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Physical properties of the nematic liquid crystal, APAPA
Physica 132B (1985) 153--164 North-Holland, Amsterdam
P H Y S I C A L P R O P E R T I E S OF T H E N E M A T I C L I Q U I D CRYSTAL, A P A P A DISCUSSION OF THE MEASURED VISCOELASTIC PROPERTIES IN RELATION
TO CURRENT THEORIES Jitze P. V A N D E R M E U L E N and Rijke J.J. Z I J L S T R A Physics Laboratory, Fluctuations Phenomena Group, University of Utrecht, Princetonplein 5, NLo3584 CC Utrecht, The Netherlands Received 23 January 1985 Measured viscoelastic properties of nematic APAPA are discussed in relation to currentlY available theories. The comparison is made in terms of reduced Frank elastic constants, reduced Migsowicz and Leslie coefficients and the temperature dependences of these parameters. The data used were calculated from light scattering data reported in an earlier paper [1]. It turns out that the measured elastic properties are in reasonable agreement with current theoretical predictions. However, current theories on the viscous bebaviour are scarce and ambiguous. No satisfactory comparison of the experimental data with these theories can be made.
1. I n t r o d u c t i o n
In a previous p a p e r [1] we described the appropriate optical configurations needed to determine viscoelastic properties of nematics from light scattering experiments. It was shown that three viscoelastic ratios as well as the ratio of two Miesowicz viscosity coefficients can be obtained by c o m p u t e r analysis of the observed line broadening data of the scattered light. In addition we d e m o n s t r a t e d how the ratios of the t h r e e distortion elastic constants can be derived from m e a s u r e m e n t s Of the angular distribution of the scattered light. In this p a p e r we shall c o m p a r e experimental results with some current theories concerning the elastic and viscous properties of nematics. T o that end we used experimental data of the compound A P A P A (anisylindene-p-aminophenylacetate), as presented in our previous p a p e r [1]. In section 2 the elastic properties are treated. In addition a theoretical interpretation of the t e m p e r a t u r e behaviour of the splay/bend elastic ratio is presented. T h e viscous properties are discussed in section 3. In order to be able to c o m p a r e our results with current molecular theories and with results pertaining to other
compounds, we first had to calculate reduced values for the Mi~sowicz and Leslie viscosity coefficients from experimental light scattering data. Relations between the viscous properties and temperature, derived from phenomenological considerations, will be discussed briefly.
2. Elasticity
2.1. Theory Various authors [2-10] have published systematic theoretical studies on the Frank elastic constants [11]. For a m o r e detailed review of these studies we refer to refs. [10, 12]. Taking the molecules as cylindrically symmetrical rods of length L and width W, some of these authors [2, 10, 13] find for a dense nematic liquid the following simple relation:
KI : K2 : K a = I : I : ( L / W ) 2,
(1)
where K1, K2 and K 3 are the Frank elastic constants associated with the splay, twist and bend deformations, respectively.
0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ZP. van der Meulen and R . Z Z Ziflstra I Properties of the nematic liquid crystal, A P A P A
154
On the basis of a mean field theory, derived by Maier and Saupe [14], Saupe and Nehring [2] calculated that
A=
(i = 1, 2, 3),
(2)
where ( P , ) denotes the mean value of the order parameter of the nth rank. v represents the packing fraction, i.e. v = Pn " v0, where Pn is the number density of molecules and v0 is the molecular volume; c~ is a constant, which depends on molecular properties. On the basis of this mean field theory they calculated Cl :c2:c3, from which it followed that K ] / K 2 = 5/11 = 0.45,
(3) K 1 / K 3 = 1.
Recently the same result was also obtained by Vertogen e t al. [8] from calculations based on the continuum theory of liquid crystals. Priest [4], as well as van der Meer et al. [15] extended the theory of Saupe and Nehring by considering in more detail the effect of intermolecular interactions. They derive the following expressions for the reduced distortion elasticity, K~ - (K, -
g(a))/g(a)
(i = 1, 2, 3) ;
(4)
and
Jo - Ixo( T/ Tni) A'=
K i = ci(P2)21j -713
J l - I'LI ( T / TNI)
(7)
J2- 1~2( T/ TNI) Jo- go(Y/T~,) '
where the dimensionless coefficients ]i and /zi (i = 0, 1, 2) are related to attractive and repulsive interactions, respectively. Priest derived expressions for A and A' on the basis of the Onsager model for the special case where spherocylinders interact only via a hard core repulsion (HCR). He found Ah~ _ / Z l - [ 2R 2 - 2 /zo \ 7 R 2 ~- 2 0 ] '
(8)
, =/*_22= __.9 { 3 R 2 - 8 ~ Ah~ /xo 16 \ 7 R 2 + 2 0 / '
(9)
where L-W
where L and W stand for the overall length and width of the spherocylinders respectively. Faber [9] developed a continuum theory to describe a relation between the fluctuating distortions of the director, the elastic properties and the orientational order of nematics. He predicted
K ~ = A - 3A '((P4)/(P2)), K i ~- ai(P2) L4s +/3,(P4) "s K ~ = - 2 A - A '((P4)/(P2)),
(i = 1, 2, 3),
(10)
(5)
K ~ = A + 4A '((1)4)/(1:'2)) .
/((a) stands for the arithmetical mean value of the K{s, for which they predicted
where a~ and/3,, are constants that are not known a priori. In order to verify the equations of both Priest and Faber, one needs to know the values of both (P~) and (P4)2.2. Interpretation of experimental results
g ( ~ = 1/3 E K, --- ,,2(&>~.
(6)
i
To obtain explicit expressions for A and A' one still has to choose a specific form for the intermolecular interactions. Van der Meer et al. [15] derived expressions for A and A', using pair interaction of non-axially symmetrical molecules and a mean field approximation. They found
Leenhouts [16] calculated the average values of L and W from molecular dimensions of Schiff's bases. From these values it follows for A P A P A that R =2.25 and ( L / W ) 2~- 10.6. Thence eq. (1) predicts K 1 : K 2 : K 3 = 1 : 1 : 10.6. If we compare our experimental results (cf. e.g. fig. 13 of ref. [1] and fig. 1) with this relation we observe a large discrepancy.
J.P. van der Meulen and R.J.J. Ziflstra / Properties of the nematic liquid crystal, APAPA l
2
I
I
I
I
_'
I
I
'
I
'
I
'
t
155 '
I
0.5
--
"----. •
"S----- ..........%
APAPA
0
0
0
kPkPk
.I
0
l
0.9t.
~
l/T,a
l
I
0.96
I 0.98
1
1.00 -0.5 0.94
Fig. 1. The bend/splay elastic ratio K3/K1 versus the reduced temperature TITm. Fig. 2 shows t h e reduced values of the distortion elastic constants K~, (i = 1, 2, 3), as a function of temperature. T h e values of K/R were calculated with the help of eq. (4) from o u r data for K J K 2 and K 2 / K a in ref. [1]. For comparison we have added to fig. 2 the results obtained by Leenhouts [16]. W e conclude that although our data differ from the data of Leenhouts, for reasons that are not obvious, as discussed in ref. [1], the temperature dependences are in excellent agreement. Since our elastic ratios were obtained in the nematic range up to T = 0.9995 x T m we are able to estimate accurately by extrapolation the values of K~ at Tr~~. We find a rather symmetric result: K~ = 0 ,
K~ = - 0 . 3 3 - - 1/3
and
KR3= 0.33 ~-- 1/3. From the values of K~ as a function of temp e r a t u r e we can determine A and A ' . ((P4)/(P2)) as a function of temperature with the help of eq. (5). W e obtain 1
d' (P4)
and 1
a
(11)
(12)
'TI "l'Ml
0.96
0-98
1.00
Fig. 2. The reduced values of the three distortion elastic constants: K~,m(K~ - gt.))/g(o) versus the reduced temperature T/Tm, as obtained from the light scattering experiments (full curves). The dashed curves are results obtained from Fr6ededcksz transitions (Leenhouts et al.
[161). Fig. 3 shows the results for A and A'(p4)/(p2). Since accurate values for (P~ are not available at present we cannot determine A' itself. W e fitted the values for A to eq. (7) which l e d to the following results: JJJo = 0.2101,
/,~l/Jl
=
0.9867, (13)
/z~//z o = 0.2115,
tZo/Jo = 0.9802.
As fig. 3 shows, eq. (7) fits in very well with the experimental values for our set of ]i and /zivalues. From eq. (7) it follows that there is-a singularity in A for (T/TNI)= (/zo/jo)-t. Since in the nematic phase T < Tr~ we expect /z0/j0< 1. Our measured value of #o/Jo is in agreement with this prediction but it is very close to 1. One might therefore expect that the behaviour of K~ should show signs of this divergence at temperatures close to TNr However, the tendency of K~ is strongly suppressed because /g/J1 =/zo/Jo. T h e temperature dependence of A, as shown in fig. 3, is due to competition between the temperature dependences of the ratios /%/Jo and
J.P. van der Meulen and R.J.J. Ziflstra / Properties of the nematic liquid crystal, A P A P A
156 I
'
I
I
I
~
I
'
1
'
1
i
1
6.0
, I ~ 1 1
•
1.2
10"fl
0.2
I
J
1.0
5.5 /
0.1 O
O0
0.8
O"
--
APAPA
1
(P2)
0.94
L
I
T/THI
t
i 0.97
,
I
t
I
t
~5.0 100
APAPA 0.0
,
0.94
I
;
I
~ 0.96 TITs,
,
I
,
I
0.98
J
I
,
1.00
Fig. 3. Results of the calculations of A and A'. ((P4)/(P2)), according to eqs. (11) and (12), The solid curve represents the best fit of values of A according to eq. (7), with parameters as given in eq. (13).
~JjO/(lZo/Jo), which are associated with repulsive and attractive forces. If the repulsive forces dominate (HCR: /z~->]~), then A and A' will be represented by the simple forms of eq. (8). If we substitute the above-mentioned value of R = 2.25 into eq. (8) we get A = 0.147. Althougfi the latter value is of the same order of magnitude as our measured values, it still differs from them by about 30% (except near TNI). Although no accurate values for (P4) are available at present, we can estimate (P4) with the help of the mean field theory of Maier and Saupe [14]. For values of (P2)= 0.4 to 0.6, it follows that (P4)= 0.10 to 0.25. With R = 2.25 and with the help of eq. (9) we find that A ' = 0.073. Hence it follows that A '(P4)/(P2) = 0.018 to 0.030. Fig. 3 shows that we found values of 0.06 to 0.12; these are approximately 5 times larger than the predicted values. Another point of interest in connection with eq. (6) is the relation between the physical mean value of the distortion elastic constants [9]: g(P) = 2 { [ K z ( K 3 - / ( 2 ) ] -v2 a r c t g [ ( K 3 - K2)/K2] v2
+ [ g l ( r 3 - r l ) ] -la arctg[(g 3-- Kt)/K1]Vz} -1
(14) and/~(~).
Fig. 4. The ratio of the arithmetic mean value of the distortion elastic constant/((*) and the product of the squares of the packing fraction v and the time averages value of the second rank order parameter (P2), i.e. /((')/(v2(P2)2), as a function of the reduced temperature TITm (short dashed curve; from l.,eenhouts [16]). The value of/f@)/0,2(Pz)2) can be calculated (long dashed curve) using the results of the calculations of the ratio of the physical mean value of our K, /~), and/~(a), i.e./~)//~c,~ (see full curve). Obviously/(@)
~2(t"2)z.
From our results for K 1 / K 2 and K 3 / K 2 we can calculate the ratio/~,)//~(a), which is shown in fig. 4 as a function of the reduced temperature. Apparently/~0,) __./~(,). Although it has already been observed that eq. (6) is in agreement with experimental results [16], we show in fig. 4 that a l s o / ~ P ) - z,2(Pz)2. In this figure we used our data for/((P)//~(') and the data for/~(')/v2(P2) 2 given in ref. [16]. 2.3. Concluding remarks
Comparing our experimental results with current theories we conclude that our results are in reasonable agreement with the theoretical results of Priest and van der Meer et al. From an analysis of the terms in eq. (5), containing (P4)/(P~, we conclude that these terms need to be modified. Since higher order terms are neglected in eq. (5), any modification has first of all to take these terms into account. Expressions derived by taking only hard core repulsion into account are in poor agreement with our data.
J.P. van tier Meulen and R.J.Z Ziflstra / Properties of the nematic liquid crystal, A P A P A
3. V
osity
According to eqs. (16), (18)-(20) and the Onsager-Parodi relation [19]:
In our earlier paper [1] we pointed out that the three viscosities due to pure splay, pure twist and pure bend distortions can be written in terms of Mi~sowicz viscosity coefficients [17] and Leslie coefficients [18]:
we have 1
r/splay = '}11- O~21r/2 ,
(15)
r/t,a,, = Y,,
(16)
r/~,,, = 3 ' 1 - a ~ / r / , ,
(17)
(21)
Of2 -I- ~ 3 = ~ 6 -- 0~5 = r/2 -- r/1 ,
- a 2 = ~(r/.,~ + r/~ - r / 2 ) ,
and
1
a 3 = ~ ( r / ~ , t + r/2 - r / l ) .
where in terms of Leslie coefficients a~:
(18)
+
,'5),
(19)
1
r/2 = ~(as + a4 + a6).
O)
® L
vi-eo-ity
"~,
tt, llt.llt t I t I}1~
a 2/r/2 ,
r/bend = r/twist- °12 ] r/* ,
(24) (25)
it follows that
(20)
The Mi~sowicz viscosity coefficients are related to special shear flow experiments with simple geometries (see fig. 5). The Leslie coefficients are the transport coefficients appearing in the consitutive hydrodynamic equations for viscous stress tensor of a uniaxial fluid. At present the viscous properties of nematics are not understood very well. Fundamental theories based on molecular dynamics are scarce and incomplete. In order to compare the experimental data of ref. [1] with some theoretical predictions and experimental results for other nematic compounds we shall first derive the ratios of the Mi~sowicz and Leslie coefficients from the ratios of viscosity coefficients as reported in ref. [1]. We must restrict ourselves to calculating ratios, since these are the quantities that can be derived from light scattering experiments.
l
(22) (23)
Since the splay and bend viscosities are given by (cf. eqs. (15)--(17)) r/.ay = r / ~ . -
"}/1 = 0 ¢ 3 - 0~2'
r/, =
157
i
|t I I
"q,:½t-%.=,.%l
® ~
-~--.--~l .--: -.-:-.-.r
~:½t%.%.%l
: :;-- -:" -'.:'.": ." -)
r/splay = r/twist
r/~d = r/t,~t-
(r/twia + r/2- r/,)2 4r/2 , ( r / ~ s t + r/* -- 7"]2)2
4r/,
(26) (27)
If we define a reduced viscosity r/* by r/~*= r/ffr/t~,t, then the eqs. (24), (25) reduce to
r/,p,,y = 1
r/~.d* = 1
(1 + r/~ - r/]') 2 4nZ = -r/~'. H(r/~', r/~)'(28)
(1 + r/~' -- r/~) 2 = --ri~ " H(r/~, r/Z), 4r/T
(29)
where H(r/~, r/Z ) =
- ,79 = - 2(r/; + , 7 9 + 1 4r/,r/2
(30)
Note that the function H(r/7, r/~) is symmetric in r/~' and r/Z. It follows from eqs. (28), (29) that
'
Fig. 5. Viscosity coefficients of an aligned n e m a t i e liquid crystal as m e a s u r e d in a s h e a r flow e x p e r i m e n t , *Tt (i = 1, 2, 3) a n d their relations to the Leslie coefficients, ai (i = 1 . . . 6).
n,p,./r/~..d = r/i/r/2 •
(31)
Consequently we are able to calculate the ratios
J.P. van der Meulen and R.J.J. Ziflslra / Properties of the nematic liquid crystal, A P A P A
158
711"712:7] 3 from the experimentally determined values of *7~,~oy/~/be~aand .71/~/3. Note that one can find the ratios of the Mifsowicz coefficients without knowing the value of a ratio related to . 7 ~ . The ratios of the Leslie coefficients can now be determined with the exception of ratios involving 0/j = *Tn because, as was pointed out in ref. [1], it proved experimentally not feasible to defermine 7/~2. * From eqs. (24), (25) it follows that
0/~ = ~2(~twist- *Tsplay) ,
(32)
0/2 = *71(nl~st- T]bend) •
(33)
With the help of eq. (31) we now get 0/3) (*7~,iJ*7,q,,.y- 1 ) ":~ = (.7,,~,/.7,,..d-- 1)"
(34)
Consequently, only the absolute value of 0/3/0/2 can be determined from the measured values of *7~,~.y/*Tt~st and *Tbena/*7~,~t" On the basis of a molecular theory of flow alignment in pure nematics, where the molecules are considered as ellipsoids with hard cores, having length L and width W, Helfrich [20] predicted that --
=
,
(35)
0/2
which implies that the sign of a3/0/2 will be positive. From eqs. (19)-(21) and the relation 1 *73 = ~0/,t
(36)
we can derive the following relations between the Leslie coefficients and the ratios of the Mi¢sowicz viscosities: (0/5 - 0/2)/0/4 - rh/~T3- 1,
(37)
1,
(38)
(O/6 + 0/3)/0/4 = *72/*73-
(0/2 + 0/3)/a4 = (0/6- 0/5)/0/4 = (.72--
~1)/2.73
•
(39)
With the help of eqs. (34), (37)-(39) and 0/2/O'3> 0, it is now possible to calculate the ratios 0/2 : 0/3 : 0/4 : 0/5 : t~6. Note that in the relations of the viscosity coefficients presented above the sets of coefficients (%,~y, .71,-az, as) and (*Tb,,d, *72,0/3, 0/6) are interchangeable. assuming
3.2. Experimental results in terms of the M i f s o w i c z and Leslie coefficients The results of the calculations for the ratios of the Migsowicz and Leslie coefficients are given in table I, where we have used the data of ref. [1]. In fig. 6 we have plotted the results for la3/0/21, 0/6/0/5, -0/2/0/4 and as/o/4, as a function of the reduced temperature T I T m. Obviously the ratio 0/5/0/4 and 0/2/0/4 are rather independent of temperature. The ratios 0/3/0/2 and a6/0/5 are found to be small but not independent of temperature. We have assumed that 0/3/0/2>0, which is in agreement with the experimental data for I-IBBA and PAA [12]: Leenhouts [16] has estimated that at T / T m = 0.95, ( W / L ) z ~ 0.1, hence eq. (35) is in reasonable agreement with our data on 0/3/0/2" However, other equations, predicted by
Table I Calculated results of the ratios of the Migsowicz and Leslie viscosity coefficients as a function of the reduced t e m p e r a t u r e T/TN~ (Tin--- 109.2°C)
T]TNI
"01/~ (a)
172]'~ (a)
Y/3/~ (a)
O~2/~4
~3]014
O~5/O~4
O~6/a4
0.999 0.998 0.995 0.993 0.988 0.978 0.967 0.957 0.947
1.905 1.953 1.965 1.995 2.004 2.049 2.052 2.127 2.145
0.573 0.558 0.543 0.525 0.534 0.501 0.456 0.405 0.393
0.522 0.489 0.492 0.480 0.462 0.450 0.492 0.468 0.462
- 1.273 - 1.312 - 1.258 - 1.328 -1.343 - 1.440 - 1.389 - 1.620 - 1.691
0 -0.117 -0.188 -0.207 -0.252 -0.278 - 0.232 -0.220 -0.210
1.364 1.688 1.742 1.839 2.005 2.106 1.778 1.925 1.961
0.091 0.259 0.295 0.305 0.411 0.388 0.157 0.085 0.060
ZP. van der Meulen and R.ZJ. Ziflstra / Properties of the nematic liquid crystal, A P A P A I
I
[
I
I
~ / ~ - (7/,-
159
~(*))/~(').
(41)
A
04
2.0
-A x
~
X~
~4
0.2
0.0 0.94
[ ~
I 0.96
I
$--<- ~ 9 - - .
--71.0
I 0.98
0.0 1.00
[
TIT m
Fig. 6. Results of the calculations for la3/a2l (dots), a61a5 (circles), -ct2/a4 (crosses) and a5/a4 (triangles) obtained from our experimental data of ref. [1], versus T/Tm.
Helfrich [20], which relate the viscous properties to the sizes of the molecules, e.g. ~2/~/1 = (W/L) 4, conflict with our data. Some authors (G~ihwiller [21], Skarp et al. [22]), found for H B A B and 8CB a3/a2
3(7~1 at" 1']2 at" ~ 3 ) ,
Fig. 7 Shows the results for 7/~ (i = 1, 2, 3) versus the reduced temperature. We find that ~ < ~ / ~ ; this result differs considerably from results for other nematics published by various authors [12, 21-25] who found: ~/3 > 72. Since we found ~/~,Z~y= rhwia it follows that the inequality ~/~ ~ 7/~ implies, with the help of eqs. (20), (24) and (36), a 6 ~> 0. At present only a few theories are available to describe the temperature and order parameter dependence of the viscosity coefficients of nematics [26-29]. Imura et al. [28] stated that 7/~ (i = 1, 2, 3) can be written in terms of an isotropic part and a part which is linear with (P2). From their calculations it follows that 1
1
(42)
7/~o = ~(rh + r/2+ B3)- g71,
where */i~ is the extrapolated value of the viscosity of the isotropic phase. If a3/a 2 = 0, eq. (42) reduces to 1 r/iso --~ ~, (r/1 + 3r/2 + 2r/3 ) .
(43)
Imura et al. give the following relations for the viscosity coefficients: '7,-
= q " (P2),
(44)
I
I
I
I
I
APAPA i=1 A &
o R
i=3
(40)
g
Q
s
0" s s ~
i=2
where the superscript (a) denotes the arithmetical mean value of ~l (i = 1, 2, 3), we can define a reduced shear viscosity, representing the anisotropic parts of the viscous properties:
"10.94
--~
0.96
0.98
1.0o
TIT.m
Fig. 7. Results of the calculated values of ,/,gm,/d~(*)-1 (i = 1, 2, 3) versus T[Tm. Obviously ~2 = ~3-
J.P. van der Meulen and R.J.J. Zijlstra / Properties of the nematic liquid crystal, A P A P A
160 7 / 1 - 7/3 = G "
(P2),
(45)
where C~ and C 2 are constants. In order to check eqs. (44) and (45) experimentally we plotted (7/1- 7/2)/(7/a- 7/3), since we expected these ratios to be rather constant with varying temperature (see fig. 8). In fact, we find that (7/1- Wz)/(7/1- 7/3) varies by less than 10% between 0.94Tin and T m. Recent calculations by Marrucci [26] yield different relations between the Leslie coefficients and (Pz). According to Marrucci, 0/1 = C ( T ) "
[-6(P2)Zl,
a 2 = C ( T ) . [-6(P2)(1 + 2(P2))/(2 + (P2))], 0/3 = C ( T ) . [-6(P2)(1 - (Pz))/(2 + (P2))],
(46)
34 = C ( T ) . [2(1 - (P2))],
(50)
7/t~t = 9 (1 +(P2) ½(P2))" C ( T ) ,
(51)
7/b,,.d = O.
(52)
Marrucci's theory inspires little confidence as it leads to the unphysical prediction 7/~,d = 7/~,fsy= 0. Moreover, we found that the predicted ratios of 7/i ( i = 1,2,3) deviate strongly from our experimental results. Hence the fact that this theory correctly predicts a 6 = 0 should be considered fortuitous. Now we shall go on to check the isotropic part of 7/i (see eq. (42)). 7/isois independent of (P2) and corresponding with the extrapolated values of the viscosity in the isotropic phase, 7/(T; T >
T I).
a s = C ( T ) " [6(ez) ] , 0/6=0
7/splay = O ,
Kneppe et al. [23] suggested that (cf. eq. (40))
,
7/i~o= ~(a), where C ( T ) = const., k s T / 6 D r > O, and D~ is a rotational diffusion constant. From these relations it is possible to derive the Mi~sowicz and the distortion viscosities; one gets (cf. eqs. (15)(20) and (36))
7/1
=
(1 + 2(P2)) 2 (1 + ½(P2))
(1 - (P2)) = 7/2 = (1 + ½(P2)) 7/3 = (1 - (P2))" 1.1
C(T),
(47)
C(T),
(48)
C(T), t
(49) t
=,
i
i
which contradicts the theoretical results of Imura et al. [28] (see eq. (42)). As can be seen from fig. 8 we find typically a 20% deviation from eq. (53). Since we are unable t o measure the viscosity of a nematic above its clearing temperature T m by light scattering, we cannot determine 7/(T; T > Try1). In order to estimate an expression for 7/~o we calculated the physical mean value of the viscosity, which is related to the sum of the dissipative parts of the viscous stress tensor, defined as [12, 21] )du
t
(53)
1
t
t
7/'(0, ~b dx -=~(o'= + O'z,,),
(54)
APAPA
a
1-0
averaged over all angles 0 and 4~ (cf. fig. 9). We obtained for the physical mean value of 7/':
ql"q2 q~-93 o!9 . •
•
17 ,(p)
a•
0.8
1 1 1 = ]'~7/12 + ~(7/1 + Y/2+ 7/3)-- ?~')"1 = l o t I + 6(0/5 1 + 0/6) + ]13 4 "
I 0.9/.
) 0.96
I
I
(55)
I
0.98
1.00
I/T m
Fig. 8. The calculated results of (71 -- 7 2 ) / ( 7 1 -- 7 3 ) (triangles) and 7~0/~ (') (dots) versus T/TIn, w h e r e 7~,0 ~-- - 6171 (see eqs. (42) and (53)).
~{')
Note that this equation is similar to the expression for 7/i~o in eq. (42), except for the term 1 ~7/12, which is assumed to be negligible compared to the other viscosity coefficients in eq. (55)
ZP. van der Meulen and R . Z Z Ziflstra / Properties of the nematic liquid crystal, A P A P A
91
.
161
I
/
Z /" ,i
t
/ eele* I I
/
I
0
L
~ //.~
-
~
i /
i=2
Fig. 9. Definition of the orientation of the director ri0 with respect to the shear plane.
NI
I 0.0
-1
[12,23,24]. We can now try to describe the temperature dependence on a phenomenological basis, as was suggested by Kneppe et al. [23]. According to these authors the following relation is approximately satisfied: 7,((P2)) = @(~)(a,+
bi(P2) )
(i = 1, 2, 3).
(56)
In the isotropic phase, however, where (P2) = 0, ~/i reduces to ~(~) (i = 1, 2, 3) which implies that a i = 1 (i = 1, 2, 3). Consequently, because fi(~)= ~E~ 7~ it follows that Xi b~ = 0. Since 7~ > 0 (i = 1, 2, 3) for reasonable values of (P2) (0.3 < (P2) < 1) it follows that b i > - 1 . In fig. 10 we have plotted 7~ = (7,./~(')) - 1 versus (P2>, the latter being from Leenhouts [30]. Obviously eq. (56) is satisfied, but not with a~ = 1, since the lines do not intersect at the origin. One might expect that a better fit would be obtained if @(')in eq. (56) is replaced by 7too (cf. eq. (42)). In fact with this substitution our experimental data points fall on a curve which is shown in fig. 10. Clearly, there is not even a hint of a linear relation between these data points and (P2)Thus far we have only been considering ratios of viscosities. If we take the values of the twist elastic constant K2, as determined by Leenhouts [16], we are able to calculate the absolute values of 7 ~ or y~ from the twist viscoelastic ratio, and hence obtain absolute values for all other viscosities. The absolute values of 7~ (i = 1, 2, 3), are shown i n fig. 11. It follows from this figure that to a good approximation
/
-0"
1
""
I 0.5
1.0
(P2) Fig. 10. The reduced values r/~ m (~lil@('))- 1 (i = 1, 2, 3; d o t s , pluses, crosses respectively) and (~h/~/~)- 1 (triangles) versus the order parameter (/'2).
7, = 7 ° +/3~(P2),
(57)
with 7~ = -0.068 Pa s,
fll = 0.214 Pa s,
7~ = -0.007 Pa s,
/32 = 0.029 Pa s,
7] = -0.012 Pa s,
/33 ---- 0.042
Pa s.
By extrapolating the values for 7i given in fig. 11 one finds a)
71 = 72 = 73
Pct.s 0.1
for (/'2) = 0.33
~ /
APAPA
**
rli I
P'"
0.0 0.0
~ •
(P2 >
, NI
0.5
2 1.0
Fig. 11. The calculated absolute values of the shear viscosities ~ (i = 1, 2, 3; circles, dots, crosses, respectively) versus the order parameter (P2).
J.P. van der Meulen and R.J.J. Zi]lstra I Properties of the nematic liquid crystal, APAPA
162
instead of the expected value of (1)2) = 0, b)
~7i < 0
"3 /
I
I
i
I
i
I
for (/)2)< 0.3. ~
:(a)
Note that our results for eq. (57) are in agreement with the suggestion of Kneppe et al. [23], namely that the Mi¢sowicz viscosities rh are related to each other according to
•
q
~
/~6"J-~
~
l
~o
~
O ~
~o .....
o
o
,YLo m
Ai~ i = 0.
-s L
(58) tn ~(a)
i
From our results for ~o and /3~ it follows that A 1 : A 2 : A 3 = 0.114 : 0.608 : - 1 . These results are in reasonable agreement with the results of Kneppe et al. [23] who found mean values of 0.08 : 1.1 : - 1 for the compounds of N 4, M B B A , E M and 5CPB. W e are, however, not aware of any convincing theoretical argument for the existence of such a linear relation between the Migsowicz coefficients. W e shall now go on to consider the absolute values of ~/~o and ~ ) , which we expect to be rather independent of (P2) and related to the temperature, according to (cf. Imura et al. [28]) 71i~o, ~(~)~ e x p ( e o / k a T ) .
3.3. Concluding remarks conclude
that
APAPA
¢ I
-6
2.62
the
behaviour
of
the
I
I
I
2.70
11 T
I
I
2.78
"10 "3 K -1
Fig. 12. The calculated absolute values of In ~i~o(circles)and In ~(a) (dots) versus the inverse temperature. -3
I
I
I
I
i
I
X / fl= twist / "~-"
X//
,~
r?~
11=spMy
" o~
~ * .---
oo I -6
~NI 1-0
X
/'O
[}:bend In
/ 10
/5o "/
"/~
-4
(59)
In order to investigate eq. (59) we have plotted in fig. 12, In ~1~o, In ,i (a) versus lIT. T h e figure shows that the assumption that 7/i~o and ~(a) are independent of (P2) is not correct. In a previous publication (ref. [31]), we found that for O H M B B A In 3,1-(P2)/T. With this result in mind we have plotted In 7?~,~ay,In ~/~t and In ~/~nd versus (P2)/T in fig. 13. From the figure it can be seen that to a good approximation: , / . ( T ) = C..exp(e.(P2)/kaT), ( t z ~ s p l a y , twist, bend), where C, and e. are constants. Note that the above relation suggests that these distortion coefficients will not tend to zero when (P2) goes to zero, i.e. in the isotropic phase, although it is not obvious that such an extrapolation is justified.
We
In q i s o
./-
•"
t l
o I
(P2)/T
o I~°
APAPA I 1.2
I
I
I
1.4 .10.3K.1
I %.6
Fig. 13. Results of the calculations of the natural logarithm of the distortion viscosities, In ~?~,~y (circles), In ~?t~ (crosses) and In ~Tb~ (dots), versus (P2)IT. Note that ~?t,,~ = yl.
Migsowicz coefficients, as determined by light scattering, is in qualitative agreement with data obtained from flow experiments done with other compounds. T h e r e is however one difference, specific in that ~/2 = ~3 or a6/as<~0.2. Flow experiments on other compounds (MBBA, P A A and H B A B ) find rather large negative values for
ZP. van der Meulen and R.ZJ. Ziflstra / Properlies of the nematic liquid crystal, A P A P A
4
I
I /
APAPA
I /I /
2_ _ / 2K]1~ KI*K2| O/ 0
1
i
and simulating discussions. We are also grateful to Sheila McNab for making linguistic improvements and to Wendelies van der Meulen for typing the manuscript. This work was performed as part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (ZWO).
oo
I
I
'~" 2 (~h+~3)/2~1z
1
I
4
Fig. 14. The calculated ratios 2K3/(KI + 1(2) versus (~1 + ~73)/ 272. The curve refers to the prediction of Lin-Liu et al. [32] (see eq. (60)).
this ratio (typically -0.5). It is not obvious whether this difference is real, in the sense that the properties of A P A P A are rather different from the above-mentioned compounds, or that it is an artifact due to the differences in experimental techniques. In this context it should be noted that flow experiments are performed under conditions which inquire rather large perturbations (although supposedly still negligible) on the nematic. In contrast, light scattering experiments probe the relaxation of spontaneous fluctuations in equilibrium. Finally, we would point out that there is a similarity between the results for K]~ (fig. 2) and ~ (fig. 7). Although a linking of the static and dynamic properties seems to be rather tentative, Lin-Liu et al. [32] predicted on a molecular basis, taking the molecular sizes into account, that
2K3/(KI +/(2) =
071 + ¢/3)/2¢/2 •
163
(6o)
Fig. 14 shows that the elastic and viscous properties do indeed seem to be related. The experimental data satisfy eq. (60) reasonably well. On the basis of this result we suggest that more theoretical studies should be done on the relation between the elastic and viscous properties of nematics and that in these studies the sizes of the molecules should be taken into account.
Acknowledgments W e wish to thank D r . D . Frenkel for valuable
References [1] J.P. van der Meulen and R.J.J. Zijistra, J. Physique 45 (1984) 1627. See also, J.P. van der Meulen, Ph.D. Thesis, University of Utrecht (1983), Ch. 3. [2] A. Sanpe, Z. Naturforsch. A15 (1960) 810. J. Nehring and A. Saupe, J. Chem. Phys. 56 (1972) 5527. [3] R.G. Priest, Mol. Cryst. Liq. Cryst. 17 (1972) 129. [4] R.G. Priest, Phys. Rev. A7 (1973) 720. [5] B.W. van der Meet, F. Postma, A.J. Dekker and W.H. de Jeu, Mol. Phys. 45 (1982) 1227. [6] J.P. Straley, Phys. Rev. A8 (1973) 2181. [7] J. Stecki and A. Poniewierski, Mol. Phys. 41 (1980) 1451. A. Poniewierski and J. Stecki, Phys. Rev. A25 (1982) 2368. [8] G. Vertogen, S.D.P. Flapper and C. Dullemond, J. Chem. Phys. 76 (1982) 616. G. Vertogen, Physica 117A (1983) 227. [9] T.E. Faber, Proc. R. Soc. Lond. A353 (1977) 247; A375 (1981) 579. [10] W.H. de Jeu, Mol. Cryst. Liq. Cryst. 63 (1981) 83. [11] F.C. Frank, Disc. Faraday Soc. 25 (1958) 19. [12] W.H. de Jeu, Physical Properties of Liquid Crystalline Materials (Gordon and Breach, New York, 1980); see also, Phys. Lett. 69A (1978) 122. [13] W.H. de Jeu, F. Leenhouts and F. Postma, in: Advances in Liquid Crystal Research and Applications, ed. L. Bata (Pergamon Press, Oxford, 1980), p. 499. [14] W. Maier and A. Saupe, Z. Naturforsch. 13a (1958) 564; 14a (1959) 882; 15a (1960) 287. [15] B.W. van der Meer and G. Vertogen, Phys. Lett. 71A (1979) 486. See also, B.W. van der Meer, Ph.D. Thesis, Univ. of Groningen (1979). B.W. van der Meer and G. Vertogan, in: The Molecular Physics of Liquid Crystals, eds. G.R. Luckhurst and G.W. Gray (Academic Press, London, 1979), Ch. 6, p. 159. [16] F. l.,eenhouts and A.J. Dekker, J. Chem. Phys. 74 (1981) 1956. See also, F. Leenhouts, Ph.D. Thesis, Univ. of Groningen (1979). [17] M. Miesowicz, Bull. Intern. Acad. Poion. (Sci. Math. Nat.) Ser. A. (1936) 228; Nature 158 (1946) 27; note that ~1 and 72 are interchanged.
164
J.P. van der Meulen and R.J.J. Zijlstra / Properties of the nematic liquid crystal, A P A P A
[18] F.M.I.~slie, Quart. J. Mech. Appl. 19 (1966) 357; see also, in: Advances in Liquid Crystals, vol. IV, ed. G.H. Brown (Academic Press, New York, 1979), p. 1. [19] O. Parodi, J. Physique 31 (1970) 581. [20] W. Hclfrich, J. Chem. Phys. 50 (1969) 100; 53 (1970) 2267. [21] Ch. (3iihwiller,/Viol. Cryst. Liq. Cryst. 20 (1973) 301. [22] K. Skarp, T. Carlsson, S.T. Lagerwall and B. Stebler, Mol. Cryst. Liq. Cryst. 66 (1981) 199. [23] H. Kneppe, F. Schneider and N.K. Shazma, Bet. Bunsenges. Phys. Chem. 85 (1981) 784. [24] W.W. Beens and W.H. de Jeu, J. Physique 44 (1983) 129.
[25] H. Kneppe and F. Schneider, Mol. Cryst. Liq. Cryst. 65 (1981) 189. [26] G. Marrucci, Mol. Cryst. Liq. Cryst. l.,¢tt. 72 (1982) 153. [27] S. Hess, Z. Naturforsch. 30a (1975) 1224. [28] H. Imura and K. Okano, Jpn. J. Appl. Phys. 11 (1972) 1440. [29] A.O. Tseber, Magnetohydrodynamics 14 (1978) 267. [30] F. Leenhouts, W.H. de Jeu and A.J. Dekker, J. Physique 40 (1979) 989. See also, F. l_,eenhouts, Ph.D. Thesis, Univ. of Groningen (1979). [31] J.P. van der Meulen and R.J.J. Zijlstra, J. de Physique 43 (1981) 411. [32] I.R. Lin-Liu, I.M. Shih and C.-W. Woo, Phys. Lett. 57A (1976) 43.
P H Y S I C A L P R O P E R T I E S OF T H E N E M A T I C L I Q U I D CRYSTAL, A P A P A DISCUSSION OF THE MEASURED VISCOELASTIC PROPERTIES IN RELATION
TO CURRENT THEORIES Jitze P. V A N D E R M E U L E N and Rijke J.J. Z I J L S T R A Physics Laboratory, Fluctuations Phenomena Group, University of Utrecht, Princetonplein 5, NLo3584 CC Utrecht, The Netherlands Received 23 January 1985 Measured viscoelastic properties of nematic APAPA are discussed in relation to currentlY available theories. The comparison is made in terms of reduced Frank elastic constants, reduced Migsowicz and Leslie coefficients and the temperature dependences of these parameters. The data used were calculated from light scattering data reported in an earlier paper [1]. It turns out that the measured elastic properties are in reasonable agreement with current theoretical predictions. However, current theories on the viscous bebaviour are scarce and ambiguous. No satisfactory comparison of the experimental data with these theories can be made.
1. I n t r o d u c t i o n
In a previous p a p e r [1] we described the appropriate optical configurations needed to determine viscoelastic properties of nematics from light scattering experiments. It was shown that three viscoelastic ratios as well as the ratio of two Miesowicz viscosity coefficients can be obtained by c o m p u t e r analysis of the observed line broadening data of the scattered light. In addition we d e m o n s t r a t e d how the ratios of the t h r e e distortion elastic constants can be derived from m e a s u r e m e n t s Of the angular distribution of the scattered light. In this p a p e r we shall c o m p a r e experimental results with some current theories concerning the elastic and viscous properties of nematics. T o that end we used experimental data of the compound A P A P A (anisylindene-p-aminophenylacetate), as presented in our previous p a p e r [1]. In section 2 the elastic properties are treated. In addition a theoretical interpretation of the t e m p e r a t u r e behaviour of the splay/bend elastic ratio is presented. T h e viscous properties are discussed in section 3. In order to be able to c o m p a r e our results with current molecular theories and with results pertaining to other
compounds, we first had to calculate reduced values for the Mi~sowicz and Leslie viscosity coefficients from experimental light scattering data. Relations between the viscous properties and temperature, derived from phenomenological considerations, will be discussed briefly.
2. Elasticity
2.1. Theory Various authors [2-10] have published systematic theoretical studies on the Frank elastic constants [11]. For a m o r e detailed review of these studies we refer to refs. [10, 12]. Taking the molecules as cylindrically symmetrical rods of length L and width W, some of these authors [2, 10, 13] find for a dense nematic liquid the following simple relation:
KI : K2 : K a = I : I : ( L / W ) 2,
(1)
where K1, K2 and K 3 are the Frank elastic constants associated with the splay, twist and bend deformations, respectively.
0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ZP. van der Meulen and R . Z Z Ziflstra I Properties of the nematic liquid crystal, A P A P A
154
On the basis of a mean field theory, derived by Maier and Saupe [14], Saupe and Nehring [2] calculated that
A=
(i = 1, 2, 3),
(2)
where ( P , ) denotes the mean value of the order parameter of the nth rank. v represents the packing fraction, i.e. v = Pn " v0, where Pn is the number density of molecules and v0 is the molecular volume; c~ is a constant, which depends on molecular properties. On the basis of this mean field theory they calculated Cl :c2:c3, from which it followed that K ] / K 2 = 5/11 = 0.45,
(3) K 1 / K 3 = 1.
Recently the same result was also obtained by Vertogen e t al. [8] from calculations based on the continuum theory of liquid crystals. Priest [4], as well as van der Meer et al. [15] extended the theory of Saupe and Nehring by considering in more detail the effect of intermolecular interactions. They derive the following expressions for the reduced distortion elasticity, K~ - (K, -
g(a))/g(a)
(i = 1, 2, 3) ;
(4)
and
Jo - Ixo( T/ Tni) A'=
K i = ci(P2)21j -713
J l - I'LI ( T / TNI)
(7)
J2- 1~2( T/ TNI) Jo- go(Y/T~,) '
where the dimensionless coefficients ]i and /zi (i = 0, 1, 2) are related to attractive and repulsive interactions, respectively. Priest derived expressions for A and A' on the basis of the Onsager model for the special case where spherocylinders interact only via a hard core repulsion (HCR). He found Ah~ _ / Z l - [ 2R 2 - 2 /zo \ 7 R 2 ~- 2 0 ] '
(8)
, =/*_22= __.9 { 3 R 2 - 8 ~ Ah~ /xo 16 \ 7 R 2 + 2 0 / '
(9)
where L-W
where L and W stand for the overall length and width of the spherocylinders respectively. Faber [9] developed a continuum theory to describe a relation between the fluctuating distortions of the director, the elastic properties and the orientational order of nematics. He predicted
K ~ = A - 3A '((P4)/(P2)), K i ~- ai(P2) L4s +/3,(P4) "s K ~ = - 2 A - A '((P4)/(P2)),
(i = 1, 2, 3),
(10)
(5)
K ~ = A + 4A '((1)4)/(1:'2)) .
/((a) stands for the arithmetical mean value of the K{s, for which they predicted
where a~ and/3,, are constants that are not known a priori. In order to verify the equations of both Priest and Faber, one needs to know the values of both (P~) and (P4)2.2. Interpretation of experimental results
g ( ~ = 1/3 E K, --- ,,2(&>~.
(6)
i
To obtain explicit expressions for A and A' one still has to choose a specific form for the intermolecular interactions. Van der Meer et al. [15] derived expressions for A and A', using pair interaction of non-axially symmetrical molecules and a mean field approximation. They found
Leenhouts [16] calculated the average values of L and W from molecular dimensions of Schiff's bases. From these values it follows for A P A P A that R =2.25 and ( L / W ) 2~- 10.6. Thence eq. (1) predicts K 1 : K 2 : K 3 = 1 : 1 : 10.6. If we compare our experimental results (cf. e.g. fig. 13 of ref. [1] and fig. 1) with this relation we observe a large discrepancy.
J.P. van der Meulen and R.J.J. Ziflstra / Properties of the nematic liquid crystal, APAPA l
2
I
I
I
I
_'
I
I
'
I
'
I
'
t
155 '
I
0.5
--
"----. •
"S----- ..........%
APAPA
0
0
0
kPkPk
.I
0
l
0.9t.
~
l/T,a
l
I
0.96
I 0.98
1
1.00 -0.5 0.94
Fig. 1. The bend/splay elastic ratio K3/K1 versus the reduced temperature TITm. Fig. 2 shows t h e reduced values of the distortion elastic constants K~, (i = 1, 2, 3), as a function of temperature. T h e values of K/R were calculated with the help of eq. (4) from o u r data for K J K 2 and K 2 / K a in ref. [1]. For comparison we have added to fig. 2 the results obtained by Leenhouts [16]. W e conclude that although our data differ from the data of Leenhouts, for reasons that are not obvious, as discussed in ref. [1], the temperature dependences are in excellent agreement. Since our elastic ratios were obtained in the nematic range up to T = 0.9995 x T m we are able to estimate accurately by extrapolation the values of K~ at Tr~~. We find a rather symmetric result: K~ = 0 ,
K~ = - 0 . 3 3 - - 1/3
and
KR3= 0.33 ~-- 1/3. From the values of K~ as a function of temp e r a t u r e we can determine A and A ' . ((P4)/(P2)) as a function of temperature with the help of eq. (5). W e obtain 1
d' (P4)
and 1
a
(11)
(12)
'TI "l'Ml
0.96
0-98
1.00
Fig. 2. The reduced values of the three distortion elastic constants: K~,m(K~ - gt.))/g(o) versus the reduced temperature T/Tm, as obtained from the light scattering experiments (full curves). The dashed curves are results obtained from Fr6ededcksz transitions (Leenhouts et al.
[161). Fig. 3 shows the results for A and A'(p4)/(p2). Since accurate values for (P~ are not available at present we cannot determine A' itself. W e fitted the values for A to eq. (7) which l e d to the following results: JJJo = 0.2101,
/,~l/Jl
=
0.9867, (13)
/z~//z o = 0.2115,
tZo/Jo = 0.9802.
As fig. 3 shows, eq. (7) fits in very well with the experimental values for our set of ]i and /zivalues. From eq. (7) it follows that there is-a singularity in A for (T/TNI)= (/zo/jo)-t. Since in the nematic phase T < Tr~ we expect /z0/j0< 1. Our measured value of #o/Jo is in agreement with this prediction but it is very close to 1. One might therefore expect that the behaviour of K~ should show signs of this divergence at temperatures close to TNr However, the tendency of K~ is strongly suppressed because /g/J1 =/zo/Jo. T h e temperature dependence of A, as shown in fig. 3, is due to competition between the temperature dependences of the ratios /%/Jo and
J.P. van der Meulen and R.J.J. Ziflstra / Properties of the nematic liquid crystal, A P A P A
156 I
'
I
I
I
~
I
'
1
'
1
i
1
6.0
, I ~ 1 1
•
1.2
10"fl
0.2
I
J
1.0
5.5 /
0.1 O
O0
0.8
O"
--
APAPA
1
(P2)
0.94
L
I
T/THI
t
i 0.97
,
I
t
I
t
~5.0 100
APAPA 0.0
,
0.94
I
;
I
~ 0.96 TITs,
,
I
,
I
0.98
J
I
,
1.00
Fig. 3. Results of the calculations of A and A'. ((P4)/(P2)), according to eqs. (11) and (12), The solid curve represents the best fit of values of A according to eq. (7), with parameters as given in eq. (13).
~JjO/(lZo/Jo), which are associated with repulsive and attractive forces. If the repulsive forces dominate (HCR: /z~->]~), then A and A' will be represented by the simple forms of eq. (8). If we substitute the above-mentioned value of R = 2.25 into eq. (8) we get A = 0.147. Althougfi the latter value is of the same order of magnitude as our measured values, it still differs from them by about 30% (except near TNI). Although no accurate values for (P4) are available at present, we can estimate (P4) with the help of the mean field theory of Maier and Saupe [14]. For values of (P2)= 0.4 to 0.6, it follows that (P4)= 0.10 to 0.25. With R = 2.25 and with the help of eq. (9) we find that A ' = 0.073. Hence it follows that A '(P4)/(P2) = 0.018 to 0.030. Fig. 3 shows that we found values of 0.06 to 0.12; these are approximately 5 times larger than the predicted values. Another point of interest in connection with eq. (6) is the relation between the physical mean value of the distortion elastic constants [9]: g(P) = 2 { [ K z ( K 3 - / ( 2 ) ] -v2 a r c t g [ ( K 3 - K2)/K2] v2
+ [ g l ( r 3 - r l ) ] -la arctg[(g 3-- Kt)/K1]Vz} -1
(14) and/~(~).
Fig. 4. The ratio of the arithmetic mean value of the distortion elastic constant/((*) and the product of the squares of the packing fraction v and the time averages value of the second rank order parameter (P2), i.e. /((')/(v2(P2)2), as a function of the reduced temperature TITm (short dashed curve; from l.,eenhouts [16]). The value of/f@)/0,2(Pz)2) can be calculated (long dashed curve) using the results of the calculations of the ratio of the physical mean value of our K, /~), and/~(a), i.e./~)//~c,~ (see full curve). Obviously/(@)
~2(t"2)z.
From our results for K 1 / K 2 and K 3 / K 2 we can calculate the ratio/~,)//~(a), which is shown in fig. 4 as a function of the reduced temperature. Apparently/~0,) __./~(,). Although it has already been observed that eq. (6) is in agreement with experimental results [16], we show in fig. 4 that a l s o / ~ P ) - z,2(Pz)2. In this figure we used our data for/((P)//~(') and the data for/~(')/v2(P2) 2 given in ref. [16]. 2.3. Concluding remarks
Comparing our experimental results with current theories we conclude that our results are in reasonable agreement with the theoretical results of Priest and van der Meer et al. From an analysis of the terms in eq. (5), containing (P4)/(P~, we conclude that these terms need to be modified. Since higher order terms are neglected in eq. (5), any modification has first of all to take these terms into account. Expressions derived by taking only hard core repulsion into account are in poor agreement with our data.
J.P. van tier Meulen and R.J.Z Ziflstra / Properties of the nematic liquid crystal, A P A P A
3. V
osity
According to eqs. (16), (18)-(20) and the Onsager-Parodi relation [19]:
In our earlier paper [1] we pointed out that the three viscosities due to pure splay, pure twist and pure bend distortions can be written in terms of Mi~sowicz viscosity coefficients [17] and Leslie coefficients [18]:
we have 1
r/splay = '}11- O~21r/2 ,
(15)
r/t,a,, = Y,,
(16)
r/~,,, = 3 ' 1 - a ~ / r / , ,
(17)
(21)
Of2 -I- ~ 3 = ~ 6 -- 0~5 = r/2 -- r/1 ,
- a 2 = ~(r/.,~ + r/~ - r / 2 ) ,
and
1
a 3 = ~ ( r / ~ , t + r/2 - r / l ) .
where in terms of Leslie coefficients a~:
(18)
+
,'5),
(19)
1
r/2 = ~(as + a4 + a6).
O)
® L
vi-eo-ity
"~,
tt, llt.llt t I t I}1~
a 2/r/2 ,
r/bend = r/twist- °12 ] r/* ,
(24) (25)
it follows that
(20)
The Mi~sowicz viscosity coefficients are related to special shear flow experiments with simple geometries (see fig. 5). The Leslie coefficients are the transport coefficients appearing in the consitutive hydrodynamic equations for viscous stress tensor of a uniaxial fluid. At present the viscous properties of nematics are not understood very well. Fundamental theories based on molecular dynamics are scarce and incomplete. In order to compare the experimental data of ref. [1] with some theoretical predictions and experimental results for other nematic compounds we shall first derive the ratios of the Mi~sowicz and Leslie coefficients from the ratios of viscosity coefficients as reported in ref. [1]. We must restrict ourselves to calculating ratios, since these are the quantities that can be derived from light scattering experiments.
l
(22) (23)
Since the splay and bend viscosities are given by (cf. eqs. (15)--(17)) r/.ay = r / ~ . -
"}/1 = 0 ¢ 3 - 0~2'
r/, =
157
i
|t I I
"q,:½t-%.=,.%l
® ~
-~--.--~l .--: -.-:-.-.r
~:½t%.%.%l
: :;-- -:" -'.:'.": ." -)
r/splay = r/twist
r/~d = r/t,~t-
(r/twia + r/2- r/,)2 4r/2 , ( r / ~ s t + r/* -- 7"]2)2
4r/,
(26) (27)
If we define a reduced viscosity r/* by r/~*= r/ffr/t~,t, then the eqs. (24), (25) reduce to
r/,p,,y = 1
r/~.d* = 1
(1 + r/~ - r/]') 2 4nZ = -r/~'. H(r/~', r/~)'(28)
(1 + r/~' -- r/~) 2 = --ri~ " H(r/~, r/Z), 4r/T
(29)
where H(r/~, r/Z ) =
- ,79 = - 2(r/; + , 7 9 + 1 4r/,r/2
(30)
Note that the function H(r/7, r/~) is symmetric in r/~' and r/Z. It follows from eqs. (28), (29) that
'
Fig. 5. Viscosity coefficients of an aligned n e m a t i e liquid crystal as m e a s u r e d in a s h e a r flow e x p e r i m e n t , *Tt (i = 1, 2, 3) a n d their relations to the Leslie coefficients, ai (i = 1 . . . 6).
n,p,./r/~..d = r/i/r/2 •
(31)
Consequently we are able to calculate the ratios
J.P. van der Meulen and R.J.J. Ziflslra / Properties of the nematic liquid crystal, A P A P A
158
711"712:7] 3 from the experimentally determined values of *7~,~oy/~/be~aand .71/~/3. Note that one can find the ratios of the Mifsowicz coefficients without knowing the value of a ratio related to . 7 ~ . The ratios of the Leslie coefficients can now be determined with the exception of ratios involving 0/j = *Tn because, as was pointed out in ref. [1], it proved experimentally not feasible to defermine 7/~2. * From eqs. (24), (25) it follows that
0/~ = ~2(~twist- *Tsplay) ,
(32)
0/2 = *71(nl~st- T]bend) •
(33)
With the help of eq. (31) we now get 0/3) (*7~,iJ*7,q,,.y- 1 ) ":~ = (.7,,~,/.7,,..d-- 1)"
(34)
Consequently, only the absolute value of 0/3/0/2 can be determined from the measured values of *7~,~.y/*Tt~st and *Tbena/*7~,~t" On the basis of a molecular theory of flow alignment in pure nematics, where the molecules are considered as ellipsoids with hard cores, having length L and width W, Helfrich [20] predicted that --
=
,
(35)
0/2
which implies that the sign of a3/0/2 will be positive. From eqs. (19)-(21) and the relation 1 *73 = ~0/,t
(36)
we can derive the following relations between the Leslie coefficients and the ratios of the Mi¢sowicz viscosities: (0/5 - 0/2)/0/4 - rh/~T3- 1,
(37)
1,
(38)
(O/6 + 0/3)/0/4 = *72/*73-
(0/2 + 0/3)/a4 = (0/6- 0/5)/0/4 = (.72--
~1)/2.73
•
(39)
With the help of eqs. (34), (37)-(39) and 0/2/O'3> 0, it is now possible to calculate the ratios 0/2 : 0/3 : 0/4 : 0/5 : t~6. Note that in the relations of the viscosity coefficients presented above the sets of coefficients (%,~y, .71,-az, as) and (*Tb,,d, *72,0/3, 0/6) are interchangeable. assuming
3.2. Experimental results in terms of the M i f s o w i c z and Leslie coefficients The results of the calculations for the ratios of the Migsowicz and Leslie coefficients are given in table I, where we have used the data of ref. [1]. In fig. 6 we have plotted the results for la3/0/21, 0/6/0/5, -0/2/0/4 and as/o/4, as a function of the reduced temperature T I T m. Obviously the ratio 0/5/0/4 and 0/2/0/4 are rather independent of temperature. The ratios 0/3/0/2 and a6/0/5 are found to be small but not independent of temperature. We have assumed that 0/3/0/2>0, which is in agreement with the experimental data for I-IBBA and PAA [12]: Leenhouts [16] has estimated that at T / T m = 0.95, ( W / L ) z ~ 0.1, hence eq. (35) is in reasonable agreement with our data on 0/3/0/2" However, other equations, predicted by
Table I Calculated results of the ratios of the Migsowicz and Leslie viscosity coefficients as a function of the reduced t e m p e r a t u r e T/TN~ (Tin--- 109.2°C)
T]TNI
"01/~ (a)
172]'~ (a)
Y/3/~ (a)
O~2/~4
~3]014
O~5/O~4
O~6/a4
0.999 0.998 0.995 0.993 0.988 0.978 0.967 0.957 0.947
1.905 1.953 1.965 1.995 2.004 2.049 2.052 2.127 2.145
0.573 0.558 0.543 0.525 0.534 0.501 0.456 0.405 0.393
0.522 0.489 0.492 0.480 0.462 0.450 0.492 0.468 0.462
- 1.273 - 1.312 - 1.258 - 1.328 -1.343 - 1.440 - 1.389 - 1.620 - 1.691
0 -0.117 -0.188 -0.207 -0.252 -0.278 - 0.232 -0.220 -0.210
1.364 1.688 1.742 1.839 2.005 2.106 1.778 1.925 1.961
0.091 0.259 0.295 0.305 0.411 0.388 0.157 0.085 0.060
ZP. van der Meulen and R.ZJ. Ziflstra / Properties of the nematic liquid crystal, A P A P A I
I
[
I
I
~ / ~ - (7/,-
159
~(*))/~(').
(41)
A
04
2.0
-A x
~
X~
~4
0.2
0.0 0.94
[ ~
I 0.96
I
$--<- ~ 9 - - .
--71.0
I 0.98
0.0 1.00
[
TIT m
Fig. 6. Results of the calculations for la3/a2l (dots), a61a5 (circles), -ct2/a4 (crosses) and a5/a4 (triangles) obtained from our experimental data of ref. [1], versus T/Tm.
Helfrich [20], which relate the viscous properties to the sizes of the molecules, e.g. ~2/~/1 = (W/L) 4, conflict with our data. Some authors (G~ihwiller [21], Skarp et al. [22]), found for H B A B and 8CB a3/a2
3(7~1 at" 1']2 at" ~ 3 ) ,
Fig. 7 Shows the results for 7/~ (i = 1, 2, 3) versus the reduced temperature. We find that ~ < ~ / ~ ; this result differs considerably from results for other nematics published by various authors [12, 21-25] who found: ~/3 > 72. Since we found ~/~,Z~y= rhwia it follows that the inequality ~/~ ~ 7/~ implies, with the help of eqs. (20), (24) and (36), a 6 ~> 0. At present only a few theories are available to describe the temperature and order parameter dependence of the viscosity coefficients of nematics [26-29]. Imura et al. [28] stated that 7/~ (i = 1, 2, 3) can be written in terms of an isotropic part and a part which is linear with (P2). From their calculations it follows that 1
1
(42)
7/~o = ~(rh + r/2+ B3)- g71,
where */i~ is the extrapolated value of the viscosity of the isotropic phase. If a3/a 2 = 0, eq. (42) reduces to 1 r/iso --~ ~, (r/1 + 3r/2 + 2r/3 ) .
(43)
Imura et al. give the following relations for the viscosity coefficients: '7,-
= q " (P2),
(44)
I
I
I
I
I
APAPA i=1 A &
o R
i=3
(40)
g
Q
s
0" s s ~
i=2
where the superscript (a) denotes the arithmetical mean value of ~l (i = 1, 2, 3), we can define a reduced shear viscosity, representing the anisotropic parts of the viscous properties:
"10.94
--~
0.96
0.98
1.0o
TIT.m
Fig. 7. Results of the calculated values of ,/,gm,/d~(*)-1 (i = 1, 2, 3) versus T[Tm. Obviously ~2 = ~3-
J.P. van der Meulen and R.J.J. Zijlstra / Properties of the nematic liquid crystal, A P A P A
160 7 / 1 - 7/3 = G "
(P2),
(45)
where C~ and C 2 are constants. In order to check eqs. (44) and (45) experimentally we plotted (7/1- 7/2)/(7/a- 7/3), since we expected these ratios to be rather constant with varying temperature (see fig. 8). In fact, we find that (7/1- Wz)/(7/1- 7/3) varies by less than 10% between 0.94Tin and T m. Recent calculations by Marrucci [26] yield different relations between the Leslie coefficients and (Pz). According to Marrucci, 0/1 = C ( T ) "
[-6(P2)Zl,
a 2 = C ( T ) . [-6(P2)(1 + 2(P2))/(2 + (P2))], 0/3 = C ( T ) . [-6(P2)(1 - (Pz))/(2 + (P2))],
(46)
34 = C ( T ) . [2(1 - (P2))],
(50)
7/t~t = 9 (1 +(P2) ½(P2))" C ( T ) ,
(51)
7/b,,.d = O.
(52)
Marrucci's theory inspires little confidence as it leads to the unphysical prediction 7/~,d = 7/~,fsy= 0. Moreover, we found that the predicted ratios of 7/i ( i = 1,2,3) deviate strongly from our experimental results. Hence the fact that this theory correctly predicts a 6 = 0 should be considered fortuitous. Now we shall go on to check the isotropic part of 7/i (see eq. (42)). 7/isois independent of (P2) and corresponding with the extrapolated values of the viscosity in the isotropic phase, 7/(T; T >
T I).
a s = C ( T ) " [6(ez) ] , 0/6=0
7/splay = O ,
Kneppe et al. [23] suggested that (cf. eq. (40))
,
7/i~o= ~(a), where C ( T ) = const., k s T / 6 D r > O, and D~ is a rotational diffusion constant. From these relations it is possible to derive the Mi~sowicz and the distortion viscosities; one gets (cf. eqs. (15)(20) and (36))
7/1
=
(1 + 2(P2)) 2 (1 + ½(P2))
(1 - (P2)) = 7/2 = (1 + ½(P2)) 7/3 = (1 - (P2))" 1.1
C(T),
(47)
C(T),
(48)
C(T), t
(49) t
=,
i
i
which contradicts the theoretical results of Imura et al. [28] (see eq. (42)). As can be seen from fig. 8 we find typically a 20% deviation from eq. (53). Since we are unable t o measure the viscosity of a nematic above its clearing temperature T m by light scattering, we cannot determine 7/(T; T > Try1). In order to estimate an expression for 7/~o we calculated the physical mean value of the viscosity, which is related to the sum of the dissipative parts of the viscous stress tensor, defined as [12, 21] )du
t
(53)
1
t
t
7/'(0, ~b dx -=~(o'= + O'z,,),
(54)
APAPA
a
1-0
averaged over all angles 0 and 4~ (cf. fig. 9). We obtained for the physical mean value of 7/':
ql"q2 q~-93 o!9 . •
•
17 ,(p)
a•
0.8
1 1 1 = ]'~7/12 + ~(7/1 + Y/2+ 7/3)-- ?~')"1 = l o t I + 6(0/5 1 + 0/6) + ]13 4 "
I 0.9/.
) 0.96
I
I
(55)
I
0.98
1.00
I/T m
Fig. 8. The calculated results of (71 -- 7 2 ) / ( 7 1 -- 7 3 ) (triangles) and 7~0/~ (') (dots) versus T/TIn, w h e r e 7~,0 ~-- - 6171 (see eqs. (42) and (53)).
~{')
Note that this equation is similar to the expression for 7/i~o in eq. (42), except for the term 1 ~7/12, which is assumed to be negligible compared to the other viscosity coefficients in eq. (55)
ZP. van der Meulen and R . Z Z Ziflstra / Properties of the nematic liquid crystal, A P A P A
91
.
161
I
/
Z /" ,i
t
/ eele* I I
/
I
0
L
~ //.~
-
~
i /
i=2
Fig. 9. Definition of the orientation of the director ri0 with respect to the shear plane.
NI
I 0.0
-1
[12,23,24]. We can now try to describe the temperature dependence on a phenomenological basis, as was suggested by Kneppe et al. [23]. According to these authors the following relation is approximately satisfied: 7,((P2)) = @(~)(a,+
bi(P2) )
(i = 1, 2, 3).
(56)
In the isotropic phase, however, where (P2) = 0, ~/i reduces to ~(~) (i = 1, 2, 3) which implies that a i = 1 (i = 1, 2, 3). Consequently, because fi(~)= ~E~ 7~ it follows that Xi b~ = 0. Since 7~ > 0 (i = 1, 2, 3) for reasonable values of (P2) (0.3 < (P2) < 1) it follows that b i > - 1 . In fig. 10 we have plotted 7~ = (7,./~(')) - 1 versus (P2>, the latter being from Leenhouts [30]. Obviously eq. (56) is satisfied, but not with a~ = 1, since the lines do not intersect at the origin. One might expect that a better fit would be obtained if @(')in eq. (56) is replaced by 7too (cf. eq. (42)). In fact with this substitution our experimental data points fall on a curve which is shown in fig. 10. Clearly, there is not even a hint of a linear relation between these data points and (P2)Thus far we have only been considering ratios of viscosities. If we take the values of the twist elastic constant K2, as determined by Leenhouts [16], we are able to calculate the absolute values of 7 ~ or y~ from the twist viscoelastic ratio, and hence obtain absolute values for all other viscosities. The absolute values of 7~ (i = 1, 2, 3), are shown i n fig. 11. It follows from this figure that to a good approximation
/
-0"
1
""
I 0.5
1.0
(P2) Fig. 10. The reduced values r/~ m (~lil@('))- 1 (i = 1, 2, 3; d o t s , pluses, crosses respectively) and (~h/~/~)- 1 (triangles) versus the order parameter (/'2).
7, = 7 ° +/3~(P2),
(57)
with 7~ = -0.068 Pa s,
fll = 0.214 Pa s,
7~ = -0.007 Pa s,
/32 = 0.029 Pa s,
7] = -0.012 Pa s,
/33 ---- 0.042
Pa s.
By extrapolating the values for 7i given in fig. 11 one finds a)
71 = 72 = 73
Pct.s 0.1
for (/'2) = 0.33
~ /
APAPA
**
rli I
P'"
0.0 0.0
~ •
(P2 >
, NI
0.5
2 1.0
Fig. 11. The calculated absolute values of the shear viscosities ~ (i = 1, 2, 3; circles, dots, crosses, respectively) versus the order parameter (P2).
J.P. van der Meulen and R.J.J. Zi]lstra I Properties of the nematic liquid crystal, APAPA
162
instead of the expected value of (1)2) = 0, b)
~7i < 0
"3 /
I
I
i
I
i
I
for (/)2)< 0.3. ~
:(a)
Note that our results for eq. (57) are in agreement with the suggestion of Kneppe et al. [23], namely that the Mi¢sowicz viscosities rh are related to each other according to
•
q
~
/~6"J-~
~
l
~o
~
O ~
~o .....
o
o
,YLo m
Ai~ i = 0.
-s L
(58) tn ~(a)
i
From our results for ~o and /3~ it follows that A 1 : A 2 : A 3 = 0.114 : 0.608 : - 1 . These results are in reasonable agreement with the results of Kneppe et al. [23] who found mean values of 0.08 : 1.1 : - 1 for the compounds of N 4, M B B A , E M and 5CPB. W e are, however, not aware of any convincing theoretical argument for the existence of such a linear relation between the Migsowicz coefficients. W e shall now go on to consider the absolute values of ~/~o and ~ ) , which we expect to be rather independent of (P2) and related to the temperature, according to (cf. Imura et al. [28]) 71i~o, ~(~)~ e x p ( e o / k a T ) .
3.3. Concluding remarks conclude
that
APAPA
¢ I
-6
2.62
the
behaviour
of
the
I
I
I
2.70
11 T
I
I
2.78
"10 "3 K -1
Fig. 12. The calculated absolute values of In ~i~o(circles)and In ~(a) (dots) versus the inverse temperature. -3
I
I
I
I
i
I
X / fl= twist / "~-"
X//
,~
r?~
11=spMy
" o~
~ * .---
oo I -6
~NI 1-0
X
/'O
[}:bend In
/ 10
/5o "/
"/~
-4
(59)
In order to investigate eq. (59) we have plotted in fig. 12, In ~1~o, In ,i (a) versus lIT. T h e figure shows that the assumption that 7/i~o and ~(a) are independent of (P2) is not correct. In a previous publication (ref. [31]), we found that for O H M B B A In 3,1-(P2)/T. With this result in mind we have plotted In 7?~,~ay,In ~/~t and In ~/~nd versus (P2)/T in fig. 13. From the figure it can be seen that to a good approximation: , / . ( T ) = C..exp(e.(P2)/kaT), ( t z ~ s p l a y , twist, bend), where C, and e. are constants. Note that the above relation suggests that these distortion coefficients will not tend to zero when (P2) goes to zero, i.e. in the isotropic phase, although it is not obvious that such an extrapolation is justified.
We
In q i s o
./-
•"
t l
o I
(P2)/T
o I~°
APAPA I 1.2
I
I
I
1.4 .10.3K.1
I %.6
Fig. 13. Results of the calculations of the natural logarithm of the distortion viscosities, In ~?~,~y (circles), In ~?t~ (crosses) and In ~Tb~ (dots), versus (P2)IT. Note that ~?t,,~ = yl.
Migsowicz coefficients, as determined by light scattering, is in qualitative agreement with data obtained from flow experiments done with other compounds. T h e r e is however one difference, specific in that ~/2 = ~3 or a6/as<~0.2. Flow experiments on other compounds (MBBA, P A A and H B A B ) find rather large negative values for
ZP. van der Meulen and R.ZJ. Ziflstra / Properlies of the nematic liquid crystal, A P A P A
4
I
I /
APAPA
I /I /
2_ _ / 2K]1~ KI*K2| O/ 0
1
i
and simulating discussions. We are also grateful to Sheila McNab for making linguistic improvements and to Wendelies van der Meulen for typing the manuscript. This work was performed as part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (ZWO).
oo
I
I
'~" 2 (~h+~3)/2~1z
1
I
4
Fig. 14. The calculated ratios 2K3/(KI + 1(2) versus (~1 + ~73)/ 272. The curve refers to the prediction of Lin-Liu et al. [32] (see eq. (60)).
this ratio (typically -0.5). It is not obvious whether this difference is real, in the sense that the properties of A P A P A are rather different from the above-mentioned compounds, or that it is an artifact due to the differences in experimental techniques. In this context it should be noted that flow experiments are performed under conditions which inquire rather large perturbations (although supposedly still negligible) on the nematic. In contrast, light scattering experiments probe the relaxation of spontaneous fluctuations in equilibrium. Finally, we would point out that there is a similarity between the results for K]~ (fig. 2) and ~ (fig. 7). Although a linking of the static and dynamic properties seems to be rather tentative, Lin-Liu et al. [32] predicted on a molecular basis, taking the molecular sizes into account, that
2K3/(KI +/(2) =
071 + ¢/3)/2¢/2 •
163
(6o)
Fig. 14 shows that the elastic and viscous properties do indeed seem to be related. The experimental data satisfy eq. (60) reasonably well. On the basis of this result we suggest that more theoretical studies should be done on the relation between the elastic and viscous properties of nematics and that in these studies the sizes of the molecules should be taken into account.
Acknowledgments W e wish to thank D r . D . Frenkel for valuable
References [1] J.P. van der Meulen and R.J.J. Zijistra, J. Physique 45 (1984) 1627. See also, J.P. van der Meulen, Ph.D. Thesis, University of Utrecht (1983), Ch. 3. [2] A. Sanpe, Z. Naturforsch. A15 (1960) 810. J. Nehring and A. Saupe, J. Chem. Phys. 56 (1972) 5527. [3] R.G. Priest, Mol. Cryst. Liq. Cryst. 17 (1972) 129. [4] R.G. Priest, Phys. Rev. A7 (1973) 720. [5] B.W. van der Meet, F. Postma, A.J. Dekker and W.H. de Jeu, Mol. Phys. 45 (1982) 1227. [6] J.P. Straley, Phys. Rev. A8 (1973) 2181. [7] J. Stecki and A. Poniewierski, Mol. Phys. 41 (1980) 1451. A. Poniewierski and J. Stecki, Phys. Rev. A25 (1982) 2368. [8] G. Vertogen, S.D.P. Flapper and C. Dullemond, J. Chem. Phys. 76 (1982) 616. G. Vertogen, Physica 117A (1983) 227. [9] T.E. Faber, Proc. R. Soc. Lond. A353 (1977) 247; A375 (1981) 579. [10] W.H. de Jeu, Mol. Cryst. Liq. Cryst. 63 (1981) 83. [11] F.C. Frank, Disc. Faraday Soc. 25 (1958) 19. [12] W.H. de Jeu, Physical Properties of Liquid Crystalline Materials (Gordon and Breach, New York, 1980); see also, Phys. Lett. 69A (1978) 122. [13] W.H. de Jeu, F. Leenhouts and F. Postma, in: Advances in Liquid Crystal Research and Applications, ed. L. Bata (Pergamon Press, Oxford, 1980), p. 499. [14] W. Maier and A. Saupe, Z. Naturforsch. 13a (1958) 564; 14a (1959) 882; 15a (1960) 287. [15] B.W. van der Meer and G. Vertogen, Phys. Lett. 71A (1979) 486. See also, B.W. van der Meer, Ph.D. Thesis, Univ. of Groningen (1979). B.W. van der Meer and G. Vertogan, in: The Molecular Physics of Liquid Crystals, eds. G.R. Luckhurst and G.W. Gray (Academic Press, London, 1979), Ch. 6, p. 159. [16] F. l.,eenhouts and A.J. Dekker, J. Chem. Phys. 74 (1981) 1956. See also, F. Leenhouts, Ph.D. Thesis, Univ. of Groningen (1979). [17] M. Miesowicz, Bull. Intern. Acad. Poion. (Sci. Math. Nat.) Ser. A. (1936) 228; Nature 158 (1946) 27; note that ~1 and 72 are interchanged.
164
J.P. van der Meulen and R.J.J. Zijlstra / Properties of the nematic liquid crystal, A P A P A
[18] F.M.I.~slie, Quart. J. Mech. Appl. 19 (1966) 357; see also, in: Advances in Liquid Crystals, vol. IV, ed. G.H. Brown (Academic Press, New York, 1979), p. 1. [19] O. Parodi, J. Physique 31 (1970) 581. [20] W. Hclfrich, J. Chem. Phys. 50 (1969) 100; 53 (1970) 2267. [21] Ch. (3iihwiller,/Viol. Cryst. Liq. Cryst. 20 (1973) 301. [22] K. Skarp, T. Carlsson, S.T. Lagerwall and B. Stebler, Mol. Cryst. Liq. Cryst. 66 (1981) 199. [23] H. Kneppe, F. Schneider and N.K. Shazma, Bet. Bunsenges. Phys. Chem. 85 (1981) 784. [24] W.W. Beens and W.H. de Jeu, J. Physique 44 (1983) 129.
[25] H. Kneppe and F. Schneider, Mol. Cryst. Liq. Cryst. 65 (1981) 189. [26] G. Marrucci, Mol. Cryst. Liq. Cryst. l.,¢tt. 72 (1982) 153. [27] S. Hess, Z. Naturforsch. 30a (1975) 1224. [28] H. Imura and K. Okano, Jpn. J. Appl. Phys. 11 (1972) 1440. [29] A.O. Tseber, Magnetohydrodynamics 14 (1978) 267. [30] F. Leenhouts, W.H. de Jeu and A.J. Dekker, J. Physique 40 (1979) 989. See also, F. l_,eenhouts, Ph.D. Thesis, Univ. of Groningen (1979). [31] J.P. van der Meulen and R.J.J. Zijlstra, J. de Physique 43 (1981) 411. [32] I.R. Lin-Liu, I.M. Shih and C.-W. Woo, Phys. Lett. 57A (1976) 43.