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SHEAR FLOW OF CHOLESTERICS NORMAL TO THE HELICAL AXIS
U. Kini
To cite this version:
U. Kini. SHEAR FLOW OF CHOLESTERICS NORMAL TO THE HELICAL AXIS. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-62-C3-66. �10.1051/jphyscol:1979314�. �jpa-00218710�
SHEAR FLOW OF CHOLESTERICS NORMAL TO THE HELICAL AXIS
U. D. KINI
Raman Research Institute, Bangalore 560006, India
Abstract. — Analytical and numerical calculations of the steady state shear flow of a cholesteric between two plane parallel plates which are normal to the helical axis and at a fixed distance h apart are presented on the basis of Leslie's continuum theory. The material chosen is nematic MBBA in which a twist is assumed to be imposed by the addition of chiralic impurities with no appreciable change in its elastic and viscous properties. The director is assumed to be anchored at the plates in such a way that the twist in the structure is undistorted in the absence of flow. The apparent viscosity t], the orientation and velocity profiles are studied as functions of pitchy and shear rate s.
For very small s, r\ is analytically shown to be a function of/> and h but independent of s, in quali- tative agreement with the observations of Candau et al. on the Poiseuille flow of a twisted nematic.
When/> is varied r\ exhibits oscillations which are pronounced when/> ~ h. For large p, r\ asymptoti- cally approaches a lower limit. For p < h, the oscillations are quite negligible and r\ for different p saturates to almost the same value for low s, again in agreement with the results of Candau et al.
For given p and h, numerical calculations show that on increasing s from a low value r\ decreases from a constant upper limit »j0 (given exactly by the previous analytical considerations) tending to a lower limit for large s. The shear rate *c at which r\ starts decreasing from rjc is predictably higher for lower p.
1. Introduction. — The flow of cholesterics normal to the helical axis was investigated by Leslie [1] on the basis of the continuum theory. He considered the case of shear flow between two plane parallel plates, with one plate moving relative to the other with a constant velocity V, the helical axis of the cholesteric being normal to the plates. He set up the fundamental differential equations and obtained analytical solu- tions for high shear rates. He was able to conclude that (i) the apparent viscosity r\ depends on the pitch, shear rate and sample thickness, (ii) rj approaches a value which is independent of the shear rate at high shear rates, (iii) at low shear rates the twist in the
structure persists throughout the gap, (iv) at any non-zero shear rate there is secondary flow (i.e., flow normal to the plane of applied shear) which is a direct consequence of the intrinsic twist in a choles- teric, (v) a temperature gradient can be set up in the fluid as a consequence of thermomechanical coupl- ing which is unique to cholesterics. Leslie imposed the usual no-slip boundary condition on the velocity but assumed that the twist in the structure goes over to its value in the absence of distortion, with the director parallel to the plates.
Candau et al. [2] studied the Poiseuille flow of a nematic (viz., Merck IV) twisted by adding choles- JOURNAL DE PHYSIQUE Colloque C 3 , supplément au n° 4, Tome 40, Avril 1979, page C3-62
Résumé. — Des calculs analytiques et numériques de l'écoulement stationnaire en cisaillement d'un cholestérique entre deux surfaces planes parallèles maintenues à distance fixe h sont présentés sur les bases de la théorie de Leslie du milieu continu. On a étudié la phase nématique du MBBA dans laquelle la torsion est induite par addition d'impuretés chirales ; on admet que cette addition ne change pas de manière appréciable les propriétés élastiques et visqueuses du milieu. Le directeur est supposé ancré sur les surfaces de telle manière, qu'en l'absence d'écoulement, la torsion dans la structure ne soit pas perturbée. La viscosité apparente rj, l'orientation et les profils d'écoulement sont étudiés en fonction du pas de l'hélice p et de la vitesse de cisaillement .s.
Pour s petit, on montre que rç est une fonction analytique de pet h et est indépendante de s, en accord qualitatif avec les observations de Candau et coll. sur l'écoulement de Poiseuille d'un nématic chiral.
Lorsque h varie r\ présente des oscillations prononcées pour/? ~ b. Pour/? grand, t\ tend asymptoti- quement vers une limite inférieure. Pour p -4 h les oscillations sont quasiment négligeables et r\
sature pratiquement aux mêmes valeurs que pour s petit pour les différentes valeurs de />, encore en accord avec les résultats de Candau et coll.
Apeth donnés, les valeurs numériques montrent qu'en augmentant s depuis les faibles valeurs r\
décroît à partir de la limite supérieure rjc (prévue exactement par les considérations analytiques précé- dentes) vers une limite inférieure pour s large. La vitesse de cisaillement sc à laquelle r\ commence à décroître depuis r\a est prévue plus élevée pour/; petit.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979314
SHEAR FLOW OF CHOLESTERICS NORMAL TO HELICAL AXIS C3-63
teryl propionate in different quantities to produce samples of different pitches. Their optical observa- tion with polarised light showed that the helical axis of the structure was everywhere normal to the tube axis with the different layers folded around the tube axis to form coaxial cylinders. They found that in general for a given pitch with increasing shear rate q decreases from a constant value at low shear rates and approaches a lower-limit at high shear rates. The initial constant value of q itself depends on the pitch. The critical shear rate at which q starts decreasing from its initial value depends linearly on the square of the pitch.
In this paper the differential equations set up in [l]
for flow between parallel plates have been solved numerically for the case of twisted MBBA and PAA ignoring thermomechanical coupling. The apparent
viscosity has been studied as a function of shear rate.
The results reproduce trends observed by Candau et al. [2]. Orientation and velocity profiles have been presented for different shear rates. The differential equations have been solved analytically for low shear rates where q is independent of the shear rate but exhibits interesting oscillations with change in pitch.
2. Differential equations. - The cholesteric is assumed to be sheared between two plane parallel plates occupying the planes z = f h/2, the plate z =
+
h/2 moving relative to the plate z = - h/2 with a constant velocity V along the x axis. In the hydrostatic limit the cholesteric is assumed to have its helical axis parallel to z. Following [l] solutions are sought for the director and velocity fields in the formn, = cos 8(z) cos cp(z) , n, = cos 8(z) sin cp(z)
,
n, = sin 8(z)v x =
4.4 ,
0, = v(z>,
u z = o .On ignoring temperature effects one has for steady state flow the following differential equations :
[HI
+
Hz cos2 cp] u'+
H z sin cp cos cpv' = 2 a (2.1) [HI+
Hz sin2 cp] v'+
H 2 sin cp cos cpu' = 0 (2.2)f l 0"
+
(8')' (dfl/d0)/2 - (cp')2(df2/d8)/2 - 2 k2 sin 8 cos 8cp'+
+ ( ~ , + A 2 c o s 2 8 ) ( u r c o s c p + v ' s i n c p ) / 2 = 0 (2.3)
f 2 cp"
+
(df2/d8) 0' cp'+
2 k2 sin 8 cos 88'+
(A, - A,) sin I3 cos 8(ur sin cp - v' cos q)/2 = 0 (2.4)where
f i = k l cos2 6 f k 3 3 sin2 8 , f 2 = (kz2 cos2 8
+
k33 sin2 8) cos2 8 ,H , = HI(@ = p4
+
(p5 - p2) sin2 8 , Hz = Hz(@ = (2 p, sin2 8+
p3+
p6) cos2 8 , A 1 = p 2 - p 3 , A 2 = p 5 - p 6 , u l = d u / d z , etc.,(( a >> is the shear stress applied to the fluid, k2 and kii the elastic constants of a cholesteric and the pk the six viscosity coefficients.
One can show from eqs. (2.1)-(2.4) that 8 and v are even functions of z while cp and u
-
V/2 are odd.Leslie [l] has suggested boundary conditions which are such that
where p is the pitch and q the wave vector of the unperturbed cholesteric. The derivative of cp is equat- ed to q at the boundaries instead of fixing the twist angle cp. This implies that at any shear rate the choles- teric retains its twist in the vicinity of the plates. So far it has not been possible to determine experimen- tally the precise boundary conditions that can be imposed on the cholesteric director in practical situa-
tions involving flow. In an experiment as in [2] a twisted nematic is used as a substitute for a choles- teric. As is well known the orientation of the nematic director can be strongly influenced by proper surface treatment of the plates. Hence for investigating the flow of twisted nematics a more reasonable set of boundary conditions would be
which fixes cp rather than dcpldz at the boundary.
At very low shear rates dcpldz will be close to q in the entire gap, but at high shear rates its value can differ appreciably from q even in the vicinity of the plates.
However as emphasized later the nature of the solu- tions is qualitatively the same for both types of boundary conditions.
3. Low shear rates. - In the region of low shear rates eqs. (2.1)-(2.4) can be solved analytically because of negligible distortion in the cholesteric structure.
We can write cp = qz
+
4(z), where $I is small.Second and higher powers OT 8, u, v, and their derivatives are assumed to be negligible. Noting that
cos q w cos qz -
4
sin qz andsin q w sin qz
+ 4
cos qz we can write eqs. (2.1) and (2.2) in the formu' = 2 a(H1
+
Hz sinZ qz)/[H,(H,+
Hz)] (3.1) v' = 2 aHz sin qz cos qz/[H,(H,+
H,)] (3.2)where H1 = p4 and H2 = p 3
+
p6.Solving (3.1) and (3.2) with (2.6) one gets
~ ( z ) = a[(2 H ,
+
Hz) (h/2+
z) -- H2(sin 2 qz
+
sin qh)/2 q]/[H,(H,+
Hz)]~ ( z ) = aH2(cos 2 qz - cos qh)/[2 qH,(H,
+
Hz)].
The velocity V of the plate z = h/2 is V = u(hl2) = a[h(2 H I
+
H,) -The apparent viscosity v] = ah/V is given by
v] = q(qh)= Hl(H,
+
H2)/[2 H1 + H z -Hz sin qhlqh](3.3) Thus y is a function of the sample thickness and pitch and is independent of the shear rate at low shear rates.
Whereas v is an even function of z, u - V/2 is an odd function.
When q + 0 (the case of an untwisted nematic) (sin qh)/qh -, 1
and
corresponding to a nematic flowing with the director oriented along the flow direction in the entire sample.
On the other hand when q becomes extremely large (as it would for a highly twisted cholesteric) (sin qh)/qh
=
0 andA plot of y(q)/v](O) as a function of the pitch for a 100 pm sample is given in figure 1 for MBBA and PAA.
Values for the viscosity coefficients of MBBA have been taken from Gahwiller [3] while those of PAA have been chosen from Tseng et al. [4]. It is assumed
FIG. 1. - Variation of apparent viscosity q with pitch p(= 2 xlq) for low shear rates. Plot of q(q)/q(O) versusp for (a) twisted MBBA, (b) twisted PAA ; h = 100 pm. The horizontal dashed lines corres-
pond to ?(co)lri(O).
that these values are unaffected by the presence of a small quantity of the chiral additive. The function tends to the value 1 for large pitches but in the region of small pitches it oscillates around the value q(co)/v](O).
As the pitch increases from a low value the extrema become more widely spaced and differ more and more from y(oo)/v](O). When the pitch attains the value p, the function attains the highest maximum. On increas- ing the pitch further the function decreases conti- nuously attaining the value 1 at very large pitches.
The extrema of y(q) can be shown to be the roots of the transcendental equation tan qh = qh. The trivial and lowest root q = 0 corresponds to the case of the untwisted nematic. The next higher root occurs when q, h = 4.485 and corresponds to the highest maximum so that p, = 2 nh14.485. Also since v](qh) involves the p, the strength of oscillation can be diffe- rent for different compounds as is evident from figure 1 . Experimental study on the variation of v] with pitch at low shear rates has not been reported so far in the literature. To the order of the approximation used here one finds from eqs. (2.3) and (2.4) that 4(z) = 0 and that
8(z) = a(A,
+
A2) [cos (qz) - cosh (kz) xx cos (qh/2)/cosh (kh/2)]/[(k2
+
q2)with k2 = k 3 3 q2/k1
,.
Thus at low shear rates 0, v and u are all directly proportional to the applied shear stress a. It is satis- factory to note that all the above results can also be arrived at on the basis of the boundary conditions
SHEAR FLOW O F CHOLESTERICS NORMAL TO HELICAL AXIS C3-65
(2.5). Thus at low shear rates one can obtain identical results on the basis of either of the boundary condi- tions specified earlier. It is of course obvious that for a given pitch the apparent viscosity should exhibit similar oscillations with change in sample thickness.
The above results are in qualitative agreement with the experimental observations of [2]. But quantitative comparison is difficult because of the following rea- sons : Firstly the geometries are different ; secondly Candau et al. [2] have not gone to sufficiently low shear rates for any pitch and thirdly they have used tube radii of the order of 500 pm and pitches of the order of 10 pm so that even at low shear rates the y values for different pitches may not differ appreciably.
(The oscillations in y damp out quickly as we go to lower pitch values.)
4. General solution. - The non-linearity of the eqs. (2.1)-2.4) prevents analytical solution at any general value of the shear rate. A model calculation has been performed for MBBA with the boundary conditions (2.6). The eqs. (2.1)-(2.4) have been numerically solved by the orthogonal collocation method [5] using the zeroes of the 24th Legendre polynomial as collocation points and double preci- sion arithmetic. The elastic constant values reported in [6] have been used. For every value of (( a )) there corresponds a velocity V of the moving plate from which the apparent viscosity can be calculated : y = ah/V. A plot of y as a function of V for different pitches is shown in figure 2a for a sample thickness of 50 pm. The curves are in good qualitative agree- ment with those presented in [2]. For pitches of 10, 41 and 70 pm y is nearly constant (y,) at low V values. As V increases y decreases finally approaching the value y, z 0.262 poise, y, being the apparent viscosity of an untwisted nematic which is flowing with the director aligned at an angle of 8, = 112 COS-' ( - 1 , / A 2 ) :
For smaller pitches y starts decreasing from y, at higher V values since higher energies are required to distort a structure of lower pitch. At low V values y, is highest for p = 70 pm and has lower values for p = 10 pm and p = 41 pm in this order. This is a direct consequence of the pitch variation of y at low shear rates discussed earlier. However for p = oo (corresponding to the untwisted nematic) we find a different behaviour. At low V values y is constant at a value of (p3
+
p4+
p6)/2 (= 0.238 poise) corres- ponding to the apparent viscosity of a nematic flow- ing with the director aligned parallel to the flow.As V increases y also increases approaching the value y, at high shear rates. (Since 8, is zero for PAA there would be no change in y with shear rate for this nematic.) On the other hand ifp were zero (corres- ponding to a highly twisted cholesteric) y would remain constant at the value y(m) at all finite shear rates and
FIG. 2. - a) Apparent viscosity ;rl of twisted MBBA as a function of the velocity V of the moving plate for p (p) = (1) 70 (2) 10 (3) 41 (4) m. The dashed horizontal line is for p = 0. b) Plot of the twist p at the boundaries ( z =
*
h/2) as a function of V forh = 5 0 p n a n d p = l o p .
the plot of y vs. V would be the dashed horizontal line in figure 2a. However no simple relation between pitch and the shear rate at which y starts decreasing from y, have been found in the present calculations.
The plots such as those for p = 41 pm and 70 pm cannot be found in the experimental results of [2]
because as mentioned earlier they have not used pitches of the order of the sample thickness.
In the light of boundary conditions (2.6) it is interesting to plot Q = q ' ( z = h/2) as a function of shear rate. It is found that at low shear rates
Q
remains almost constant at the value q (Fig. 2b).
However on increasing the shear rate,
Q
starts increas- ing the increase at high shear rates being as high as 30%.
This is just the consequence of fixing angle cp at the plates. As the shear rate is increased the struc- ture starts unwinding near the centre of the gap which is farthest from the plates and consequent to the angle cp being fixed at the platesQ
the twist in the vicinity of the plates increases. This indicates that the solution of eqs. (2.1)-(2.4) with boundary condi- tions (2.5) can yield quantitatively different results since the solution would then be sought in such a way thatQ
= q at all shear rates. But the qualitative nature of the results are not likely to change.FIG. 3. -Orientation and velocity profiles for p = 41 pm and h = 50 pm plotted as functions of = 2 z/h for different V. a) tilt, angle 0 ; V (cm s- ') = 1) 0.317,2) 0.097 4,3) 0.030 4 ; b) primary velocity u ; V (cm s-') = 1) 1.043, 2) 0.0974, 3) 0.962 x lo-', 4) 0.961 x 5) 0.961 x c) sin cp ; V (cm s-') =
1) 0.175 3, 2) 0.097 4, 3) 0.030 4 ; d ) secondary velocity v ; V(cm s-') = 1) 0.030 4, 2) 0.097 4, 3) 0.175 3.
sin cp has been plotted since it can reflect the twist in the director orientation. The primary velocity u
(Fig. 3b) shows the simplest profile. At any shear rate u increases from zero at the stationary plate to the value V at the moving plate. The profiles of sin 9, u and 8 are more interesting. Since cp is antisymmetric sin cp always becomes zero at the centre of the gap (Fig. 3c). The distortion in the twist with increasing shear rate can be clearly made out from figure 3c.
But the general nature of the profile is not lost even at shear rates where the apparent viscosity starts approaching its lower limit; the maxima are not appreciably shifted as compared to the low shear rate. The profiles of 8 and v are symmetric and the number of zeroes of these functions in the sample is determined by the ratio of pitch to sample thickness.
The peak values up of u increase in general with shear rate though at any shear rate up/V
x
lo-'. Thus net secondary flow is considerably smaller than the pri- mary flow. The peak values of 8 are found not to increase beyond about 8, = 112 cos-'
(- 1 , / A 2 )( z 0.15 radians for MBBA) even with increase of shear rate. At extremdy high shear rates if steady state flow prevails Leslie [l] predicts that the twist and secondary velocity vanish and 8 = O0 in the sample except in thin layers very close to the bounda- ries. Calculations for such high shear rates have not been attempted here.
Orientation and velocity profiles have been pre- Acknowledgment. - The author thanks Professor sented in figure 3 for a pitch of 41 pm and a sample S. Chandrasekhar for encouragement and useful thickness of 50 pm. Since cp changes over large values discussions.
References
[l] LESLIE, F. M., Mol. Cryst. Liquid Cryst. 7 (1969) 407. [4] TSENG, H. C., SILVER, D. L., FINLAYSON, B. A., Phys. Fluids
- -
15 (1972) 1213.
L2] CANDAU, S., P.3 DEBEAUV*1S3 F.3 C'R' Hebd' [5] FINLAYSON, B. A., The Method of Weighted Residuals and SPan. Acad. Sci. A 277 (1973) 769. Variational Principles (Academic Press) 1972, Chapter 5.
131 GAHWILLER, C., Phys. Lett. A 36 (1971) 311. [6] HALLER, I., J, Chern. Phys. 57 (1972) 1400.
