Hydrodynamic properties of the smectic A phase formed in colloidal solutions

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Hydrodynamic properties of the smectic A phase formed in colloidal solutions

Xin Wen, Robert B. Meyer

To cite this version:

Xin Wen, Robert B. Meyer. Hydrodynamic properties of the smectic A phase formed in colloidal solutions. Journal de Physique, 1989, 50 (19), pp.3043-3054. �10.1051/jphys:0198900500190304300�.

�jpa-00211123�

Hydrodynamic properties of the smectic A phase ^{formed in}

colloidal solutions

Xin Wen and Robert B. Meyer

Martin Fisher School of Physics, ^Brandeis University, Waltham, ^MA 02254, ^U.S.A.

(Reçu ^{le 2 mai 1989,} accepté ^{le 2} juin 1989)

Résumé.

Nous étudions les propriétés hydrodynamiques ^{de la} phase smectique ^A formée dans des suspensions ^de particules colloïdales. Nous avons trouvé

mode diffusif

rapport

avec l’écoulement relatif entre le solvant et les particules colloïdales. Dans la direction

perpendiculaire ^{à la couche,}

^{mode est} couplé

^la perméation ^entre couches. Le temps ^de relaxation du mode diffusif dépend ^à la fois des composantes parallèle ^et perpendiculaire ^{à la}

couche du vecteur d’onde. On trouve que le deuxième

^est atténué par le cisaillement

hydrodynamique, ^la perméation ^{et le mouvement} visqueux ^entre le solvant et les particules

colloïdales. Dans cette théorie, le nombre total de modes hydrodynamiques ^est toujours égal

nombre de relations de conservation,

qui ^est

^accord

la théorie générale ^de Martin,

Parodi et Pershan. Nous montrons aussi, par

simple estimation d’un système ^réel, que le second

est suramorti par l’effet de cisaillement hydrodynamique, ^et que le mode de diffusion relative est observable par diffusion quasi élastique de la lumière.

Abstract.

The hydrodynamic properties of the smectic A phase ^{formed in} suspensions ^of

colloidal particles

studied. We have found

diffusive mode related to the relative flow between the solvent and colloidal particles. Along ^the layer ^normal direction, this mode is

coupled ^{to the} permeation ^between layers. The relaxation time of the diffusive mode depends

the wavevector components ^both along ^and perpendicular ^{to the} layers. The second sound is found to be damped by hydrodynamic shear, permeation, and the viscous motion between the solvent and colloidal particles. The total number of hydrodynamic modes in this theory ^is always equal ^to the number of conservation relations, ^{which is} consistent with the general theory proposed by ^Martin, Parodi and Pershan. We also show by simple estimation in

real

experimental system that the second sound is overdamped by ^the hydrodynamic shear effect and the mode of relative diffusion is observable by ^the quasi-elastic light scattering technique.

Classification

Physics ^Abstracts

61.30

47.35

82.70D

1. Introduction.

There is currently great ^interest [1-6] ^{in the} lyotropic smectic A phase ^formed by rigid particles which interact mainly through ^steric exclusions. The phase ^is entropically ^{driven and}

has properties substantially different from the thermotropic smectic A phase. Although

several works [2-6] have considered the formation mechanism of the phase, ^its dynamic properties ^{have not} yet ^been investigated. Experimentally, ^the lyotropic ^{smectic A} phase ^is

observed in suspensions of colloidal particles, e.g. solutions [7] of tobacco mosaic virus

(TMV). Studying ^the hydrodynamic properties ^{of these} systems ^is fundamental for our

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500190304300

understanding ^{of the} lyotropic ^{smectic A} phase ^{as well as} the other ordered phases ^observed

in colloidal solutions.

The general hydrodynamic theory [8] in condensed matter was proposed by Martin, ^Parodi

and Pershan (MPP) in 1972. In the landmark work, ^MPP explicitly ^defined hydrodynamic

modes in a many body system ^{as the few}

long ^lived

collective modes that decay ^{in times} proportional ^{to some} power of their wavelengths. ^{MPP also} argued that the number of

hydrodynamic ^{modes is} always equal ^to the number of the conserved quantities, ^{or « the} independent degrees ^{of freedom}

^{In a} single-component isotropic liquid, ^for example, ^there

are five hydrodynamic ^{modes : a} thermal diffusive mode, ^{two diffusive} transverse shear

modes, ^{and a} pair ^of propagating sounds. The number of hydrodynamic modes is determined

by ^{the five conserved} quantities characterizing ^the phase, namely, the energy density, ^the

mass density, and the three components ^{of the momentum} density. As the continuous

symmetries ^{of the} isotropic liquid ^are broken, ^new independent degrees of freedom will appear, resulting ^{in new} hydrodynamic modes. For the nematic ordering ^{in which} anisotropic particles ^{orient in a} preferred direction, ^{the two} possible orientational fluctuations represent

two new hydrodynamic modes. The free energy ^terms associated with the two orientation fluctuations are proportional ^to the squares of the spatial gradients ^{of the director}

distribution. Consequently the related relaxation times are proportional ^to the square of the

wavelength of the fluctuations.

In a smectic A ordering, the continuous translational symmetry is broken into periodic layers. ^{Due to the} symmetry requirement, ^{directors are bound to the} layer ^{normal direction.}

As a result, the director fluctuations relax in

microscopic ^times

^{which are} determined by microscopic binding forces and which remain finite in the long wavelength ^limit. Clearly, ^the

director fluctuations in the smectic A phase ^{are no} longer hydrodynamic. ^{The extra}

«

independent degree of freedom

in the smectic A phase ^{is the} layer displacement [9, 10]

whose propagation ^{forms a new} hydrodynamic mode, ^the second sound wave. Similar to the director fluctuations in the nematic phase, the fluctuations of the layer displacement ^are

associated with a free energy increase determined only by ^the spatial gradient ^{of the.} layer displacement. ^The characteristic time of the fluctuation, namely ^the period ^{of the second} sound, ^is consequently proportional ^{to the} wavelength. ^For the smectic A phase ^{formed in} binary systems, ^{there is an} additional hydrodynamic mode associated with the mass

conservation of the second component. ^One special system of the latter _case, the lamellar

phase ^formed by lipids ^{and water, was studied} [11] by Brochard and de Gennes in 1973. In this

work, ^we attempt ^to understand the hydrodynamics of the smectic A phase ^formed by particles suspended in solutions.

A solution of colloidal particles ^{consists of two} ingredients : the solvent and the particles.

The particles usually interact with some types ^of repulsive ^forces, either coulombic repulsion

or steric exclusion in origin, ^{so that the} particles ^are separated from each other and can be

suspended by ^thermal agitations. ^{If the} particles ^are charged, ^the strength ^{of the} repulsion

between the particles ^can be modified by the ionic concentration and the pH value in the solution. When the smectic A order forms, ^the particles distribute in a stack of layers ^and

orient preferentially ^{in the} layer normal direction. In the example of the TMV solution, ^a

virus particle [12] ^{is an} electrically charged rigid cylinder, ^{3 000} Â in length ^{and 180} Â in diameter. The virus [13] ^{has a molecular} weight ^{of 4 x} 107 g/M ^{and a} specific ^{volume of}

0.73 cc/g. The smectic A phase [7] forms with layer thickness about 3 300 Â ^at concentration around 175 mg/ml (corresponding ^{to a} particle ^{volume fraction of} 0.20) ⁱⁿ pH ^{8.5 and 50 mM}

borate buffered solution. By simple estimation, ^{one finds the} in-layer ^{area fraction} [3] ^{of the}

particles ^{to be about 0.18, which} gives ^{the mean} distance between axes of particles ^within

layers ^to be about 400 Â, ^{or a little more than two virus diameters.}

Having ^an idea of the physical ^structure of real colloidal systems, ^{we should} always

remember that it is the conserved quantities that determine the types ^{of the} hydrodynamic

modes. Quantities describing ^the sizes, ^the physical shape, ^or the detailed form of interactions of the particles, however, only affect the magnitudes of the relaxation times of the

hydrodynamic ^modes. Comparing ^{with a} single-component ^{smectic A} system, ^{we have in the} colloidal solutions a new hydrodynamic variable, ^{the local} concentration of the solvent, ^{and a}

related conservation relation, ^{the mass} conservation of the solvent. From the argument ^of MPP, ^{one would} naturally expect ^an additional hydrodynamic mode in the smectic A phase ⁱⁿ

colloidal solutions. Generally speaking, ^{a new} hydrodynamic mode is added in two possible

situations : 1) ^{there is one} additional diffusive mode ; 2) combining ^{with an existed} mode,

there can be a pair of acoustic modes. The addition of the second component may also introduce modes which are not hydrodynamic ^{in nature. For} example, the second component in a binary crystal gives ^{rise to} non-hydrodynamic optical modes which vibrate at frequencies

determined by the interactions between the two types ^{of atoms. In the} following ^{sections, we}

will show that the new hydrodynamic mode in the smectic A phase ^formed in colloidal solutions is related to the relative diffusion between the particles and the solvent. We also

study the effects of solvent on the properties of the other hydrodynamic ^modes.

2. The dynamic equations.

Consider a smectic A system of colloidal particles which is in equilibrium ^at temperature ^{T in}

a fixed volume. For studying hydrodynamic properties, ^{we define three} dynamic variables : 1)

0 (r) the bulk dilation of the solution defined as 0 (r) ^{=1- p} (r )/p , ^{in which p} (r) is the local

mass density of the solution and _p the average of p (r) ; 2) ^y

aulaz the derivative of the

layer displacement u (r) along

^the layer ^normal direction ; 3),6 (r) ^the relative dilation of the local solvent fraction defined as 6 (r) ^{=1- p} S(r )/p ^{s, in which p} S(r ) is the solvent mass in a

unit volume of solution and p ^{S the} average of ^p S(r).

With the defined variables, the conservations of the total mass and the solvent ^{fraction can} be written as :

where v and v’ are the velocities of the total mass and the solvent.

The free energy density [11] consists of the compressional (dilational) ^elastic energies ^and

the curvature elastic energy :

where el = (J, £2

y, E3

8, Cij ^are the elastic coefficients associated with the three

variables, and K is the splay elastic coefficient. Of the two côntributions to the free energy, the splay ^{elastic term is of} higher ^{order in} spatial gradients and is therefore a small contribution for long-wavelength fluctuations. As we will see, this term is important only

when wavevectors are parallel ^to layers.

Three forces [11] ^can be derived from the free energy expression : the pressure

the restoring ^{force on the} layers along ^the

^direction

and the osmotic force on the solvent (the negative gradient of the osmotic pressure)

The acceleration equation, ^{or the momentum} conservation, ^can then be written as :

where Q’ is the viscous stress tensor.

The entropy ^source consists four dissipative processes :

which include four fluxes : Vv the shear rate tensor, vP - ^{ù the} velocity ^of particles ^{relative to}

the layer displacement ^{in the}

direction, ^{vs - vP} the relative velocity of the solvent to the

particles, ^J the heat flux density, ^{and their} respective conjugate ^{forces :} a’, g, f, ^and

E

VT/T. ^{In this work, we} neglect any temperature gradient and associated heat- transport processes, since ^{we are} primarily interested in the hydrodynamic ^modes specifically

attributed to the binary lyotropic smectic A phase which is determined [7] mainly by

concentration instead of temperature.

Since fluxes are small for long wavelength hydrodynamic fluctuations, linear relations may be assumed between the forces and the fluxes :

(1) ^the symmetry of the smectic A phase requires ^{Q’ and Vv to} only couple ^to each other.

At the incompressible limit, ^the general ^relations [10] ^{between Q’ and Vv are} [11] simplified

as :

where i and j ^{denote x or} y, and 17e, 17t, and 17c ^{are the} longitudinal, transverse and ^cross

viscosities ;

(2) along ^{the z} direction, ^the permeation and the relative diffusion are generally coupled,

which is a new physical property of the smectic A ordering ^formed by particles in colloidal solution. When particles permeate through layers, this also involves a flow of particles ^relative

to the solvent in the same direction. The restoring ^{force on} layers ^is always opposite ^{to the} particle ^{movement :}

in which Q22 ^{is the same as the} permeation coefficient in an ordinary ^{smectic A} phase.

vg, ^the velocity ^of particles ^{in the z} direction is related to the other velocities by ^the

conservation of momentum : p vz

p S vi ⁺ p p vp ^{in which} p P

^{p -} p S ^{is the mass of} particles

in a unit volume of the solution. When an osmotic force is established on the solvent in the

direction, ^either externally ^{or due to thermal} fluctuations, ^the solvent is driven to flow in the

same direction relative to the particles. ^{At the same time, the} particles ^{are also} dragged by ^the

solvent to permeate through layers. The force-flux relation of this process is described ^{as :}

According ^{to the} physical explanations ^{of the two} processes, the viscous coefficients

Qij ^{are all} positive. Equations (10) ^and (11) ^{can be} easily transformed into a mathematically

convenient form :

where P22 = Q22, P32 = Q32, P23 = (P Q23 ⁺ pSQ22)/pP ^and P33 = (P Q33 - pSQ32)/pP.

Since the solvent should flow relative to the total fluid in the same direction with

f z, ^{we have} P33

0 or p Q33

p S Q32· ^{As we will see later, results can be} simplified ^{to more} physically transparent forms, in the limit of weak coupling between the permeation ^{and the}

relative diffusion, namely, Q23, Q32 = 0, ^or P32 = ^{0 ;}

(3) ^{due to} the uniaxial symmetry ^{of the smectic A} order, ^we may write in-layer ^force-flux

relations only in the x direction :

in which L is the viscosity of the relative flow in lateral directions.

So far, ^{we have} included six conservation relations : the two mass conservations (Eqs. (1)

and (2)), the three components ^{of the momentum} conservation (Eqs. (7)), ^{and the}

conservation relation for ù (Eq. (10)). Together with the energy conservation which is related to thermal relaxation, ^{we have a total of seven} conservation relations. From the theory [8] ^of MPP, ^we expect ^six hydrodynamic modes besides the ignored thermal relaxation mode.

3. The no dissipation ^limit.

In this section, ^the hydrodynamics is studied with the simplification [9] ^of ignoring ^{all the} dissipations. Mathematically, ^this approximation ^is equivalent ^to neglecting ^the imaginary part ^{in the} frequencies of all the hydrodynamic modes. In this way, ^{wa can} easily acquire ^an

overall picture of all the hydrodynamic modes in the system. ^{In the next} section, ^{the second}

sound and other slow modes will be studied with all the viscous processes. Without

dissipation, ^we may take P22, P33 ^{and L in} equations (10a), (lla) ^and (12) ^to infinity ^and obtain vz

ù, ^{and v}

^v’. The viscous shear term in equation (7) may also be neglected. ^With

v

v’, ^one easily obtains from equations (1) ^and (2) ^à

^{6. Since we are} interested in modes with amplitudes ^oc exp (i q . ^{r +} iw t ), ^this equation ^thus requires ^{either w}

^{0 or 5}

^{0. The}

first _case, shown in the next section, is related to the purely damped ^mode of the relative diffusion between the particles and the solvent. In the rest of this section, ^we consider modes that satisfy ⁸

^8.

Ignoring ^{the small curvature} elastic energy, F in equation (3) ^{is now} simplified ^{as :}

in which C il = C 11 ⁺ C 33 ^{+ 2} C 13,

CÍ2 = Cl2 ⁺ C 23, ^and C 22 = C 22. With this free energy

expression, ^{f can} be combined into - Vp ⁱⁿ equation (7). ^So, within the difference in the elastic coefficients, ^the dynamics has become effectively ^{the same as} the smectic A phase [11]

in a single-component system.

With the rotational symmetry around the

axis, ^we may choose q = (q,,, ^0, qz).

Equation (7) ^{now reads :}

Equation (16) gives ^a zero-frequency mode which will be related in the next section to the

hydrodynamic shear motion. Recognizing ÿ

a,ù

azvz, equation (14) ^becomes

Using axUx = 9 - aZvz = 8 - 00FF, equation (15) ^becomes

For nonzero solutions, the determinant of equations (17) ^and (18) ^{should vanish :}

Solving equation (19), ^{one can} easily obtain si, the speed ^{of the} ordinary ^{acoustic wave, and}

s2, the speed of the second sound. Since Cl,, ^the elasticity of the average density, ^{is much} larger ^{than the osmotic} compressibilities C22 ^and C33 (Sect. 5), ^{one can} obtain the simplified

results :

in which 0 ^{is the} angle ^between q and the z axis. Except for the small angular modulations,

s2 ^is isotropic ^and approximately equal ^to C11/p which is the expression ^{for an} isotropic liquid. s2 ^is exactly ^{the same as the} expression [10] for the smectic A phase ^{with one} component.

.

To conclude this section, ^{we have six} hydrodynamic ^modes (in ^{addition to} the thermal diffusion mode) ^{in the no} dissipation ^{limit : two} pairs of acoustic modes, ^{and two zero-} frequency modes. The velocities of the propagating ^{modes are} basically ^{the same as those of a} single-component ^{smectic A} phase. The additional variable 6 does not bring ^{in new} propagating modes. When dissipations ^are considered, ^{the two} zero-frequency ^{modes are to}

be related to the hydrodynamic shear motion and relative diffusions.

4. The low frequency ^modes.

It is shown in section 5 that the oscillation of the bulk dilations is much faster than the oscillation of the layer compressions and the relaxation of the solvent dilation. The first sound

can therefore be decoupled, ^{within a} good approximation, from the other hydrodynamic

modes. With this consideration, ^the first sound may be ignored by imposing 0

0, ^{as we}

concentrate on the properties ^{of the rest} of the four slow modes. We will look for modes

oc exp (i q - r + i w t ) ^{and set} qy = 0 ^without losing generality. Equations (1) ^and (3) ^then

become :

where the elastic coefficients C’s are isothermal relative to the bulk dilations of the first sound.

4.1 MODES WITH OBLIQUE WAVEVECTORS. - We start from the acceleration equations

For the motion in the y direction, ^one easily derives from equation (26) :

which is associated with the transverse hydrodynamic shear mode that propagates ^{in the}

(x, z ) plane ^and oscillates in the y direction.

In the following derivation of the two relations between y and 8, ^{we will} only keep ^the

q4 ^{terms that are the} lowest q power ^terms when q is parallel ^or perpendicular ^{to the} layers.

Using equations (25) ^and (22), ^we substitute p ⁱⁿ equation (24) and obtain :

with

To express v, ^with y and 6, ^we get ^{rid of} vs ⁱⁿ equations (10a) ^and (12) ^with equations (2) ^and (22) :

and obtain

Substituting qZ v, ⁱⁿ equation (28), ^{we have :}

where

p w - i q 2 ~ .

Another relation between y and ⁸ is obtained by ^first deleting ^{û -} v,, ⁱⁿ equations (10a) ^and (lla) :

then substituting uz - ^{v, in} equations (33) ^and (12) ^with equations (2) ^and (22) :

Then we express g and f, by ^{y and 8 with} equations (5) ^and (6) :

where

In the limit of P32

⁰ (see ^Sect. 2), it is easy ^{to see} P1= P2

P33-

Nonzero solutions require the determinant of equations (32) ^and (35) ^{to be zero, which} gives ^after ignoring ^the Kq 2 qx ^{term in} equation (32) :

We obtain a purely damped ^{mode at low} frequency :

which is the frequency ^{of a new} type ^of hydrodynamic mode, the relative diffusion between the colloidal particles and the solvent. At the small P32 limit, ^this frequency ^{becomes :}

The relaxation time is a function of both qx and qz. This is reasonable because the relative flow between the colloidal particles and solvent can be both parallel ^and perpendicular ^{to the}

smectic layers. ^{In contrast, the} slip motion in the lamellar phase ^formed by lipids ^{and water is}

confined in the plane ^of lipid layers, ^{and as a} result, ^the relaxation time [11] only depends ^on

qx. The hydrodynamic mode of relative diffusion should generally exist in other orderings

formed in colloidal solutions, e.g., the nematic [15] and columnar [16] phases observed in the solutions of tobacco mosaic virus. In the case of the columnar phase, the relative diffusion

perpendicular ^to the columns should couple ^{with the} permeation process between columns.

Neglecting high power q terms, it is easy ^to obtain the other two roots of equation (37) :

which are the frequencies ^of the second sound. The real part ^{is the same as} the result obtained in the no dissipation ^limit (Eq. (21)). ^The damping part consists of three effects : the

hydrodynamic shear, the relative diffusion, ^{and the} permeation. In the limit of C23 --> 0, ^the damping part is reduced to the form of a single-component ^{smectic A} phase comprising only

the permeation ^{and the} hydrodynamic ^{shear :} i 2- 1 (C22 p- 22 1 q2 z + îîp - q2), This may be

expected ^since C 23 ^{is the} coupling coefficient between the layer compressions ^{and the}

dilations of the solvent fraction.

4.2 MODES WITH WAVEVECTORS PARALLEL TO LAYERS.

When _q

(q,,, ^0, 0), equation (27) gives w 1 = i p - ~ t qx, which describes the transverse shear modes oscillating ⁱⁿ the y direction. The relaxation time of this mode is characterized by q, t the transverse viscosity

in the smectic layers. With qz

0, Cd34 in equation (40) ^becomes purely imaginative ^{and the} pair of second sounds become diffusive. To find the relaxation times of the diffusive modes,

we write the counterpart ^of equation (37) by setting qZ

0 in equations (32) ^and (35) :

One immediately ^{obtains the} frequency ^{for the relative} diffusive mode

and the frequencies ^{for the} coupled ^{modes of} hydrodynamic shear and the in-layer ^undulation

When Klq,, « qlp, ^{the two types} of motions become decoupled :

CJ)3 corresponds ^to the shear mode oscillating ^{in the}

direction, which is characterized by ^the

cross viscosity ^{related to the z} and lateral directions. _w4 is the frequency ^{of the} in-layer

undulation mode which oscillates with constant layer distance and constant mass distributions.

The frequencies ^{of the} undulation mode and the two hydrodynamic shear modes are of the

same form as [10] ^{that of a} single-component ^{smectic A} phase.

4.3 MODES WITH WAVEVECTORS PERPENDICULAR TO LAYERS.

When wavevectors are

normal to the smectic layers, ^the hydrodynamic shear motions in the x and y directions become degenerate ^and give ^{the same} relaxation times (Eqs. (25) ^and (26)) ^described by ^the

cross viscosity :

The dispersion relations of the rest of the modes can be obtained similar to our approach

for the in-layer ^modes by solving ^the parallel ^of equation (37) with qx

^{0 in} equations (32)

and (35). However, ^we will choose a more illustrative path by deriving ^this directly ^{from the}

basic dynamic equations. ^{We first} simplify ^{the two mass} conservation equations ^with

qx

0: dzVz

^{0 and} dzV; ^=03B4 (Eqs. (22) ^and (2)). The former relation immediately gives

vz

^{0. We then} apply these relations to equations (10a) ^and (11a), take the derivative over z on both sides, express g and f z by y and ^{5 with} equations (5) ^and (6), ^and finally ^{arrive at the} following equations :

The determinant should vanish for nonzero solutions, ^which gives

Solving ^this equation ^we easily ^{obtain two} modes related to the coupled motion of the

permeation [10, 14] and the relative flow between the solvent and particles. ^{If one} imposes complete decoupling between the two variables, i.e., C23, P23, P32

0, ^{one has the} simplified

results :

describing respectively ^the interlayer permeation and the relative diffusion perpendicular ^to

the layers.

5. Experimental considerations.

In this section, ^{we discuss the} possibilities ^of experimental observation of the hydrodynamic

modes studied in this work. Specifically, ^{we estimate the} relaxation times and acoustic velocities in the smectic A phase [7] formed in solutions of tobacco mosaic virus. In addition,

the experimental observation is discussed in terms of the quasi-elastic light scattering (QELS) technique, ^{which is} commonly ^used [17] ^for observing dynamic fluctuations. The « slip

mode » in the lamellar phase, ^for example, has been observed [18, 19] ^{with this} technique. ^A typical QELS set-up can measure fluctuations of wavelength ^{from a} few thousand angstroms

to a few microns and in the time range from about 0.1 ps ^to seconds.

Due ^to lack of experimental data, little is known about the values of the elasticities and the viscosities of the smectic A phase ^{in TMV} solutions. Exact calculation of these quantities requires the consideration of detailed mechanical and structural properties ^{of the} ordering,

which is beyond the scope of the present ^{work. For this} reason, ^we will only ^{make the order-} of-magnitude estimation of the frequencies ^{of the} hydrodynamic modes. Since the density ^{of a}

TMV solution is about 1.05 g cm- 3 for a virus concentration of 0.175 g cm- 3, ^the velocity ^of

the first sound in the TMV system should be similar to the acoustic velocity in pure ^water

(1483 ^{m/s at} 20 °C), i.e. in the order of 103 m/s. At the wavelength ^{of a} few thousand angstroms, ^the period of the acoustic oscillations is in the order of 0.1 ns, which is too fast for the QELS technique. ^The hydrodynamic shear modes in the smectic A phase ^can couple ^to

the change in the refractive index when they propagate along and oscillate perpendicular ^to

the layers. ^{Due to the} periodic distribution of the refractive index in the direction

perpendicular ^{to the} layers, the shear motion normal to the layers ^{should cause refractive}

index fluctuations. The relaxation time of the shear mode in pure ^water is in the order of ns in

a QELS experiment. ^{Since the shear} viscosity q,, in ^{the TMV smectic A} phase ^is higher ^than

the viscosity of pure ^water (10- 2 dyn cm- 2 ^{s at} 20 °C), ^the corresponding relaxation time

(Eq. (44)) in the TMV smectic A phase ^{therefore can not be observed} by ^{the QELS} technique.

The hydrodynamic ^modes specifically attributed to the smectic A phase formed in colloidal solutions are the second sound and the mode of relative diffusion. The maximum speed ^{of the}

former is (C22/p )1/2 (Eqs. (21) ^and (40)). The relaxation time of the latter is approximately

determined by C33/L ^{in the x direction and} by C33/P33 ^{in the z direction} (Eqs. (39), (42) ^and (51)). ^For simplicity, ^we only estimate the relaxation times in the ^z direction. The osmotic

inter-layer compressibility C22 ^{and the} isotropic ^osmotic compressibility C33 may be estimated with the findings [20] of Crandall and Williams who measured osmotic compressibilities ^{of the}

colloidal crystals ^of polystyrene spheres ^with gravitational ^effect. They found that the order of

magnitude ^{of the} compressibilities is determined mainly by the number density ^{of the} crystal

instead of by the detailed form of the interaction between particles. ^For example, ^an ordinary

metallic crystal ^of density 1023 cm- 3 ^has compressibilities in the order of 1012 dyne cm- 2 ;

whereas for a colloidal crystal ^of density 1012 cm- 3, the ^osmotic compressibilities ^that

Crandall and Williams found are about 1 dyne cm- 2. Compressibilities approximately ^{scale to}

the number density. ^{For the smectic A} phase ^{in TMV} solutions, the number density ^{is in the}

order of 1015 cm- 3. We estimate by simple interpolation ^that C22 ^and C33 ^{in TMV smectic A} phase ^are in the order of 103 dyne cm- 2. ^{As the mass} density ^of TMV solutions is about 1.05 g cm- 3,

the maximum velocity of the second sound (Eqs. (21) ^and (40)) is then in the range of 0.1 -- 1 m S-I, ^which is much lower than that of the first sound. In the wavelength

range of ^a few thousand angstroms ^to microns, this maximum velocity corresponds ^to periods

in the order of 1 _03BCS, which is within the detection range of the QELS technique. ^The

observability of the second sound also depends ^{on the} damping ^effects (Eq. (40)) involved in

the process. According ^{to our} previous estimation, ^the hydrodynamic shear motion

corresponds ^{to a} time scale of 0.1 ~ 1 _ns, which is much faster than the oscillation of the second sound. The second sound is overdamped by ^the hydrodynamic ^shear effect and is

probably ^{difficult to observe.}

P33, ^the viscosity ^{of relative flow} along ^{the z} direction, may be estimated in analogy ^{to the}

calculation [11] ^{of the} slip coefficient in the lamellar phase. Ignoring ^the irregular shape ^{of the}

flow channels and the water molecules bound to the virus surfaces, ^{we make two crude} approximations ^{for the} order-of-magnitude estimation : 1) ^the viscous flow of water relative

to the TMV particles ^can be described by ^the macroscopic viscosity ^{of water} TI W ; 2) ^water

flows along cylindrical channels that are in contact with TMV rods separated by ^{the mean}

distance between particles. We have estimated in section 1 the average distance between TMV particles ⁱⁿ layers ^{to be about 400 Â.} With the second approximation, ^one easily ^obtains

with simple geometric analysis ^{the radius} of flow channels RS = 142 Â. When considering ^the

relative flow, ^we may assume zero velocity for the TMV particules- Since flows are weak at the

long wavelength limit, Poiseuille flow can be assumed in these channels :

in which vi ^{is the} velocity of solvent along ^{the z} direction, ^r is the distance to the axis of a flow

channel, ^and fz ^{is the osmotic force} causing the flow. From this expression, it is easy ^to obtain the mean solvent speed along ^{the z} direction: v’ z

fz Rs21 (8 ^{q W)} and the average solution

velocity along ^the

^direction vz

(p y? ) vi

fz R 2 _{P Y (8 ~w} p ). Assuming ^small P32 ^limit (see ^Sect. 2), equation (lla) gives :

One easily ^obtains P33

^{2.4 x} 1011 dyn ^{s cm- 4} for the TMV smectic A phase. ^{For the} wavelengths ^{of a} few thousand angstroms, the relaxation time of the mode of relative diffusion is estimated to be in the order of 1 - 10 millisecond, ^{which can} easily be measured by

the QELS technique.

6. Conclusions.

Hydrodynamic properties of the smectic A ordering formed in solutions of colloidal particles

have been studied. We have found a new purely damped mode associated with the relative diffusion between colloidal particles and the solvent. The dispersion relations of the mode of relative diffusion as well as the first and second sounds are calculated in a frame that is strictly

consistent with the general theory [8] ^of hydrodynamics. By simple estimation in a real

experimental system, ^we have shown that the mode of relative diffusion should be observable

by ^the quasi-elastic light scattering technique. We have also found that the second sound is

overdamped by ^the hydrodynamic ^{shear effect and is} probably ^{difficult to be observed. It is} expected ^{that a} similar mode of relative diffusion also exists in the other ordered phases

observed in solutions of colloidal particles.

Acknowledgments.

We thank K. Migler, ^{Dr. R.} Hentschke, ^{and M. P.} Taylor ^for illuminating discussions and critical reading ^of the paper. This work is supported ⁱⁿ part by the National Science

Foundation, through ^{Grant No.} DMR88-03582, ^and by ^{the Martin Fisher School of} Physics ^at

Brandeis University.

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