Dynamics of collapse of a smectic bubble

DYNAMICS OF COLLAPSE OF A SMECTIC BUBBLE P. Oswald*

The University ^of Chicago, ^The James Franck Institute, 5640 Ellis Avenue, Chicago, ^Illinois 60637, ^U.S.A.

(Reçu ^{le 23 janvier 198?’, accepte le 8} avril 1987)

Résumé.- L’effondrement d’une bulle smectique ^est analysé théoriquement ^et expérimentalement. ^De

cette étude est déduite la première ^{mesure directe du} coefficient de perméation.

Abstract.- The dynamics ^of collapse ^{of a} smectic bubble is analysed theoretically ^and experimentally.

From this study ^{the first direct} measurement of the permeation coefficient is deduced.

Classification

Physics ^Abstracts

61.30J

Introduction

In the same way ^{as a} fluid can flow inside a porous

medium, the molecules of a smectic A liquid crystal ^can

flow through ^its layers. ^The originality here is that the smectic acts as the _porous medium ^as well as the fluid.

This flow, ^called permeation and introduced for the first time by W. Helfrich [1] ^{in 1969, is described by}

the following phenomenological equation [2] :

u is the displacement ^{of the} layers, ^{vz the} velocity ^{in a}

direction normal to the layers ^{and G the elastic force}

associated with the deformation of the layers (free ^en- ergy F) :

Ap ^{is called the} permeation coefficient. Although ^it plays ^an important ^{role is smectic} dynamics, ^{it has}

never been measured in a direct _way.

In this article we show that by measuring ^{the col-} lapse ^{of a} bubble, ^one gets ^{a direct} measurement of

Ap*

*

Adresse permanente : Universite de Paris-Sud,

Laboratoire de Physique ^des Solides, ^Bat. 510, ⁹¹⁴⁰⁵ Orsay Cedex, ^France

Two basic experiments have been tried prior ^to

the one we present ^{in this} paper.

-

The first one consists in applying ^a pressure gra- dient p’ ^{normal to the} layers (Fig. la). ^A permeation

flow results from it [3] :

This expression ^{assumes that the} layers ^{are anchored}

on the walls (du/dt

0). Experimentally, ^{this con-}

dition is not satisfied, ^the layers gliding easily ^{on the}

walls through defect motion (glide ^{of edge} dislocations),

and all measurement of the permeation velocity ^be-

comes impossible.

-

The second _one, namely ^the creep by compres- sion normal to the layers (Fig. Ib) ^is governed by per- meation only ^{under the} assumption that the solid sur-

faces act as perfect ^{sinks for the} layers [3]. Experimen- tally, this condition is not satisfied because molecules

are strongly ^{anchored on the} walls, ^{and the} permeation

flow which is theoretically foreseen does not appear.

In fact, ^the creep is governed by the climb of edge

dislocations [4] ^{and by their mobility m under stress.}

Let -(1 be the applied ^{stress and v the} velocity ^{of a}

dislocation. By definition, ^we put :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004806089700

Fig.la.- Theoretical permeation ^{flow under a pressure} gradient ^{normal to the} layers. ^The velocity ^{anneals on}

the wall through ^{a thin} boundary layer ^{of thickness A.}

Fig.lb.- Creep ^under compression ^{normal to the} layers

and flow lines when the smectic layers ^{are eliminated}

on the walls.

In reality, ^{it is} possible ^to express the mobility ^of

the edge dislocations as a function of the permeation

coefficients Ap ^{and a} typical viscosity t7 [3,5] :

{3 ^{is a} numerical coeflicient assumed to be close to 1, but has not been exactly calculated. The previous equation ^{assumes that the} hydrodynamic equations ^are

valid at the scale of a dislocation. This hypothesis ^is generally ^well accepted. Experimentally, ^the mobility

of the dislocations has been measured and a value of the permeation coefficient can be deduced from the above equation. ^{In the} thermotropic systems (Sm ^A phases) ^{one finds typically} Ap N 10- 13CM2 /poise [4,6].

This value is in good agreement ^{with the} assumption usually made that the length :

is of the order of magnitude ^{of a molecular} length [7].

One should note that much smaller values of Ap, ^{of the}

order 10-30cm2/poise, ^have also been measured in a

lyotropic system (La ^lamellar phase), ^{which is a very} puzzling ^result [8].

In fact, ^{this method of} measurement of Ap ^{via the}

dislocations is _very indirect, ^even fairly inaccurate ({3

in Eq.(5) ^is unknown) ^and above all subject ^{to the}

restrictions made concerning ^{the use of the} hydrody-

namical equations ^{at the} molecular scale.

In this article, ^we propose ^a direct method to mea- sure this coefficient. The experiment consists in deflat-

ing ^{a smectic bubble} through ^{a thin} capillary ^{and mea-} suring ^its collapse time. Before describing ^this experi-

ment we study theoretically ^{the formation of a smectic}

film and its dynamical ^{behaviour under} stretching ^with

the help ^{of the} dynamical equations.

1. The dynamics ^of stretching ^{of a} smectic film : theoretical aspect

In this paragraph, ^we consider the case of a planar

and of a spherical ^film successively.

1.1 PLANAR FILM.- Let us consider a planar ^{film be-}

tween two holders (Fig.2). ^{Let e} be the thickness of this film and L its width along ^the >axis. The film is treated as though infinite in the y-direction. ^The

smectic layers ^are perpendicular ^{to the .axis. Let us}

call F the force (per ^unit area) applied ^to the film. The

equations ^{to solve are the} following ^{ones :}

where P is the _pressure and A

82 j8x2 ⁺ 82 /8z2.

Fig.2.- Smectic film between two holders.

The incompressibility ^{condition must be added :}

In the above situation, ^the layers ^{are fixed and} du/dt=O (the convection terms are negligible). ^{By us-}

ing equations (1,8) ^can be rewritten in the form :

The solution of equations (7), (9) ^and (10) ^{can be}

easily found. It is given by :

where v and C are two numbers ( v ^{is the velocity of}

the _upper plate, ^{the lower} being supposed immobile)

to be determined by using ^the boundary conditions :

and

7 is the surface tension of the film and a is the z- component ^{of the} stress tensor :

B is the compressibility modulus of the layers. ^In gen-

eral, the viscous part ^{of the stress is} wholly negligible

in comparison with the elastic one.

In fact, ^{it can be} easily ^{seen that the conditions}

(14) ^and (15) ^{are not} sufficient to determine the two

parameters ^{v and c. A} further relation must be intro-

duced, ^a constitutive equation ^on the surfaces :

nz is the z-component of the unit vector normal to the surface and oriented towards the outside of the film. This phenomenological equation (of ^{the same}

nature ^as the permeation Eq.) ^{describes the ability}

of the molecules to quit ^{the free surface. Notice that}

the dissipated energy by ^{unit area is} equal ^to (l/ç) (du/dt - vz)2 and so £ ^{must be} positive. Furthermore

£ ^{must be related to the} density p of the surface dis- locations (expressed ⁱⁿ cm/cm 2) ^{and to their mobility}

m (defined ^{as in Eq.} (5)) :

We can now calculate explicitly ^the two constant numbers v and Cwith the help ^of equations (14), (15)

and (17). ^It yields :

and

It is of interest to calculate the _energy which is dissi-

pated per unit area of film during ^the stretching. ^{It is} given by

In view of the fact that v

dL/dt ^{and that the}

volume V

eL of the film is constant, ^{we obtain}

In this formula, ^{the first term} represents ^{the bulk} dissipation ^{and the second one the surface} dissipation.

It is interesting to compare these two contributions.

Using equations (5), (6) ^and (18), ^{it can be} easily ^seen

that the surface term is negligible ^if

For thick films (typically ^some micrometers) ^this

condition must be widely satisfied. In what follows, ^we

will neglect the surface dissipation. ^{This means that}

the layers ^can easily disappear ^{on the free surface.}

1.2 GENERALIZATION TO A SPHERICAL FILM.- Let us

consider a thin spherical ^smectic sheet, ^{a bubble.} By changing ^the pressure inside the bubble, its radius will increase or decrease depending ^{on the} sign ^{of the} pres-

sure variation. The volume of the sheet staying ^con- stant, its thickness must vary, leading ^{to a} permeation

flow through the smectic layers ^{similar to that de-}

scribed in the preceding paragraph. The resolution of the hydrodynamical equations ⁱⁿ spherical coordinates leads to the following ^{law for the} dissipation ^{inside the}

film :

Notice here that the numerical’coefficient is slightly

different from the one found in the planar ^case (1/16

instead of 1/12).

Experimentaly, ^a free smectic bubble is _very dif- ficult to produce. ^{For this} reason, ^{we are} going ^to

describe a situation which is easier to carry out exper-

imentally.

2. Description ^{of the} experiment : theoretical

predictions

The experimental set up is shown in figure ^{3. At}

the top ^{of a hollow} cylinder ^a planar ^{smectic film is}

stretched. At the bottom, ^a piston, pushed by ^{a mi-} crometer, changes ^the pressure inside the cylinder ^and

thus the curvature of the film. The experiment ^consists

in deflating the so-formed bubble through ^{a thin} cap-

illary ^{tube and in} recording ^the collapse ^{of this bubble}

with a video tape recorder. This awkward technique ^is

used to deflate slowly ^the bubble, which otherwise will

collapse ^{in a very short time t}

^{0.1 s} [9].

Before describing ^the experimental results, ^{let us}

see how to calculate the dynamics ^{of the} collapse.

Let us first introduce ^our notation figure ^{3. We}

call r the inner radius of the cylinder, ^{R the curvature}

radius of the smectic film (R varies from r to infinite),

S its total _area, V the volume of the spherical cap, ^a the radius of the capillary tube and L its length (with-

out possible confusion with the notation of the first

paragraph).

In this problem, ^the driving ^{force is} the surface tension. The equation ^{which governs} the relaxation of the film can be obtained by writing the balance be- tween the work done by ^the driving force and the dissi-

pated energy by permeation ^{in the film and} by ^viscous

flow of the air (dynamical ^viscosity ’1a) ^{in the capillary}

tube. It yields :

In fact it is more convenient to use as a variable the distance x between the top ^{of the} bubble and its

base figure ^3. Using ^the geometrical relations :

Fig.3.- Experimental ^set-up.

Equation (25) ^{becomes :}

where

is a characteristic time of relaxation related to the dis-

sipation ^{in the} capillary ^{tube and}

a characteristic time related ^to the dissipation ^{in the}

film (eo is the thickness of the planar film, ^{before de-}

formation).

Experimentally tc is well known and it is conve-

nient to use it as unit time. Let us introduce the new

dimensionless variables :

Equation (27) ^{can be} rewritten in the form :

Integrating ^this equation with the initial condition X

=

1 at T

0 leads to the following relaxation law for the film :

Figure ^{4 shows some} theoretical curves X

X(Il ^for

different values of the ratio R. Let us see now how these

curves can compare with the experimental ^ones.

3. Experimental ^results

The experiment ^has been carried out with the

4n-octyl-4’-cyanobiphenyl (8CB) ^{at room} temperature

(26°C). ^{A video} tape ^{recorder was used to measure} the collapse ^{fo the} bubble. The distance x is mea-

sured directly ^{on the screen} picture ^after picture (30

pictures ^per second). The film thickness is estimated

by ^direct observation of its interference colour. In _gen-

eral, the film is thinner in its centre just after stretch-

ing ^{and a recovery of some hours is} necessary to get ^a uniform thickness. If ^a non-uniform film is deformed,

it breaks rapidly ^{because the thinner} region ^becomes

thinner and thinner. This observation _agrees with the fact that the dissipated energy is smaller for thinner parts of the film according ^to equation (24). ^{This also}

shows that the bulk permeation is indeed the dominat-

ing ^{process as it can be seen from} equation (22).

Fig.4.- Theoretical curves (from ^Eq. (32)) calculated for different values of the parameter ^R.

In order to avoid the film exchanging ^material

with the bulk phase ^{around the} edge ^{of the} cylinder during expansion ^or collapse, ^the following ^{trick has}

been employed : ^the pressure is slowly decreased in- side the cylinder. ^{In this} way, the film glides ^down slowly along the wall of the cylinder. When the film is far enough ^{from the} top (typically ⁵ mm) ^{the edge}

of the cylinder ^is wiped very carefully ^{with an} opti-

cal _paper in order to remove the excess material. The

following step consists in increasing ^the pressure again

until the film has returned to its initial position.

Finally, ^we have verified that the colour of the film (and consequently ^its thickness) ^{does not} change

after each experiment ^{and is} always perfectly ^uniform.

That means that the film does not expel ^{material to-}

wards the edge ^{of the} cylinder during collapse. Thus,

the volume of the film remains unchanged, ^{which is a}

crucial piont ⁱⁿ testing ^our theoretical model of _per- meation flow.

In figure ^{4 we have} reported experimental ^curves

obtained with the same film for four different values of the capillary ^{time to. This} parameter ^{can be} easily

varied by changing ^the length ^{of the} capillary ^tube.

In this example, the chosen values of the geometrical parameters ^{are the} following :

Taking t7,,,

^{1.8 x 10-4} poise [9] ^{and 7=23} dyne/cm (obtained by measuring ^the pressure difference between the inside and the outside of the bubble) ^{we find for}

the capillary ^{time :}

and thus for the preceding ^{values of L :}

Quantitatively, ^{it can be seen in} figure ^{5 that the}

mean slope ^{of the} experimental ^curves decreases when tc decreases. This point is consistent with the the- ory. More accurately, ^{it is} reasonably possible ^{to fit}

these curves with the help ^{of the} theoretical law (32)

by choosing tp - ^0.8s (full ^{line curves in Fig.} 5). ^In

this experiment, the initial thickness of the film was

approximatively ^{eo - 2} p,m. Using equation ^{29 we find}

for the permeation coefficient :

This value (the ^{first one} obtained in a direct way)

is in good agreement ^{with the one deduced from mea-}

surements of dislocation mobility [4,6].

4.Concluding ^remarks

In our model, ^{we have not taken the} bulk disloca- tions into account. In fact we think that the film is free of edge dislocations because they ^are strongly ^attracted by ^{the free surfaces.} Concerning ^{the screw} dislocations,

their main effect is to renormalize the permeation ^co- eflicient, ^{as we} have shown recently [10]. Nevertheless this effect is _very small in thermotropic liquid crystals

and can be neglected ^here. Finally, ^{we have} performed

the same experiment ^{with a} soap bubble (water ⁺ liq-

uid detergent ^{10 % +} glycerol ^5% ). ^No significant

deviation from the curve R

0 (see Fig. 4) ^{has been}

observed experimentally ^{which means} that the dissi-

pation ^{in such a} soap film is _very small, ^without any

doubt because of the big quantity ^{of water inside.}

Fig.5.- Experimental results : L=250 cm (measurement ^{every three} frames) ; ^{L=100 cm} (measurement ^{every three} frames) ; ^{L=50 cm} (measurement ^each frame) ; ^{L=25 cm} (measurement ^each frame). ^{The full line curves have been}

calculated from equation (32) ^by taking tp

0.8s.We have indicated the corresponding ^{values of the} parameter ^R.

Acknowledgments

I thank A. Libchaer for fruitful criticism of the

manuscript and for his support ^{under contract NSF} DMR 8316204 and NSF INT A883728 exchange ^award.

References

[1] HELFRICH, ^W. Phys. Rev. Lett. 23 (1969) ^372.

[2] MARTIN, P.C., PARODI, O., PERSHAN, ^P.S. Phys.

Rev. A6 (1972) ^2401.

[3] Orsay Group ^on Liquid Crystals, ^J. Physique ^Col- loq. ³⁶ (1975) ^C1-305.

[4] OSWALD, P., C.R. Heb. Séan. Acad. Sci., ²⁹⁶

Série II (1983) ^1385.

[5] DUBOIS-VIOLETTE, E., GUAZELLI, ^{E. and} PROST,

J. Philos. Mag. ⁴⁸ (1983) ^727.

[6] OSWALD, ^{P. and} KLEMAN, ^{M. J.} Physique ^Lett.

45 (1984) ^L-319.

[7] ^DE GENNES, P.G., ^The Physics of Liquid Crys- tals, (Oxford) ^1974.

[8] CHAN, ^{W.K. and} WEBB, ^{W.W. J.} Physique ⁴²

(1981) ^1007.

[9] ^We have recorded the collapse ^{of the} bubble with-

out capillary tube. The measured time is of the or-

der of three frames (taking ^{into account the screen} remanence) ^and consequently of the order of 0.1

s. In fact, ^{this time can} be calculated by ^solv- ing equation (27) ^and making tc

^O.It yields tcollapse =(7/48) tp ~ tp/7 ^whence tp ~ ^{0.7s. This}

values is in good agreement ^with that found later.

[10] OSWALD, P., ^to appear in C.R. Heb. Séan. Acad.

Sci. 1987.