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Macroscopic dynamics of the isotropic-nematic transition in polymeric liquid crystals
Helmut Brand, Kyozi Kawasaki
To cite this version:
Helmut Brand, Kyozi Kawasaki. Macroscopic dynamics of the isotropic-nematic transition in polymeric liquid crystals. Journal de Physique II, EDP Sciences, 1992, 2 (10), pp.1789-1795.
�10.1051/jp2:1992234�. �jpa-00247766�
Classification
Physics Abstracts 6470M-0570-6130
Short Communication
Macroscopic dynamics of the isotropic-nematic transition in
polymeric liqmd crystals
Helmut R. Brand (~) and Kyozi Kawasaki (~)
(~) FB 7, Physik, Universitit Essen, D 4300 Essen 1, Germany and
Department of Electrical Engineering, Kyushu Institute of Technology Tobata, Kitakyushu, 804, Japan
(~) Department of Physics, Kyushu University, Fukuoka, 812, Japan and IFF, KFA J61ich, Postfach 1913, D5170 Jfilich, Germany
(Received 22 July 1992, accepted 31 August 1992)
Abstract. We present the macroscopic dynamics near the isotropic to nematic transition in liquid crystalline side chain polymers. We suggest experiments to observe the transient
polymeric network via a qualitative change in the dynamic behaviour of the birefringence above the isotropic to nematic transition.
1 Introduction.
Since thermotropic liquid crystalline side-chain polymers have been synthesized first [Ii in 1978, this field has been growing rapidly (compare e-g- Refs.[2, 3] for reviews). A particularly inter-
esting sub-field in recent years has been the domain of liquid crystalline elastomers, materials that combine the properties of liquid crystals and of elastomers [4-7]. The static properties of these materials are strongly influenced by the coupling between the nematic order parameter and the strain field, in particular in the vicinity of the nematic-isotropic transition [8-12]. The
effects encountered due to this coupling are large, because the associated term in the general-
ized energy is linear in the nematic order parameter and in the strain. More recently it has become clear [13-16], that liquid crystalline elastomers, especially cholesteric and chiral smectic elastomers, show interesting electromechanical properties including piezoelectricity.
Based on our previous experience with the physical properties of liquid crystalline elastomers,
we address here a related, but slightly different question: we consider the macroscopic dynamics of the isotropic-nematic transition in liquid crystalline polymers. The macroscopic dynamics of usual polymer melts using the strain as macroscopic variable has been given recently iii]
and it has been pointed out that for frequencies above a typical Maxwell frequency shear waves should propagate, whereas for lower frequencies only vorticity diffusion as a in a simple liquid
1790 JOURNAL DE PHYSIQUE II N°10
is expected. These predictions are based on the picture of a transient network, which gives
the polymer a finite shear modulus for frequencies higher than
a typical Maxwell frequency.
Recently this approach has been generalized to nematic polymers [18].
The key question we want to address in the present note is the following. Which parts of the novel static effects due to the coupling between the network and the mesogenic units in permanently cross-linked elastomers can be expected to occur in liquid crystalline polymers for
frequencies above the Maxwell frequency in the vicinity of the isotropic-nematic transition? In addition
we will ask whether there are dynamic cross-couplings, which might influence both, the dynamics of liquid crystalline polymers and elastomers.
In the following we will present the macroscopic dynamic equations applicable to the vicin-
ity of the nematic-isotropic transition in liquid crystalline side-chain polymers. Aside from the strain field, which was recognized as an important dynamic variable for sufficiently high frequencies above the Maxwell frequency before [Iii, we have in the vicinity of the isotropic-
nematic transition jig, 20] the nematic order parameter Q;j, a symmetric traceless tensor, as an additional macroscopic variable. We will give the complete set of linearized equations and then discuss in
a separate section how the cross-coupling between backbone and side-chain could influence significantly experimental results in the vicinity of this transition. In addition to the static coupling already mentioned, we find that there is also a dissipative dynamic cross- coupling between motions of the backbone and the dynamics of the nematic order parameter.
No information about the strength of this novel dissipative cross-coupling is available at this time. From thermodynamic considerations we can only give an upper bound for its magni-
tude. It turns out that the method of choice to investigate this phase transition as a function of frequency is sound propagation of longitudinal and transverse sound. Quite recently there
has been a first experiment on the isotropic nematic phase transition in liquid crystalline side-chain polymers using ultrasound of a fixed frequency [21].
2. Linear macroscopic equations.
Generalizing our earlier result on polymers iii], to the vicinity of the nematic isotropic tran-
sition, we have for the Gibbs relation, the local formulation of the first law of thermodynamics
Tda=de-~dp-v.dg-E.dP-4l;jde;j-S;jdo;j (1)
where we have as macroscopic variables the entropy density a, the energy density e, the density
p, the density of linear momentum g, the polarization density P, the strain tensor e;j and,
as a new macroscopic variable close to the nematic-isotropic transition, the order parameter
Q;j. The thermodynamic conjugate quantities are defined via equation ii) and correspond to
temperature T, chemical potential ~, velocity v, the electric field E and the conjugates to the strain and the nematic order parameter, 4l;j and S;j, respectively. Since we are dealing with a local formulation, all quantities in equation (I) are fields.
To keep things as simple as possible, we will focus throughout the present note on linearized
equations. For the generalized energy we have (E = J dve)
E # Eo +
j
dV[)C;jklfijEkl + iii(X~~P + X~~°) + (ijklfiji7kPI + )XijPi I
+ )AQ;jQ;j + (~ v;Qj~v;Qj~ + (2 v;Q;~vjoj~ (2)
+ uQ;je;j jeaE;EjQ;j + ((6;~6ji + 6;16j~)Eiv~o;ji
where A
= a(T T*). Eo contains all the terms present in a simple liquid, c;jki is Hooke's tensor of elasticity and contains two independent elastic moduli in an isotropic medium, xP and
x~ are static susceptibilities, (;jki is of the same structure as the elastic tensor in an isotropic
medium [22], and x;j is the electric susceptibility. All the terms containing these coefficients have already been discussed in reference [Iii. Then we have in addition, and most importantly
in the present context, the terms containing the nematic order parameter. The contributions
involving only Q;j, its gradients and the electric field, are isomorphic to those at the nematic to isotropic transition in low molecular weight materials [19, 20, 23]. The most important cross-coupling term, which will be of interest in the following, is the contribution
oc lf, which is linear in the strain and in the nematic order parameter. The relevance of this contribution for
elastomers, which
are crosslinked permanently and have a finite static shear modulus, has been studied in detail experimentally and theoretically for static situations [8-12]. As it will become clear in the following, this term has also important consequences for the dynamic behaviour of
liquid crystalline polymers for frequencies above the smallest Maxwell frequency.
The thermodynamic forces introduced via equation (I) can be obtained from the energy
(Eq.(2)) by taking variational derivatives with respect to one variable while keeping all other variables fixed. We find for example for S;j, the thermodynamic conjugate to the order pa- rameter Q;j
~~ ~~j ~~~ ~~~~~ ~ ~~~~ ~~~~~
(~) ((6;k6Ji + 6;i6jk)v~Ei Liv~v~o;J ) v~(v;Qkj
+ v~ok;)
where the indicate that all other variables are kept at a fixed value while the variational derivative is taken with respect to Q;j.
For the balance equations for the macroscopic variables
we have
p+V;g;=0 j; + Via;j # 0
b + V;jf = R/T (4)
I;j + X;j # 0
fi;+J;~=0
Q;j+(j=0
where a;j is the stress tensor, jf the entropy current, R/T, the source term in the equation for the entropy density, the entropy production, Jf the polarization current, and X;j and l§j
are the quasi-currents for the strain tensor and the order parameter, respectively.
To close the system of macroscopic equations [24, 25] we have to expand the currents and
quasi-currents in equations (4) into the thermodynamic forces obtained by taking variational derivatives of equation (2). This procedure is split into two parts. First we give the reversible parts of the currents and quasi-currents, which lead to no entropy production (R % 0). We
obtain
gt
= Poui
°~ " P~ij 4~ij ~ijklskl
1792 JOURNAL DE PHYSIQUE II N°10
jf~
# aoU;
Jf~
= ° (5)
~R ~
ij ij
(~
" ~ijkiAki
where po and ao are the equilibrium values of density and entropy density, p is the hydrostatic
pressure and, A;j is the symmetric velocity gradient A;j = )(V;uj + Vjn;). The tensor ~;jki takes the form ~;jki " ~(6;k6ji + 6;16jk). It is of the same form as for the nematic-isotropic
transition in low molecular weight compounds [19, 20] and it is the analogue of the flow
alignment term in the nematic phase [25].
To derive the dissipative parts of the currents and quasi-currents in equations (4), we expand
the dissipation function R into the thermodynamic forces and then obtain the dissipative
currents by taking variational derivatives with respect to the forces. We have for R:
R = Ro +
f
dV[4l;j4l;j + a~E;E; + US;
jS;j + ~l4l;jS;j
2T 2 2 (6)
+ ~I~S;jV;Ej + (Vj4l;j )((°~V;T + (~E;)]
where Ro contains all the contributions already present in a simple fluid [2]. In equation (6) T is the relaxation constant of the network (we have discarded the self-diffusion of the strain field in comparison with relaxation, compare reference [Iii for a detailed discussion),
a~ is the electric conductivity and v the relaxation constant of the order parameter. ~l is
a dissipative cross-coupling between the order parameter and the transient network and can
be regarded as the dissipative dynamic analogue of the static cross-coupling lf, coupling e;j
and Q;j. While nothing is known about the value of this cross-coupling term, that is neither
magnitude nor sign, one can nevertheless give an upper bound which follows from the condition that the dissipation function must be positive: [~l[~ < v/T. The contributions oc (~ and (~ are dynamic cross-coupling terms between the transient network and electric fields or temperature gradients, respectively, which exist already in usual polymer melts iii]. ~l~ finally describes
a dissipative cross-coupling between the order parameter and inhomogeneous electric fields, a
term whose existence has already been discussed close to the nematic to isotropic transition in low molecular weight materials [26].
We obtain for example for the quasi-current ():
if =
~~
[.. = US;> + ~l4l;j + ~~V;Ej (7)
6Sij
where the indicate, that all other thermodynamic forces have been kept fixed while taking
the variational derivative with respect to S;j. All other dissipative currents and quasi-currents
can be obtained from equation (6) by the same procedure,
We close this section by pointing out that nonlinear terms can be incorporated on a case to case basis as needed into the statics and into the dynamics along the lines indicated in references [13, 27]. They are crucial in obtaining non mean-field values for dynamic exponents as they
have been observed recently in different types of side-chain polymers [21, 28]. A mode-coupling
calculation undertaken to get these dynamic exponents deviating from mean-field values will be described elsewhere [29].
3 Experimental considerations.
In the following we will describe two possible dynamic experiments in the vicinity of nematic to
isotropic transitions, which will give a different result, depending on whether they are performed
in a low molecular weight material or a polymeric material below the Maxwell frequency on
one hand or in a polymeric material above the Maxwell frequency, on the other hand. Firstly
we note that in the linear regime we do not expect any qualitative difference in the behaviour of low molecular weight materials and in polymeric materials for frequencies less than the
relaxation frequency of the transient network. Therefore both experiments we will discuss in
the following can be done on polymeric materials by simply varying the frequency from a value below to one above the Maxwell frequency.
Looking through the dynamic phenomena studied for low molecular weight liquid crys- tals, the case of flow birefringence seems especially well suited for a comparison between low frequency behaviour and frequencies above the Maxwell frequency. We start by considering stationary situations (alai % 0) using ordinary longitudinal sound as it exists in all condensed systems. In this case one finds for low molecular weight materials or for liquid crystalline polymers at sufficiently small frequencies for the flow-induced order parameter
Qij " -())Aij (8)
whereas one has for polymers above the Maxwell frequency
Q>j " -( ~~(~~~~
)A>J (9)
with C
= 2c2 for shear (in the notation of Ref.[li]). Equations (8) and (9) give one immediate possibility to study the influence of the transient network. Comparing equations (8) and (9)
one finds a variation in the degree of induced nematic order and thus in birefringence, if one monitors the changes in the refractive index An as a function of frequency for frequencies
from below to above the Maxwell frequency. In writing down equation (9) we have discarded for simplicity contributions involving the dynamic dissipative cross-coupling ~l, which would
make the result even more involved, as one can see when doing the calculations using equations (1-7).
A more drastic effect can be expected if one is performing an experiment using shear waves incident on a sample of a liquid crystalline polymer in the isotropic phase close to the nematic-
isotropic transition. Below the Maxwell frequency polymers do not support shear waves, but
only vorticity diffusion is possible. This changes above the Maxwell frequency, where the
polymer starts to support propagating shear waves due to the presence of the transient network.
This transition could now be studied very conveniently optically by monitoring the changes in the index of refraction close to the isotropic to nematic transition. Depending on the speed
of the shear waves, a continuous wave (CW) mode or the propagation of pulses could be the
preferred mode of operation.
For frequencies for which only vorticity diffusion is possible, the information about nematic order carried by An is decaying away from the CW source and the distance Ax over which
nematic order spreads grows diffusively: < (Ax)~ >oc i. If pulses are used, they only widen
diffusively as a function of time, but do not propagate.
This situation changes qualitatively if one goes to frequencies that support shear waves. In this case a pulse does not only become wider as a function of time due to diffusion, but in
1794 JOURNAL DE PHYSIQUE II N°10
addition it propagates away from the location where it was imposed. If the speed of shear wave
propagation is sufficiently high, one might perform an especially simple CW experiment: before the sound wave is switched on, the sample is in the isotropic phase and thus transparent, after
switching on the shear wave generator, the sample becomes turbid, since the sound induces a transition to the nematic state, if the sample is sufficiently close to the phase transition or the
intensity of the shear wave is sufficiently high.
We note, that instead of varying the frequency to get the phenomena outlined above, one
could alternatively study mixtures of a low molecular weight liquid crystal with a polymeric liquid crystal for fixed external frequency. By varying the concentrations of the mixtures one
can then get both cases, namely mixtures with a Maxwell frequency higher than the applied frequency as well as mixtures with a Maxwell frequency lower than the applied frequency.
Thus we predict that one can detect the transition from vorticity diffusion to propagating
shear waves in
a liquid crystalline side-chain polymer in a straightforward way optically in the
vicinity of the nematic-isotropic transition.
In closing we point out that the considerations given in this note do not only apply to the
isotropic-uniaxial nematic transition, but also hold for the isotropic-biaxial nematic transition, which has been found to occur for side-on liquid crystalline side-chain polymers in a number of compounds recently [30-32].
Acknowledgements.
KK thanks Heiner Mfiller-Krumbhaar annd Walter Zimmermann for their hospitality in Jfilich, where this paper was completed. Support of HRB'S work by the Deutsche Forschungsgemein-
schaft is gratefully acknowledged. He also thanks the Kyushu Institute of Technology for partial support through the award of the KIT fellowship 1991. KK thanks the Humboldt Foundation for the award of a Humboldt Research Prize that made his stay in Jiilich possible.
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