r, biaxiality b and angle ’ for ˙ = 1:5; # = 0; = 0; k = 1:4.
Fig. 22. Dynamics of system (38) – (40) in the “simple” aligning regime for k ¿ aeq . Order parameters p, q, r, biaxiality b and angle ’ for ˙ = 0:5, # = 0, = 0, k = 1:6.
work in connection with the Doi theory [28,10,5], the domains are thought to be independent of each other, and the ensemble average is replaced by an average over a tumbling=wagging period for the nonstationary case. Using the parameters # = 0, k = 1:0 and = 0:4 (eq = 0:8), one obtains a plot of N1 and N2 vs. ˙ as shown in Fig. 24. The functional relation is, for lower shear rates, similar to experimental results and theoretical predictions using the Doi theory [3,5 –7]. In particular, it contains one sign change in the normal stress di erences near the tumbling-wagging transition at ˙ = 1:89. On the other hand, both N1 and N2 tend asymptotically towards a constant value for high ˙ .
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Fig. 23. Critical shear rates ˙c1 , ˙c2 of system (38) – (40) for the tumbling–wagging and wagging–aligning transitions vs. equilibrium tumbling parameter eq for # = 0 and = 0.
Fig. 24. Normal stress di erences N1 and N2 computed from the symmetric-traceless part of the pressure tensor vs. shear rate ˙ for # = 0, k = 1:0 and = 0:4.
4. Conclusions A nonlinear relaxation equation for the alignment tensor [11–13] was used to investigate the spatially homogeneous dynamics of rod-like low molecular weight and polymeric nematic liquid crystals in a steady shear ow. Both a uniaxial ansatz with variable order parameter and a symmetry-adapted tensorial ansatz led, depending on the equilibrium tumbling parameter eq , either to simple shear aligning with a positive alignment angle and an enlarged order parameter or to a
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cascade from tumbling via wagging to shear aligning with a negative alignment angle and a diminished order parameter for increasing shear rates. In both cases, the liquid crystal becomes aligned in the ow direction for high shear rates. In contrast to the uniaxial description, the tensorial ansatz showed that, in general, the steady and transient states are biaxial, i.e., the orientational distribution is biaxially distorted by the driving ow eld. Depending on the parameters, a low or medium degree of biaxiality was observed in the numerical calculations carried out. Biaxial behavior except for isolated parameter values was recently shown also for the steady-state solutions of a Doi-type theory using the quadratic and the rst Hinch–Leal closure approximations [29]. The normal pressure di erences are calculated here as a by-product by averaging over oscillation periods (in the case eq ¡ 1). At lower shear rates, the results are in qualitative agreement with experiments and with previous theoretical results of the Doi theory as far as the prediction of the rst sign change in N1 and N2 near the tumbling-wagging transition point is concerned. However, they disagree with both experiments and the Doi theory for higher shear rates. Also, the alignment angle does not become positive again in the high shear limit, in contradistinction to the results of the Doi theory. The di erences in the high shear rate behavior are likely to be due to higher-order terms in the order parameter and back ow e ects in the solution of the dynamic equations, both of which were neglected here. However, both the in-plane uniaxial and symmetry-adapted tensorial cases of the present approach show qualitatively the same dynamical regimes as constitutive equations derived from the molecular kinetic theory (Fokker–Planck equation) in two dimensions by Marrucci and Ma ettone [28,30], and in three dimensions by Larson [10], as well as a continuum tensorial theory by Farhoudi and Rey [16]. The simple approach presented here gives rise to rather complex dynamic behavior which agrees with experimental observations for small and intermediate shear rates. Further investigations using all ve tensor components and including spatial derivatives of the alignment tensor [12] (Frank elasticity terms) are desirable and will lead to more insight into fully three-dimensional alignment and textures. Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich “Anisotrope Fluide” (SFB335) is gratefully acknowledged. References [1] [2] [3] [4] [5]
C. Gahwiller, Mol. Cryst. Liq. Cryst. 20 (1973) 301. P. Pieranski, E. Guyon, Phys. Rev. Lett. 32 (1974) 924. G. Kiss, R.S. Porter, J. Polym. Sci. Polym. Symp. 65 (1978) 193. G. Kiss, R.S. Porter, J. Polym. Sci. Polym. Phys. Ed. 18 (1980) 361. J.J. Magda, S.-G. Baek, K.L.D. Vries, R.G. Larson, Macromolecules 24 (1991) 4460.
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S.-G. Baek, J.J. Magda, S. Cementwala, J. Rheol. 37 (1993) 935. S.-G. Baek, J.J. Magda, R.G. Larson, J. Rheol. 37 (1993) 1201. M. Doi, J. Polym. Sci. Polym. Phys. Ed. 19 (1981) 229. G. Marrucci, P.L. Ma ettone, Macromolecules 22 (1989) 4076. R.G. Larson, Macromolecules 23 (1990) 3983. S. Hess, Z. Naturf. 30a (1975) 728. S. Hess, I. Pardowitz, Z. Naturf. 36a (1981) 554. C. Pereira Borgmeyer, S. Hess, J. Non-Equilib. Thermodyn. 20 (1995) 359. J.L. Ericksen, Trans. Soc. Rheol. 5 (1961) 23. F.M. Leslie, Arch. Rat. Mech. Anal. 28 (1968) 265. Y. Farhoudi, A.D. Rey, J. Rheol. 37 (1993) 289. A.P. Singh, A.D. Rey, J. Phys. II France 4 (1994) 645. O. Hess, S. Hess, Physica 207A (1994) 517. S. Hess, Z. Naturf. 31a (1976) 1034. D.R. Baalss, S. Hess, Z. Naturf. 43a (1988) 662. J. Feng, C.V. Chaubal, L.G. Leal, J. Rheol. 42 (1998) 1095. W. Maier, A. Saupe, Z. Naturf. 13a (1958) 564. P.P. Karat, N.V. Madhusudana, Mol. Cryst. Liq. Cryst. 36 (1976) 51. P.G. De Gennes, J. Prost, The Physics of Liquid Crystals, second ed., Clarendon, Oxford, 1993. K. Skarp, S.T. Lagerwall, B. Stebler, Mol. Cryst. Liq. Cryst. 60 (1980) 215. H. Kneppe, F. Schneider, N.K. Sharma, Ber. Bunsenges. Phys. Chem. 85 (1981) 784. S. Hess, Z. Naturf. 31a (1976) 1507. G. Marrucci, P.L. Ma ettone, J. Rheol. 34 (1990) 1217. Q. Wang, J. Rheol. 41 (1997) 943. P.L. Ma ettone, S. Crescitelli, J. Rheol. 38 (1994) 1559.
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Orientational dynamics of nematic liquid crystals under shear ow G. Rienacker ∗ , S. Hess Institut fur Theoretische Physik, TU Berlin, Hardenbergstr. 36, 10623 Berlin, Germany Received 9 December 1998
Abstract The orientational dynamics of low molecular weight and polymeric nematic liquid crystals in a ow eld is investigated, based on a nonlinear relaxation equation for the second rank alignment tensor. Various approximations are discussed: Assuming uniaxial alignment with a constant order parameter, the results of the Ericksen–Leslie theory are recovered. The detailed analysis to be presented here for plane Couette ow concerns (i) uniaxial alignment with a variable degree of order and (ii) the tensorial analysis involving the three symmetry-adapted components of the ve components of the alignment tensor. The transitions between tumbling, wagging and aligning behavior observed in polymeric liquid crystals and described by the Doi theory of rod-like nematic polymers are recovered. Consequences for the rheological behavior c 1999 Elsevier Science B.V. All rights reserved. are indicated. PACS: 61.30.Gd; 05.70.Ln; 62.15.+i; 05.60.+w Keywords: Nematic liquid crystals; Nematic polymers; Order parameter; Orientational dynamics; Shear ow
1. Introduction The nonsteady, oscillating response of a physical system to a steady driving “force” is a fascinating phenomenon. In nematic liquid crystals undergoing a shear ow, such behavior can occur when the anisotropy of the viscosity does not allow steady ow alignment of the director. This e ect is referred to as director tumbling [1,2]. In low molecular weight liquid crystals, the transition from ow aligned to tumbling behavior can occur with decreasing temperature. In polymeric liquid crystals, such as PBLG (poly- -benzyl L-glutamate) and HPC (hydroxypropylcellulose), still richer behavior of ∗
Corresponding author. E-mail address: [email protected] (G. Rienacker)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 6 9 - 4
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the molecular orientation has been observed via the resulting changes of the rheological properties with increasing shear rate [3–7]. Solutions of the Doi equation for rod-like liquid-crystalline polymers [8] showed that shear induced transitions between tumbling, wagging and ow aligned regimes are responsible for the unusual behavior of the rst and second normal stress di erences [9,10]. It is the purpose of this article to present theoretical results for the dynamics of the molecular orientation based on a nonlinear relaxation equation for the second rank alignment tensor [11–13] which characterizes the molecular orientation in a ow eld, which is treated here as an external variable. Various approximations are discussed. Assuming uniaxial alignment with a constant order parameter, the results of the well-known Ericksen–Leslie theory [14,15] are recovered. The detailed analysis to be presented here for plane Couette ow concerns (i) uniaxial alignment with variable degree of order and (ii) the tensorial analysis involving the three symmetry-adapted components of the ve components of the alignment tensor. Consequences for the rheological behavior are indicated. Compared to those of previous theoretical studies [16,17], the constitutive equations given here are relatively simple and contain a smaller set of parameters. The steady states can be calculated analytically and some conclusions concerning stability and, in case (ii), biaxial distortions of the steady states can be drawn. We believe that the qualitative dynamical behavior for low and medium shear rates can be captured by this model. Because terms higher than third order in the second-rank alignment tensor are neglected, deviations are to be expected for high shear rates. Further restrictions are given by the in-plane assumption and the spatially homogeneous ansatz for the alignment tensor, and the disregarded back ow e ects in the solution of the equations. Spatially inhomogeneous ow phenomena of liquid-crystalline polymers, such as ow instabilities, banded and striped textures, are beyond the scope of this model. This article is divided into four sections. Section 2 introduces the second-rank alignment tensor, discusses uniaxial and biaxial alignment, and the constitutive equations for the alignment tensor and pressure tensor. In Section 3, stationary and transient solutions of the alignment tensor equation are presented (i) for the in-plane uniaxial and (ii) symmetry-adapted tensorial ansatz mentioned above. The stability of the stationary solutions is investigated, the tumbling, wagging and aligning regimes are discussed, and the critical shear rates for the transition between these regimes are calculated. Using ansatz (ii), also the biaxiality of the alignment is discussed and consequences for the normal stress di erences are indicated. Concluding remarks are given in Section 4.
2. Theory 2.1. Alignment tensor properties (1) Orientational distribution function and alignment tensor. The orientation of a nematic liquid crystal, consisting of e ectively uniaxial molecules with the molecular
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direction given by the unit vector u, can be described by the one-particle orientational distribution function (u): 1 (1 + 2 a : + higher-order terms) : (1) = 4 The second rank alignment tensor a = 2 h
i;
(2)
sometimes also denoted by Q or S, is directly connected with mechanical and electromagnetic properties of the liquid crystal and, p time. R in general, depends on space and The average h:R : :i is de ned as h i:= S 2 d 2 u, and the constant 2 = 15=2 is chosen so that S 2 d 2 u = 1. The expression := 12 (b + bT ) − 31 (tr b) denotes the symmetric-traceless (irreducible) part of the tensor b where is the unit tensor. When possible, the compact notation (bold italic C for vectors and bold upright t for second rank tensors) is used where s · t and s : t are de ned by s t and s t respectively. By convention, summation over equal Greek indices is assumed. (2) Uniaxial ansatz, biaxiality. If in the nematic phase the orientational distribution is uniaxial, a has two equal eigenvalues 1 = 2 6= 3 and can be written as p (3) a = 3=2a with the nematic director n being the direction corresponding to the third eigenvalue 3 and the symmetry axis of the orientational distribution. The scalar order parameter a with a2 = a : a is related √to the commonly used Maier-Saupe order parameter S2 = h 32 (n · u)2 − 12 i by S2 = a= 5. In general, however, the alignment tensor has three di erent eigenvalues and leads to a biaxial orientational distribution. A measure of biaxiality is given via the parameter b de ned by [18] b2 = (I23 − I32 )=I23
(4)
with the scalar invariants I2 = tr(a · a) = a : a; 2
I3 =
√
6tr(a · a · a) =
√
6(a · a) : a :
(5)
2
The cases b = 0 and b = 1 correspond to uniaxial and planar biaxial alignment, respectively. 2.2. Constitutive equations for alignment and pressure (1) Alignment tensor. Constitutive equations for the dynamics of alignment tensor a and friction pressure tensor p in a ow eld C have been derived within the framework of irreversible thermodynamics [11–13] and describe the isotropic and nematic phase of a liquid crystal. The equation for the alignment tensor can be written for the spatially homogeneous case as √ ap da − 2 + −1 : (6) −2 a =− 2 dt a
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The tensors = 12 ((∇C)T +∇C) and = 12 ((∇C)T −∇C) are the symmetric traceless and asymmetric parts of the velocity gradient ∇C. In the case of the shear ow considered, C = ˙ ye x (plane Couette ow), with the shear rate ˙ , they are given by = 12 ˙ (e x ey + ey e x )
and
1
= ˙ (e x ey − ey e x ): 2
(7)
The constants a and ap are phenomenological relaxation times with a ¿ 0 and ap having either sign, and is a coecient explicitly describing the change of alignment caused by the deformation rate . It is needed to obtain the full anisotropy of the viscosity in the nematic phase [13]. In Eq. (6), = @=@a is the derivative of the Landau–de Gennes potential (a) = 12 Aa : a −
1 3
√
6B(a · a) : a + 14 C(a : a)2
(8)
which is proportional to the free energy of the alignment. The coecients A = A0 (1 − T ∗ =T ); B and C are constrained by the conditions A0 ¿ 0; B ¿ 0; C ¿ 0 and B2 ¿ 92 A0 C. The pseudo-critical temperature T ∗ is somewhat below the nematic-isotropic transition temperature Tk . Eq. (6) can also be obtained from a Fokker–Planck type equation for ellipsoidal particles [19]. From the latter equation, coupled nonlinear equations for the second and fourth rank alignment tensors a = a(2) and a(4) were derived. The coupling to alignment tensors of rank six and higher was ignored, and a(4) was expressed in terms of a(2) , based upon the assumption that the relaxation of a(4) is considerably faster and the in uence of the velocity gradient on it is considerably weaker than on a(2) . This molecular approach yields the result that and ap = a are related to a “nonsphericity” parameter R which is positive for prolate, p zero for spherical and negative for oblate particles by = 37 R and ap = a = − 3=5R. The ane transformation model for perfectly aligned uniaxial ellipsoidal particles with the axis ratio Q suggests R = (Q2 − 1)=(Q2 + 2) [20]. The Doi theory [8–10] is based on the same Fokker–Planck equation, but uses a di erent decoupling scheme (closure) for the elimination of the higher rank tensors. Recently, a variety of other closure approximations have been discussed in [21]. It should be mentioned that a relaxation equation with the same mathematical structure as (6) was suggested by Hess and Hess [18] for the friction pressure tensor, which describes the non-Newtonian behavior of the viscosity and the normal pressure di erences of uids showing shear thickening. (2) Pressure tensor. The full pressure tensor p consists of a hydrostatic pressure p = 13 trp, an asymmetric part pa = 12 (p − pT ) and the symmetric traceless part referred to as friction pressure tensor. The latter can be split into an “isotropic part” and a part explicitly depending on the alignment tensor: = −2iso
+
(9)
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with iso = kB T p m =
kB T m
2ap 1− a p
! ;
√ ap 2 − 2 a
(10) :
(11)
The constant p is another phenomenological relaxation time, and there are two more restrictions p ¿ 0 and a p ¿ 2ap for the relaxation times. In general, there is also an antisymmetric part of the pressure tensor (see [13]). However, it is not needed for the calculation of the normal stress di erences and is therefore omitted. (3) Dimensionless form. With the scalings t = t ∗ a =Ak , C = C∗ Ak = a and a = a∗ ak , Eq. (6) can be transformed into a dimensionless form r √ 3 da∗ ∗ ∗ ∗ ∗ k ∗ ; −2 − 2 + (#a − 3 6 + 2(a : a )a ) = ∗ dt 2 (12) where Ak = A0 (1 − T ∗ =Tk ) = 2B2 =9C, ak = aeq (Tk ) = 2B=3C is the (nonzero) equilibrium value of the scalar order parameter a at the transition temperature Tk , 1 2 √ ap k = − 3 ; (13) 3 a a k ∗
and ∗ are the parts of the dimensionless velocity gradient and T∗ T∗ 1− #= 1− T Tk
(14)
is a reduced temperature variable chosen so that # = 0 for T = T ∗ and # = 1 for T = Tk . For the sake of simplicity, the asterisks for the rescaled variables are omitted in the following.
3. Orientational dynamics in a plane Couette ow 3.1. Dynamics of director and order parameter Inserting (3) into (12), applying the operations · = 13 + 29 and n · n=1, d(n · n)=dt =0, for a and n da = (a) : − a (a) ; dt 1
: and n ×n· and using the relations : = 23 leads to coupled equations (15)
Assuming the critical order parameter Sk = S(Tk ) ≈ 0:4, one has ak ≈ 0:9. Measurements of S yielded, e.g., Sk = 0:4 for 4:40 -di-methoxy-azoxybenzol [22], Sk = 0:28 for 5CB and Sk = 0:32 for 8CB [23].
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dn = · n + (a)( · n − : nnn) : (16) dt with the abbreviations k 3 (17) (a) = a + k and (a) = + 2 3 a and the derivative @ = (#a − 3a2 + 2a3 ) (18) a (a) ≡ @a of the dimensionless Landau–de Gennes potential (a) for the uniaxial case. If no ow is present, the order parameter a reaches an equilibrium value aeq , which in the nematic phase is given by √ (19) a (aeq ) = 0; aeq = 34 + 14 9 − 8# : (1) Relation to the Ericksen–Leslie theory. In the low shear rate limit ˙ 1, the in uence of the ow on the order parameter vanishes and one can identify a with aeq . Eq. (16) then reduces to the spatially homogeneous case of the Ericksen–Leslie equation (see, e.g., [24]) for the director, usually written as dn − · n + 2 · n = 0 ; (20) n × 1 dt which says that the viscous torque on the director has to vanish. The Leslie coecients
1 and 2 are related to the coecients of (6) by 2 √
1 = 3 kB T (ak aeq )2 a and 2 = kB T (2 3(ak aeq ) ap − (ak aeq )2 a ): (21) m m The tumbling parameter = − 2 = 1 of the Ericksen–Leslie theory is then given by k : (22) = (aeq ) = + 3 aeq It is positive (negative) for nematics built from rod-like (disk-like) molecules. In a steady shear ow with the shear rate ˙ , for ||¿1, the torque at the director vanishes when it aligns in the shear plane at a speci c angle ’ = 12 cos−1 (1=)
(23) 3
to the ow direction ( ow-aligning). In the case || ¡ 1, no stationary solution with zero torque is possible and the director, when it initially lies in the shear plane, rotates continuously in this plane (tumbling). 4 The tumbling period is given by 2 p 1 − 2 : (24) T=
˙ For an out-of-plane director, the movement is more complicated, but the projection of n to the shear plane still rotates with the period T . 2
Note that ak appears here explicitly because aeq is given in units of ak . Low molecular ow-aligning substances are, e.g., p0 -methoxybenzylidene-p-n-butylaniline (MBBA) with ◦ ◦ = 1:02 for T = 22 C [1] and 4-n-pentyl-40 -cyanobiphenyl (5CB) with = 1:16 for T = 32:5 C [25]. 4 The substance 4-n-octyl-40 -cyanobiphenyl (8CB) with = 0:27 for T = 34:0◦ C [26] is an instance of a low molecular tumbling nematic. 3
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The ow-aligning or tumbling behavior described by (20) is completely determined by the tumbling parameter , i.e., independent of the shear rate. This is practically true for low molecular weight nematics, where the relaxation times a , ap are very small, and therefore the dimensionless shear rate ˙ ∗ = ( a =Ak )˙ is much smaller than one and the order parameter is not de ected much from its equilibrium value. For nematic polymers, the situation is di erent. Because of their high relaxation times, 5 even small experimental shear rates lead to a strong coupling of (15) and (16). The e ective tumbling parameter (a) now becomes time dependent. (2) In-plane ansatz. To investigate the dynamics of scalar order parameter and director in a plane Couette ow C = ˙ ye x with the dimensionless shear rate ˙ , the most simple ansatz for the director is the restriction to the shear plane n = cos ’ e x + sin ’ ey
(25)
with the director angle ’. Insertion of (25) into (15) and (16) leads to the dynamical system a˙ = 12 ˙ (a)sin 2’ − a ;
(26)
’˙ = 12 ˙ ((a)cos 2’ − 1) :
(27)
In the following, we only consider liquid crystals composed of elongated particles (k ; ¿ 0) and positive shear rates. 3.1.1. Steady states As a rst step, the stationary solutions a(˙ ), ’(˙ ) are computed by setting a˙ and ’˙ to zero. It is more convenient to solve ˙ as a function of a in (26), where sin 2’ can be expressed through (a) with the help of (27): cos 2’ = (a)−1 ;
˙ (a) =
2a 2(#a − 3a2 + 2a3 ) p : =± (a)sin 2’ (a) 1 − (a)−2
(28)
The solutions are plotted in Fig. 1 for the temperature variable # = 0, the coecient = 0 and characteristic values of k . For # ¡ 0 and a¿0, the numerator a of ˙ (a) has two zeros 0 and aeq and is negative for 0 ¡ a ¡ aeq . The denominator is de ned for a ¡ amax = k =(1 − =3) and tends to zero for a → amax . Depending on the ratio of amax and aeq there are three cases: (i) amax ¡ aeq , (ii) amax =aeq and (iii) amax ¿ aeq . They correspond to eq ≡ (aeq ) ¡ 1, =1 and ¿ 1, the tumbling and aligning cases of the Ericksen-Leslie theory. In the rst case, one has a ¡ aeq and ’ ¡ 0. This means the alignment angle is negative here. In the second case, one has solutions with a ¡ aeq and ’ ¡ 0 for low shear rates and an additional solution a = aeq and ’ = 0 for arbitrary ˙ . The third 5
The viscosities of liquid-crystalline polymers are often some orders of magnitude larger than those of low molecular weight nematics.
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Fig. 1. Stationary values of a and ’ in system (26), (27) for # = 0 and = 0 (amax = k ). Top: ’(a=k ) (positive branch). Bottom: ˙(a) for (a) k = 1:4 (eq = 0:93), (b) k = 1:5 (eq = 1) and (c) k = 1:6 (eq = 1:07). For a ¡ aeq = 3=2, the alignment angle ’ is negative.
case again permits a ¡ aeq and ’ ¡ 0 for low shear rates, but also has solutions with amax ¿ a ¿ aeq and positive ’ for arbitrary ˙ . In cases (i) and (iii), the order parameter a tends to amax and the alignment angle ’, to zero for high shear rates. Although the parameters and k are not accessible by experiment, the tumbling parameter eq ≡ (aeq ) can be obtained from the Leslie coecients 1 and 2 . When (a) and (a) are expressed by eq and , aeq 3 ; (a) = aeq eq + a − 2 2
(a) =
aeq aeq eq + 1− ; a 3 a
(29)
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one can see that, for a given eq , the in uence of a realistic 6 is relatively small and one can expect that the essential dynamics is already captured in the simpler case = 0, where (a) = k =a, (a) = 32 k and amax = k . Indeed, for, say, = 3=7, the same qualitative behavior of the dynamical systems occurs for slightly di erent ˙ (see Fig. 12). (1) Stability analysis. A linear stability analysis reveals whether the steady states calculated are stable against small perturbations. For the trace and determinant of the Jacobian matrix, 1 0
˙ (a)sin 2’ − aa
˙ (a)cos 2’ ; (30) J (a;’) = 2 1 0 ˙ (a)cos 2’ −˙ (a)sin 2’ 2 one obtains, using the relations ˙ (a) and ’(a), = tr J (a; ’) =
= det J
(a; ’)
a ( − 2(a)) − aa (a)
2a = (a)
(31)
k a (a)a − + (a)aa a2 ((a)2 − 1) (a)
= (a)sin 2’ a ˙ 0 (a);
(32)
where aa ≡ @2 =@a2 and ˙ 0 (a) ≡ @˙ =@a. The xed points of (26), (27) are hyperbolic and asymptotically stable (of type focus or node) if ¿ 0 and ¡ 0. The case ¡ 0 leads to a saddle point. Eq. (32) shows a relation between saddle point behavior and the shape of ˙ (a). Because both sin 2’ and a are positive (negative) for a ¿ aeq (a ¡ aeq ), a saddle point occurs when ˙ 0 (a) is negative. The stability regions are discussed for the reduced temperature # = 0 and coecient = 0. In this case, is independent of k and negative for a ¿ as ≈ 1:154. The real and imaginary parts of the eigenvalues 1 , 2 of J (a; ’) for k = 1:4 (eq = 0:93), 1:5 (eq = 1) and 1:6 (eq = 1:07) are shown in Figs. 2– 4. • For k = 1:4, one has ¿ 0 (no saddle points) throughout. When ˙ ¿ ˙ c2 ≈ 1:55, a is bigger than as and the stationary state with negative alignment angle is stable. Fig. 2 shows that the imaginary part of the eigenvalues is nonzero for approximately a ¡ 1:29 and a ¿ 1:347. This means one has a focus in these cases. In particular, at a = as , the real part crosses zero with a nonvanishing slope indicating a supercritical Hopf bifurcation (birth of a stable limit cycle for a ¡ as or ˙ ¡ ˙ c2 ). • If k = aeq = 1:5, one has ¿ 0 for a ¡ 1:171 (left of the local maximum of ˙ (a)). That means the steady states in the small interval as ¡ a ¡ 1:171 are stable. In the special case a = aeq and ’ = 0, the relation for ˙ (a) is invalid and, therefore, and have to be taken directly from (30): = −aa = −# + 6aeq − 6a2eq ¡ 0
and
= − 12 ˙ 2 (a)0 (a) = 34 ˙ 2 ¿ 0 : (33)
6
The ane transformation model, e.g., admits ||63=7.
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Fig. 2. Real and imaginary part of the eigenvalues 1 , 2 of the Jacobian matrix of system (26), (27) for # = 0, = 0 and k = 1:4 (eq = 0:93).
Fig. 3. Real and imaginary part of the eigenvalues 1 , 2 of the Jacobian matrix of system (26), (27) for # = 0, = 0 and k = 1:5 (eq = 1).
Consequently, this state is asymptotically stable for all ˙ 6= 0. The type of the xed point changes at approximately ˙ ¿ 2:6 where 2 =4 ¡ from node to focus causing a relaxation to the steady state via damped oscillations. • For k = 1:6 (aligning case), is positive when a ¿ aeq = 1:5 or a ¡ 1:122. In this case, one has a stable shear aligning solution with a ¿ aeq and ’ ¿ 0 for all
˙ ¿ 0. For a ¡ 1:122, one has ¿ 0, i.e., an unstable focus or node. The eigenvalue plot, Fig. 4, shows that the type of the xed point changes from node to focus for a ¿ 1:588 (˙ ¿ 3:02) indicating the presence of damped oscillations of a and ’ towards the steady state.
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Fig. 4. Real and imaginary part of the eigenvalues 1 , 2 of the Jacobian matrix of system (26), (27) for # = 0, = 0 and k = 1:6 (eq = 1:07).
No shear aligning state is possible for amax = k ¡ as , because then is positive for all values of a. 3.1.2. Dynamical behavior of the system To investigate the dynamical behavior of (26), (27), the system is numerically integrated for the temperature # = 0 and coecient = 0 using the initial conditions a = aeq = 1:5 and ’ = 0. The results are shown in Figs. 5 –11. For k = 1:4, the system shows three dynamical modes: • When ˙ ¡ ˙ c1 ≈ 1:44 tumbling occurs. The director rotates in the x − y plane and the order parameter is periodically de ected from its equilibrium value. The periodic minima of a become deeper when ˙ is increased (Figs. 5 and 7). • For ˙ in the small interval between ˙ c1 and ˙ c2 ≈ 1:55, the xed point a(˙ ) is still unstable but the tumbling is replaced by wagging [10], an oscillation of the director angle. Increasing ˙ leads to a shift of the center of the oscillation towards a negative value and to a decrease of the amplitude (Figs. 6a–c and 8). • For ˙ ¿˙ c2 , the xed point is stable and the oscillations of a and ’ are damped. The director aligns to a negative alignment angle that tends towards zero with further increase of the shear rate (Figs. 6d and 9). ◦ For k =1:5, the xed point is at a=aeq and ’=0 and, therefore, the initial conditions are chosen di erently. At higher shear rates, a and ’ undergo damped oscillations around their stationary values as indicated (Fig. 10). For k = 1:6, the director aligns at a positive angle which tends towards zero when
˙ is increased. Again, damped oscillations occur at higher shear rates (Fig. 11). In the rst case, k = 1:4 (tumbling regime for low shear rates), the system described
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Fig. 5. Dynamics of system (26), (27) for # = 0, = 0 and k = 1:4. Top: a(˙ t) and bottom: ’(˙ t) for (a)
˙ = 0:1, (b) ˙ = 0:5, (c) ˙ = 1:0, (d) ˙ = 1:2, (e) ˙ = 1:3, (f) ˙ = 1:4.
by (26), (27) shows at low and medium shear rates qualitatively the same dynamics as reported, e.g., by Larson [10] for the (direct) solutions of the Doi equation. As the shear rate increases, the order parameter deviates more and more from its equilibrium value and a transition from tumbling to wagging occurs. Note that the solutions of the Doi equation are actually no longer uniaxial and, therefore, Larson introduces the concept of the birefringence axes as a generalization of the director. With further increase of the shear rate, the oscillations become damped, leading to a steady state with slightly negative alignment angle and diminished uniaxial order parameter. The di erence to the results obtained here is that the alignment angle again becomes positive and the order parameter exceeds its equilibrium value for even higher shear rates.
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Fig. 6. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Top: a(˙ t) and bottom: ’(˙ t) for (a)
˙ = 1:45, (b) ˙ = 1:5, (c) ˙ = 1:55, (d) ˙ = 1:6.
Coupled dynamical equations for the in-plane director and scalar order parameter dynamics based upon a phenomenological continuum theory for the alignment tensor have also been derived by Farhoudi and Rey [16]. They predict, in analogy to the results presented here, a complex orientation mode in simple shear ow at suciently high nematic potentials which exhibits tumbling, wagging and stationary regimes depending on the shear rate. (2) Critical shear rates. The critical shear rates ˙ c1 and ˙ c2 of the tumbling–wagging and wagging–aligning transition are plotted vs. the equilibrium tumbling parameter eq in Fig. 12 for various values of temperature # and coecient . The values decrease with eq . For low eq , rst the aligning and then the wagging state vanishes.
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Fig. 7. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Plot of a(t) vs: ’(t) for ˙ = 1:4 (tumbling regime, just below ˙c1 ).
Fig. 8. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Plot of a(t) vs: ’(t) for ˙ = 1:45 (wagging regime).
Plots a, b for # = 0; = 0:4 and plots c, d for # = 0; = 0 show that has not much in uence on the dynamics. For lower # (plots e, f), ˙ c1 and ˙ c2 are shifted towards higher values. 3.2. Dynamics of the symmetry-adapted alignment tensor After the investigation of the uniaxial in-plane system, now the dynamics of the alignment tensor itself is considered and compared to that of system (26), (27). Because
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Fig. 9. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:4. Plot of a(t) vs: ’(t) for ˙ = 1:56 (aligning regime, just above ˙c2 ).
the driving ow eld ∇C is biaxial, the resulting alignment is also expected to be biaxially distorted in the nematic phase. (1) Basis tensors. The tensorial equation (12) is equivalent to ve equations for ve linearly independent scalars because a is symmetric traceless. For the plane Couette
ow considered, a special geometry-adapted ansatz with three components can be made [27] which includes, e.g., those uniaxial states with the director either lying in the shear plane (x–y) or in the vorticity direction (z): a = a0 T0 + a1 T1 + a2 T2
(34)
with the three orthonormalized symmetric traceless tensors r √ 3 1√ x x 0 ; T1 = 2(e e − ey ey ); T2 = 2 T = 2 2
:
(35)
The following relations are satis ed Ti : Tk = ik
(i; k = 0; 1; 2) ;
(36)
a20 + a21 + a22 = a2 = a : a :
(37)
(2) Equations for three tensor components. Ansatz (34) leads to three equations for the three components a0 ; a1 and a2 √ a˙0 = −#a0 − 3(a21 + a22 − a20 ) − 2a2 a0 − 13 3˙ a2 ; (38) ˜ 1; a˙1 = ˙ a2 − #a ˜ 2+ a˙2 = −˙ a1 − #a
(39) 1 2
√
3k ˙ −
1 3
√
3˙ a0 ;
(40)
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Fig. 10. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:5 (from initial conditions a = 1:6 and ◦ ’ = 45 ). Top: a(˙ t) and bottom: ’(˙ t) for (a) ˙ = 0:1, (b) ˙ = 0:5, (c) ˙ = 1:0, (d) ˙ = 5:0, (e) ˙ = 10.
with #˜ = # + 6a0 + 2a2 :
(41)
(3) Equations for one angle and two order parameters. Using the de nitions a1 = p cos 2’; 2
2
2
a2 = p sin 2’; 2
r =a =p +q ;
p2 = a21 + a22 ;
q = a0 ;
(42)
Eqs. (38) – (40) are equivalent to three equations for one angle ’ and two order parameters p and q [27,18] √ √ (43) p˙ = −(# + 6q + 2r 2 )p + ˙ ( 12 3k − 13 3q) sin 2’ ;
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Fig. 11. Dynamics of system (26), (27) for # = 0; = 0 and k = 1:6. Top: a(˙ t) and bottom: ’(˙ t) for (a) ˙ = 0:1, (b) ˙ = 0:5, (c) ˙ = 1:0, (d) ˙ = 5:0, (e) ˙ = 10.
√ √ ’˙ = 12 ˙ (p−1 ( 12 3k − 13 3q) cos 2’ − 1) ; q˙ = −#q − 3(p2 − q2 ) − 2qr 2 −
1 3
√
3˙ p sin 2’ :
(44) (45)
With the de nitions (42), the alignment tensor has two eigenvectors within the shear plane enclosing the angle ’ √ and =2 + √ ’ with the ow direction, which correspond to the eigenvalues 1; 2 = (± 3p −pq)= 6, and a third eigenvector in the vorticity direction with the eigenvalue 3 = 2=3q. There are two uniaxial cases: (a)√ n = e z for p = 0 and q = r and (b) n = cos’ e x + sin’ ey (in-plane case) for p = 12 3r and q = − 12 r. For the general, biaxial case, three mutually orthogonal directors (Larson: birefringence axes) exist, which are parallel to the principal axes.
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Fig. 12. Critical shear rates ˙c1 ; ˙c2 of system (26), (27) for the tumbling-wagging and wagging-aligning transitions vs. equilibrium tumbling parameter eq for a, b: #=0; =0:4; c, d: #=0; =0; e, f: #=−1; =0.
Inserting the in-plane case (b) into (43) – (45), one does not recover the uniaxial system (26), (27) (Eqs. (43) and (45) are not compatible for, e.g., = 0). Therefore trajectories of (43) – (45) are, in general, not uniaxial for ˙ 6= 0, even when they start from uniaxial initial conditions. 3.2.1. Steady states In the following, = 0 is assumed for the sake of simplicity: √ ˜ + ˙ 1 3k sin 2’ ; p˙ = −#p 2 1 √ k 1 3 cos 2’ − 1 ; ’˙ = ˙ 2 2 p q˙ = −#q − 3(p2 − q2 ) − 2qr 2 :
(46) (47) (48)
Again, as a rst step, the stationary solutions are obtained by setting p; ˙ ’˙ and q˙ to zero. The procedure is similar to the previous case: ˙ is expressed as a function of p; ’ and q in (46). Then ’ and p are eliminated using (47) and (48). As in the uniaxial case, only positive k will √ be considered. With the abbreviation k = 12 3k and the assumption k ¿ 0 one gets p2 (q) = −
#q − 3q2 + 2q3 ; 3 + 2q
(49)
cos 2’(q) = p(q)=k ;
˙ (q) =
(50)
˜ ˜ #(q)p(q) #(q)p(q) ; = ±q k sin 2’(q) 2 − p2 (q) k
(51)
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Fig. 13. Stationary solutions of system (46) – (48). Order parameters p(q) and r(q) for (a) # = −1, (b) # = 0, (c) # = 1.
with #˜ expressed by q 3(# + 6q + 8q2 ) ˜ : #(q) = # + 6q + 2q2 + 2p2 (q) = 3 + 2q
(52)
Relations p(q) and r(q) are plotted in Fig. 13 for various reduced temperatures #, the rst plot is similar to the corresponding gure in [27]. The shear rate dependency ˙ (q) of the steady states is plotted in Fig. 14 for # = 0 and various k . The domain of q is limited by the condition 06p(q)6k , i.e., for # = 0, one has q ¡ 1:5 independent of k and q¿ − 0:719; q¿ − 0:75, and q¿ − 0:779 for k = 1:4; 1:5 and 1:6, respectively.
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Fig. 14. Stationary solutions of system (46) – (48). Shear rate ˙(q) for # = 0 and (a) k = 1:4, (b) k = 1:5, (c) k = 1:6.
For q ¡ 0, Fig. 14 looks similar to the plot of ˙ (a) in Fig. 1 when q is replaced by −q. In the interval 0 ¿ q ¿ qeq = −0:75, the alignment angle is negative. If qmin denotes the minimum value of q, such that p(q) ¡ k , then, for k = 1:4; k = 1:5 and k = 1:6, one has qmin ¿ qeq ; qmin = qeq and qmin ¡ qeq . In the rst case, only steady states with ’ ¡ 0 exist for high shear rates, whereas for k = 1:6 also states with ’ ¿ 0 are possible. If the uniaxiality condition p2 = 3q2 = 34 r 2 (see below) is assumed, one obtains
˙ (r) =
#r − 3r 2 + 2r 3 q ; k2 − r 2
(53)
which is, except for the factor 3=4, the same as ˙ (a) in (28). (1) Uniaxiality and biaxiality. The biaxiality can be generally expressed as a function of p and q. With I2 = r 2 = p2 + q2 and I3 = q3 − 3qp2 one has b2 = 1 −
I32 p2 (p2 − 3q2 )2 = 3 (p2 + q2 )3 I2
and
1 − b2 =
I32 q2 (q2 − 3p2 )2 = : 3 (p2 + q2 )3 I2
(54)
If one excludes the isotropic case r 2 = p2 + q2 = 0, the uniaxiality condition b2 = 0 is equivalent to p(p2 − 3q2 ) = 0 :
(55)
With use of the stationarity relation p2 (q) (49) and under the assumption q ¿ − 32 , condition p = 0 is ful lled for √ (56) q = 34 ± 14 9 − 8# :
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Fig. 15. Biaxiality b(q) of the stationary solutions of system (46) – (48) for (a) # = −1, (b) # = 0, (c) # = 1.
Because of p2 − 3q2 = −q(# + 6q + 8q2 )=(3 + 2q), the second condition p2 − 3q2 = 0 holds for √ (57) q = − 38 ± 18 9 − 8# : Both cases lead to the order parameter √ r = 34 ± 14 9 − 8# :
(58)
Furthermore, using (52), one can see that the uniaxiality condition (55) is equivalent to ˜ = 0. Consequently, all equilibrium states (steady states for ˙ = 0) are uniaxial. The #p reverse is also true: except for p =k (’=0), all uniaxial steady states are equilibrium states. The special case p = k is reached for #˜ = 0. Then the states for any ˙ are uniaxial. The planar biaxial states correspond to b2 = 1, i.e. q(q2 − 3p2 ) = 0 :
(59)
For q = 0, one obtains the isotropic case p = r = 0. Because of q2 − 3p2 = q(3# − 6q + 8q2 )=(3 + 2q), condition q2 = 3p2 leads to √ (60) q = 38 ± 18 9 − 24# : The corresponding order parameter is given by r 1√ 1√ 4 3± q= 3 − 8# : r= 3 4 4
(61)
The biaxiality √ b(q) of the steady states is plotted in Fig. 15. For # = 0, the √ states q = − 34 , p = 34 3 and q = 32 ; p = 0 are uniaxial whereas the state q = 34 ; p = 14 3 is planar biaxial.
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Fig. 16. Real part of the eigenvalues 1 ; 2 ; 3 of the Jacobian matrix of system (38) – (40) for # = 0; = 0 and k = 1:4.
(2) Stability analysis. The steady states are stable against small perturbations when the real parts of all eigenvalues of the Jacobian matrix are negative. The Jacobian matrix is given by −6p2 − 6q − 2q2 − # 2k ˙ cos 2’ −p(6 + 4q) (p; q; ’) : = −(k =2p2 )˙ cos 2’ −(k =p)˙ sin 2’ J 0 2 2 −p(6 + 4q) 0 −2p + 6q − 6q − # (62) q ˜ and k ˙ cos 2’ = |#|p ˜ 2 = 2 − p2 , using the fact Relation (51) yields k ˙ sin 2’ = #p k ˜ that ’ and # are of the same sign for p ¿ 0. Inserting the steady states into (62) yields for the case # = 0 the following intervals of q where the steady states are stable: −0:719 ¡ q ¡ − 0:62 or q ¿ 1:17 for k = 1:4 and −0:779 ¡ q ¡− 0:75 or q ¿ 1:18 for k = 1:6. Consequently, for q ¡ 0, the steady states for k = 1:4 are stable when ˙ is suciently high, whereas for k = 1:6 stable steady states exist for any ˙ as in the uniaxial case. √ In the case k = 1:5, the state p = 34 3, ’ = 0; q = − 34 with p = k is inserted directly into (62), and one obtains stability for all ˙ . The real parts of the eigenvalues of (62) are shown in Figs. 16 –18. Fig. 14 shows that for low shear rates, there are also steady states with positive q. In this case, q must be to the right of the local maximum of ˙ (q) to ensure stability. 3.2.2. Dynamical behavior To investigate the dynamical behavior, system (38) – (40) is integrated numerically and the values p; q; r; b and ’ are extracted from a0 ; a1 , a2 . As initial conditions, uniaxial √ equilibrium states are chosen with the director oriented in the ow direction: p = 12 3aeq ; q = − 12 aeq ; ’ = 0. The results for # = 0; = 0; k = 1:4; 1:6 and various
˙ are shown in Figs. 19 –22.
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Fig. 17. Real part of the eigenvalues 1 ; 2 ; 3 of the Jacobian matrix of system (38) – (40) for # = 0; = 0 and k = 1:5.
Fig. 18. Real part of the eigenvalues 1 ; 2 ; 3 of the Jacobian matrix of system (38) – (40) for # = 0; = 0 and k = 1:6.
Again, for k ¡ aeq , the system is of the tumbling type for low shear rates and switches to wagging and aligning at higher ˙ . The alignment angle is negative and the order parameter r is smaller than aeq . The biaxiality b is nonzero and becomes considerably large when the order parameters p and r are low. If k ¿ aeq , aligning occurs for all ˙ with ’ ¿ 0 and r ¿ aeq . The biaxiality is low but nonzero (see Section 3.2.1). In both cases, the alignment angle tends towards zero for high ˙ . (1) Critical shear rates. For # = 0; = 0 and k = 1:4, the critical shear rates of the tumbling-wagging-aligning transitions are ˙ c1 ≈ 1:35 and ˙ c2 ≈ 1:45. Other parameter values lead to di erent ˙ c1; 2 . The values of ˙ c1 and ˙ c2 for # = 0, = 0 and various k are plotted vs. the equilibrium tumbling parameter eq ≡ k =aeq in Fig. 23.
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Fig. 19. Dynamics of system (38) – (40) in the tumbling regime for k ¡ aeq . Order parameters p; q; r, biaxiality b and angle ’ for ˙ = 1:2; # = 0; = 0; k = 1:4.
Fig. 20. Dynamics of system (38) – (40) in the wagging regime for k ¡ aeq . Order parameters p; q; r, biaxiality b and angle ’ for ˙ = 1:4; # = 0; = 0; k = 1:4.
3.2.3. Consequences for the normal stress di erences of liquid-crystalline polymers The normal stress di erences N1 and N2 are de ned as N1 = xx − yy = pyy − pxx and N2 = yy − zz = pzz − pyy . They can be related to the components of in the tensor basis (35): r 3 1√ 1√ p0 ; 2p1 ; N2 = 2p1 + (63) N1 = − 2 2 2 where is calculated from the alignment tensor using Eq. (11). Because samples of liquid-crystalline polymers have a domain structure, i.e., the alignment tensor is spatially inhomogeneous, measured normal stress di erences are always the result of spatial averaging of the individual domains. As done in previous
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Fig. 21. Dynamics of system (38) – (40) in the aligning regime for k ¡ aeq and higher shear rates. Order parameters p; q; r, biaxiality b and angle ’ for ˙ = 1:5; # = 0; = 0; k = 1:4.
Fig. 22. Dynamics of system (38) – (40) in the “simple” aligning regime for k ¿ aeq . Order parameters p, q, r, biaxiality b and angle ’ for ˙ = 0:5, # = 0, = 0, k = 1:6.
work in connection with the Doi theory [28,10,5], the domains are thought to be independent of each other, and the ensemble average is replaced by an average over a tumbling=wagging period for the nonstationary case. Using the parameters # = 0, k = 1:0 and = 0:4 (eq = 0:8), one obtains a plot of N1 and N2 vs. ˙ as shown in Fig. 24. The functional relation is, for lower shear rates, similar to experimental results and theoretical predictions using the Doi theory [3,5 –7]. In particular, it contains one sign change in the normal stress di erences near the tumbling-wagging transition at ˙ = 1:89. On the other hand, both N1 and N2 tend asymptotically towards a constant value for high ˙ .
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Fig. 23. Critical shear rates ˙c1 , ˙c2 of system (38) – (40) for the tumbling–wagging and wagging–aligning transitions vs. equilibrium tumbling parameter eq for # = 0 and = 0.
Fig. 24. Normal stress di erences N1 and N2 computed from the symmetric-traceless part of the pressure tensor vs. shear rate ˙ for # = 0, k = 1:0 and = 0:4.
4. Conclusions A nonlinear relaxation equation for the alignment tensor [11–13] was used to investigate the spatially homogeneous dynamics of rod-like low molecular weight and polymeric nematic liquid crystals in a steady shear ow. Both a uniaxial ansatz with variable order parameter and a symmetry-adapted tensorial ansatz led, depending on the equilibrium tumbling parameter eq , either to simple shear aligning with a positive alignment angle and an enlarged order parameter or to a
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cascade from tumbling via wagging to shear aligning with a negative alignment angle and a diminished order parameter for increasing shear rates. In both cases, the liquid crystal becomes aligned in the ow direction for high shear rates. In contrast to the uniaxial description, the tensorial ansatz showed that, in general, the steady and transient states are biaxial, i.e., the orientational distribution is biaxially distorted by the driving ow eld. Depending on the parameters, a low or medium degree of biaxiality was observed in the numerical calculations carried out. Biaxial behavior except for isolated parameter values was recently shown also for the steady-state solutions of a Doi-type theory using the quadratic and the rst Hinch–Leal closure approximations [29]. The normal pressure di erences are calculated here as a by-product by averaging over oscillation periods (in the case eq ¡ 1). At lower shear rates, the results are in qualitative agreement with experiments and with previous theoretical results of the Doi theory as far as the prediction of the rst sign change in N1 and N2 near the tumbling-wagging transition point is concerned. However, they disagree with both experiments and the Doi theory for higher shear rates. Also, the alignment angle does not become positive again in the high shear limit, in contradistinction to the results of the Doi theory. The di erences in the high shear rate behavior are likely to be due to higher-order terms in the order parameter and back ow e ects in the solution of the dynamic equations, both of which were neglected here. However, both the in-plane uniaxial and symmetry-adapted tensorial cases of the present approach show qualitatively the same dynamical regimes as constitutive equations derived from the molecular kinetic theory (Fokker–Planck equation) in two dimensions by Marrucci and Ma ettone [28,30], and in three dimensions by Larson [10], as well as a continuum tensorial theory by Farhoudi and Rey [16]. The simple approach presented here gives rise to rather complex dynamic behavior which agrees with experimental observations for small and intermediate shear rates. Further investigations using all ve tensor components and including spatial derivatives of the alignment tensor [12] (Frank elasticity terms) are desirable and will lead to more insight into fully three-dimensional alignment and textures. Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich “Anisotrope Fluide” (SFB335) is gratefully acknowledged. References [1] [2] [3] [4] [5]
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