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Thermal instabilities in long pitch cholesteric liquid crystals
C. Fraser
To cite this version:
C. Fraser. Thermal instabilities in long pitch cholesteric liquid crystals. Journal de Physique, 1984, 45 (6), pp.973-980. �10.1051/jphys:01984004506097300�. �jpa-00209841�
973
Thermal instabilities in long pitch cholesteric liquid crystals
C. Fraser
Department of Mathematics and Computer Studies,
Dundee College of Technology, Bell Street, Dundee DD1 1HG, U.K.
(Reçu le 21 juin 1983, révisé le 24 janvier 1984, accepte le 17 fevrier 1984)
Résumé. 2014 Nous étudions l’apparition d’une convection thermique de type cellulaire dans un cristal liquide cholestérique à long pas soumis à un gradient de température vertical, appliqué entre deux plaques horizontales et
infinies. L’alignement initial considéré correspond à la configuration hélicoïdale normale où l’axe de l’hélice est
perpendiculaire à la direction des plaques. Le gradient de température critique est déterminé numériquement en
fonction du pas du cholestérique et de la torsion initiale que nous faisons subir à l’échantillon. Nos valeurs calculées devraient permettre une comparaison utile entre théorie et expériences.
Abstract. 2014 This paper uses continuum theory to investigate the onset of a particular roll-type thermal convection when a long pitch cholesteric liquid crystal confined between two infinite horizontal flat plates is subjected to a
vertical temperature gradient. The initial alignment considered is the characteristic helical configuration with
the axis of the helix perpendicular to the plane of the plates. Employing a numerical procedure to determine the temperature gradient at which convection occurs, the variation of this critical value with the pitch of the cho- lesteric and the initial twist imposed on the sample are investigated Our predictions appear to provide a useful opportunity for comparison between theory and experiment.
LE JOURNAL DE
PHYSIQUE
J. Physique 45 (1984) 973-980 JUIN 1984,
Classification
Physics Abstracts 47.25Qi - 61.30
1. Introduction.
During the last ten years there has been considerable
activity in the study of thermally induced flow phe-
nomena in liquid crystals, mainly on account of the
novel features peculiar to these anisotropic fluids.
This interest began with certain qualitative predictions by Dubois-Violette [1] and Currie [2] based on conti-
nuum theory. In particular, they predicted the onset
of such convection at much lower temperature dif- ferences than for a comparable isotropic liquid with
the same gap width, and also the possibility that one
could induce motion by heating from above. Subse- quent experimental investigations soon confirmed
these predictions, and this has led to several more
detailed theoretical studies of thermal instabilities in nematic liquid crystals (references are available in
the review by Leslie [3]). More recently, the subject
has received further stimulus from the study of oscil- latory instabilities (see, for example, Guyon et al. [4]).
By way of contrast, rather less consideration appears to have been given in the same period to similar phenomena in cholesteric liquid crystals. In some
ways this is rather surprising, since cholesteric liquid
crystals in several respects seem to offer more scope for novel thermal effects on account of the greater
degree of thermomechanical coupling in the conti-
nuum theory. Indeed, the first observations of ther-
mally induced motions in liquid crystals were by
Lehmann [5] some eighty years ago in experiments
with cholesteric liquid crystals, an explanation of
which was given by Leslie [6] in his first application
of continuum theory for cholesteric liquid crystals.
Apart from a further paper on this topic by Leslie [7]
and approximate investigations of B6nard convection
in cholesterics with relatively small pitch by Dubois-
Violette [8], Parsons [9] and Pleiner and Brand [10],
very little more appears to have been attempted either
theoretically or experimentally on this topic for this
class of liquid crystal. Two factors peculiar to cho-
lesteric continuum theory certainly inhibit further theoretical progress. One is the more complex orienta-
tional configurations inherent in such materials due to their intrinsic helical arrangements, and the other is the aforementioned thermomechanical coupling
included in the theory which also leads to more
complex calculations.
In a recent paper Fraser [ 11 considers a problem
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004506097300
974
involving thermal convection which evades some of the complications just mentioned He considers the
special case of cholesterics with long pitch uniformly aligned between parallel horizontal plates and exa-
mines the extent to which weak chirality influences
the corresponding results for nematic liquid crystals.
Such uniform alignments of cholesteric liquid crystals
may occur naturally between parallel plates when the pitch of the sample is sufficiently long or the gap width is sufficiently small [12]. The present paper
develops this study further and investigates the more
difficult problem when a temperature gradient is applied across a strongly anchored cholesteric exhi-
biting the characteristic twisted or helical configu-
ration between parallel horizontal plates, the axis
of the helix being vertical. In this study, like Dubois- Violette [8], we make some simplifying assumptions
in the theory proposed by Leslie [3], principally choosing to ignore certain of the thermomechanical
coupling terms. One reason for doing this is the
absence of experimental evidence and data regarding
these coupling terms, and another is the arguments
by Prost [13] that they are small. The present investi- gation generally differs from those by Dubois-Vio- lette [8], Parsons [9] and Pleiner and Brand [10] in
that we consider the situation in which the cholesteric
pitch is of the same order as the gap width, whereas they discuss cases in which the pitch is small compared
with the gap width. However, there are other dif- ferences which we return to later.
For a given cholesteric liquid crystal we select particular uniformly twisted orientation patterns as possible equilibrium solutions of the field equations employing an energy argument This essentially
restricts the range of values of initial twist imposed
on the sample on stability grounds. The governing equations are then linearized about this equilibrium
state for a class of roll-type perturbations, and solved numerically to determine the critical threshold gra- dient for certain permissible combinations of values for the pitch and imposed twist In contrast to the
corresponding results for twisted nematic liquid crystals [14], the introduction of additional twist can
lead to significant variations in the temperature difference necessary for convection. However, a some- what surprising further result is the following. For
certain long pitch cholesterics, the minimum tem- perature difference occurs when the imposed twist is greatest, relative to the sample’s natural twist, but as
the chirality increases there is a transition to the other extreme in which the minimum occurs when the
imposed twist is least Intuitively one would expect the imposition of twist relative to the natural pitch
of the cholesteric to result in physically less stable
states, the effect being greatest at the most extreme values for the imposed twist However, there appears to be no obvious physical explanation for the tran-
sition from greatest to least imposed twist leading to
the minimum absolute temperature difference. These
predictions clearly offer the possibility of some interest- ing comparisons between theory and experiment.
Of particular interest, experimental confirmation of the above would lend support to the view that the thermomechanical coupling terms are insignificant
in this problem, whereas disagreement could indicate
a need for some reassessment of the importance of
these terms in the present context, or possibly even
some re-appraisal of the theory itself. Alternatively, however, experimental evidence could conceivably
show that our choice of a roll-type instability moti-
vated by nematic studies is inappropriate.
2. Formulation of the problem,
In this paper we assume that the equations governing
the behaviour of an incompressible cholesteric liquid crystal are those proposed by Leslie [3], and proceed
to examine the stability of a cholesteric sample
confined between two horizontal infinite flat plates
in the presence of a vertical temperature gradient,
the upper and lower surfaces being held at constant
temperatures TU and TL respectively. For convenience
we choose Cartesian co-ordinates so that the upper
plate occupies the plane z = h and the lower plate
z = 0, the z-axis being directed vertically upwards.
Also, the x-axis is chosen to coincide with the ali- gnment at the lower plate. Initially one looks for
static solutions of the form
where n is the unit vector specifying the alignment of
the liquid crystal and T is the temperature. In this
event the governing equations yield
where 00 denotes the total twist in the sample, p is the pressure, po an arbitrary constant, p the density and g
the acceleration due to gravity, this assuming the forms
of body force and body couple proposed by Ericksen [15]. The scalar function y is the Lagrange multiplier arising from the constraint that n is a unit vector, and K2 and T are material parameters, the latter being
related to pitch P of the cholesteric sample by
Following the energy argument of Dafermos [16],
the above solution appears more likely to occur in preference to others exhibiting more or less twist
between the plates, provided
ir qr
which restricts the imposed twist 00 in relation to the
sample thickness for a given cholesteric. The ensuing analysis assumes that the above inequality is satished If the equilibrium state is slightly disturbed, we consider a small amplitude velocity v, and associated perturbed director n + d, a temperature T + s, a
pressure p + p and a director tension y + y. Employ-
ing the usual Boussinesq approximation [17] in which
all variations of the material parameters with tempe-
rature except where associated with gravity are ignored, linearization of the governing equations
about the equilibrium state results in the system
where C(3’ Q, K3, Yl, Ki, rc2 are further material parameters, C is the specific heat at constant volume and k
is the unit vector in the z-direction. Appendix I lists the coefficients Aijk’ Bijkm5 Cijk’ Dijkm, Eijk’ Fijk5 Ci jkm and Hijk which are functions of the director n and its gradients. For the reasons given earlier the thermomechanical
coupling parameters a, C(7’ a8, y3, K3 and x4 in Leslie’s original equations [7] have been set equal to zero.
An excellent description of the physical mechanism leading to convective instabilities in liquid crystals is given by Dubois-Violette [8, 18], and it does not seem possible to gain further insight into the mechanisms for this problem, given the mathematical complexity of the equations employed Consequently, we proceed to a
detailed mathematical analysis of the above linear equations (2.6)-(2.9) which govern the instability. This paper considers roll-type perturbations with a direction of roll-axis arbitrary in the (x, y) plane and takes disturbances of the form
where vi, di, s, j5 and y are functions of z alone. Since static rigid boundaries maintained at steady temperatures confine the sample, the usual boundary conditions pertain for the velocity, director and temperature fields,
so that the perturbations satisfy
Our aim is to find the smallest absolute value of the temperature gradient 0 at which these disturbances become unstable for all values of the wavenumbers I and m. In line with other similar studies we adopt the prin- ciple of exchange of stabilities and assume that for each value of 1 and m the onset of instability is characterized
by w being identically zero. To facilitate the solution of the linearized system of equations, it is convenient to follow Brochard [19] and Barratt and Sloan [14] and make the change of dependent variables
Straightforwardly equation (2. 6)2 shows that di is identically zero. Elimination of p and y and the substitutions
976
eventually yield the linear system
where the prime denotes dldç. The coefficients appearing in equations (2.18)-(2.23) are defined in Appendix II.
At this stage it is interesting to note that the further
transformation of variables
reveals that for fixed values of 00 and hi the product /Jh4 is a universal function of the non-dimensional wavenumbers a and b. This conclusion that /Jh4 is a
universal function essentially agrees with that of Dubois-Violette [8] for the reasons given by Fraser [11].
However, the substitutions (2.24) appear to accelerate the rate at which computational difficulties occur, and so we do not incorporate these additional trans- formations into the formulation of the problem.
3. Numerical solution.
The structure of the system of equations (2.18)-(2.23)
appears to be too complex to employ the Fourier
series method of solution used in a number of similar Benard convection problems, in particular the ana- lysis of uniformly aligned cholesterics [11]. Conse- quently, we follow Barratt and Sloan [14] and refor-
mulate the problem as a set of first order linear diffe- rential equations with variable coefficients. However, the direct and natural formulation as a set of twelve first order equations appears to give rise to singula-
rities in certain coefficients and, in order to describe
these difficulties and clarify the way in which they
were overcome, the prescription as a system of first order equations is presented in some detail. Firstly, equations (2.20)-(2.23) are readily written as a set of
first order equations, and in a relatively straight-
forward manner equation (2.23) and its derivative eliminate v3 and v3 from equation (2.19). The pair of
simultaneous equations (2.18) and (2.19) can then
be regarded as a fifth order system for u and v. In one approach u"’ can then be eliminated by subtracting
the product of Q 1 with the derivative of equation (2.18)
from the product of P 1 with equation (2.19). Equation (2.18) together with this modified version of (2.19)
then constitute a set of equations for u" and v"’ res-
pectively which can be formulated as a system of first order equations for u, u’, v, v’ and v". However, this
procedure entails using equation (2.18) both as a
first order equation for u’ and also to eliminate u"
from the modified version of equation (2.19), and
each role results in terms which involve quotients of Pl, a coefficient which vanishes within the range of values of a, b and 0 considered in this paper. An alternative approach involves a similar use of the
derivative of equation (2.18) to eliminate v"’ from
equation (2.19), but using equation (2.18) as a first
order equation for v’ and to eliminate v" from this
modified version of equation (2.19) subsequently
results in quotients of P2 which can again vanish for
certain values of a, b and 0. Finally, a judicious com-
bination of both of the above methods over regions in
which each is valid appears infeasible as it would add
considerably to the computational difficulties.
To overcome this problem of singularities in
coefficients we retain both versions of equation (2.19)
obtained by eliminating u"’ and v"’ in turn with the
aid of the derivative of equation (2.18), but not includ- ing the subsequent substitutions for u" or v" using equation (2.18). The resulting two equations, together
with equations (2.20)-(2.23) are then reformulated
as a set of thirteen first order equations of the form
where L is a (13 x 13) matrix and f is a thirteen
component vector function with components [u, v, V3’ d 1.’ d3, S, u’, u", v’, v", dl, d3, s’]. Twelve relevant
boundary conditions are readily derived from equa- tions (2 .14) as
together with the further condition
obtained from equations (2.18) and (3.2). Clearly, it
suffices to use the latter condition at one end only
and it proves convenient in the numerical method
employed to use it at the upper plate, i.e. ç = 1.
For details of the procedure to integrate this system of equations for specific values of a, b, 00 and hr we
refer the reader to Barratt and Sloan [14].
To obtain quantitative results for the above for- mulation it is necessary to prescribe values for a, b, 90
and h as well as the various material parameters arising in the theory. For the numerical integration
we assume that the sample thickness h is 1 mm and for
a specified value of the initial twist 00 relevant values for the material parameter i are restricted by equa- tion (2. 5). In the absence of cholesteric data for all of the remaining parameters, and since cholesterics with sufficiently long pitch are usually produced by incorporating optically active additives in a sample
of nematic liquid crystal, we follow other similar studies [8, 11] and adopt the available mechanical data for a typical nematic liquid crystal, namely
MBBA. Thus the viscosities are those given by Gahwil-
ler [20], thermal conductivities and elastic constants as given by Dubois-Violette [18] and Haller [21],
and we take p’ g = - 1 g cm - 2 s - 2 OC-’. For these fixed values of the material parameters, one specifies
values for Bo, r, a and b and then proceeds to find the
value of 0 with smallest modulus for which a non-
trivial solution of equation (3.1) subject to boundary
conditions (3.2) and (3.3) exists. As a and b vary 0
traces out a surface in the (a, b, 0) Cartesian co-
ordinate system, and the required threshold gradient 0
is that value having smallest modulus on this surface.
Finally, we note that the numerical results indicate that, for fixed values of 00 and hr, the threshold gradient 0 possesses symmetry about the origin in the (a, b) plane, i.e.
this result being confirmed by appropriate changes
of sign in the variables v3’ sand d1. in equations (2.18)- (2.23). Consequently, the search for the smallest absolute value of 0 can be confined to values of a and b in the upper half of the (a, b) plane.
Before listing the numerical results we note that the two computational problems listed by Barratt
and Sloan [14] were again encountered in this similar
analysis. Firstly, it was necessary to use Crout decom-
position with partial pivoting to avoid the growth of rounding errors in the evaluation of the (7 x 7)
determinant derived from the boundary conditions at ç = 1. Secondly, when integrating the system of first-order differential equations to obtain seven linearly independent solutions, the solution set was
reorthonormalized when the element of largest modu-
lus in the seven vectors exceeded a prescribed amount
to prevent the introduction of a numerical linear
dependence in the solutions. As remarked earlier, this
latter problem becomes more severe when the depen-
dence of the original system of perturbation equations
on the gap width h is reduced.
4. Numerical results and comments.
Table I displays the computed values for the threshold
gradient 0 and the associated values of the non-
dimensional wavenumbers a and b for various values of hi and 8o when the sample thickness is 1 mm.
Table I. - Threshold gradient 0 °C cm-1 and the
associated values of a and b for some values ofhr and 00
when h = 1 mm.
Threshold gradients for other gap widths can be obtained from the fact that /Jh4 is a universal function of hT and 90. The neutral stability surfaces in the
(a, b, 0) co-ordinate system are very flat near the minima and therefore the values of a and b may be in error in the least significant digit. The results for r
zero are in agreement with those of Dubois-Violette
[18] and Barratt and Sloan [14] for a nematic liquid crystal, and the results for 00 zero agree with Fraser [11] ]
for a uniformly aligned planar sample of cholesteric
liquid crystal. From equations (2.18)-(2. 23) it follows that, for negative values of 6a and hr in the range of the constraint (2.5), the threshold gradient 0 and its
location in the (a, b) plane can be derived from the relationship
Figure 1 also illustrates graphically the variation of 0
with 9o and hr. For each value of the inherent twist hi
we plot the minimum value of 10 over the applied
twists 00 which satisfy condition (2.5). The figure also
indicates the value of (90 - hr) for this threshold gradient of minimum modulus.
It is of interest to note several aspects of the com-
puted results, confining attention to non-negative
values of his as symmetry arguments can be used to derive similar results for negative values of this parameter. For the values of MBBA adopted in this
paper the coefficient x2 is always negative, and in all
cases considered the instability occurs when heating
is from below, this being in agreement with the results of Dubois-Violette [8]. However, it is interesting to
note that Dubois-Violette and Parsons [9] indicate
that convective instabilities will occur when the
sample is heated from above if the anisotropic thermal conductivity x2 is positive. For the listed values of
pitch P( = 2 7E/T) and applied twist 80, table I and
