The thermal fluctuations of diluted emulsion

Advances in Colloid and Interface Science 108 – 109 (2004) 23–27

The thermal fluctuations of diluted emulsion T.Yu. Tchesskayaa,*, A.V. Zatovskyb a Information Technology Department, Odessa State Environmental University, Lvovskaya 15, Odessa 65016, Ukraine Theoretical Physics Department, I.I. Mechnikov Odessa National University, Dvoryanskaya 2, Odessa 65026, Ukraine

b

Abstract The collective excitations of the compressible diluted emulsion are studied. The diluted emulsion is consisted of molecules with intrinsic momentum. The motion equation for fluctuation fields of the velocity, density, intrinsic momentum and pressure were chosen here in the form of linearized Navier–Stokes equations with constant shear and volume viscosity, and diffusion coefficient of momentum, spontaneous stresses. For simplicity in the calculations, temperature fluctuations were not taken into account. We assume that the emulsion possesses constant surface tension. For hydrodynamic fields excited by spontaneous sources, continuity conditions on the surface tangents of the velocity and stress tensor components should be satisfied at the interface. The boundary conditions for the normal force components comprise the excess Laplace pressure and the random surface force determined by the difference of the radial values of the spontaneous volume stress densities. Because of the smallness of the surface displacements, all the boundary conditions should be applied on sphere. To construct the spectral densities of the thermal fields caused by random surface forces, we used the fluctuation-dissipate theorem. The spectral densities of the thermal amplitude fluctuations of the hydrodynamic currents are expressed in terms of susceptibilities. The spectrum of fluctuations contains all the collective excitations of the emulsion associated with fluctuating motions of the surface and with the properties of near-surface currents. Their analysis, based on strict inequalities between the drop’s emulsion size, the penetration depth of a viscous wave, and the wavelength of sound, essentially depends on the order of magnitude of the boundary transitions. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Emulsion; Thermal fluctuations; Spectral density fluctuations

1. Introduction Now dynamic properties of disperse systems, which the various objects are related to, are actively studied. The dynamic light scattering was used at study of suspension dynamics of polymers particles w1x, micellar solutions w2x, polymers diffusion in porous glasses w3x. The intensity of scattered light of liquid crystals in pores was measured in work w4x, the research of dynamics of microemulsions particles w5x and polymers w6x was carried out by a method of the neutron spin echo, the neutron scattering on gels is considered in Ref. w7x. Microemulsions, colloids, polymers solutions are complex as all these experiments indicate. Besides the equilibrium and dynamical properties of liquids in smallrestricted volumes like emulsion droplets significantly differ from those in large volumes w8–11x. The experiments show that the shear viscosity of some liquids as *Corresponding author. E-mail addresses: [email protected] (T.Yu. Tchesskaya), [email protected] (A.V. Zatovsky).

a phenomenological parameter loses its physical meaning of a material constant and near a solid surface becomes an effective quantity that depends on the distance from the wall w12x. In the classical Navier– Stokes hydrodynamics there is no difference between the internal friction of the liquid shells with one another and the external friction on the solid boundary. The boundary must, however, cause an anisotropy of the properties of the liquid being in contact with it. The ordering of the liquid can be connected either with the orientation of the molecules at the boundary or with the change of their translational and rotational mobility. The account for such effects is possible in the following way: we shall assume that the liquid, in addition to the density of the kinetic moment connected with the translational motion of the molecules, contains an intrinsic ™ that moment of momentum—the spin moment ™ssmyI is due to the rotation of the molecules (I is the volume density of the inertia moment of the molecules). For the first time the necessity of the account for the rotation of the molecules in liquids has been shown by Ya.I.

0001-8686/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2003.10.001

T.Yu. Tchesskaya, A.V. Zatovsky / Advances in Colloid and Interface Science 108 – 109 (2004) 23–27

24

Frenkel w13x. Later w14,15x, the hydrodynamics of liquids with additional degrees of freedom has been built. The derivation of the extended set of equations of such a liquid that in addition to the shear and bulk viscosity contains the rotational viscosity, has been described in detail in the work w15x. Much of the experimental data given in Refs. w8,12x are well explained on the basis of the system of hydrodynamic equations taking into account the local rotational velocity and appropriate boundary conditions in which a new kinetic coefficient is present: the boundary viscosity. In this paper the hydrodynamic fluctuations of a compressible viscous liquid with internal rotation inside the emulsion droplet are studied. The liquid itself is placed in a rigid spherical droplet, which is imbedded into another compressible viscous liquid with internal rotation. The linearized equations of motion for a compressible fluid with internal rotations are B ≠v™i hir E ri0 syc2i =driqChiq FDv™i D ≠t 2 G hi hir E ™ qCjiq y F=div ™viqhirrot ™siqfiv, D 3 2 G

i

velocity, respectively, is1, 2 for internal and external fluid correspondingly, ri0 is the equilibrium mass density ™ of the liquid, ci is the velocity of sound, fiv,s are external fluctuational forces, and mi is the diffusion coefficient of the internal momentum. The coefficients hir, hi, and ji denote the rotational, shear, and bulk viscosity, respectively. The closed system of hydrodynamic equations includes also the continuity equation and the assumption of non-divergency of the rotational velocity field: ≠dri qri0div v™is0, div s™is0. (2) ≠t At the boundary of the emulsion droplet the translational velocity of the liquid vanishes and the rotational velocity is connected with the vortex through the linear expression w12x: ™v s0, v™ s0, v™ sv™ , s™ s0, s™ s0, 2n

1t

2t

1n

hirqhiyhib , rsR, is1, 2, hir

As a result the equations for the transverse components of the translational velocity and the field of rotation separate from the rest of equations, ri0

B ≠v™iH hir E ™ syChiq Frotrotv™iHqhirrot ™siHqfivH. D ≠t 2 G

(5) In spherical coordinates with the origin in the center of the droplet the solutions of Eq. (1) can be found using the expansion in the basis functions obeying the vector Helmholtz equations

(6)

These systems of solutions should be taken to be finite at rs0. They will be determined as the products of spherical harmonics and spherical Bessel functions w16,17x, 1 B E Lim,nD™rGs =ŽYmnŽu,w.zinŽkr.., k C

F

B E B E Mim,nD™rGsrotD™rYm,nŽu,w.zinŽkr.G, C

F

C

F

1 1 B E B E Nim,nDr™Gs rotMim,ns rotrotD™rYm,nŽu,w.zinŽkr.G. k k C

F

C

F

(7)

zin(x) — are spherical Bessel functions of first order jn(x) and second order yn(x) for is1 and 2 correspondingly. The expansion for the translational and rotational velocity can be thus expressed as follows: w ˜ B™E N ™ ˜ B™Ez| ™ E ™v B x M™ iHDr,tGs8yuilMilDrGquilNilDrG~, C

F

C

F

C

F

l

2n

™˜ ™ E ™v B L i±±Dr,tGs8uilŽt.LilŽr., C

™s s l1 rotv™ , ™s s l2 rotv™ , 1t 1 2t 2 2 2 lis

(4)

™ ™ rotrotNisk2Ni.

™ B1 E ≠s™i mi ™ ri0 symirotrots™iqhirC rotv™iys™iFqfis, s™is , (1) D2 G ≠t Ii where ™v and s™ are the translational and rotational

1n

™v sv™ qv™ , div v™ s0, rot v™ s0. i i±± iH iH i±±

™ ™ ™ ™ =divLisyk2Li rotrotMisk2Mi,

B

i

where hib is the viscosity at the interface between the two media (the liquid and the surface). The second boundary condition takes into account the difference between the bulk and surface viscosities w12x. The velocity and the random forces in Eq. (1) can be divided into the longitudinal and transverse parts,

F

l

(3)

w ˜ B™E N ™ ˜ B™Ez| ™s B ™ E x M™ iHDr,tGs8ysilMilDrGqsilNilDrG~, C

F

C

l

F

C

F

(8)

T.Yu. Tchesskaya, A.V. Zatovsky / Advances in Colloid and Interface Science 108 – 109 (2004) 23–27

where we have introduced the normalized functions ˜ B E ™B™E (™2(1y2 (™2( ™ ™™2B ™E aDr™Gsa , a s dra DrGy a DrG. C

F

C

|

F

C

F

V

From the conditions Eq. (3) it is easy to obtain transcendental equations that determine the eigenvalues kl of the expansions of the fluctuating fields of translational and angular velocities. The normalizing factors are calculated taking into account the properties of the spherical harmonics and Bessel functions, and with allowance for Eqs. (3) and (7) they take the form

(™a2l(sqlLal, qls LM ls

Žnym.

4p 1 , 2ydm0 2nq1 Žnqm.

R3 nŽnq1.z2nq1Žbnl., 2

25

where Dssm1 y(r10I1), tsI1r10 y h1r. Analogous expressions take place for the component of the rotational velocity. The spectral densities of the expansion amplitudes of the fluctuating fields are found using the fluctuation– dissipation theorem w18x:

NjjjkMvs

ikBT Žajkya*jk. 2pv

(12)

The spectral density Njjjk*Mv is the Fourier transformation of the correlation function NjjŽt.jk*Ž0.M and ajk is the generalized susceptibility matrix that relates the random fields jj to the corresponding generalized forces f k: jjŽv.s8ajkfkŽv..

(13)

k w z 1 1 LLlsR3x znŽx.z9nŽx.q z2nŽx.|, 2 yx ~

w z 1 1 LlNsR3nŽnq1.x znŽx.ŽxznŽx..9y zn2Žx.|, xsgnl, 2 yx ~

(9)

where dij is the Kroneker symbol. The index l is the root number and the collective index l stays for the set of three numbers n, m, and l. We expand the longitudinal ™ ™ and transverse parts of the random forces fv and fs in the same way as in Eq. (8), with the expansion coefficients fail where asL, M, N for isn, and asM, N for iss. Rewriting Eqs. (1), (2) and (5) in the Fourier representation in time and taking into account the orthogonality of the eigenfunctions corresponding to different eigenvalues, we obtain algebraic equations for the Fourier components of the expansion coefficients, ulLs

n±±s

yivfLvl(v)yr0 2

2 2 l

2 l ±±

yv qc k yivk n

|

(14)

os Here sos ln and gln are the external tensors of momentum and angular velocity flows. Eq. (14) can be rewritten distinguishing the divergent contributions from the external sources. As a result the final expression for the energy dissipation is

S W B1 E ≠E T ™ ™™t y 1 rotsf ™™aT ™ t ™ ™Ff™aqsf X sy dVU rotvys TvfvyC v s s T, D2 G ≠t V 2 Y

(15)

(10)

nHshyr0, 1 = r0 w z 1 1 2 M,N fN,M k nl Žv.xyivq qDskl|qfsl y t ~ t

Ez

(11)

B B hr hr Ez yv2qk2lnHx qkl2 DsC1q F|yivx qkl2 CDsqnHq F| D D yt 2h G~ yt 2r G~ w1

≠E os s dVµsos lvŽ=lvnysmenml.qgln=lsn∂. ≠t

|

,

E 1 B4 C hqjF, G r0 D 3

uN,M l s

The correspondence between the quantities jj and f k is found from the expression for the mean power dissipated in the system. The energy dissipated in the viscous compressible fluid with internal rotation can be written as follows w14,15x:

w1

,

t os,t where ftvlsdivsos,t are true vectors, ln , fslsdivgln a os,a a os,a fvlselnmsln , fslselnmsln are pseudovectors and elnm is the Levi–Civitta tensor. With the help of Eq. (15), the Langevin equations Eq. (11), and the fluctuation–dissipation theorem Eq. (12), the spectral densities of the thermal fluctuations of the expansion amplitudes of the hydrodynamic fields can be determined and then the spectral densities of the velocity correlation functions can be obtained. We shall give here the transverse component that is connected, as distinct from the longitudinal one, with the internal moment of the liquid inside the spherical droplet:

T.Yu. Tchesskaya, A.V. Zatovsky / Advances in Colloid and Interface Science 108 – 109 (2004) 23–27

26

Fig. 1. The local diffusion coefficient of a liquid inside the spherical droplet: 1–the liquid without an internal moment, 2–the rotational viscosity is hrs0.1, 3– hrs0.5, 4– hrs1 poise.

kBT 2nq1 ™ E ™ B™ E N™vHBDr,t Re 8 GvHDr,t9GMvs 3 C

F

C

F

pr0

n)0,I

4pR

S

T 2nŽnq1.j2nŽblx. =U A b q2AŽgl. Ž . l T j2 b nq1

V

Ž l.

w

z

z2 W

w

nŽnq1.yjnŽglx.yŽglx.~2qyŽglxjnŽglx..9yŽglx.~ T X T, j2 g w1qgy2 " 2nq1 q1 z x

|

n

Ž l.y x

l

Bk E

w

B

RG

y

D

AC D

l

x

Ž Ž

Fsxyivqk2lCnHq

|

.

.~ |

Y

hr E 1 hrk2l Fy 2r0 G t 2r0

B

Ey1zy1 1 Cyivq qDsk2lF | , xsryR. D G t ~

(16)

Using the result Eq. (16), the local coefficient of diffusion for the liquid inside the droplet can be evaluated as follows: DŽr.s

1 3

`

1 ™B™ E™B™ E ™ E™B™ E ™vB Dr,tGvDr,0GMdts NvDr,tGvDr,0GMv. 3

|N 0

C

F

C

F

C

F

C

F

(17) The summation over the roots of Bessel functions is carried out up to the root not exceeding the value amaxs pRya, where a is the interatomic distance, and the summation over the numbers of Bessel functions is from the first number to such a number nmax, for which the first root of the Bessel function becomes larger than amax.

Fig. 2. The autocorrelation function of the transverse velocity component of a liquid inside the spherical droplet at different rotational viscosities: 1– hrs0.01, 2– hrs0.1, 3– hrs0.5, 4– hrs1 poise.

On Fig. 1 some results of the calculations of the local diffusion coefficient are presented showing its dependence on the coordinate r inside the droplet of radius Rs 100 A. The rest of the parameters of the liquid are Dss 10y5 cm2 ys, ts10y10 s, r0s1 gycm3, and as5 A. Fig. 2 demonstrates the frequency dependence of the autocorrelation function (normalized to its value at the zero frequency) of the transverse velocity component for various rotational viscosities hr at the distance 0.5 R from the center of the droplet. The parameters of the liquid are the same as above. The presented theory generalizes the previously obtained results of the work w17x to the case of the fluctuations of liquids with internal moment in the droplet of emulsion. The confined space significantly influences the thermal hydrodynamic fluctuations of the compressible liquid. Besides this, the proper internal moment leads to the decrease of the local diffusion of the liquid. References w1x W. van Megen, R.H. Ottewill, S.M. Owens, J. Chem. Phys. 82 (1985) 508. w2x S. Walrand, L. Belloni, M. Drifford, J. Physique 47 (1986) 1565. w3x Yi. Guo, K.H. Langley, F.E. Karasz, Macromolecules 23 (1990) 2022. w4x F.M. Aliev, G.Yu. Vershovskaya, L.A. Zubkov, JETF 99 (1991) 1512, (in Russian). w5x J.S. Huang, S.T. Milner, Phys. Rev. Lett. 59 (1987) 2600. w6x D. Richter, B. Stuhn, ¨ B. Ewen, Phys. Rev. Lett. 58 (1987) 2462. w7x H.D. Middendorf, F. Cavatorta, A. Deriu, U. Steigenberger, Physica B 156–157 (1989) 456. w8x V.N. Churaev, Colloid J. 58 (1996) 725. w9x K.P. Travis, B.D. Todd, D.J. Evans, Phys. Rev. E 55 (1997) 4288.

T.Yu. Tchesskaya, A.V. Zatovsky / Advances in Colloid and Interface Science 108 – 109 (2004) 23–27 w10x K.P. Travis, D.J. Evans, Phys. Rev. E 55 (1997) 1566. w11x I. Bitsanis, J.J. Magda, M. Tirrell, H.T. Davis, J. Chem. Phys. 87 (1987) 1733. w12x E.L. Aero, N.M. Bessonov, A.N. Bulygin, Colloid J. 60 (1998) 446. w13x Ya.I. Frenkel, Kinetic Theory of Liquids, Oxford University Press, 1946. w14x S.D. de Groot, P. Mazur, Nonequilibrium Thermodynamics, North-Holland, Amsterdam, 1962.

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w15x M.I. Sliomis, JETP 24 (1967) 73, (in Russian). w16x P. Morse, H. Feshbach, Methods of Theoretical Physics, 2,, McGraw-Hill, New York, 1953. w17x A.V. Zvelindovsky, A.V. Zatovsky, Nuovo Cimento 19 (1997) 725. w18x L.D. Landau, E.M. Lifshits, Statistical Physics, 2,, Oxford, Pergamon, 1980.