Homogeneous instability in the gravity flow of nematic liquid crystals

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Homogeneous instability in the gravity flow of nematic liquid crystals

U.D. Kini

To cite this version:

U.D. Kini. Homogeneous instability in the gravity flow of nematic liquid crystals. Journal de Physique,

1984, 45 (12), pp.1875-1881. �10.1051/jphys:0198400450120187500�. �jpa-00209931�

Homogeneous instability ^{in the} gravity flow of nematic liquid crystals

U. D. Kini

Raman Research Institute, Bangalore, ^{560 080, India}

(Reçu ^{le 17 mai 1984,} accepté le 21 août 1984)

Résumé. 2014 Le seuil de l’instabilité homogène (HI) ^d’un cristal liquide nématique

écoulement le long ^d’un plan

incliné est étudié dans le cadre d’un modèle simple développé récemment. Pour

nématique s’alignant ^dans l’écoulement, ^{le taux} de cisaillement seuil de l’HI est déterminé

fonction de l’anisotropie de tension superficielle

03C3a ^et de l’intensité du champ magnétique appliqué. ^{On montre} que l’HI peut également

produire ^dans

^néma- tique qui

s’aligne pas à condition d’appliquer

champ magnétique déstabilisant suffisamment fort dirigé

suivant l’écoulement

perpendiculairement à la surface libre de la couche fluide. On étudie le champ ^seuil

fonction du taux de cisaillement déstabilisant et de 03C3a.

Abstract.

Homogeneous instability (HI) threshold for the flow of

nematic liquid crystal ^down

^inclined plane ^is investigated using

simple model which

recently developed. ^For

flow-aligning nematic, ^{the HI} shear rate threshold is studied

function of surface tension anisotropy 03C3a and applied magnetic ^field strength.

It is shown that HI may

in non-flow-aligning nematics in the presence of

sufficiently strong destabilizing

field which is applied along ^{the flow}

^{normal to} the free surface of the layer. The HI field threshold is studied

function of stabilizing ^{shear rate and} 03C3a.

Classification

Physics ^Abstracts

61.30G - 47.20

1. Introduction.

The continuum theory ^of liquid crystals [1-5] ^has

been useful for understanding ^a variety of instabilities which occur in shear flow and plane Poiseuille flow of nematic liquid crystals, with the director initially

oriented normal to the shear plane [6-13]. ^{In all these}

cases, the nematic is confined between two flat plates

and subjected ^to shear either by moving .one plate

relative to the other or by subjecting ^{the fluid to}

pressure gradient. ^{While HI is not} ordinarily possible

for a non-flow-aligning ^nematic [12], it has been shown

on the basis of the continuum theory [14] ^{that HI}

may ^occur in such materials when the stabilizing

effect of shear against homogeneous perturbations ^is

counteracted by applying ^a destabilizing magnetic

field either parallel ^{to the flow or} along ^the primary velocity gradient. This has also been extended to the

case where the shearing ^is produced by ^{free convec-}

tion [15].

Recently Chilingaryan et ^al. [16] have studied the

gravity ^{flow of a} flow-aligning ^nematic (such ^as MBBA) ^{down an inclined} plane, ^on the basis of a

simple mathematical model. They have shown that the initial orientation of the director, which is normal to the shear plane may get disrupted ^{above a HI} threshold; they have considered two extreme values

of _Qa and have also studied

possible orientational

optical non-linearity ^{in the} vicinity of the HI threshold.

In this communication, ^{the model} developed ⁱⁿ [16]

has been used to make a more detailed investigation

of the HI threshold, ^{as a} function of _a. and an applied magnetic field. It is again shown that HI _may occur

in

non-flow-aligning ^nematic (such ^as HBAB)

when the stabilizing effect due to shear is overcome

by

sufficiently strong destabilizing field which is

applied along ^{the flow or normal to} the free surface of the layer. The field threshold is investigated ^{as a}

function of _Ca and stabilizing ^{shear rate.} Orientational

optical non-linearity ^{is not} considered in the present work.

2. Mathematical model, differential equations ^and boundary conditions.

Except for the presence of

magnetic ^field along ^some specific direction and the absence of an electromagne- tic wave the model is the same as that used in [16];

the same symbols have been used ^as far as possible.

However, it is restated for completeness. ^{The flow}

is assumed to occur along + y ^{direction on the} xy

plane

^{= 0} which makes an angle 4/ with the hori- zontal. In steady state, ^{the fluid} layer ^has

thickness h

along

axis and width I along x.‘ The volume rate

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450120187500

1876

of flow is Q ^{cm 3 s-1.} The nematic director is assumed to be strongly ^anchored along x

the inclined

plane and is also assumed to be oriented uniformly along

throughout ^the sample ^{in the} steady ^state.

Thus in the steady state, the director and velocity

fields

described by

with Vyo(z)=pgz(2 h - z) (sin §)/2 fJa’ Q = pglh3(sin §)/

3 11a. The vanishing of the shear stress at the free sur-

face

h and the no-slip ^condition [17] ^{have been}

used in deriving ^the expression ^for _vyo. Perturbations

are

imposed ^{on n° and vo such that}

where n, nz, vx ^are functions of

alone. Time depen-

dence is ignored ^for investigating ^{the HI threshold}

which is non oscillatory in character. Perturbation of primary flow has also been ignored ⁱⁿ anticipation [16]. Using the continuum theory ^one gets ^the following

system of linearized equations :

where r is the total torque ^{exerted on the} director,

S

S(z)

pg(h - z) (sin §)/ ’1a

dvyoldz ^{the velo-} city gradient ^{of the} unperturbed ^{flow, xa the diamagne-}

tic susceptibility anisotropy and Hx, Hy, Hz ^the compo- nents of

magnetic ^{field H which is assumed to be}

directed either along

^or along y

along ^{z, but not} along any general direction. Equations (3)-(5) ^have

been so written in order to avoid writing separate equations ^{for the case of} Hx ^or Hy ^{or HZ. The boundary}

.conditions for the perturbations ^are [16]

dvx/dz

K1 dnz/dz ⁺ (1a nz

The vanishing ^of perturbations ^at

⁰ corresponds

to the no-slip condition and firm director anchoring

on the inclined plane. ^The vanishing ^{of the} gradient

of the velocity perturbation vx reflects the vanishing

of the shear stress at the free surface [17]. ^{This has}

been expressed ⁱⁿ equation (5) ^{also. The two} remaining

conditions on the director gradients arise from the finiteness of the anchoring energy ^at the free surface

[18, 19]. It is convenient to transform to the dimen- sionless variable Z

z/h. Equations (3)-(5) ^{can be}

written

The boundary conditions (6) ^become

As _per the convention described earlier, A

= 0

A 3

for the field-free case; A1, A3 ^{0 for}

stabilizing

field Hx ; A 1 > 0, A3

^{0 for}

destabilizing ^field Hz ; A 1

0, A3 > 0 for a destabilizing ^field Hy. ^{In the case}

of a stabilizing ^field Hx, it is natural to use the quantity

A

(A1 A3)1/2 ^for measuring ^the stabilizing ^effect

of the field [9]. ^For

given case, the compatibility ^of equations (7), (8) ^and (11) gives the threshold. Relative

or normalized profiles ^{of the} perturbations ^{can also}

be calculated using equations (7-11).

In [16], solutions have been presented ^{in terms of} Airy functions. Keeping in mind numerical results,

a

series solution method has been used in the present work. Care has been taken to retain sufficient number of terms in the series to ensure convergence. As the HI mechanism is well understood [6-13], only ^the

results have been presented.

A unique feature of gravity flow is the in built asymmetry ^{which stems from an} asymmetry ^{in the} primary ^flow itself, ^{which is zero at the bound surface}

and reaches a maximum value at the free surface.

The primary ^{shear rate} is maximum at the bound surface and decreases to zero at the free surface.

This asymmetry ^is reflected in the variation of the

perturbations ^{with z, as} also in their boundary ^condi-

tions. Thus there is only ^{one mode in} gravity ^flow.

(Due ^{to the} symmetry ^of primary ^flow in shear flow and plane Poiseuille flow, ^{there exist more than one} uncoupled mode, ^{out of which} only ^{one will be} gene-

rally favourable [9, 11 ]. Under certain circumstances,

an unfavourable mode can become more favourable,

as for instance in plane Poiseuille flow [14. 20]). ^Thus

the lowest value of the destabilizing entity ^{needs to be}

taken as the threshold It is possible, ^{under a} given situation, ^{to find a} higher value for the destabilizing entity ^{at threshold But this} threshold, ^{which corres-} ponds ^to greater distortions of the perturbation pro- files (higher harmonics, ^{if one} may ^use the term) ^is

found to be generally higher and hence unfavourable.

Another unique feature of gravity ^{flow is that the}

average shear rate S

pg(sin §) h/2 fla ^{is propor-}

tional to the layer ^thickness h, ^{and not} inversely proportional, ^{as in the case} of shear flow. This can

lead to significant differences between the two flows, especially ^{in the} presence of magnetic ^fields.

3. Results for a flow-aligning ^nematic (MBBA).

MBBA has been chosen

the model liquid. ^The

material constants have been taken to have the

following ^values (see [9] for relevant literature) :

K 1

⁶

^{10 -’} dyne, K2

³

^{10 -’} dyne,

xa

1.15

10-’ _{cgs, a2}

0.775 poise,

a3 = - 0.012 poise, fla = 0.416 poise, fib =0.248 poise.

For

given value of field and _Qa, h is varied till the threshold condition is fulfilled for h

h,. ^{It is conve-}

nient to measure the destabilizing effect of shear at threshold by ^the parameter [16]

For _6a

0 and for _{(Ja »} 0, one recovers the results of [16]. ^{For MBBA} parameters, me has been studied

as

function of magnetic field and _{(J a.} The surface tension anisotropy ^{is a material} property ^{of the} liquid-

gas interface and should therefore have

fixed value.

However it is known that interfacial properties depend strongly ^{on various factors, such as the} pre-

sence of impurities. ^Hence

variation of _Qa is not

completely irrelevant, especially ⁱⁿ

model calcula- tion, where it is being ^{used as a} parameter ^for studying

its effect on the threshold. It is also known [16] ^that

aa

0 implies ^no effect of the free surface on the director orientation; ^if Qa > 0, the free surface tends to orient the director parallel ^{to its} plane; ^if

6a 0, ^{the free surface tends to} orient the director normal to itself. While it is natural to investigate ^the positive range of _{(J a} values, ^{it is} tempting ^{to see the}

effects of negative values of the surface tension ani- sotropy

^{the HI threshold.}

Figure ¹ illustrates the variation of me ^with magnetic

field and _{(J a. In} general mc increases with _Qa [16]. ^The

variation of _me with Hx (Fig. 1 a), ^with Hy ^{(Fig. 1 b)}

and with Hz (Fig. 1 c) ^is qualitatively ^{clear. However,}

as compared ^to shear flow and plane Poiseuille flow

[14, 20], ^{there are two} important differences. For H,

mc decreases towards zero when A2 -+ ^2.47 (approxi- mately n2 /4), regardless of the value of the surface tension anisotropy. ^{In the case of} Hz, mc tends to zero

for different values of A1, depending upon the value of _6a. In the case of shear flow ^or plane Poiseuille flow,

the HI threshold decreases ^{to zero} when Hy ^{or Hz}

attains the corresponding Freedericsz value (when

the corresponding dimensionless

vector A 1

A3 attains the Freedericsz value n2 ). Also in the present case, the limit of the HI threshold 0 (i.e.

me -+ 0) ^is

purely mathematical one, possible only

in the mathematical limit of infinite Hy

^{H,,, since}

Fig. 1.

^Variation of HI threshold shear rate with magnetic

field and surface tension anisotropy ^for

flow-aligning

nematic (MBBA). (a) ^{Plot of} me vs. A for

stabilizing ^field H;xe (1a = (1)1 0 ^{(2) - 5}

^10-’ dyne cm -1;

^for higher positive values of a.

close to

1 and have not been shown. (b) ^Plot of me

A3 ^for

destabilizing ^field Hy. Ua

(1)0.023(2)0(3) - ⁵

^10-4 dyne cm-’. (c) ^Plot

of me vs. A, ^for

destabilizing ^field Hz. (1a = (1) ^0.023 (2)4.3

^10-4 (3) 0 (4) - ^1.44

^10-4 dyne cm -1. (d) ^Plot

of me vs. a. for different stabilizing values of Hx ^field.

Hx

(1) ⁰ (2) ¹⁰⁰⁰ (3) 1 500 gauss.

the shear rate is directly proportional ^{to the} sample

thickness.

The differences between the Hy ^and Hz ^{cases in the}

limit of low _me or high field values can be qualitatively

understood to be a consequence of equations (6) ^or (11). ^Consider

static, horizontal (§

0) ^nematic layer of thickness h bounded by the free surface

z

= h, ^with the nematic director aligned parallel ^to

throughout ^the sample. ^The coupling ^between ny

and nz which exists in the presence of shear rate, is ^no

longer present. ^Thus equations (3) ^and (4) ^become (ignoring Hx ^{for the} present)

with boundary conditions selected from equation (6).

For ny ^one gets

1878

The critical field H F ^is independent Of Ua ^{and corres-} ponds ^to A3F

qz h2

^{n’/4 ;:z: 2.47. In the case of}

nz, ^one finds that

where X2 ^is again

^{constant. Thus for} Ua

0, ç = n/2 ^or A1F

n2/4. ^For large, positive (fa’ ç = n, A1F

n2. However, when ca ^takes

negative values, ç decreases from n/2, reaching ^{zero when}

Qa

1. Thus for Qa - K 1/ h, the surface exerts

sufficient destabilizing ^{influence on the} homogeneous

director orientation, ^{so that even} in the absence of

destabilizing H, ^field,

splay distortion of the director field becomes possible. These considerations show

why ^the Hy ^{and H, cases} differ for gravity ^{flow in} respect ^of (fa variation. Before going ^{over to the next} part ^{of the} discussion, it may be worth mentioning

that in the presence of ^a stabilizing ^field H X’ ^the

twist distortion ny ^{cannot exist.} But nz ^{can come into}

existence for negative Qa, when the strength ^{of the} stabilizing field falls below

value HxF, ^where

However, ^{this case} may be only of academic interest.

The above considerations, being for the static case are also valid for

non-flow-aligning ^nematic.

Profiles of perturbations ^at threshold have been

presented ⁱⁿ figures ² and 3. The profiles ^{show the}

effects of both field and _Ua. The effect of applied ^fields

is not very evident for a.

0.023 dyne ^{cm -1} (Fig. 3, corresponding ^to

fairly high positive ^{value of} aj.

The strong anchoring ^{is found to} quench ^the splay perturbation ^{near the free surface; the} profiles ^are

similar to those in [16], for the field-free case. However, the effect of applied ^{field is more} marked in the case

of Qa

⁰ (Fig. 2), especially ^{with a} stabilizing Hx

field which quenches ^both the ny ^and the nz perturba-

tions in the vicinity ^{of the free surface} where the desta-

bilizing effect of shear rate approaches ^zero.

Before passing ^{over to the next} section, ^{one wonders}

what might happen ^{for the case of a} flow-aligning

nematic with X. ^{0. For such a} material, Hy ^{and Hz}

act as stabilising fields and one finds the expected

increase of _me with I A1 I or I A3 1. ^{However, the case}

of the Hx field, ^{which is now} destabilizing, ^{is more} interesting. ^{The field} couples ^to both the director

perturbations; me decreases as A

(A 1 A3)1/2 ^increa-

ses. However, ^{as the twist} perturbation ny is unaffected

by ^{the free surface} directly [Eq. (6)], ^for sufficiently large Hx fields, when me ^{becomes small, A tends to}

the same limit AF regardless of the value of _a.. A

preliminary calculation, (by taking xa

^1.15

10-’ cgs) ^{shows that} AF ~ rr.2/4J2.

Fig. ^2.

Perturbation profiles ^{at HI} threshold for

flow-

aligning ^nematic (MBBA). cr.

^0. (a), (b), (c) The field-free case. me

7.84. (d), (e), (f) Stabilizing ^field Hx

2 400 gauss;

me

76 ; A

^34. (g), (h), (i) Destabilizing ^field Hy

2 100 gauss; me

1.3; A3

^2.4. (j), (k), (1) Destabilizing

field Hz

³ 600 gauss; me

0.74; A

2.5.

Fig. ^3.

Perturbation profiles ^{at HI threshold of}

^flow- aligning ^nematic (MBBA). a.

^0.023 dyne ^cm-1. (a), (b), (c) Field-free case; me

12.7. (d), (e), (f) Stabilizing ^field Hx

2 400 gauss; me

77 ; A

^34. (g), (h), (i) ^Destabi- lizing ^field Hv

^{2 100} gauss; me

1.3; A3

^2.4. (j), (k), (1) Destabilizing ^field Hz

3 600 gauss; me

4.6;

A

^8.3.

4. Results for

non-flow-aligning ^nematic (HBAB).

HBAB has been chosen

the model fluid to represent this class. The material constants have been assumed to have the following ^{values :} K1

^8.44

^{10 -’} dyne, K2 ^{= 4.78}

^10-’ dyne, xa ^{= 0.745}

^10-’ cgs, a2 = -0.327 poise, CX3 =0.0034 poise, fla =0.1373 poise,

fib = 0.0881 poise (see [21] for relevant literature). ^For

a

given value of shear rate (or m) ^and Ua, the field Hy

or

Hz is varied till the threshold condition is realized

Naturally, the threshold value of the field is higher

than the corresponding ^{static threshold} given by equation (15)

(16).

Figure ⁴ illustrates the variation of the HI field threshold with m and _Qa. The increase of Alc

A3c

with m is to be expected. ^{However, it is} again possible

to see two differences between the present ^{case and} that of shear flow [14, 20]. For small m (near ^the hydrostatic limit) ^the Hy threshold has nearly ^the

same value regardless ^of Ca (Fig. 4a); ^the Hz ^field

threshold has different values in the same limit (Fig. 4b).

This difference has already been dealt with in section 3.

For sufficiently large ^m, (high values of shear rate),

the Hy threshold increases with decreasing ^{Ua; the}

reverse is true for the Hz threshold. The behaviour is different from that of the flow aligning ^{case, described}

in the previous ^case. Figures ^{5 and 6} give ^a sample

of the perturbation profiles ^at threshold. As expected,

Fig. ^4.

^Variation of HI field threshold with stabilizing

shear rate (m) and surface tension anisotropy oe. (a) ^Plot

of A3e

^for Hy ^{field; (Ja}

^{(1) 0.1, (2) 4}

^{10-4 (3) 0 (4)}

- 2

10-4 dyne ^cm-1. (b) ^Plot of A1e

^for Hz field;

(Ja = (1) ^0.1 (2) ⁴

^10-4 (3) ⁰ (4) - ⁴

^10-4 dyne ^{cm -1.}

(c) ^{Plot of} A3e at Hy threshold for different shear rates (m)

vs.

(Ja; m

(1) ³³ (2) ¹⁴ (3) ^4. (d) ^{Plot of} A1c

(Ja for diffe- rent shear rates (m) ^at Hz threshold, m

(1) ⁶⁴ (2) ⁸ (3) ^0.5.

Fig. ^5.

Perturbation profiles ^{at HI} field threshold for

non-flow-aligning ^nematic (HBAB). oa

0. (a), (b), (c)

m ⁼

0.014; A3c

^2.47. Hy ^field. (d), (e), (f)

8; A3c = 4.7 ; Hy ^{field. (g), (h), (i)}

^{0.9; A1c}

^{2.5; Hz field. (j),} (k), (1)

14.0; A1e

8.1 ; Hz ^field.

Fig. ^6.

Perturbation profiles ^{at HI} field threshold for

non-flow-aligning ^nematic (HBAB) Qa = 0.1 dyne ^cm-1.

(a), (b), (c)

0.014; A3c

2.47; Hy ^field. (d), (e), (f)

m ⁼

8; A3c

^3.3; Hy ^field. (g), (h), (i)

0.9; A1c

9.8 ; Hz field; (j), (k), (1)

14.0 ; A1c

21.0; Hz ^field.

1880

the Hy field has

pronounced ^effect

^the

twist fluctuation than

nz. The H., field affects the

splay perturbation

Figures 4c and 4d show the variation of the field threshold with _{(J a} over

small _range of positive ^and negative values, for diffe- rent shear rates (or m). ^{There is} again ^seen

^marked

difference between the Hy ^and H,, thresholds. When (Ja decreases for

given shear rate, the Hy ^threshold

increases and attains

limiting value when _{(J a} becomes

negative (Fig. 4c). ^The Hz ^{threshold on the other} hand,

decreases and tends to zero when _Qa attains a critical

negative ^value (Jac which depends

the value of m

(Fig. 4d). Figure ^4d prompts ^{one to} consider the

possibility ^{of HI} threshold when _(Ja becomes suffi-

ciently negative ^{and the} destabilizing effect of the free surface counteracts the stabilizing effect of the shear rate. A calculation shows that _(Jac takes on large negative values for small shear rates (i.e. ^{for thin} layers) where the stabilizing effects of the elastic torques arising ^from strong ^director anchoring ^at

the bound surface, ^are prominent. ^{When m increases,}

(J ac increases and tends ^to

limiting negative ^value,

via oscillations, when the shear rate attains large

values (thick layers). If the material has xa ^0,

the Hy ^{and H,, fields are of no interest, as they stabilise}

the orientation against homogeneous perturbations.

The H, ^{field has} destabilizing effect and preliminary

calculations show that in the limit of low shear rate,

Ac ^{tends to the same limit} n’/4.,, ,./2, which is inde-

pendent ^of (Ja. This is again probably ^{due to the fact}

that Hx couples ^to both the director perturbations,

one of which is not directly ^affected by the free surface.

5. Concluding remarks ; limitations of the model used in the present ^work.

In conclusion, ^it is necessary ^to remark some of the salient limitations of the model used in the present

work. Firstly, the director orientation is assumed to be uniform throughout ^the sample, ^{in the} steady

state. It is known (see for instance [22-25]) ^{that the}

angle ^made by the director ^at the free surface can

vary from

nematic to another; while for PAA the director orients almost parallel ^{to the free} surface,

for MBBA the director orientation at the free surface is known to be almost vertical. If the director is ^not oriented parallel ^{to the free surface, director} gradients

can exist in the sample, ^{even in the} steady ^{state. One}

should also not forget ^the possibility of surface dis- clinations. Secondly, ^{it is} implicitly assumed that flow can occur for _any thickness of the fluid layer.

This _may not be valid when the fluid layer ^is very thin and adheres to the inclined plane. ^The present model should of course be better valid for sufficiently

thick layers. A critical layer thickness, beyond ^which

flow can occur will have to be incorporated ^{into the} present model inorder to bring the calculations closer to reality. ^For

flow-aligning ^nematic, the critical value of m determines the HI threshold. For a given

value of _mc, the layer ^{thickness at threshold will be} larger ^when §, ^the angle of the incline is smaller

[Eqs. (10) ^and (12)]. ^The present model should be better applicable ^{for such a case.} Lastly, ^{the model}

will not be valid when roll instability (RI) ^{is more}

favourable. RI is known to occur in flow-aligning

nematics when a sufficiently strong stabilizing ^field (Hx in ^the present case) ^is applied [7]. ^{In the field-}

free _case, RI is known to occur in non-flow-aligning nematics, ^when

sufficiently large ^{shear rate is} imposed ^{on the} sample [12]. Hence, ^{the curves in} figure ¹

for MBBA and those in figures ^{4a and 4b}

for HBAB _may not be valid for large enough ^values

of Hx ^and

respectively. ^It should be interesting

to study ^the possibility ^of occurrence of RI in the

present ^{case, for} gravity ^flow.

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