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DYNAMICS OF THE ISOTROPIC-CHOLESTERIC TRANSITION : DIFFERENCES BETWEEN THE
NORMAL AND BLUE CHOLESTERIC PHASES
P. Keyes, C. Yang
To cite this version:
P. Keyes, C. Yang. DYNAMICS OF THE ISOTROPIC-CHOLESTERIC TRANSITION : DIFFER- ENCES BETWEEN THE NORMAL AND BLUE CHOLESTERIC PHASES. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-376-C3-379. �10.1051/jphyscol:1979374�. �jpa-00218770�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 4, Tome 40, Avril1979, page C3-376
DYNAMICS OF THE ISOTROPIC-CHOLESTERIC TRANSITION
:DIFFERENCES BETWEEN THE NORMAL AND BLUE CHOLESTERIC PHASES
P. H. KEYESUniversity of Massachusetts at Boston, Boston, MA 02125, U.S.A.
C. C. YANG
Xerox research center of Canada, Mississauga, Ontario, Canada
RBsumk. - Nos experiences de diffusion de lumihre et nos mesures de viscositk nous renseignent sur la dynamique de la transition entre la phase isotope et la phase cholestkrique. On trouve que la relaxation des fluctuations du param&tre d'ordre ne peut se dkcrire simplement avec un seul taux de dbcroissance. On propose une explication thkorique qui devrait aussi s'appliquer aux nkmatiques.
On precise ainsi les differences entre les liquides donnant lieu a une phase cholestkrique normale et ceux qui forment une phase bleue.
Abstract. - We report on the dynamics of the transition from the isotropic to the cholesteric phase as revealed through light scattering and viscometry. We find that the order parameter fluctua- tions fail to relax with a single decay rate. A theoretical explanation is proposed which should also apply to nematics. The differences between those liquids which form a normal cholesteric and those which form a blue phase are emphasized throughout.
The cholesteric blue phase remains to this day one of the most poorly understood mesophases. Its appearance is radically different from the normal cholesteric, one being transparent and the other turbid. And yet, the microscopic organization of the two types of cholesterics must not be very different since the phase transition from one type to the other is accompanied by very small latent heat and volume change [l]. In this article we will show that the two types of cholesterics also exhibit significant differences in their respective isotropic phases in the temperature regime where pretransitional effects are evident.
For the purposes of this discussion we will choose as prototypes the two liquid crystals which we have studied the most thoroughly : cholesteryl2-(2-ethoxy- ethoxy) ethyl carbonate (CEEEC) which forms a normal cholesteric [2] and cholesteryl olyel carbo- nate (COC) which, when cooled from the isotropic phase, forms a blue phase [3] (upon further cooling the blue phase is converted to a normal cholesteric).
Attention has already been drawn to the fact that the isotropic-blue phase transition leads to a logarith- mic divergence of the shear viscosity y [4]. It has also been noted that the temperature dependence of the linewidth
r
at this transition is decidedly nonlinear [3].We now believe that this latter effect is related to an anomaly in the viscosity coefficient v similar to the anomaly in y. The relationship between
r
and v hasbeen given by de Gennes [5] as
r
= Alv with A = Ao(T - T*)and A, constant. We expect this equation to be valid for vertical polarization of the incident and scattered beams and at small angles where q dependent corrections are unimportant. Therefore, we use mea- surements of
rvv
taken at a small angle (6 = 28O).Using the value T* already obtained from the inten- sity measurements [3] we can calculate v to within an undetermined multiplicative constant :
V = Ao(T - T*)/Tv, = A, V ' ,
which defines v'. The resulting values of v' are plotted in figure 1 together with the previously reported data
I O O O / T (K-9
FIG. 1. - The vlscoslty coefficients and v vs 1 000/T.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979374
DYNAMICS OF ISOTROPIC-CHOLESTERIC TRANSITION C3-377
for
v.
These results strongly suggest that there is an anomalous decrease in v' which goes hand in hand with the increase in y. Extracting the anomalous portion of v' according to the method of reference [2], we find that it also appears to have a logarithmic temperature dependence (Fig. 2).FIG. 2. - The anomalous portion I A V I of v vs T .
We now turn our attention to a most interesting feature of the transition revealed in the autocorrelation function (or spectrum) of the scattered light. Much to our surprise, we have found that the order parameter fluctuations fail to relax with a single decay rate. This effect is evident for the normal cholesteric only very near the transition temperature, whereas in the material which forms the blue phase it is found throughout the entire pretransitional region of the isotropic phase. We have extended the de Gennes theory [5] in a way which seems to account for this result in the case of the normal cholesteric. It will be seen that a similar effect, but not so large, is predicted for the nematic as well.
The starting point for the calculation is the set of equations advanced by de Gennes [5] relating the forces and fluxes appropriate to the pretransitional dynamics. There are two forces, the viscous stress tensor o,, and
where F is the free energy and Q,, the order parameter.
The fluxes conjugate to these forces are the shear rate tensor
e,, = davp
+ a,~,
(2)and the rate of the change of the order parameter
aQap
R,, = -
at
.
These forces and fluxes are coupled through the phe- nomenological equations :
and through Euler's equation
Finally, the free energy is given by
X aaQay apQay +L1 40 Eaby Qap a y Qpp (7) where L , and L, are constants and go is the wave number for the cholesteric pitch. For a nematic go should be set equal to zero. Following reference [5]
we omit from eq. (7) all terms higher than second order in Q.
Our calculation differs from that which has been presented previously only in that we have included the gradient terms in eq. (7). To perform the derivative indicated in eq. (1) we must recall that F is actually a free energy density and the variation in Q must be taken within the integral according to the methods of the calculus of variations. This has the effect of replacing the derivative in eq. (1) by the operation which we may write symbolicaly as :
[h]
= - -a
effective
a~
V .[&I
-By Fourier analyzing the fluctuations we may make the substitutions : a/at + io, d/dx + 0, d/ay + 0, 8/82 + iq, taking q to be in the z direction.
The dynamical eq. (1-6) may then be represented in the matrix notation :
P. H. KEYES A N D C. C. YANG
iw + a
+
l, + - l 2 12 - ic
1
Q") + id[:) = 0 (1 1)
ic iw
+
a+
l ,+;;;E2
Q,,- g ~ 2 Q , , - (iw
+
f)v, = 0 , a = x or y . (12)The following shorthand notation has been used
d = ~ q l v , g = 2 pqlv and f = qq2/p
.
It should be noted that only the component Q,, is a normal mode relaxing with a single decay rate.
Q, and Q,, are not normal because of the coupling through v. This point has already been discussed by de Gennes [5]. What appears as a new feature in our calculation is the coupling between Q,,, Q,,, and Q,, owing to the gradient terms. For the sake of conci- seness we will discuss the effect of this coupling on the power spectrum S(q, o ) of Q, only. This is the component selected by the vertical polarizations employed in our experiment.
S(q, w) may be calculated according to the method employed by de Gennes [5] : an external perturbation (rx, H, H,, 0,O) with X, = 112 A is applied to the right handside of eq. (9) and then the fluction-dissipa- tion theorem is employed to give
After considerable algebra we obtain
where
The correlation lengths
5,
and5 ,
have been defined through5:
= L,/v and5:
= L2/v.The spectrum is thus seen to consist of three Lorent- zians (if the autocorrelation function is measured instead, it should correspondingly consist of a sum of three exponentials). In the case of a nematic, 4, = 0, the second and third lines collapse into one. It is instructive to calculate the spectrum proceeding from the Laplace transform rather than from the Fourier
transform method. In this case it is seen [2] that the amplitude of the first lorentzian is proportional to ( Q ) and the amplitudes of the second and third are proportional to ( P2 ), where Q and P are, respecti- vely, the uniaxial and biaxial order parameters intro- duced by de Gennes [5]. The appearance of more than one linewidth is therefore seen to be a consequence of competing order parameter fluctuations [6].
Far from the transition, where the correlation lengths are small, the three linewidths will not differ appreciably from each other and the spectrum will appear to be a single Lorentzian with linewidth T.
Closer to the transition the splitting of the line into its components may become evident. The detection of all three lines would be considerably difficult, however, unless all three widths are vastly different. In particu- lar, it should be most difficult to resolve the two lines with widths Ti and T2, from each other. In CEEEC, as the experiment was performed, we had q = 2.4 q, resulting in extremely inauspicious conditions for the mutual resolution of these two lines.
It need not be so difficult to separate the line with width
r2,
from the other two, however. Indeed, the data for CEEEC clearly show the presence of at least two decay rates for temperatures sufficiently close to T*. In figure 3 we show on a semilogarithmic plot the autocorrelation function for the scattered intensity ( I(z) I(0) ) vs z for a temperature such thatThe background value ( I(z) I(0) ), z %- T has been subtracted away. At 'this temperature, although the data could be fit to a single line, the presence of a second decay rate is just barely discernible. The data in figure 4 taken at T - T* = 0.239 OC show quite clearly evidence for at least two decay rates. Fitting the data for large z to one line we can then peel away this exponential to deduce the decay rate of a second one. We find that the two linewidths differ by factor of 3.88. To estimate how this large difference could come about, we will assume that the two widths are given by T2, and T,, of eq. (14). It turns out that this would require a value of
5 ,
q, = 0.65. We recall that5 ,
q, = 0.5 is the value for which the Bragg fringes [2] first become evident and5,
q, = 1 is thevalue at which the isotropic phase becomes comple- tely unstable. These results are, therefore, quite reasonable.
DYNAMICS OF ISOTROPIC-CHOLESTERIC TRANSITION C3-379
FIG. 4. - The autocorrelation function
<
Z(T) Z(0) ) vs r forT - T* = 0.239 OC.
FIG. 3. -The autocorrelation function
<
I(?) Z(0)>
vs r forT - T* = 0.753 'C.
The resui~s for COC are clearly not in accord with the theory which has been presented here. For this liquid we find two linewidths which at 8 = 90" differ from each other by about a factor of five. This large splitting persists over a range of at least 3 OC above T*. In fact, the splitting is so large that in reference [3]
the lower line was not even noticed. Details of the temperature and angular dependence of these lines will be presented in the future.
It is not altogether surprising that the theory should fail in this case since it allows for but one type of
cholesteric to begin with. It has been shown by Brazovskii and Dmitriev [7] that when one relaxes some of the assumptions which have gone into the de Gennes theory, in particular the restriction to quadratic terms in the free energy, then three different types of cholesteric are permitted. It is not known which, if either, of these two new cholesterics might be the blue phase, nor have the dynamical equations been developed which would permit comparison to our experimental data. One encouraging sign, however, is that these new cholesterics are characterized as having short pitch, which is also the hall-mark of those cholesterics which form blue phases.
References
[l] ARMITAGE, D., PRICE, F. P., J. Physique Colloq. 36 (1975) [4] KEYES, P. H., AJGAONKAR. D. B., PI?I'.s. Lett. 64A (1977) 298.
Cl-133 ; J. Appl. Phys. 47 (1976) 2735. [S] DE GENNES, P. G.. M O / . Crj..rt. Liq. C,:I.J/. 12 (1971) 193.
[2] YANG, C. C., Phys. Rev. Lett. 28 (1972) 955; thesis Harvard [6] KEYES, P. H., Phys. Lett. 67A (1978) 132
University (1972) unpublished. [7] BRAZOVSKII, S. A., DMITRIEV, S. G., SOV. Phys. JETP 42 (1976) [3] MAHLER, D. S., KNES, P. H., DANIELS, W. B., Pliys. Rev. Lett. 497.
36 (1976) 491.
