Magnetohydrodynamic effects in the nematic mesophase

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G.R. LUCKHURST and H.J. SMITH

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of~Chemistry,

7% Urrivenity. Southm&ton.

SO9 5hW. UK

Received 11 October 1971

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theory is employed

to investigate

the behaviour

of G-I; director.when

a nematic mesophase

is sub-

” __&xed to d mg&tetic tield while the sample tube is spun, The p&dictions of the theory are tested, with the aid of ,’ ~elcctron rkionktce’spectroscopy, and found to be in complete accord with experiment. In additioir the director is .: .: -. observed to b.ehaire in 3 manner not predicted by the theory. : .. . : . . .‘, : ,.. ,. .. _:‘. : : ‘..‘.. . . ‘-. _. .-. The application polarity.is absent, d and .--c! are’ihdistinguishable. of a.stiqng magnetic field to the 2. mesopha.se of a netiatogen gives rise eo a’ torque upon When the alignment of the director is uniform, the -continuum equatipns reduce to [S, 61: the pref&d inolecular orientation whfch tends to ‘, akgn ii parallel to the field (see, for example ref. [I].). vi i = 0 , If ,tie tube _con~ai~ng ,th& mesophase is no-W spun abou’t its Iting axis, which is drthogonal to the field, pi;‘F,ft.. a.i ‘. -the pref&red’arient&on experiences a second torque ‘. .. :. and .:. 6n acc&.mt of its motion reiative to the mesophase: Zyetkoff (21 and Lippmann [3] adopt a rather crvde a;ii$$+gi. ,. :.. ‘(I>.. :m.estimate for.this latter torque based on the concept -These e&&ions have beeri,siniplified by using,the’ of swanin theory; in order to predict the behaviour of ,.

F.-’th~.pref~r~ed.orientation-in the.rotat~ng sample. Their .. tilculations ignore entirely the influetice of the tt&e .G:walls~u$&i theg:eferred orientation, &ce, the’y argue,. this iS impo&rit otity in.the’inimediate vitiiriity of the ‘,Ic ..ti~lls.:In .$I$ @ter we present. the.corresp’onding anal-‘. :

+is based on. th&&tipium the&~~&f.the nematic .-::m~sop&ati~[4j i a&j:find per&s stirprisingly that

_- .’ ‘.,

cartesian tensor convention bf s~rn~ing over repeated subscripts, $z icitill aS.the comma notation which @II- ” plies diff&x&tion with respect td a ,spatial coordinete. The ielo&ty vector of the bulk material is de: noted by v,‘~ & the d&i,ty of the-fluid and is assumed ‘to be cans&+~ The in&t&-constknk asso&tted with .the diiector is. tabelled u. The tionstitutive relatioris,,

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Volume 13, number 4

CHEMICALRH’YSICS LEnERS

1972.

15 .h&ti

in which.

where p0 is an arbitrary co&nt:BcIow‘a ‘criticalanY velocity -Q, the.diffeiential equation (11) yields the simple solution. : ,1 : .. _,- ..- ,-_,’

Di =cii -‘Wijdj,

sin 2lj = slji2,

gj=ydi+XIDitX2Aikdk,

gular

where W is the local rotational fluid

wi;= (Vi. i-vj_i)/2 ,

velocity

tensor

of the

:lT& = -A&/2$

,

(5)

In eq. (2) p is the hydrostatic pressure and like y in eq. (3) is an arbitrary scalar. The constant Coefficients A, and X2 haire the dimension (and magnitude) of and are related

to the p’s by

Theoretically the coefficient Xl is predicted to be negative [6]. IVhen the mesophase is subject to a uni-

..

(14)

tan ((sl’-~f)“2(t~t,)}

form magnetic field B the body force F and the external body force C take the forms [7] F,=-$

_,

& other words, the .diiecior makes a constant‘angle q with the-magnetic field: However, when the speed of rotation is greater than flc eq..(l I) does nothave a steady state soiution. Often the-molecular inertia in the system is taken to be’negligibTe and so the term involving u in eq. (t 1) may be neglected_ If this is so. then the differential equation reduces to that obtained by Zwetkoff [Z] and for rotation speeds greater. than 52, the solution is

(6)

Aii = (vi if~j i)/2 . I ,

(13)

where

and A is the shear rate tensor

viscosity

;

= -

.I ,

-.

and

((C2tS2c)j(S24Zc))“’

ran@-n/4).,

(19

where r,-, is an arbitrary constant. The director is therefore predicted to rotate with a mean angular velocity w @ven by w = ($$_fif)1’2

.*

(16)

Here + is the gravitational potential and Ax is the difference in the components of the diamagnetic sus-

It is different to analyse how the interaction behveenthe surface of the sample tube and the meso-ceptibility parallel and per_pendicular to the director. phase affects these predictions. However, in a recent In the experiment with which we are concerned”fhe paper [S] de Gennes considers the related problem for s&n&e is rotated with constant angular velocity Sz ‘. a nematic mesophase in contact with a plane solid about thex ads, while the field is applied along the z; - .bbundary. He discusses the production and migration. axis. The components of Y and B in the spatial coordiof inversion walls from the surface when a magnetic nate system defined by these two directions are then field rotates about the normal to the boundary. For the present problem de Gennes’calculations suggest .‘. = vy=.s-k: v;=-iy, v 0, .x that the influence of the boundaries may extend- into and the bulk of the sample, even when the coherence ‘: ‘(9) length [l ] is smali compared with the diameter of, the B,=B: B,=O, By=O’ tube. However, for rotational’speeds less than R,, the .Under these conditions eq. (i) has the solution experimental results described below. are in good-agree: ment .with our predictions. On the other- hand; for’. .. dv = sin&r) ,’ dz = cos$(r) , (10) : dx’O, rotational speeds greater-than.this‘cri:icaI value the’ .. provided the time dependent angle cp satisfies behaviour of Ihe’mesophase.differs from that. ‘pre- -: dieted byeq; (lS).This discie&incy-‘may stem from ’ sj’~~Xl~(&SZ)fAx& sinlpcosg=O,’ (Ilj ,_ an effect s.imilar,to~that discussed by de Cennes: .A& 1 .: ternatively, there may be a solutiortof t&5’continu&t~ .: and equatioriS~corres@ondmg to.a non-uniform .aligi&knt ., ., -. ,‘_, _p+_+=p~ (12). :: .-, ,._ ,_ ,:: ;..

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15 March 1972

CHEXIJCAL PHYSICS LETTERS

: volumi l~;‘n&nbcr 4 :

bf the diiector.

ti@pared

Clea{ly, if the ttibe.diameter

is large

u$h _the’Coherencelength [I] then the pre-

d&ions should still .be valid except close to the wall; However, when the speed of rotation is greater than S2c there ma be solutions of the continuum equations in whitih the alignment of the director is non-uniform. Indeed, as‘we shaU see tine such solution is encountered

expkrti.entally. -en

a paramagnetic

probe is dissolved

in a nema-

ing point

of 16°C and a nematic-isotropic

transition

at 76°C. The electron resorknce spectrum of the probe was measured as a function of the rotation speed at 21°C. Below the critical speed the spacing between the three nitrogen hyperfine lines increases with increasing R. The value of the nitrogen coupling

con-

was used, in conjunction with eq. (17), to determine the angle cp.The dependence of cpon the ro-

stant

tation

speed is accurately

fitted

by eq. (13) and so

tic.mesophase the observed hyperfine coupling con-

confirms the first theoretical prediction. Our results

stant Z differs from its isotropic value u because of the partial alignment and the anisotropy in the hyperfine tensor [9]. In addition, the hyperfine coupling in the mesophase depends on the orientation of the director with respect to the magnetic field [lo] . Jf ‘p is the angle between the Geld and the director then

are therefore in accord with the conclusions of earlier, tl- _gh less precise, nuclear magnetic resonance ex+riments [3, 1 l] . The analysis of our results gives 51, as 0.122 Hz which means that Ax/h1 is equal to -2.24 X 10B8 cm set g-l for Phase IV at 21”C, unfortunately no other determinations of thiS ratio are available. Clearly this technique provides a convenient method for determining AdX1 over the entire range of the mesophase and such measurements are in progress. When the sample is rotated above the critical speed the spacing between the hyperfine lines is seen to oscillate in time, which implies that the orientation of

where c and x are the parklly averaged g and hyperfine tensors’ [lo]. Because the ordering potential is cylindrically symmetric with respect to the director both Z and x also possess cylindrical symmetry. The component of the tensor parallel to the.director is given by

(18) and the perpendicular

component

is (19)

There are similar expressions for the components of the; tensor 1. The prime in eqs. (18) and (19) denotes the anisotropic part of the total tensor and 0 is 1 thtq drdering matrix which indicates the extent of -solute alignment .[9] . Since the parameters involved in eq. (17) can be determined experimentally the .&agn.it&e of Z(p) can be employed to measure the :-a&e which the director makes with the magnetic field_- ._ :. ,.

YThe paramagnetic probe used in our experiments WaS 2,2,6,6-tetramethylpiperidine decanoat-4, oxyl-1 aiid the ne’kgttigen was.Phase IV *, which has a melt: .: .: * &k ner+&en, Ph&e Iv is supplied byE. Merck, Dan& ’ -.stad!; and-is ihouj$t to be a mixture of the -iwo isomers 4~ : :: methoxyl4’-n-bu~l-NNO-azoxybe~ene jnd 4-mcthoxy4’-: ,.(,n-buiyi-oNk&&ybe~~ne( ,,, .y_- .._: i,.;_ I ,_... : ?,o. _::: -, ,: ‘-. ,,I.-. :;__:,:: ::. ” .. . ‘,.~‘_... :_‘, ..I .,.. : ‘..’ -..‘:’., j : :: -:

the director

is time dependent.

Further,

the periodi-

city of these oscillations is the inverse of approximately twice the rotation speed of the sample tube. This behaviour is in accord with the qualitative predictions of our theory for the angular velocity, w, of the director is approximately equal to that of the sample tube and since Z(p) is an even function of +J the periodicity of Z should be about (252)-l. The following experiment permits a more preciese test of eq. (16). The magnetic field is held constant and equal to the resonkt field for one of the hyperfine lines when the tube is stationary. The sample is then rotated w!th a speed greater than Q, and the output from the electron resonance spectrometer is recorded as a function of time. This mode of presentation, which might be called a time spectrum, is shown for a rotation speed of 0.333 Hz in fig. I; Since the resonant fields are also even functions of 4 the periodidty of the time spectrum should be (ZW)-~. We have obtained values for the rotation speed of the director as’a function of R : by measuring the periodicity from the initial regions of the time spectra._ The expe;rimental dependence of. .’ w on the sample rotaiiori spied is well &presented by .: eq. (16) and so ctii~tirms our second predictipn. The time spectrum showti.kifig. 1.demonstrates.. .. ,, ._ .. . . . ..,.__

‘:

Volume ij, A&W

:

CHEMICAL PHYSIC& LE’ITERS

4

15 h&l;.i972.

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spectrum

obtained

from a paramagnetic

that rotation of the director is not .the stable state of the system. Presumably the interaction of the dirkctor with the surface of the tube forces the system to adopt a static solution in which the director is not uniformly oriented with respect to the magnetic field. In this situation the observed spectrum is the sum of spectra from all orientations

of the director

which then’cor-

responds to a polycrystailine spectrum. Such a polycrystalline spectrum is observed and we are attempting to anatyse this spectrum in order’to determine the angular distribution function of the director. The spectrum given in fig. 1 also exhibits a pr& ‘nounced beat pattern which is observed at other rotation.speeds and for the mesopha’se of other nematrigens. Of course, a beat pattern is to be expected if the director rotates, not with a.constant angtllar velocity, but with several allowed speeds. We have investigated this possibility by Fourier analysing the time spectrum and the Fourier transform does indeed contain several peaks co~esponding to different rotation speeds Of the director. Ihree’ p&s occur at a frequency of about 252, atid the& are responsible for the beats in the time’spec‘trum. We believe that the presence of several frequencies close to 2R.is associated with the way in which the rotation speed is suddenly c@&ed from’zero to the final speed. indeed, if the’. “sample is first rotated with a speed less than 51, and then akelergted to a speed abc&the critical speed, _ then

the Fo-Jrier

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essentially

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transform

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probe dissolved in Phase IV and rota&d at 0.333 Hz.

peak at frequency 2Q. In,addition to the expected frequencies the Fourier transform also indicates the existence of higher ha~onics_ Their presenc.e is of considerable interest since they indicate a non-linear response of .the system to the sudden application of the viscous torque, and we are now investigating this behaviour in greater depth. ‘. We are grateful to Dr. J. Roberts and Mr. P.D. . Francis for their assistance in computing the Fourier transforms. We also wish to thank the Science Research Council for the award of a Studentship to N.J. Smith and for a grant towards’the cost of the equipment.

References [ 1) :2] [3] [4]

P.G.de Gennes, Mol. Cryst Liquid’Cry&7. (1969) 32.5. V.Zwetkov. Acta Physicohim. USSR IO (1939) 55.7. ” .H.Lippmann, A~n:Physik 1 (!9>8) 157. F.hf.Leslie. Arch. Rat. hlech. A.naf. 28 (1968) 265. ” -[S] J.L.Eri+sen, Arch..Rat hfech. Anal. 4 (1960) 231. ‘, .’ [6] F.hf.Leslie, QuaH. J. hfech. AppL Math. 19 (1966) 357:. [7] J.L.Ericksen, Arch. Rat.:hfech. An& 23 (1966) 266. .l [ 81 P.G;dk Gennes. I. Phys (Paris). to be publisheb .. [9].H.R.FzUe and G.&Luc&hur&.i. hfyy. Resonke i : (1970, 161. : [IO] G.R.Ltickburs~ and F.&d~ol+, rfdl.;Phys. 21 {lqii)::.,,:I .: : : 349. [ 11’1 ,P.Diehl and C.L.Kh&apal, hiol; P’h$% 14 (1967) 2k)i’., 1

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