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Two-alignment tensor theory for the dynamics of side chain liquid-crystalline polymers in planar shear flow
J. Non-Newtonian Fluid Mech. 134 (2006) 2–7
Two-alignment tensor theory for the dynamics of side chain liquid-crystalline polymers in planar shear flow夽 Patrick Ilg a,b,∗ , Siegfried Hess a a
Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, PN 7-1, Hardenbergstr. 36, D-10623 Berlin, Germany b D´ epartement de Physique des Mat´eriaux, Universit´e Claude Bernard Lyon1, F-69622 Villeurbanne, France Received 28 June 2005; received in revised form 1 September 2005; accepted 27 September 2005
Abstract A two-alignment tensor model for the dynamics of side chain liquid-crystalline polymers is applied to the case of planar shear flow. The predictions of the model are worked out for the case of stationary and oscillatory shear flow. In stationary shear, a flow induced isotropic-to-nematic phase transition is observed as well as a stress plateau for intermediate shear rates. In oscillatory shear flow, a low frequency elastic modulus is found. The model predictions agree qualitatively with experimental results. © 2005 Elsevier B.V. All rights reserved. PACS: 36.20.Ey; 64.70.Md; 83.50.Ax Keywords: Side-chain liquid crystal polymers; Shear flow; viscoelastic moduli; Isotropic-to-nematic transition
1. Introduction Shear-flow-induced isotropic-to-nematic phase transitions have been observed in a variety of different side chain liquidcrystalline polymers [1]. Rheological measurements in planar shear flow, both stationary [1,2] and oscillatory [3–6] yield additional information on the dynamics of side chain liquidcrystalline polymers. While the effect of different mesogen lengths, cross-linker geometry and molecular weight on the dynamical mechanical response of side-chain liquid-crystalline polymers have been studied in [5,4,6], an interesting low frequency elastic plateau was observed in [3]. Very recently, we have proposed in [7] a “two-alignment tensor” theory of shear-flow-induced isotropic-to-nematic phase transition in side chain liquid-crystalline polymers. This model extends previous theories of the flow-induced isotropic-tonematic phase transition [8–10] which are applicable to ordinary thermotropic nematics, to lyotropic liquid crystals and to main chain liquid-crystalline polymers.
An alternative approach to the dynamics of side chain liquidcrystalline polymers in extensional flow has been proposed in [11]. Here, we study the predictions of the two-alignment tensor model for the dynamical behavior of side chain liquid-crystalline polymers in stationary and oscillatory shear flow. Qualitative agreement with the experimental observations in [1–3] are observed. The paper is organized as follows. In Section 2, the twoalignment-tensor model is reviewed. The dynamic and constitutive equations are given explicitely. Analytical solutions for the low and high frequency limit in small amplitude oscillatory shear flow are presented. In Section 3, the model predictions for the orientational and rheological properties are obtained from the numerical solution of the model equations for several parameter values. Both, stationary and oscillatory shear flow is considered. Qualitative comparison to the experimental data of [1–3] is made. Finally, some conclusions are offered in Section 4. 2. Model equations
夽 Talk presented at the AERC 2005.
2.1. Dynamics of alignment tensors
∗
Corresponding author. E-mail addresses: [email protected] (P. Ilg), [email protected] (S. Hess). 0377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2005.09.003
In order to make the paper self-contained, we briefly review the two-alignment tensor model introduced in Ref. [7].
P. Ilg, S. Hess / J. Non-Newtonian Fluid Mech. 134 (2006) 2–7
For more details the reader is referred to to the original reference. Let the alignment of the liquid-crystalline side groups be described by the second rank, symmetric, traceless alignment tensor a. Similarly, the orientation of the polymer back bone is characterized by the corresponding second rank tensor b. The normalization is chosen such that a2 = a : a where a is the scalar order parameter in case of uniaxial symmetry, a = a(3/2)1/2 nn. The symbol x denotes the symmetric traceless part of the matrix x. The parameter a is therefore proportional to the Maier–Saupe √ order parameter S2 = a/ 5 = (3/10)1/2 a : nn. We assume in the sequel, that the isotropic-to-nematic phase transition is linked with the liquid-crystalline side groups described by a but not with the polymer back bone, b. Assuming further a bilinear coupling of a and b of strength D, a simple extension of the usual Landau-de Gennes potential [12] to the case of side chain liquid crystals was proposed in [7], Φ = Φ(a, b), viz. √ 1 6 1 Φ = A(T )a : a − B(a · a) : a + C(a : a)2 2 3 4 1 + Da : b + Ab b : b (1) 2 The coefficient A(T ) = A0 (1 − T ∗ /T ) changes sign at the pseudo-critical temperature T ∗ , which is below the nematic– isotropic transition temperature TK , T ∗ < TK . The coefficients A0 , B, C, D, Ab (with C < 2B2 /(9A0 )) are linked with measurable quantities and can also be related to molecular properties within the framework of a mesoscopic theory [9,10]. Here, these coefficients are treated as phenomenological parameters. A harmonic potential of strength Ab is assumed for the polymer back bone orientation. The time evolution equations for the alignment tensor a in the presence of a flow field v read [7,13,14]: √ τap ∂a − 2 × a − 2κ · a + τa−1 a (a, b) = − 2 . ∂t τa
(2)
The analogous equation for the tensor b is [7]: √ τbp ∂b − 2 × b − 2κb · b + τb−1 b (a, b) = − 2 . ∂t τb
(3)
The symmetric traceless tensors introduced in Eqs. (2) and (3) are defined as √ ∂Φ = Aa − 6Ba · a + Ca a : a + Db a (a, b) ≡ (4) ∂a and b (a, b) ≡
∂Φ = Ab b + Da. ∂b
(5)
The symbols and denote the symmetric traceless part of the velocity gradient tensor (strain rate tensor) ≡ ∇v, and the vorticity ≡ (∇ × v)/2, respectively. In the case of a pla˙ x in x-direction, gradient in y-direction, nar shear flow v = γye and vorticity in z-direction, to be considered throughout the x y
˙ e and following analysis, these quantities simplify to = γe
3
˙ z . The unit vectors parallel to the coordinate axes = −(1/2)γe are denoted by ex , ey , ez . The equilibrium orientation satisfies a = 0 and b = 0. From Eq. (5) one obtains b = −(D/Ab )a, i.e. the alignment of the side chains induces orientation of the backbone. For D > 0 the orientation of the backbone is perpendicular to the side chains and parallel for D < 0. Both conformations have been observed experimentally (see e.g. [1] and references therein). Eq. (2) has been extended to inhomogeneous systems by changing the time derivative from a partial to a substantial one, and by adding a term ∝ a characterizing inhomogeneous systems [15], see also [16,17] for related works. The equations of motion contain the relaxation times τa > 0 and τb > 0, as well as τap and τbp which determine the strength of the coupling between the alignment and the pressure tensor or the velocity gradient, as well as the dimensionless coefficients (shape factor) κ, κb . The special values 0 and ±1 for the coefficient κ in (2) correspond to corotational and codeformational time derivatives. From the solution of the generalized Fokker– Planck equation one finds, for long particles, κ ≈ 3/7 ≈ 0.4 [9]. 2.2. Constitutive equations The backreaction of the alignment on the flow properties is characterized by the friction pressure tensor. The full pressure tensor p consists of a hydrostatic pressure p, an antisymmetric part, and the symmetric traceless part p referred to as friction pressure tensor [13]. The latter splits into an ‘isotropic’ contribution as already present in fluids composed of spherical particles or in fluids of non-spherical particles in an perfectly ‘isotropic state’ with zero alignment, and a part explicitly depending on the alignment tensor: p = −2ηiso + pal , with the alignment contribution [7,14] √ τap a ρ pal = kB T 2 − 2κa · a m τa √ τbp b b + 2 − 2κb b · . τb
(6)
(7)
In equilibrium one has a = 0, b = 0 and consequently pal = 0. The occurrence of the same coupling coefficients τap and τbp in (7) as in (2) is due to Onsager symmetry relations. For studies on the rheological properties in the isotropic and in the nematic phases with stationary flow alignment, following from (2) and (7) for the case where one has b = 0 see [13,14]. Here, the main attention is focused on the behavior in the vicinity of the isotropic–nematic phase transition. For the case of oscillatory shear γ(t) = γ0 sin(ωt) with amplitude γ0 and frequency ω, the tensor is time-dependent, x y ˙ ˙ = γ0 ω cos(ωt). Within the linear re(t) = γ(t)e e , with γ(t) sponse regime and for long times, the resulting shear pressure
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P. Ilg, S. Hess / J. Non-Newtonian Fluid Mech. 134 (2006) 2–7
oscillates with the same frequency ω and can be expressed as −
pyx (t) = G (ω) sin(ωt) + G (ω) cos(ωt) γ0
(8)
where G and G denote the storage and loss modulus, respectively. In the case of stationary shear flow, a constant shear rate ˙ γ˙ is applied. The shear viscosity is defined as η = −pyx /γ. In the flow-aligning regime, a stationary state is reached for sufficiently long times, such that the viscosity becomes time independent. 2.3. Analytical results Consider the high temperature regime where the equilibrium state is isotropic, aeq = beq = 0. Analytical results for the steady state in case of stationary shear flow are presented in [7]. For small amplitude oscillatory shear flow, γ0 1, the alignment tensor components can be expanded in a power series in γ0 . Neglecting initial transient phenomena, the alignment tensors a and b are assumed to oscillate with the same frequency ω as the shear flow. In the low frequency limit, τref ω 1, we find G (ω) = Σ2 ω2 ,
G (ω) = η0 ω,
τref ω → 0
(9)
with the effective zero-shear viscosity η0 = ηiso + (ρkB T/m) 2 /τ + τ 2 /τ ]. Note, that η is identical to the effective zero[τap a 0 bp b shear viscosity in stationary shear flow. The reference time τref is given by τref = τa 9C/(2B2 ). The coefficient Σ2 is given in terms of the model parameters by 2 τbp νab 2 −1 τbp −1 Σ2 = Gal ϑb . (10) + ϑeff 1 − τap τap ϑb Gal = (ρkB T/m)(τap /τa )2 δK A0 is the shear modulus associated with the alignment contribution and δK = 1 − T ∗ /TK = 2B2 /(9A0 C). From experiments it is known that δK ranges typically between 0.001 and 0.1. In Eq. (10), we have defined the reduced temperatures ϑb = 9Ab C/(2B2 ) and ϑeff = 9Aeff C/(2B2 ), where Aeff = A − D2 /Ab . The reduced coupling strength is defined by νab = 9DC/(2B2 ). In the high frequency limit, τref ω 1, we find G (ω) = G∞ ,
G (ω) = ηiso ω,
τref ω → ∞
with the high-frequency modulus 2 τ τbp τbp τa a τa G∞ = Gal ϑ + 2νab 1+ 2 + ϑb τap τb τb τap τb
(11)
(12)
From Eqs. (9) and (11) we observe that within the present model the elastic moduli of side chain liquid-crystalline polymers show the same low and high frequency behavior as the corresponding main chain liquid-crystalline polymers. The coupling to the polymer back bone merely changes the values of the coefficients η0 , Σ2 and G∞ .
3. Rheological properties 3.1. Parameters and numerical solution The following results are obtained from numerical solutions of the dynamical equations (2),(3) with a Runge–Kutta method using an adaptive time step. Initial conditions are chosen close to a random orientation with small but non-zero values for a and b. For stationary shear flow, time averages are collected after a stationary state has been reached. Shear stress and shear viscosity, for example, are calculated from Eq. (6). The total integration time trun is chosen such that the total shear defor˙ run 1. In the case of oscillatory shear mation is large, γ = γt flow, after an initial transient period, data are extracted for times n1 tT < t < n2 tT , where tT = 2π/ω is the period of the shear oscillations. For intermediate frequencies, n1 = 50, n2 = 150, while for high frequencies n2 = 500 and for low frequencies n1 = 10, n2 = 20 was chosen. Storage and loss moduli are calculated from fits of the resulting shear stress to Eq. (8). For the parameters investigated here, the stress oscillates indeed with the frequency of the shear to a very good approximation. The choice of the model parameters deserves some discussion. It is assumed that the essential effects associated with the variation of the temperature are due to the variation of ϑ. The temperature dependence of all other parameters, including that of ϑb is disregarded. The simple choice νab = 1.0 and ϑb = 1.0 is made. This means ϑeff = ϑ − 1 and hence, in equilibrium, the isotropic-nematic phase occurs at the higher temperature ϑ = 2 and not at ϑ = 1 where it would happen without the coupling between the mesogenic side groups and the back bone segments. The parameters pertaining to the a-alignment can be chosen such that the uncoupled case would correspond to a typical flow aligning nematic substance, more specifically the values λK = 1.25, κ = κb = 0 are used, in√analogy with [18,19]. The parameter λK is defined as λK = − 3Cτap /(Bτa ) For these parameters, the flow alignment angle χ is about 18◦ , just below the transition temperature. Furthermore, it is assumed that the orientational relaxation of the polymeric back bone is considerably slower than that of the side groups, so τa /τb = 1/100 is used for the ratio of the relaxation times. On the other hand, a likely guess is that the ratio between the non-diagonal and diagonal relaxation time coefficients are of comparable order of magnitude for the a and b tensors, viz. |τap /τa | ≈ |τbp /τb |. For simplicity, λbK = 1.0 is chosen, where √ λbK = − 3Cτbp /(Bτb ). As a side remark, here rod-like mesogenic side groups have been assumed tacitly. For disc-like side groups one would have to consider the case λbK ≈ −λK . A reasonable choice for the κ parameter to be made in the following is κb = κ. Finally, for the computation of rheological properties, the value ηiso = 0.1ηref is chosen, as in [20]. This means that the ratio between the first and second Newtonian viscosity in the isotropic phase is about 10 when no coupling with the polymer back bone exists. We emphasize that this choice of parameters is by no means adapted for a specific substance but is proposed for a typical side chain liquid-crystalline polymer.
P. Ilg, S. Hess / J. Non-Newtonian Fluid Mech. 134 (2006) 2–7
5
rameters are chosen as in Fig. 1. For low shear rates, τref γ˙ ˙ The orientational 10−2 , S2 increases linearly with increasing γ. order parameter reaches a plateau value around τref γ˙ = 10−1 , before it attains its maximum value for τref γ˙ 100 which more ˙ or less remains constant with further increasing γ. 3.3. Oscillatory shear flow
Fig. 1. The dimensionless shear stress σyx /Gal = −pyx /Gal is shown as a func˙ Squares and circles correspond the reduced tion of the reduced shear rate τref γ. temperatures ϑ = 3.0 and 2.5, respectively.
3.2. Stationary shear flow The shear stress in the isotropic regime is shown in Fig. 1 as a function of the shear rate. Note that a double logarithmic scale is used. The parameters are chosen as described above with λK = 1.25 and κ = κb = 0. The reduced temperature has been chosen as ϑ = 2.5 and 3.0, respectively. For very low (γ˙ 10−3 ) and very high (γ˙ 10) shear rates, a linear increase of the shear stress σyx = −pyx with increasing shear rate is found, in agreement with the theoretical results [7]. In addition, we observe also a linear increase of σyx for intermediate shear rates, a stress plateau for 1 τref γ˙ 10, and a region, where ˙ The latter result indicates a σyx decreases with increasing γ. mechanical instability which typically leads to the formation of shear bands [21,22]. Therefore, if spatial variations of the alignment tensor are included, the stress is expected to form a plateau also in this region. The existence of stress plateaus in side chain liquid crystals has been observed experimentally in [1,2]. The Maier–Saupe orientational order parameter S2 is shown in Fig. 2 as a function of the reduced shear rate. The same pa-
Fig. 2. The Maier–Saupe orientational order parameter S2 defined in the text as a function of the reduced shear rate. Squares, circles and diamonds correspond to reduced temperatures ϑ = 3.0, 2.5 and 2.125, respectively.
In Fig. 3, the dependency of the storage and loss moduli on the frequency of the shear flow is shown. Results for different shear amplitudes γ0 are displayed. Results for amplitudes γ0 < 0.1 are indistinguishable from those for γ0 = 0.1 within the resolution of the figure. The low and high frequency limit is reached for τref ω 10−3 and τref ω 102 , respectively. We observe that the low frequency limit of G and G as well as the high frequency limit of G appear to be insensitive to changes in γ0 , while high frequency modulus G∞ decreases with increasing γ0 . For intermediate frequencies, 10−2 τref ω 100 , a plateau in G is observed. The plateau value of G decreases with increasing γ0 . These results are in qualitative agreement with experimental observations of a side chain liquid crystal polyacrylate [3]. The parameters were chosen as described above with λK = 1.25
Fig. 3. Storage (a) and loss (b) moduli as a function of the dimensionless frequency of the applied shear flow. Results for amplitudes γ0 = 0.1, 0.5 and 1.0 are shown by solid, dashed and dashed-dotted lines, respectively. Short dashed lines show the analytical result for low and high shear rate limit, cf. Eqs. (9) and (11). The reduced temperature was chosen as ϑ = 2.5.
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P. Ilg, S. Hess / J. Non-Newtonian Fluid Mech. 134 (2006) 2–7
It appears therefore that the present model seems to be able to describe the main experimental observations of the flow behavior of side chain liquid-crystalline polymers at least in shear flow. For a more quantitative comparison to the experimental results, a careful determination of the model parameters is needed. Further comparisons with transient behavior (e.g. [23]) are desirable. For the case of extensional flow, a comparison of the present theory with the approach proposed in [11] would be useful. Acknowledgments
Fig. 4. Storage (solid) and loss (dashed line) moduli as a function of the amplitude γ0 of the applied shear flow. The same parameters as in Fig. 3 are chosen with frequency τref ω = 10−1 .
and κ = κb = 0. For λK = 1.05 and κ = κb = 0.4 (not shown) the same qualitative dependence of G and G on the frequency ω is observed. The numerical values of G and G , however, are higher in this case compared to those in Fig. 3. For example, the value of the plateau of G at τref ω = 10−1 differs by approximately 50% for these two sets of parameters. Fig. 4 shows the dependence of the viscoelastic moduli G and G on the amplitude γ0 of the applied shear flow. The same set of parameters as in Fig. 3 are used. The frequency ω was held fix at τref ω = 10−1 where the plateau of G is observed in Fig. 3. From Fig. 4 we observe that the values of G and G remain more or less constant for shear amplitudes up to γ0 10−1 . For higher amplitudes, G starts to decrease rapidly while G varies only mildly. Also these observations are in qualitative agreement with the experimental observations made in [3]. At the moment, we do not have an explanation for the non-monotonic behavior of G for high amplitudes γ0 . This point needs to be clarified in the future. 4. Conclusions The predictions of the “two-alignment tensor model” for the non-equilibrium dynamics of side chain liquid crystal polymers introduced in Ref. [7] are studied for stationary and oscillatory shear flow. In stationary shear flow, the shear-induced isotropic-tonematic phase transition observed in the experiments [1] is reproduced within the present model. Also the stress plateau for intermediate shear rates observed in [1] is seen in the present model. For the case of small amplitude oscillatory shear flow, the low and high frequency limit of the elastic moduli are calculated analytically. In these limits, the coupling of the mesogenic side chains to the polymer back bone does not change the scaling with frequency but modifies the corresponding coefficients. For intermediate frequencies of the applied shear flow, a plateau in the elastic moduli is observed which decreases with increasing amplitude of the shear flow. These observations are in qualitative agreement with very recent experimental results [3].
The authors are grateful to L. Noirez for providing Ref. [3] prior to publication. This research has been performed under the auspices of the Sonderforschungsbereich 448 ’Mesoskopisch strukturierte Verbundsysteme’ (Deutsche Forschungsgemeinschaft). P.I. is supported by a Feodor-Lynen research fellowship from the Alexander von Humboldt foundation. References [1] C. Pujolle-Robic, L. Noirez, Observation of shear-induced nematicisotropic transition in side-chain liquid crystal polymers, Nature 409 (2001) 167–171. [2] C. Pujolle-Robic, P.D. Olmsted, L. Noirez, Transient and stationary flow behaviour of side chain liquid-crystalline polymers: evidence of a shearinduced isotropic-to-nematic phase transition, Europhys. Lett. 59 (3) (2002) 364–369. [3] L. Noirez, Origin of shear induced transitions in melts of liquid-crystal polymers, Phys. Rev. E 72 (2005) 051701. [4] S.M. Clarke, A. Hotta, A.R. Tajbakhsh, E.M. Terentjev, Effect of crosslinker geometry on dynamic mechanical properties of nematic elastomers, Phys. Rev. E 65 (2002) 021804. [5] A. Wewerka, K. Viertler, D. Vlassopoulos, F. Stelzer, Structure and rheology of model side-chain liquid crystalline polymers with varying mesogen length, Rheol. Acta 40 (2001) 416–425. [6] S.F. Rubin, R.M. Kannan, J.A. Kornfield, C. Boeffel, Effect of mesophase order and molecular weight on the dynamics of nematic and smectic sidegroup liquid-crystalline polymers, Macromolecules 28 (1995) 3521–3530. [7] S. Hess, P. Ilg, On the theory of the shear-induced isotropic-to-nematic phase transition of side chain liquid-crystalline polymers, Rheol. Acta 44 (2005) 465–477. [8] S. Hess, Pre- and post-transitional behavior of the flow alignment and flow–induced phase transition in liquid crystals, Z. Naturforsch. 31a (1976) 1507–1513. [9] S. Hess, Fokker–Planck-equation approach to flow alignment in liquid crystals, Z. Naturforsch. 31a (1976) 1034–1037. [10] M. Doi, Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases, J. Polym. Sci., Polym. Phys. Ed. 19 (1981) 229–243. [11] A.D. Rey, Macroscopic theory of orientation transitions in the extensional flow of side-chain nematic polymers, Rheol. Acta 34 (1995) 119–131. [12] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, UK, 1999. [13] S. Hess, Irreversible thermodynamics of non-equilibrium alignment phenomena in molecular liquids and in liquid crystals, I. Derivation of nonlinear constitutive laws, relaxation of the alignment, phase transition, Z. Naturforsch. 30a (1975) 728. [14] C.P. Borgmeyer, S. Hess, Unified description of the flow alignment and viscosity in the isotropic and nematic phases of liquid crystals, J. NonEquilib. Thermodyn. 20 (1995) 359. [15] S. Hess, I. Pardowitz, On the unified theory for non-equilibrium phenomena in the isotropic and nematic phases of a liquid crystal, spatially inhomogeneous alignment, Z. Naturforsch. 36a (1981) 554–558.
P. Ilg, S. Hess / J. Non-Newtonian Fluid Mech. 134 (2006) 2–7 [16] M. Kr¨oger, H. S. Sellers, A molecular theory for spatially inhomogeneous, concentrated solutions of rod-like liquid crystal polymers, in: L., Garrido, (Ed.), Complex Fluids, vol. 415 of Lecture notes in Physics, Springer, 1992. 295–301. [17] G. Sgalari, G.L. Leal, J.J. Feng, The shear flow behavior of LCPs based on a generalized Doi model with distortional elasticity, J. Non-Newtonian Fluid Mech. 102 (2002) 361–382. [18] G. Rien¨acker, M. Kr¨oger, S. Hess, Chaotic orientational behavior of a nematic liquid crystal subjected to a shear flow, Phys. Rev. E 66 (R) (2002) 040702. [19] G. Rien¨acker, M. Kr¨oger, S. Hess, Chaotic and regular shear-induced orientational dynamics of nematic liquid crystals, Physica A 315 (2002) 537– 568.
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[20] S. Hess, M. Kr¨oger, Regular and chaotic rheological behavior of tumbling polymeric liquid crystals, in: P. Pasini, C. Zannoni, S. Zumer (Eds.), Computer Simulations Bridging Liquid Crystals and Polymers, NATO, Kluwer, Dordrecht, 2005, pp. 295–334. [21] V. Schmitt, F. Lequeux, A. Pousse, D. Roux, Flow behavior and shear induced transition near an isotropic/nematic transition in equilibrium polymers, Langmuir 10 (1994) 955–961. [22] J.F. Berret, D.C. Roux, G. Porte, P. Lindner, Shear-induced isotropic-tonematic phase transition in equilibrium polymers, Europhys. Lett. 25 (7) (1994) 521–526. [23] I. Quijada-Garrido, H. Siebert, C. Friedrich, C. Schmidt, Flow behavior of two side-chain liquid crystal polymers studied by transient rheology, Macromolecules 33 (2000) 3844–3854.
Two-alignment tensor theory for the dynamics of side chain liquid-crystalline polymers in planar shear flow夽 Patrick Ilg a,b,∗ , Siegfried Hess a a
Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, PN 7-1, Hardenbergstr. 36, D-10623 Berlin, Germany b D´ epartement de Physique des Mat´eriaux, Universit´e Claude Bernard Lyon1, F-69622 Villeurbanne, France Received 28 June 2005; received in revised form 1 September 2005; accepted 27 September 2005
Abstract A two-alignment tensor model for the dynamics of side chain liquid-crystalline polymers is applied to the case of planar shear flow. The predictions of the model are worked out for the case of stationary and oscillatory shear flow. In stationary shear, a flow induced isotropic-to-nematic phase transition is observed as well as a stress plateau for intermediate shear rates. In oscillatory shear flow, a low frequency elastic modulus is found. The model predictions agree qualitatively with experimental results. © 2005 Elsevier B.V. All rights reserved. PACS: 36.20.Ey; 64.70.Md; 83.50.Ax Keywords: Side-chain liquid crystal polymers; Shear flow; viscoelastic moduli; Isotropic-to-nematic transition
1. Introduction Shear-flow-induced isotropic-to-nematic phase transitions have been observed in a variety of different side chain liquidcrystalline polymers [1]. Rheological measurements in planar shear flow, both stationary [1,2] and oscillatory [3–6] yield additional information on the dynamics of side chain liquidcrystalline polymers. While the effect of different mesogen lengths, cross-linker geometry and molecular weight on the dynamical mechanical response of side-chain liquid-crystalline polymers have been studied in [5,4,6], an interesting low frequency elastic plateau was observed in [3]. Very recently, we have proposed in [7] a “two-alignment tensor” theory of shear-flow-induced isotropic-to-nematic phase transition in side chain liquid-crystalline polymers. This model extends previous theories of the flow-induced isotropic-tonematic phase transition [8–10] which are applicable to ordinary thermotropic nematics, to lyotropic liquid crystals and to main chain liquid-crystalline polymers.
An alternative approach to the dynamics of side chain liquidcrystalline polymers in extensional flow has been proposed in [11]. Here, we study the predictions of the two-alignment tensor model for the dynamical behavior of side chain liquid-crystalline polymers in stationary and oscillatory shear flow. Qualitative agreement with the experimental observations in [1–3] are observed. The paper is organized as follows. In Section 2, the twoalignment-tensor model is reviewed. The dynamic and constitutive equations are given explicitely. Analytical solutions for the low and high frequency limit in small amplitude oscillatory shear flow are presented. In Section 3, the model predictions for the orientational and rheological properties are obtained from the numerical solution of the model equations for several parameter values. Both, stationary and oscillatory shear flow is considered. Qualitative comparison to the experimental data of [1–3] is made. Finally, some conclusions are offered in Section 4. 2. Model equations
夽 Talk presented at the AERC 2005.
2.1. Dynamics of alignment tensors
∗
Corresponding author. E-mail addresses: [email protected] (P. Ilg), [email protected] (S. Hess). 0377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2005.09.003
In order to make the paper self-contained, we briefly review the two-alignment tensor model introduced in Ref. [7].
P. Ilg, S. Hess / J. Non-Newtonian Fluid Mech. 134 (2006) 2–7
For more details the reader is referred to to the original reference. Let the alignment of the liquid-crystalline side groups be described by the second rank, symmetric, traceless alignment tensor a. Similarly, the orientation of the polymer back bone is characterized by the corresponding second rank tensor b. The normalization is chosen such that a2 = a : a where a is the scalar order parameter in case of uniaxial symmetry, a = a(3/2)1/2 nn. The symbol x denotes the symmetric traceless part of the matrix x. The parameter a is therefore proportional to the Maier–Saupe √ order parameter S2 = a/ 5 = (3/10)1/2 a : nn. We assume in the sequel, that the isotropic-to-nematic phase transition is linked with the liquid-crystalline side groups described by a but not with the polymer back bone, b. Assuming further a bilinear coupling of a and b of strength D, a simple extension of the usual Landau-de Gennes potential [12] to the case of side chain liquid crystals was proposed in [7], Φ = Φ(a, b), viz. √ 1 6 1 Φ = A(T )a : a − B(a · a) : a + C(a : a)2 2 3 4 1 + Da : b + Ab b : b (1) 2 The coefficient A(T ) = A0 (1 − T ∗ /T ) changes sign at the pseudo-critical temperature T ∗ , which is below the nematic– isotropic transition temperature TK , T ∗ < TK . The coefficients A0 , B, C, D, Ab (with C < 2B2 /(9A0 )) are linked with measurable quantities and can also be related to molecular properties within the framework of a mesoscopic theory [9,10]. Here, these coefficients are treated as phenomenological parameters. A harmonic potential of strength Ab is assumed for the polymer back bone orientation. The time evolution equations for the alignment tensor a in the presence of a flow field v read [7,13,14]: √ τap ∂a − 2 × a − 2κ · a + τa−1 a (a, b) = − 2 . ∂t τa
(2)
The analogous equation for the tensor b is [7]: √ τbp ∂b − 2 × b − 2κb · b + τb−1 b (a, b) = − 2 . ∂t τb
(3)
The symmetric traceless tensors introduced in Eqs. (2) and (3) are defined as √ ∂Φ = Aa − 6Ba · a + Ca a : a + Db a (a, b) ≡ (4) ∂a and b (a, b) ≡
∂Φ = Ab b + Da. ∂b
(5)
The symbols and denote the symmetric traceless part of the velocity gradient tensor (strain rate tensor) ≡ ∇v, and the vorticity ≡ (∇ × v)/2, respectively. In the case of a pla˙ x in x-direction, gradient in y-direction, nar shear flow v = γye and vorticity in z-direction, to be considered throughout the x y
˙ e and following analysis, these quantities simplify to = γe
3
˙ z . The unit vectors parallel to the coordinate axes = −(1/2)γe are denoted by ex , ey , ez . The equilibrium orientation satisfies a = 0 and b = 0. From Eq. (5) one obtains b = −(D/Ab )a, i.e. the alignment of the side chains induces orientation of the backbone. For D > 0 the orientation of the backbone is perpendicular to the side chains and parallel for D < 0. Both conformations have been observed experimentally (see e.g. [1] and references therein). Eq. (2) has been extended to inhomogeneous systems by changing the time derivative from a partial to a substantial one, and by adding a term ∝ a characterizing inhomogeneous systems [15], see also [16,17] for related works. The equations of motion contain the relaxation times τa > 0 and τb > 0, as well as τap and τbp which determine the strength of the coupling between the alignment and the pressure tensor or the velocity gradient, as well as the dimensionless coefficients (shape factor) κ, κb . The special values 0 and ±1 for the coefficient κ in (2) correspond to corotational and codeformational time derivatives. From the solution of the generalized Fokker– Planck equation one finds, for long particles, κ ≈ 3/7 ≈ 0.4 [9]. 2.2. Constitutive equations The backreaction of the alignment on the flow properties is characterized by the friction pressure tensor. The full pressure tensor p consists of a hydrostatic pressure p, an antisymmetric part, and the symmetric traceless part p referred to as friction pressure tensor [13]. The latter splits into an ‘isotropic’ contribution as already present in fluids composed of spherical particles or in fluids of non-spherical particles in an perfectly ‘isotropic state’ with zero alignment, and a part explicitly depending on the alignment tensor: p = −2ηiso + pal , with the alignment contribution [7,14] √ τap a ρ pal = kB T 2 − 2κa · a m τa √ τbp b b + 2 − 2κb b · . τb
(6)
(7)
In equilibrium one has a = 0, b = 0 and consequently pal = 0. The occurrence of the same coupling coefficients τap and τbp in (7) as in (2) is due to Onsager symmetry relations. For studies on the rheological properties in the isotropic and in the nematic phases with stationary flow alignment, following from (2) and (7) for the case where one has b = 0 see [13,14]. Here, the main attention is focused on the behavior in the vicinity of the isotropic–nematic phase transition. For the case of oscillatory shear γ(t) = γ0 sin(ωt) with amplitude γ0 and frequency ω, the tensor is time-dependent, x y ˙ ˙ = γ0 ω cos(ωt). Within the linear re(t) = γ(t)e e , with γ(t) sponse regime and for long times, the resulting shear pressure
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oscillates with the same frequency ω and can be expressed as −
pyx (t) = G (ω) sin(ωt) + G (ω) cos(ωt) γ0
(8)
where G and G denote the storage and loss modulus, respectively. In the case of stationary shear flow, a constant shear rate ˙ γ˙ is applied. The shear viscosity is defined as η = −pyx /γ. In the flow-aligning regime, a stationary state is reached for sufficiently long times, such that the viscosity becomes time independent. 2.3. Analytical results Consider the high temperature regime where the equilibrium state is isotropic, aeq = beq = 0. Analytical results for the steady state in case of stationary shear flow are presented in [7]. For small amplitude oscillatory shear flow, γ0 1, the alignment tensor components can be expanded in a power series in γ0 . Neglecting initial transient phenomena, the alignment tensors a and b are assumed to oscillate with the same frequency ω as the shear flow. In the low frequency limit, τref ω 1, we find G (ω) = Σ2 ω2 ,
G (ω) = η0 ω,
τref ω → 0
(9)
with the effective zero-shear viscosity η0 = ηiso + (ρkB T/m) 2 /τ + τ 2 /τ ]. Note, that η is identical to the effective zero[τap a 0 bp b shear viscosity in stationary shear flow. The reference time τref is given by τref = τa 9C/(2B2 ). The coefficient Σ2 is given in terms of the model parameters by 2 τbp νab 2 −1 τbp −1 Σ2 = Gal ϑb . (10) + ϑeff 1 − τap τap ϑb Gal = (ρkB T/m)(τap /τa )2 δK A0 is the shear modulus associated with the alignment contribution and δK = 1 − T ∗ /TK = 2B2 /(9A0 C). From experiments it is known that δK ranges typically between 0.001 and 0.1. In Eq. (10), we have defined the reduced temperatures ϑb = 9Ab C/(2B2 ) and ϑeff = 9Aeff C/(2B2 ), where Aeff = A − D2 /Ab . The reduced coupling strength is defined by νab = 9DC/(2B2 ). In the high frequency limit, τref ω 1, we find G (ω) = G∞ ,
G (ω) = ηiso ω,
τref ω → ∞
with the high-frequency modulus 2 τ τbp τbp τa a τa G∞ = Gal ϑ + 2νab 1+ 2 + ϑb τap τb τb τap τb
(11)
(12)
From Eqs. (9) and (11) we observe that within the present model the elastic moduli of side chain liquid-crystalline polymers show the same low and high frequency behavior as the corresponding main chain liquid-crystalline polymers. The coupling to the polymer back bone merely changes the values of the coefficients η0 , Σ2 and G∞ .
3. Rheological properties 3.1. Parameters and numerical solution The following results are obtained from numerical solutions of the dynamical equations (2),(3) with a Runge–Kutta method using an adaptive time step. Initial conditions are chosen close to a random orientation with small but non-zero values for a and b. For stationary shear flow, time averages are collected after a stationary state has been reached. Shear stress and shear viscosity, for example, are calculated from Eq. (6). The total integration time trun is chosen such that the total shear defor˙ run 1. In the case of oscillatory shear mation is large, γ = γt flow, after an initial transient period, data are extracted for times n1 tT < t < n2 tT , where tT = 2π/ω is the period of the shear oscillations. For intermediate frequencies, n1 = 50, n2 = 150, while for high frequencies n2 = 500 and for low frequencies n1 = 10, n2 = 20 was chosen. Storage and loss moduli are calculated from fits of the resulting shear stress to Eq. (8). For the parameters investigated here, the stress oscillates indeed with the frequency of the shear to a very good approximation. The choice of the model parameters deserves some discussion. It is assumed that the essential effects associated with the variation of the temperature are due to the variation of ϑ. The temperature dependence of all other parameters, including that of ϑb is disregarded. The simple choice νab = 1.0 and ϑb = 1.0 is made. This means ϑeff = ϑ − 1 and hence, in equilibrium, the isotropic-nematic phase occurs at the higher temperature ϑ = 2 and not at ϑ = 1 where it would happen without the coupling between the mesogenic side groups and the back bone segments. The parameters pertaining to the a-alignment can be chosen such that the uncoupled case would correspond to a typical flow aligning nematic substance, more specifically the values λK = 1.25, κ = κb = 0 are used, in√analogy with [18,19]. The parameter λK is defined as λK = − 3Cτap /(Bτa ) For these parameters, the flow alignment angle χ is about 18◦ , just below the transition temperature. Furthermore, it is assumed that the orientational relaxation of the polymeric back bone is considerably slower than that of the side groups, so τa /τb = 1/100 is used for the ratio of the relaxation times. On the other hand, a likely guess is that the ratio between the non-diagonal and diagonal relaxation time coefficients are of comparable order of magnitude for the a and b tensors, viz. |τap /τa | ≈ |τbp /τb |. For simplicity, λbK = 1.0 is chosen, where √ λbK = − 3Cτbp /(Bτb ). As a side remark, here rod-like mesogenic side groups have been assumed tacitly. For disc-like side groups one would have to consider the case λbK ≈ −λK . A reasonable choice for the κ parameter to be made in the following is κb = κ. Finally, for the computation of rheological properties, the value ηiso = 0.1ηref is chosen, as in [20]. This means that the ratio between the first and second Newtonian viscosity in the isotropic phase is about 10 when no coupling with the polymer back bone exists. We emphasize that this choice of parameters is by no means adapted for a specific substance but is proposed for a typical side chain liquid-crystalline polymer.
P. Ilg, S. Hess / J. Non-Newtonian Fluid Mech. 134 (2006) 2–7
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rameters are chosen as in Fig. 1. For low shear rates, τref γ˙ ˙ The orientational 10−2 , S2 increases linearly with increasing γ. order parameter reaches a plateau value around τref γ˙ = 10−1 , before it attains its maximum value for τref γ˙ 100 which more ˙ or less remains constant with further increasing γ. 3.3. Oscillatory shear flow
Fig. 1. The dimensionless shear stress σyx /Gal = −pyx /Gal is shown as a func˙ Squares and circles correspond the reduced tion of the reduced shear rate τref γ. temperatures ϑ = 3.0 and 2.5, respectively.
3.2. Stationary shear flow The shear stress in the isotropic regime is shown in Fig. 1 as a function of the shear rate. Note that a double logarithmic scale is used. The parameters are chosen as described above with λK = 1.25 and κ = κb = 0. The reduced temperature has been chosen as ϑ = 2.5 and 3.0, respectively. For very low (γ˙ 10−3 ) and very high (γ˙ 10) shear rates, a linear increase of the shear stress σyx = −pyx with increasing shear rate is found, in agreement with the theoretical results [7]. In addition, we observe also a linear increase of σyx for intermediate shear rates, a stress plateau for 1 τref γ˙ 10, and a region, where ˙ The latter result indicates a σyx decreases with increasing γ. mechanical instability which typically leads to the formation of shear bands [21,22]. Therefore, if spatial variations of the alignment tensor are included, the stress is expected to form a plateau also in this region. The existence of stress plateaus in side chain liquid crystals has been observed experimentally in [1,2]. The Maier–Saupe orientational order parameter S2 is shown in Fig. 2 as a function of the reduced shear rate. The same pa-
Fig. 2. The Maier–Saupe orientational order parameter S2 defined in the text as a function of the reduced shear rate. Squares, circles and diamonds correspond to reduced temperatures ϑ = 3.0, 2.5 and 2.125, respectively.
In Fig. 3, the dependency of the storage and loss moduli on the frequency of the shear flow is shown. Results for different shear amplitudes γ0 are displayed. Results for amplitudes γ0 < 0.1 are indistinguishable from those for γ0 = 0.1 within the resolution of the figure. The low and high frequency limit is reached for τref ω 10−3 and τref ω 102 , respectively. We observe that the low frequency limit of G and G as well as the high frequency limit of G appear to be insensitive to changes in γ0 , while high frequency modulus G∞ decreases with increasing γ0 . For intermediate frequencies, 10−2 τref ω 100 , a plateau in G is observed. The plateau value of G decreases with increasing γ0 . These results are in qualitative agreement with experimental observations of a side chain liquid crystal polyacrylate [3]. The parameters were chosen as described above with λK = 1.25
Fig. 3. Storage (a) and loss (b) moduli as a function of the dimensionless frequency of the applied shear flow. Results for amplitudes γ0 = 0.1, 0.5 and 1.0 are shown by solid, dashed and dashed-dotted lines, respectively. Short dashed lines show the analytical result for low and high shear rate limit, cf. Eqs. (9) and (11). The reduced temperature was chosen as ϑ = 2.5.
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It appears therefore that the present model seems to be able to describe the main experimental observations of the flow behavior of side chain liquid-crystalline polymers at least in shear flow. For a more quantitative comparison to the experimental results, a careful determination of the model parameters is needed. Further comparisons with transient behavior (e.g. [23]) are desirable. For the case of extensional flow, a comparison of the present theory with the approach proposed in [11] would be useful. Acknowledgments
Fig. 4. Storage (solid) and loss (dashed line) moduli as a function of the amplitude γ0 of the applied shear flow. The same parameters as in Fig. 3 are chosen with frequency τref ω = 10−1 .
and κ = κb = 0. For λK = 1.05 and κ = κb = 0.4 (not shown) the same qualitative dependence of G and G on the frequency ω is observed. The numerical values of G and G , however, are higher in this case compared to those in Fig. 3. For example, the value of the plateau of G at τref ω = 10−1 differs by approximately 50% for these two sets of parameters. Fig. 4 shows the dependence of the viscoelastic moduli G and G on the amplitude γ0 of the applied shear flow. The same set of parameters as in Fig. 3 are used. The frequency ω was held fix at τref ω = 10−1 where the plateau of G is observed in Fig. 3. From Fig. 4 we observe that the values of G and G remain more or less constant for shear amplitudes up to γ0 10−1 . For higher amplitudes, G starts to decrease rapidly while G varies only mildly. Also these observations are in qualitative agreement with the experimental observations made in [3]. At the moment, we do not have an explanation for the non-monotonic behavior of G for high amplitudes γ0 . This point needs to be clarified in the future. 4. Conclusions The predictions of the “two-alignment tensor model” for the non-equilibrium dynamics of side chain liquid crystal polymers introduced in Ref. [7] are studied for stationary and oscillatory shear flow. In stationary shear flow, the shear-induced isotropic-tonematic phase transition observed in the experiments [1] is reproduced within the present model. Also the stress plateau for intermediate shear rates observed in [1] is seen in the present model. For the case of small amplitude oscillatory shear flow, the low and high frequency limit of the elastic moduli are calculated analytically. In these limits, the coupling of the mesogenic side chains to the polymer back bone does not change the scaling with frequency but modifies the corresponding coefficients. For intermediate frequencies of the applied shear flow, a plateau in the elastic moduli is observed which decreases with increasing amplitude of the shear flow. These observations are in qualitative agreement with very recent experimental results [3].
The authors are grateful to L. Noirez for providing Ref. [3] prior to publication. This research has been performed under the auspices of the Sonderforschungsbereich 448 ’Mesoskopisch strukturierte Verbundsysteme’ (Deutsche Forschungsgemeinschaft). P.I. is supported by a Feodor-Lynen research fellowship from the Alexander von Humboldt foundation. References [1] C. Pujolle-Robic, L. Noirez, Observation of shear-induced nematicisotropic transition in side-chain liquid crystal polymers, Nature 409 (2001) 167–171. [2] C. Pujolle-Robic, P.D. Olmsted, L. Noirez, Transient and stationary flow behaviour of side chain liquid-crystalline polymers: evidence of a shearinduced isotropic-to-nematic phase transition, Europhys. Lett. 59 (3) (2002) 364–369. [3] L. Noirez, Origin of shear induced transitions in melts of liquid-crystal polymers, Phys. Rev. E 72 (2005) 051701. [4] S.M. Clarke, A. Hotta, A.R. Tajbakhsh, E.M. Terentjev, Effect of crosslinker geometry on dynamic mechanical properties of nematic elastomers, Phys. Rev. E 65 (2002) 021804. [5] A. Wewerka, K. Viertler, D. Vlassopoulos, F. Stelzer, Structure and rheology of model side-chain liquid crystalline polymers with varying mesogen length, Rheol. Acta 40 (2001) 416–425. [6] S.F. Rubin, R.M. Kannan, J.A. Kornfield, C. Boeffel, Effect of mesophase order and molecular weight on the dynamics of nematic and smectic sidegroup liquid-crystalline polymers, Macromolecules 28 (1995) 3521–3530. [7] S. Hess, P. Ilg, On the theory of the shear-induced isotropic-to-nematic phase transition of side chain liquid-crystalline polymers, Rheol. Acta 44 (2005) 465–477. [8] S. Hess, Pre- and post-transitional behavior of the flow alignment and flow–induced phase transition in liquid crystals, Z. Naturforsch. 31a (1976) 1507–1513. [9] S. Hess, Fokker–Planck-equation approach to flow alignment in liquid crystals, Z. Naturforsch. 31a (1976) 1034–1037. [10] M. Doi, Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases, J. Polym. Sci., Polym. Phys. Ed. 19 (1981) 229–243. [11] A.D. Rey, Macroscopic theory of orientation transitions in the extensional flow of side-chain nematic polymers, Rheol. Acta 34 (1995) 119–131. [12] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, UK, 1999. [13] S. Hess, Irreversible thermodynamics of non-equilibrium alignment phenomena in molecular liquids and in liquid crystals, I. Derivation of nonlinear constitutive laws, relaxation of the alignment, phase transition, Z. Naturforsch. 30a (1975) 728. [14] C.P. Borgmeyer, S. Hess, Unified description of the flow alignment and viscosity in the isotropic and nematic phases of liquid crystals, J. NonEquilib. Thermodyn. 20 (1995) 359. [15] S. Hess, I. Pardowitz, On the unified theory for non-equilibrium phenomena in the isotropic and nematic phases of a liquid crystal, spatially inhomogeneous alignment, Z. Naturforsch. 36a (1981) 554–558.
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