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Submitted on 1 Jan 1979
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ORIENTATIONAL ACOUSTIC NONLINEARITY IN FLUIDS WITH ANISOTROPIC PARTICLES
V. Lebedev, L. Ostrovsky
To cite this version:
V. Lebedev, L. Ostrovsky. ORIENTATIONAL ACOUSTIC NONLINEARITY IN FLUIDS WITH ANISOTROPIC PARTICLES. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-235-C8-238.
�10.1051/jphyscol:1979840�. �jpa-00219545�
ORIENTATIONAL ACOUSTIC NONLINEARITY IN FLUIDS WITH ANISOTROPIC PARTICLES V.T. Lebedev and L.A. Ostrovsky
Institute of Applied Physios of the Academy of Soienoes of the V.S.S.R., 603024 Gorky, V.S.S.B.
Abstract.- Nonlinear self-action of acoustic waves due to the orientation of anisotropic particles in a sound field (the acoustic Kerr-effect) is considered. Analysis is performed for a liquid crystal in which an ultrasound changes the order parameter, the induced transition from nematic state into iso- tropic one and vice versa being possible. Estimations are made showing the real opportunity of obser- ving different nonlinear effects (phase self-modulation, wave profile distortion, self-focusing) in a liquid crystal.
This paper considers some nonlinear acoustic effects related to orientation of anisotropic micro- particles suspended in a liquid. Orientating action of an ultrasound on the system of anisotropic par- ticles has been observed rather long ago for sus- pensions with needle-shaped crystals /l/. Similar effects acquire great interest now in connection with investigation of crystalline liquids and amor- phous polymers. For us the fact is important that orientation of particles influences the acoustic parameters of the medium that results in the non- linear self-action of a sound. In principle there is an analogy with the optical Kerr-effect here, though both the theoretical model and specific manifesta- tions of such "acoustic Kerr-effect" turn to be essentially different.
We consider here the interaction of an ultra- sound field with cigar-shaped particles (tactoids) with the length I suspended in a liquid. At the beginning we shall neglect any other orientating factors besides an ultrasound. For sufficiently small I each particle carried by the acoustic velo- city field performs oscillatory movements with the angular velocity w = - /2(6v/Sx) sin 2 0 where v(x,t) is the oscillatory velocity of fluid parti- cles in a wave and 9 is the tactoid orientation angle relative to the direction of wave propagation.
The angular distribution function f(9,t) in the given point is described by the known equation /l/ which with allowance for (1) has the form
where z = cos Q , g = &v/Sx , D is the rotatory diffusion coefficient.
The solution of (1) may be represented by a series of Legendre polynomials P (z)
Substituting this series in (2) we obtain the recur- rent system of equations for C , C„ being equal to (4 ir)" . If the wave amplitude is sufficiently small then it is easily seen that the coefficients C
n decay with the growth of n. Considering the harmonic sound field in the form v = (1/2) | A(x)£lalt + A*(x)£~ | one can find the periodic solution for C2 :
where C« is the time average of C2 proDortional to
1A
X|
2.
The mean orientation of the particle system is usually characterized by the order parameter
S = <P2> = | |3 <Cos29> - 1| = | TT C2 where < >
denotes the averaging over a. From (3) or (4) we have after averaging over a period
Hence, the orientation degree is proportional to the square of the acoustic Hach number M = |A|/C , where C is the sound velocity. To describe the JOURNAL DE PHYSIQUE Colloque C8, supplément au n°ll, tome 40, novembre 1979, page C8-235
Résumé.- Les interactions non linéaires des ondes acoustiques induites par l'orientation des p a r t i - cules anisotropes dans un champ ultrasonore (effet Kerr acoustique) sont étudiées. L'analyse est menée pour un cristal liquide dans lequel la variation du paramètre d'ordre, la transition induite de l ' é t a t nématique à l ' é t a t isotrope, et réciproquement, sont possibles. Des estimations sont faites pour différents effets non linéaires dans un cristal liquide (auto modulation de phase, distorsion du profil d'onde, auto focalisation).
Article published online by EDP Sciences and available at
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979840
JOURNAL DE PHYSIOUE
o r i e n t a t i o n n o n l i n e a r i t y i t i s necessary t o d e f i n e the r e l a t i o n between and sound v e l o c i t y i n t h e medium. This r e l a t i o n may be found o n l y w i t h i n t h e frames o f thermodynamics o f a n i s o t r o p i c f l u i d s t a - k i n g i n t o account t h e i n t e r m o l e c u l a r i n t e r a c t i o n s . E s p e c i a l l y s t r o n g n o n l i n e a r e f f e c t s a r i s e i n the case o f o r i e n t a t i o n phase t r a n s i t i o n s . Ye con- s i d e r here t h e behaviour o f a l i q u i d c r y s t a l near the p o i n t o f t r a n s i t i o n between i s o t r o p i c and neu- matic phases. I n t h i s case equation (1) must be supplemented by t h e terms r e s p o n s i b l e f o r the long- range i n t e r a c t i o n s . S t a t i s t i c a l mechanics o f neu- m a t i c i s b u i l t o n l y f o r some simple models. Themost u n i v e r s a l seems t o be t h e phenomenological t h e o r y of Fleier and Zaupe /2,3/. According t o t h i s theory, the energy o f t h i s molecule i n t h e long-range mole- c u l a r f i e l d E =
-
1 USP2(B) where 6 i s t h e angle of p a r t i c l e o r i e n t a t i o n and U i s t h e parameter de- f i n e d by Van-der-Vaals forces. The energy E must be taken i n t o account i n k i n e t i c equation ( 2 ) .I f t h e d i r e c t i o n o f the nematic o r i e n t a t i o n coincides w i t h t h e d i r e c t i o n o f u l t r a s o u n d propaga- t i o n then 6 = O and t h e k i n e t i c equation f o r f has the form
where z =
-
cos 0 , m = 3 US/2 k t .We s h a l l consider below the comparatively low frequencies when w << 6 D. I n t h i s case s e t t i n g the r i g h t p a r t o f ( 7 ) t o zero we e a s i l y f i n d t h a t
This expression i s the same as i n r l e i e r and Zaupe's t h e o r y /2,3/, b u t the parameter m' depends here on the value o f t h e a c o u s t i c v e l o c i t y f i e l d . As a r e s u l t the parameter m' o s c i l l a t e s w i t h the frequency w and, consequently, t h e s t a t e o f a l i q u i d c r y s t a l p e r i o d i c a l l y changes.
The value S i n each moment i s d e f i n e d by t h e equations /2/
L e t us use, as usual, t h e graphic i n t e r p r e t a t i o n o f equations ( 6 ) ( f i g . 1 ) . For g = 0, t h e s t a t e o f t h e system i s d e f i n e d by temperature : i n case
k T > 4.55 U, the system i s i s o t r o p i c (S = 0) ;
F i g . 1 : Graphic i n t e r p r e t a t i o n o f equations ( 6 ) c
-
t h e dependence (6 a ) , 1,2,3-
( 6 b ) a t d i f f e r e n t values o f g.
otherwise, s t a b l e i s t h e oematic phase. The minimull value o f the o r d e r parameter SC(TC) i n t h e neumatic phase i s 0.44.
The a c t i o n o f t h e u l t r a s o u n d gives r i s e t o t h e p e r i o d i c p a r a l l e l t r a n s l a t i o n s of t h e l i n e ( 6 b) i n f i g . 1 from the o r i g i n .
The most i n t e r e s t i n g i s t h e case when t h e u l - trasound a c t s near t h e t r a n s i t i o n p o i n t m = mC
,
S = SC. L e t t h e i n i t i a l value S = So i n t h e neu- matic exceed SC by a small value
5.
I n t h e presence o f t h e u l t r a s o u n d S undercoes t h e a d d i t i o n a l small v a r i a t i o nn .
One may f i n d from ( 6 a) and (6 b) t h e f o l l o w i n g r e l a t i o n between q and5
:q ( t ) =
d 2
+ Bg ( t )-
t; ( 7 )where 6 = (L)d2m/ds2)-l and t h e d e r i v a t i v e i s taken i n t h e p o i n t S = SC. Then i t i s easy t o c a l c u l a t e t h e mean value o f q
.
The formula (11) i s v a l i d i f t h e amplitude gmax i s smaller than some c r i t i c a l value g* corres- ponding t o t h e t r a n s i t i o n i n t o t h e q u a s i i s o t r o p i c s t a t e . The value g* i s equal t o
-
The corresponding mean value q i s
I n case when gmax exceeds g*
,
t h e system w i l l be thrown over between t h e nematic and q u a s i i s o - t r o p i c s t a t e s . I n t h i s case, f o r small5
one may n e g l e c t t h e change o f S o u t s i d e t h e phase t r a n s i - t i o n s ; then, approximately, one may assume S t o undergo p e r i o d i c o s c i l l a t i o n s o f a r e c t a n g u l a r form between t h e values near SC and 0V.T. Lebedev and L.A. Ostrovsky c8-237
( f i g . 2)" so t h a t t h e average value of S i s equal may be o f v a r i o u s character. Local changes o f C du-
t o r i n g each p e r i o d r e s u l t i n the deformation o f a wave
p r o f i l e up t o t h e formation o f d i s c o n t i n u i t i e s
1 9%
= SC 11
-
- a r c cos -1
Tr (9) (shocks) ( f i g . 4 ) . Here, u n l i k e t h e usual cases, t h e
g m asymptotic form of a wave p r o f i l e i s n o t a saw-tooth
Hence,
5
v a r i e s from Sc t o SC/2 ( f i g . 3) w i t h t h e one.growth o f the f i e l d ampl i t u d e .
u%++
t
F i g . 2 : Time v a r i a t i o n s o f the order parameters S i n the i n t e n s i v e harmonic a c o u s t i c f i e l d g ( t ) .
F i g . 3 : Dependence of mean value o f S on sound ampl i tude.
Now l e t us r e t u r n t o the q u e s t i o n o f t h e de- pendence o f sound v e l o c i t y C on t h e o r d e r parameter.
This dependence was considered, f o r example, i n 141.
I f the changes o f t h e t o t a l energy o f a medium due t o o r i e n t a t i o n are r e l a t i v e l y small, then
C
-
Co ( 1+
6s),
where 6 i s a temperature dependent parameter. Judging by the experimental data, f o r t h e nematic s t a t e 6-
0,l. I f gm > g*,
t h e sound ve- l o c i t y undergoes jump-1 i ke changes. According t o the experimental data 141, these jumps may achieve values o f the order oflo-'.
To o b t a i n a more con- v i n c i n g e s t i m a t i o n one must consider some a d d i t i o n - n a l f a c t o r s such as t h e f l u c t u a t i o n s o f the order parameter i n t h e v i c i n i t y o f phase t r a n s i t i o n . M o r e - over, t h e u l t r a s o u n d changes t h e temperature o f t h e medium which a l s o i n f l u e n c e s t h e c o n d i t i o n s o f phase t r a n s i t i o n . Rough e s t i m a t i o n s show t h a t i n t h e f r e - quency band 1-10 MHz t h e a n a l y s i s performed holds t r u e p r o v i d e d the a c o u s t i c Mach number exceeds 1 0 - ~ + 1 0 - ~ .The n o n l i n e a r a c o u s t i c e f f e c t s a r i s i n g here
'
To be precise, f o r g < 0 a tendency a r i s e s f o r t h e molecula-r o r i e n t a t i o n i n t h e plane perpendicular t o a x i s x ' ( i s o t r o p y i n t h i s plane bein? reserved).Such "weak" order needs a s p e c i a l i n v e s t i g a t i o n .
F i g . 4 : A scheme o f n o n l i n e a r e v o l u t i o n o f wave p r o f i l e a t gmaX > gx.
A t t h e same t i n e , due t o t h e dependence of t h e mean value o f S on t h e f i e l d i n t e n s i t y , v a r i o u s s e l f - a c t i o n e f f e c t s f o r t h e wave envelopes (amplitude, phase) may be observed. The a n a l y s i s o f these e f f e c t s may~be c a r r i e d o u t by t h e methods well-known i n t h e n o n l i n e a r o p t i c s (see a l s o t h e paper 151 d e a l i n g w i t h t h e thermal s e l f - a c t i o n o f a c o u s t i c waves). The dependence o f t h e phase s h i f t i n a wave on i t s i n t e n s i t y i s one o f t h e s i m p l e s t o f t h e ef- fects. The n o n l i n e a r phase s h i f t by ~r i s achieved a t t h e distance , xo =
XE/Z
AT, whereAT.
i s t h e change o f the mean sound v e l o c i t y i n a wave. I fAEIC -
1 0 - ~ + 1 0 - ~ a t the frequency 5 MHz,xO
=
0,2+2 cm. For a p u l s e s i g n a lAC
changes i n time t h a t y i e l d s t h e frequency modulation i n a pulse. A t distances x >> xo t h e w i d t h o f t h e p u l s e spectrum w i l l be d e f i n e d n o t by i t s d u r a t i o n T b u t t h e value Aw >> T-l. Such n o n l i n e a r broadenin? o f a spectrum due t o t h e phase s e l f - m o d u l a t i o n i s well-known i n t h e n o n l i n e a r o p t i c s161.
Of i n t e r e s t i s a l s o t h e p o s s i b i l i t y o f t h e sound s e l f - f o c u s i n g due t o t h e o r i e n t a t i o n nonlinea- r i t y . Self-focusin; occurs i f d c / d ~ 0. Usually, t h e sound v e l o c i t y i n t h e o r i e n t a t e d c r y s t a l i s g r e a t e r than t h a t i n t h e i s o t r o p i c one, i .e.
dC/dS > 0. T h i s i m p l i e s t h a t t h e a c t i o n o f t h e u l t r h sound on the i s o t r o p i c c r y s t a l r e s u l t s i n t h e defo- cusing, whereas propagation of t h e u l t r a s o u n d beam i n neumatics near t h e t r a n s i t i o n p o i n t may be ac- companied by s e l f - f o c u s i n g .
As i t i s known /7/, t h e c r i t i c a l amplitude o f
~ 8 - 2 3 8 JOURNAL DE PHYSInUE
a c y l i n d r i c a l beam, a t which s e l f - f o c u s i n g a r i s e s , i s determined by t h e c o n d i t i o n
AC/C
>, ( k D ) - l where D i s t h e diameter o f t h e beam. The d i s t a n c e xN a t which t h e c o l l a p s e o f t h e plane beam occurs i s o f t h e o r d e r o fD&c/C.
For example, i n a beam o f d i a - meter l cm a t a frequency o f 5 MHz t h e c o n d i t i o n o f t h e c o l l a p s e i s r e a l i z e d a tAC/C
> 5 . 1 0 - ~ . I fAC/C -
10-'+10-~, then xN-
3r10 cm.The damping d i s t a n c e o f sound i n l i q u i d c r y s - t a l s i s o f the same o r d e r /2,4/, t h e r e f o r e , obser- v a t i o n o f t h e s e l f - f o c u s i n ? e f f e c t (and a l l themore so, o f t h e phase self-modulation) seems t o be pos- s i b l e .
O f course, t h e o r i e n t a t i o n a l a c o u s t i c n o n l i - n e a r i t y may m a n i f e s t i t s e l f n o t o n l y i n l i q u i d c r y s t a l s . I t i s worthwhile t o consider t h e p o s s i b i - l i t y o f s i m i l a r e f f e c t s i n amorphous polymers and b i o l o g i c a l s o l u t i o n s . These e f f e c t s may be used, f o r example, f o r the d i r e c t d i a g n o s t i c s o f the par- t i c l e o r i e n t a t i o n i n nontransparent media.
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,
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AE,
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k i i zhurnal
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22 (1976).Ostrovsky, L.A., Pis'ma v ZhETF,
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(1967) 807.Karpman, V . I . , Nonlinear waves i n d i s p e r s i v e media, Nauka, Floscow (1973).
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