fulltext

Fluctuations of a fluid inside a pore

A. V. ZVELINDOVSKY(1) (2) (*) and A. V. ZATOVSKY(2)

(1_{) Department of Biophysical Chemistry, University of Groningen} Nijenborgh 4, 9747 AG Groningen, The Netherlands

(2_{) Department of Theoretical Physics, Odessa I.I. Mechnikov State University} Dvoryanskaya 2, 270100 Odessa, Ukraine

(ricevuto il 20 Luglio 1995; revisionato il 26 Agosto 1996; approvato il 13 Gennaio 1997)

Summary. — The correlation theory of the thermal hydrodynamic fluctuations of compressible viscous fluids within a spherical pore has been developed. The fluctua-tion mofluctua-tions are described by the linearized Navier-Stokes and heat transfer equations, containing spontaneous viscous stresses and heat fluxes. The exact expressions for the spectral densities of velocity and mass density correlation functions are found as well as the dynamical structure factor. The results depend on cavity size, thermodynamic and kinetic parameters of fluid and differ essentially from the case of unrestricted fluid. Long-time oscillations of velocity autocorrelation function, the frequency spectra, and the coefficient of self-diffusion are also discussed.

PACS 82.70 – Disperse systems.

PACS 05.40 – Fluctuation phenomena, random processes, and Brownian motion. PACS 47.35 – Hydrodynamic waves.

PACS 03.40 – Classical mechanics of continuous media: general mathematical aspects.

1. – Introduction

The equilibrium and dynamical properties of confined fluids significantly differ from those of bulk fluids. During last decade a lot of experimental and theoretical works have been dedicated to the study of the role of a confined geometry in determining structural, thermodynamical and transport properties of fluids inside microscopic pore spaces [1-5]. The theoretical analysis of these systems is mostly based on the computer simulation methods such as molecular-dynamics, Monte Carlo, etc. Fluid inside a pore has been used by some authors as a simple theoretical model for the description of static properties of reverse micelles in microemulsions [6].

(*) E-mail: azHdtp.Odessa.ua, a.v.zvelindovskyHchem.rug.nl

The thermal fluctuations of confined fluids are of theoretical and technological interest. The correlation functions of the molecular variables of such systems are important for explaining experiments in which pores, capillaries, dense suspensions, polymers, globular macromolecules, cells and other objects are probed by radiation. In the framework of phenomenological theory the motions of molecules within a constraining volume can be modeled as diffusion of particles within a “cage” which is considered as sphere (so-called confined diffusion) [7, 8]. In these approaches the hydrodynamic interactions are neglected. However, the possibility of applying the ordinary hydrodynamics to such objects has been indicated [9, 10] by molecular-dynamical methods of studying systems of particles as models of liquids. The applicability of hydrodynamic equations at short distances was examined in [11, 12]. The movement of the atomic-size particle with initial thermal velocity through the fluid at rest was studied by means of hydrodynamic approach. The numerical solution of Navier-Stokes equations was found to be in good quantitative agreement with the flow picture obtained from the molecular-dynamic simulation already after ten collisions of particles and at the distance of three atomic sizes [11].

Equilibrium hydrodynamic fluctuations have mostly been studied for infinite or semi-infinite spaces [9, 13, 14]. Much attention has been given to the construction of hydrodynamical models of the shape fluctuations of droplets

(

[15, 16] and references therein

)

. In [17] the simplified case of irrotational motion of a fluid within a sphere in the absence of thermal conduction has been investigated. Earlier we have briefly indicated the more general description, albeit without any detailed analysis [16, 18]. The frequency spectrum of a condensed system acts as a weight function in calculating thermodynamical quantities such as free energy, entropy, specific heat and so forth. The vibrational structure of small particles has been studied especially in the solid state [19-22]. These methods have been applied also to small liquid droplets, for which the frequency spectrum has been found by means of molecular dynamics [23] and by neutron inelastic scattering [24]. The phenomenological treatment of the above spectra has been based on the hydrodynamical liquid-drop model in the simplest case of inviscid fluid [25]. However this method cannot be directly applied to the fluid within a pore. The problem of calculating the frequency spectrum of excitations is close to the general Weyl problem of finding the distribution of wave equation eigenvalues depending on boundary conditions.

In this paper we present the most general approach to the study of the collective thermal hydrodynamic fluctuations of viscous fluid in pores. We regard the pores as rigid spheres of typical size A102–104Å. The compressibility of fluid and the thermal conduction are taken into account. The spectral densities of the correlation functions of the hydrodynamic variables are found based on the fluctuation-dissipation theorem. The results obtained (in particular, hydrodynamic correlation functions, dynamic struc-ture factor, frequency spectrum, local diffusion coefficient) differ from the ones for the infinite fluid and depend strongly on the size of pores and on the fluid characteristics.

2. – Equations of motion and boundary conditions

We will study the hydrodynamic fluctuations of a compressible viscous fluid confined by the spherical cavity of radius R. The fluctuation fields of velocity v, mass density r

and temperature T can be described by the linearized Navier-Stokes and heat transfer equations, containing spontaneous stresses f and heat fluxes div g [14, 26]

.

`

/

`

´

r ¯v ¯t 4 2˜dp 2 h rot rot v 1

g

z 1 4 3h

h

˜div v 1f , ¯ ¯tdr 1r div v40 , ¯ ¯tds 4 k rTDdT 2div g (2.1) with dp 4

g

¯p ¯r

h

T dr 1

g

¯p ¯T

h

r dT , _{ds 4}

g

¯s ¯r

h

T dr 1

g

¯s ¯T

h

r dT ,

where p is pressure, s is entropy per unit of mass, k is heat conductivity, h and z are the shear and bulk viscosities, D denotes the Laplacian, and d designates the surplus values of variables while the equilibrium ones are ar(r, t)b4r, aT(r, t)b4T, av(r, t)b 40. All kinetic coefficients and thermodynamic derivatives in (2.1) are taken to not depend on coordinates. Following the fluctuation theory based on the fluctuation-dissipation theorem, the spontaneous sources, as f and g in (2.1), are assumed to be small so that the linearized dynamic equations remain valid [14, 27]. In this theory such sources of fluctuations act as auxiliary quantities which drop out of final results for the correlation functions of fluctuating dynamical variables. In the absence of frequency dispersion of viscosities and heat conductivity the spontaneous sources f and g have white-noise spectra.

The problem can be solved in the simplest way in spherical coordinates (r , u , W) with origin at the center of the sphere. Assuming rigid-cavity surface one can apply the stick boundary conditions

v(r , t) 40 (r4R) . (2.2)

It is convenient to separate the velocity and the spontaneous stresses in (2.1) into irrotational (longitudinal) and solenoidal (transverse) parts,

v 4vll1 v», div v»4 0 , rot vll4 0 . (2.3)

the other equations

.

`

`

/

`

`

´

¯v» ¯t 4 2n»rot rot v»1 1 rf», ¯dr ¯t 4 2r div vll, ¯2r ¯t2 4 u 2_g21_{Ddr 1}

g

¯p ¯T

h

r DdT 1nll ¯ ¯tDdr 2div fll, 2¯dT ¯t 1 xgDdT 4 T cV

gg

¯s ¯r

h

T ¯dr ¯t 1 div g

h

, (2.4) where nllf

g

z 1 4

3h

h

N

r , n»fhOr, xfkO(rcp) is the temperature conductivity, u is adiabatic sound speed, g f cpOcV, and cV, cp are the specific heats at constant volume

and pressure correspondingly.

The solution of (2.4) can be found using basis functions which satisfy the vector Helmholtz equations

˜_{div L 42 k}2_{L , rot rot M 4k}2_{M , rot rot N 4k}2N . (2.5)

The three systems of solutions of (2.5) which are finite at r 40 have been thoroughly studied and are determined by differentiating products of spherical harmonics and spherical Bessel functions [28]:

.

`

/

`

´

Lmn(r) 4 1 k˜

(

Ymn(u , W) jn(kr)

)

, Mmn(r) 4rot

(

rYmn(u , W) jn(kr)

)

, Nmn(r) 4 1 krot Mmn(r) . (2.6)

We write the expansions of the velocity, mass density and the temperature in the form v»(r , t) 4

!

l

k

vlMMAl(r) 1vlNNAl(r)

l

, (2.7) vll(r , t) 4

!

l vlLLAl(r) , (2.8) dr(r , t) 4

!

l wl(t) jn(kr) Ymn(u , W) , (2.9) dT(r , t) 4

!

l Kl(t) jn(kr) Ymn(u , W) , (2.10)

where we have introduced the normalized functions a

A(r) fa(r)O

k

Va2V, Va2_{V 4}



V

a2(r) dr , (2.11)

where V is the sphere volume.

determine the eigenvalues k in (2.6) jn(bnl) 40 , kl4 bnlOR , (2.12)

N

N

N

jn8 ( mnl)

k

n(n 11) mnl jn( mnl) n(n 11) mnl jn( mnl)

k

n(n 11) mnl

(

mnljn( mnl)

)

8

N

N

N

4 0 , kl4 mnlOR . (2.13)

The index l 41, 2, . . . is the number of the root and the prime means differentiation with respect to the argument of Bessel functions. Equation (2.13) is reduced to

mnjjn8 ( mnj) 2njn( mnj) 40, mnjjn8 ( mnj) 1 (n11) jn( mnj) 40

or simply to

j_{n 61}( mnl) 40 , kl4 mnlOR .

(2.14)

Equation (2.12) gives the eigenvalues for the functions M , and eq. (2.14) for both L and N. The index l is thus a collective index, denoting the set of three numbers n , m and l. After integration with allowance for eqs. (2.6), (2.12), (2.14) and the properties of the spherical harmonics and Bessel functions [28], the normalization factors (2.11) take the form (2.15) Val2V 4 qlLla, qlf 4 p 2 2dm0 1 2 n 11 (n 2m)! (n 1m)! , (2.16) LlM4 R3 2 n(n 11) jn 11 2 _(b nl) , (2.17) LlL4 R3

k

1 mnl jn( mnl) jn8 ( mnl) 1 1 2 jn 2_{( m} nl)

l

4 4 R3jn2( mnl)

k

1 2 1 1 mnl2

g

6

g

n 1 1 2

h

2 1 2

h

l

, (2.18) LlN4 R3n(n 11)

k

1 m2nl jn( mnl)( mnljn( mnl))81 1 2 jn 2 ( mnl)

l

4 (2.18) 4 R3n(n 11)jn2( mnl)

k

1 2 1 1 m2nl

g

6

g

n 1 1 2

h

1 1 2

h

l

, where dij is the Kroneker symbol and the sign “6” corresponds to the sign in

eq. (2.14). As follows from eqs. (2.16)-(2.18) the mode n 40 is a pure compressible one, as LN0 l4 L0 lM4 0, while LL0 lc0. Therefore, we will omit the terms with n 40 in all series for N- and M-modes.

The longitudinal and transverse parts of the random viscous force f can be expanded as in eq. (2.7), (2.8) with expansion coefficients fla, a 4M, N, L, and the

It is convenient to rewrite all equations in Fourier transform x(t) 4



2Q Q x(v) exp [2ivt] dv . (2.19)

Then, using the orthogonality of the eigenfunctions corresponding to different eigenvalues, it is easy to obtain the algebraic equations for the Fourier transform of the coefficients vla vlM , N4 flM , N(v) Or 2iv 1 n»kl2 , (2.20) vlL4 iv

(

flL(v) 2gl(v) Rl

)

JlOr (2.21) with

.

`

/

`

´

Rlf (¯pO¯T)rkl2TOcV 2iv 1 kl2xg , Jl21fv22 u2g21kl21 ivkl2nll2 ivRl

g

¯s ¯r

h

T , (2.22)

where (¯sO¯r)T4 2(¯pO¯T)rOr

2

and (¯pO¯T)r4 ru

k

(g 21) g21cVOT.

The expansion coefficients for the excess density and temperature can finally be expressed in terms of vlL, for instance, for the density we have

wl(v) 42vlL rkl

iv

(

qlLl L

₎

21 O2_. (2.23)

3. – Spectral densities of fluctuations

The spectral densities of the fluctuating fields can be obtained using the fluctuation-dissipation theorem which in the classical limit reads [27]

ajjjk* bv4 2 kBT

2 pv

(

ajk2 akj*

)

, (3.1)

where the fluctuating fields j are related to the corresponding generalized forces F by means of components of the matrix of generalized susceptibility ]ajk(,

jj(v) 4

!

k

ajkFk(v) .

(3.2)

The correspondence between the generalized forces and coordinates can be found from the expression for the energy dissipation in the system averaged over time

E._{(t) f Q(t) 42}

!

j jj(t) F . j(t) . (3.3)

Using Parseval theorem for the Fourier transform one can write

Q(t) 4 Q(v), Q(v) 42 Re iv

!

j

jj(v) Fj* (v) .

(3.4)

The dissipation in the fluid has the form [14, 26]

E._{4 2}



V dV

g

s8ik ¯vi ¯xk 2 dT div g

h

, (3.5)

where sik8 is the viscous stress tensor. When the correlation functions of temperature

are not examined, the last term in (3.5) can be omitted. Then, applying the Gauss theorem and using the zero boundary condition (2.3), we obtain

E.₄



V dV vf 4



V dV(vllfll1 v»f») , f 4˜Qs8 . (3.6)

Substituting the expansions in eigenfunctions, integrating over the pore volume and averaging over time, we find

Q(v) 4Re

!

a , l

vla(v) fla *(v), a 4L, M, N .

(3.7)

Equations (3.7) and (3.4) give the Fourier components of generalized forces Fla(v) 4 fla(v) O(iv). The Langevin equations (2.20), (2.21) and fluctuation-dissipation theorem

(3.1) permit us to construct the matrix of the generalized susceptibilities ajk, which

determines the spectral densities of equilibrium thermal fluctuations for the expansion amplitudes of the hydrodynamic fields:

.

`

/

`

´

avlLvl 8L *bv4 Re kBTdll 8 pr ivJl, avlbvl 8b *bv4 Re kBTdll 8 pr(2iv1n»kl 2 ) , b 4M, N , (3.8)

where Jl is written in (2.22). This result differs from the spectral densities for an

infinite fluid only by the discrete nature of the wave numbers kl.

In the linear approximation, since the absorption at the wavelength of the perturbation is small (n AxbcOk [14, 26]), the hydrodynamic motion can be divided into solenoidal, acoustic, and temperature waves, which do not interact with each other. And the spectral density of the longitudinal component, as in the case of an infinite medium [9, 29], consists of two contributions

aNvlLN2bv4 kBTv2 pru2

y

(g 21)x v2_{1 x}2kl4 1 Gu 2_k l22 (g 2 1 ) x(v22 kl2u2) (v2_{2 k}l2u2)21 G2v2kl4

z

, (3.9) where G f (g 21)x1nll, kl4 mnlOR.

Now it is easy to construct the spectral densities of thermal fluctuations of the velocity and mass density

av(r , t) v(r8, t 8)bv4

!

l , AaNvl A N2bvAAl(r) AAl* (r8), A 4M, L, N , (3.10) adr(r , t) dr(r8, t 8)bv4

!

laNwlN 2_b vjn(klr) jn(klr 8) Ymn(u , W) Ymn(u 8, W8) , (3.11) where l 4 ]n, l, m( and aNwlN2bv4 kl2r2 v2_q lLlL aNvlLN2bv. (3.12)

In the case of coinciding spatial arguments these results simplify, since it is possible to perform the summation over m explicitly as only spherical functions depend on this number (see appendix A). Then the correlation functions depend only on the distance from the center of pore x 4rOR, for example,

av(r , t) v(r , t 8)bv4 Fll(r , v) 1F»(r , v) (3.13) with F»(r , v) 4 kBT pr Ren D0, l

!

2 n 11 4 pR3

{

1 2iv 1 n»bl2OR2 n(n 11) jn2(blx) j_{n 11}2 _(b l) O2 1 (3.14) 1 1 2iv 1 n»m2lOR2 n(n 11)[ jn( mlx) O( mlx) ]21 [

(

mlxjn( mlx)

)

8 O( mlx) ]2 jn2( ml)[ 1 O21ml22

(

6 (n 1 1 O2 ) 1 1 O2

)

]

}

, Fll(r , v) 4

!

n , l 2 n 11 4 pR3 aNvnl L N2bv n(n 11)[ jn( mlx) O( mlx) ] 2 1 [ jn8 ( mlx) ] 2 jn2( ml)[ 1 O21ml22

(

6 (n 1 1 O2 ) 2 1 O2

)

] . (3.15)

Here l 4 ]n, l(, prime denotes the differentiation with respect to the argument of Bessel functions and the sign “6” corresponds to the sign in (2.14). Note that the summation should be made over the all roots of (2.14), however the numerical analysis shows that the contribution with the lower sign is predominant. In fig. 1 the contribution of ellipsoidal mode (n 42) to the autocorrelation function of longitudinal velocity field is demonstrated for different viscosities of fluid in the case of

g f cpOcV4 1.

The results (3.14), (3.15) can be simplified, since the summation over wave numbers

kl can be performed explicitly (see appendix B). For instance, for the transverse

velocity we have F»(r , v) 4 kBT 4 p2 rRn» Re

!

n 41 Q ( 2 n 11) Q (3.16) Q

{

a1n(n11) jn(a1x) jn(a1)

(

jn(a1x) yn(a1) 2yn(a1x) jn(a1)

)

1 1 x2( 2 n 2a12)

!

j 41 2 ₍₂₁₎j 11_a j j_{n 21}(aj) Q Q

[

(

ajxjn(ajx)

)

8

(

(ajxyn(ajx)

)

8 jn 21(aj) 2

(

ajxjn(ajx)

)

8 yn 21(aj)

)

1 1n(n 1 1 ) jn(ajx)

(

yn(ajx) jn 21(aj) 2jn(ajx) yn 21(aj)

)

]

}

,

Fig. 1. – The autocorrelation function Fll(r , v) 4 avll(r , t) vll(r , t 8)bvof the longitudinal velocity of

fluid within the pore of radius R 4100 Å for n42, u41500 mOs, r4103_kgOm3_{, rOR40.5, g41} and various fluid viscosities: h 4 0.01 (1), 0.02 (2), 0.1 (3) kgO(m s); z40.75h.

where the lower sign is chosen in (2.14), a12fivR2On», a2fk2 n, and yn(z) is the

spherical Bessel function of second kind.

Note that the correlation functions of the components of the Euler velocity field, in the case of coinciding spatial arguments, are practically equal to the correlation functions of the velocity of a Lagrangian particle [9], which are a good hydrodynamic approximation for the correlation functions of the molecular variables.

4. – Dynamical structure factor

Now we can find the dynamical structure factor which is the two-fold Fourier space transform of the density-density correlation function

S(k , v) 4



dr dr8 eik(r 2r 8)_{adr(r , t) dr(r8, t 8)b} v,

(4.1)

where the integral is taken over the pore volume. Inserting (3.11) into (4.1) and using the expansion of a plane wave in spherical harmonics

eikr 4 4 p

!

l 8 m 8 il 8 q_{m 8 l 8} jl 8(kr) Yl 8 m 8(uk, Wk) Yl 8 m 8* (u , W) (4.2)

and performing the integration over u , W and summation over l 8, m 8, m we find S(k , v) 44p

!

n , l ( 2 n 11)Anl LLnl

u



0 R r2_{dr j} n(klr) jn(kr)

v

2 (4.3) with Anlf kl2r2 v2 aNvl L N2bv, (4.4) that is determined by (3.8) or (3.9).

Finally taking into account (2.14), (2.17) we obtain

S(k , v) 44p

!

n , l ( 2 n 11) AnlR 1 O21m22nl

(

6 (n 1 1 O2 ) 2 1 O2

)

3 (4.5) 3

.

`

/

`

´

k2 (kl22 k2)2 j_{n 61}2 _(kR), R2 4 jn 2_{( m} nl) , klck , kl4 k ,

where the sign “6” corresponds to the same sign in (2.14).

In the case of small wave vector of radiation kR b 1 the main contribution to (4.5), which corresponds to the lower sign in (2.14), has the form

S(k , v) Cb01 ( 3 b12 b0)(kR)2 (4.6) with bj4 4 pR3

!

l 41 Q _Ajl mjl2( m2jl2 j 2 1 ) , mjl4 p 2 ( 2 l 211j) . (4.7)

The correlator Anl has the simple form in the case of g 41 Al4 kBTr pv Re ikl2 v2_{2 u}2kl21 ivkl2nll . (4.8)

In figs. 2 and 3 the acoustic wing of the dynamical structure factor of fluid within the pore of radius 400 Å is shown for various viscosities and radiation wave number k. The plots have been calculated according to (4.5), (4.8). At some values of model parameters the spectra have only the central maximum which corresponds to pure relaxation mode and is supplemental to the usual Rayleigh peak. The condition of appearance of the shifted peak (Brillouin component) can be obtained from the equation ¯ ¯v[ (u 2_k l22 v2)21 v2nll2kl4]214 0 , (4.9) whence uR nllmnl D 1 . (4.10)

For the low viscous liquids, h A1023 _{kgO(m s), the condition (4.10) is valid for} practically any pores R D10 Å, while for the liquids such as glycerin the Brillouin peak

Fig. 2. – The dynamical structure factor of fluid within the pore of radius R 4400 Å for the various wave numbers k: kR 40.1 (1), 0.125 (2), 0.15 (3), 0.175 (4), 0.2 (5), and the viscosity h40.1 kgO(m s). The rest of the parameters are the same as in fig. 1.

Fig. 3. – The dynamical structure factor of fluid within the pore of radius R 4400 Å for the various fluid viscosities h 40.001 (1), 0.005 (2), 0.02 (3) kgO(m s) and kR40.5. The rest of the parameters are the same as in fig. 1.

Fig. 4. – The dynamical structure factor of glycerin within the pores of various radii R 4400 Å (1), 4 Q 104_{Å (2), 5 Q 10}4_{Å (3), 8 Q 10}4_{Å (4), and kR 40.5.}

will appear only in large pores of radii R D104Å. The change of shape of the Brillouin component of the dynamical structure factor of glycerin within various pores is shown in fig. 4.

5. – Time-dependent velocity autocorrelation function

The discrete nature of the wave numbers kl leads to the behaviour of correlation

functions which utterly differs from the case of infinite liquids. The sequence of wave numbers is bounded below by the minimal value klminA (n 1 C) OR , where R is the

radius of pore, n is the mode number, and C is a constant which is C  [pO2, 2 p[ for most of the modes.

It is convenient to rewrite the spectral densities of the longitudinal velocity fluctuations (3.10) into the form

aNvlLN2bvfv2cv4 kBTv2 pru2kl2 Re

k

g 21 iv 1xkl2 1 (5.1) 1(g 21) xkl 2 g_{12 g2}

g

1 iv 1g1 2 1 iv 1g2

h

1 u 2_k l2 g_{12 g2}

g

g₂21 iv 1g2 2 g1 21 iv 1g1

h

z

with gZf 1 2 Gkl 2_Z

o

1 4G 2_k l42 u2kl2.

Then performing Fourier transformation of (3.14) and cv, (5.1), one can find

av»(r , t) v»(r , 0 )b 4 kBT r

!

l [MA2_e2n»kl2t_{1 N}A2_e2n»kl2t_{] ,} (5.2) avll(r , t) vV(r , 0 )b 4

!

lL A2

g

2 ¯ 2 ¯t2

h

c(t) 4 kBT r Re

!

l L A2 Q (5.3) Q

y

(g21) x u2

g

g₂2 e2g2t_2g 1 2 e2g1t g_12g2 2xkl 2_e2xkl2t

h

1g1e 2g1t_2g 2e2g2 t g_12g2

z

.

The weights of correlators (5.2), (5.3),

!

mA

A2_{, are determined by (A.1)-(A.3), (2.16)-(2.18)} and essentially depend on the relative coordinate rOR inside the cavity.

The autocorrelation function of longitudinal velocity oscillates on the condition

GklE 2 u .

(5.4)

For the pore of radius R A100 Å and small l this condition is valid for the low viscous liquids nllE 2 Q 1025 m2 Os. On the other hand, eq. (5.4) is the cut-off condition for the phonon spectrum in liquids [9], so klmaxB 2 uOG.

For many liquids u A103

mOs and xA1027 _m2

Os, so the last term in (5.3) is predominant. Taking this fact and (5.4) into account we obtain

avll(r , t) vV(r , 0 )b 4 kBT r

!

l L A2_e2 1 2Gkl 2_t

k

cos plt 2 Gkl2 2 pl sin plt

l

, (5.5) where plf

o

u2kl22 1 4 G 2_k l4.

The zeros of eq. (5.5) are determined by

tlj04

u

arccos

o

Gkl 2 u 1 p( j 2 1 )

v

Opl, j 41, 2, R . (5.6) For r 4103 kgOm3 , u 41. 5 Q103

mOs, R440 Å, hA1023 _{kgO(m s) the first zero is}

tl10A 10212s. The value of tl10 decreases with increasing l 4 ]n, l(.

In fig. 5a) the longitudinal velocity autocorrelation functions (5.5) are shown as a function of dimensionless time yll4 tGOR2 for two values of kl: m01OR (curve 1) and

m02OR (curve 2). The functions have an oscillating shape which essentially depends on viscosity and relative coordinate x 4rOR. In fig. 5b) the longitudinal velocity autocorrelation functions are shown for the various values of dimensionless parameters

x and z f m01GO(2uR): (x; z) 4 (0.9; 0.27) (curve 1), (0.7; 0.27) (curve 2), (0.9; 0.81) (curve 3). Note that the oscillations of velocity autocorrelation functions for confined liquid are well known from the molecular-dynamic simulations [10].

Fig. 5. – The time-dependent autocorrelation functions of the longitudinal velocity of fluid within the spherical pore.

6. – Frequency spectrum

In the absence of the temperature waves we have three kinds of fluid modes inside a spherical pore: two pure relaxational ones (N- and M-modes) and the density oscillations (L-mode). The dispersion relations for these modes can be obtained from eq. (2.4) without auxiliary sources of fluctuations. They have the form (g 41)

v 94k2_n » 2 _,

v 840 for N- and M-modes

(6.1) k 4 v u

g

1 2 ivnll u2

h

21 O2 for L-mode , (6.2)

where v 4v82iv9

(

_{v 8, v9 are real, see (2.19)}

)

and k f klis determined by (2.12) for

(6.1) and by (2.14) for (6.2). For the acoustic waves in the real liquids

(

[26], Chapt. 8

)

one can rewrite (6.2) in the form

k C v

u 1 i nll 2 u3v

2

which gives the complex frequency of decaying density oscillations

v 4 iu

2

nll

] 1 2

k

1 12iknllOu( , (6.3)

where the value of square root with the positive real and imaginary parts must be chosen.

To introduce the atomicity of a real fluid into the present continuum system one cannot use the method of [22, 25] as there is no surface oscillations here. We set limitations on the wave number k supposing that the distance between zeros of Bessel function

(

see (2.9)

)

jn(kmaxri) and jn(kmaxri 11) is of order the mean

inter-atomic distance a. For kmaxr c 1 and r AR we use the asymptotic jn(kmaxr) A A (kmaxr)21cos

(

kmaxr 2p(n11)O2

)

. Therefore, the maximum wave number is determined as

kmaxA

p a

(6.4)

and does not depend on the mode number n. The minimum value of k for each mode number n is determined by (2.12), (2.14): kmin4 bn1OR (for M-mode) and kmin4 mn1OR

(for L- and N-modes).

There exists the number nmaxsuch that for any nA Dnmaxthe corresponding value of

kmin4 knA, 1 is larger than or equal to kmax4 pOa. This nmax serves as cut-off number in series. It can be estimated by means of the approximation for the first nonzero root of

n-th Bessel function [30] an , 1B

g

n 1 1 2

h

1 1 . 86

g

n 1 1 2

h

1 O3 1 1 . 03

g

n 1 1 2

h

21 O3 1 R . For example, for M-mode nmax is estimated from anmax, 1B pROa.

Thus we can calculate the total number of modes inside the pore as

N 4

!

_n

!

_a ( 2 n 11) U(nmax2 n) U(amax2 a) e(a) , where

U(x) 4./ ´

1 , _{x F0 ,}

0 , _{x E0}

is the Heaviside function, a runs over all roots of eqs. (2.12), (2.14) mnl, bnl. The

maximum root is determined by (6.4) amax4 kmaxR 4pROa. Since L-, M- and N-modes are different, we introduce the factor e(a) which is e(b0 , l) 40, e(bn c 0 , l) 4e( m0 , l) 41,

e( mn c 0 , l) 42. It turned out that the number of modes inside the pore N is less than the

total amount of degrees of freedom 3 Natom2 6 (NatomB R3O(aO2 )3). This discrepancy could be explained by the fact that some degrees of freedom in a few molecular layers near the surface are frozen due to chosen stick boundary conditions at the rigid pore shell, (2.2).

The frequency spectrum is the number of modes included in the frequency segment dv,

F(v) 4 d

dvN(v) , (6.5)

where N(v) is the number of modes whose eigenfrequencies are less than v . It is defined as

N(v) 4

!

n

!

a ( 2 n 11) U(nmax2 n) U(v 2 va) e(a)

(6.6)

Fig. 6. – The frequency spectrum (oscillating mode) of a fluid inside the spherical pore of radius R 450 Å for a44 Å, u41500 mOs, r4103_kgOm3_{, h 40.01 kgO(m s), z40.75h.}

present model there exist two kinds of spectra: pure relaxational and oscillation ones. For oscillating L-mode we have Nosc(v8) obtained from (6.6) with a4] mnl( and e(a)f1.

Here v8 and va 8maxare the real parts of (6.3) with k 4aOR and kmax, respectively. For pure relaxational M- and N-modes we have Nrel(v 9) which is found from (6.6) with a4 ]bnl, mnl( and e

(

a(n)

)

4 1 2 dn0. In this case va9 and v9maxare determined by (6.1) with

k 4aOR and kmax, respectively.

Figures 6 and 7 show these spectra Fosc(v 8) and Frel(v 9), (6.5), in which smoothing over three neighboring points is carried out. The serrate shape of the spectra is a typical feature of small particles [20, 22, 25]. The density oscillations (fig. 6) and the shear modes (fig. 7) have utterly different frequency dependance. The spectrum in fig. 6 for the fluid in a spherical pore essentially differs from the one found earlier for the liquid drop with free surface [25].

7. – Diffusion coefficient

The obtained results allow us to calculate the local coefficient of self-diffusion:

D(r) 4 1 3



0 Q av(r , t) v(r , 0 )b dt 4 1 3 av(r , t) v(r , 0 )bv 40. (7.1)

Fig. 7. – The frequency spectrum (pure relaxational modes) of a fluid inside the spherical pore. The parameters are the same as in fig. 6.

Using (3.13)-(3.15), (3.9) gives immediately for the fluid inside the spherical pore

D(r) 4 1 3F»(r , 0 ) 4 kBT 6 p2hR n 41

!

nmax ( 2 n 11)

{

!

blE amax n(n 11)jn2(blx) b2ljn 112 (bl) 1 (7.2) 1

!

mlE amax n(n 11)[ jn( mlx) O( mlx) ]21 [ ( mlxjn( mlx)

)

8 O( mlx) ]2 jn2( ml)[m2l6 ( 2 n 1 1 ) 1 1 ]

}

, where nmax and amax have been introduced in the previous section, bl, ml are

determined by (2.12) and (2.14), respectively, and the contributions from both signs “6” must be taken into account.

The result of numerical calculations of expression (7.2) is illustrated in fig. 8. The local diffusion coefficient is strongly dependent on the radial coordinate r inside the pore. It smoothly increases when moving away from the pore centre, and after reaching the maximum value at distance r A0.9R, the diffusion coefficient rapidly drops to zero in agreement with the stick boundary conditions (2.2). The spatial dispersion of the diffusion coefficient will result in the deviation of absorption or scattering functions from single Lorentzian line.

Fig. 8. – The dimensionless local coefficient of self-diffusion of a fluid within a spherical pore D

A(r) fD(r) 6p2_hRO(k

BT). The parameters are the same as in fig. 6.

8. – Conclusions

So, we have investigated the thermal hydrodynamic fluctuations of a compressible viscous fluid contained in small spherical pores. The spectral densities of fluctuations are found to depend strongly on the size of pores and fluid characteristics. The velocity autocorrelation function demonstrates significant oscillations in time. The dynamical structure factor of the confined system displays more daedal behaviour than the one for the bulk fluid. The frequency spectrum differs from the one found earlier in the liquid drop model.

The results obtained could be of considerable interest in various applications. For instance, the fluctuations of density determine the spectra of inelastic scattering. Therefore, the collective excitations of fluid in porous media have an effect on the cross-section of scattering with small energy transfer and could be studied by the slow neutron scattering [31]. In our approach the solution of the problem of diffusion motion inside confined region is expressed in terms of the velocity correlation functions obtained. The complicated coordinate dependence of the diffusion coefficient will lead to a non-Lorentzian shape of the absorption function and could be tested by Mössbauer absorption experiments.

Note that the present method of solving the problem of equilibrium fluctuations can easily be generalized to systems with temporal dispersion of the kinetic and thermodynamic coefficients. This allow us to significantly expand the region of applicability of the results, especially in the approximation of the molecular correlation functions by Lagrangian ones.

* * *

A. V. ZATOVSKY acknowledges the financial support of the International Soros Science Education Program through grant No. SPU 062022.

AP P E N D I X A

Using the presentation of spherical vector functions M, L, N (2.6) via the Legendre functions [28], and the properties of the spherical harmonics we find the following sum rules:

!

m 40 n Lmn2 (r) Oqmn4 2 n 11 4 p

g

k

1 k ¯ ¯r jn(kr)

l

2 1 n(n 11) (kr)2 jn 2_(kr)

h

_, (A.1)

!

m 40 n Mmn2 (r) Oqmn4 2 n 11 4 p n(n 11) jn 2_{(kr) ,} (A.2)

!

m 40 n Nmn2 (r) Oqmn4 2 n 11 4 p n(n 11) (kr)2

g

n(n 11) jn 2 (kr) 1

k

¯ ¯r

(

rjn(kr)

)

l

2

h

, (A.3) where qmn is written in (2.17).

Note that we use the spherical harmonics which are normalized by qmn[28]:



dV Ymn(u , W) Ym 8 n 8* (u , W) 4qmndmm 8dnn 8.

(A.4)

AP P E N D I X B

In order to perform the explicit summation of autocorrelation functions (3.14), (3.15) over the roots of (2.12), (2.14), it is necessary to calculate, for example, the sum

J f

!

j 41 Q _{2 jn( mnjx) jn( mnjX)} (z2 2 m2nj) jn2( mnj)

(

2(n 11)1mnj2

)

, (B.1) where 0 GxGXG1, z2

c 22(n 1 1 ), jn(x) is the spherical Bessel function of 1st kind

and mnj are the roots of the equation

mnjjn8 ( mnj) 2njn( mnj) 40 , j 41, 2, R .

(B.2)

Let us consider the auxiliary function F(u) 4 1 z2 2 u2 2 u2 2(n 11)1u2 jn(ux)

j_{n 11}(u) [ yn(uX) jn 11(u) 2jn(uX) yn 11(u) ] , (B.3)

where yn(u) is the spherical Bessel function of the second kind. This function (B.3) has

the simple poles at u 46mnj (when jn( mnjX) c 0), u 46z, u46i

k

2(n 11) (i is

imaginary unity), and u 40. Since function F(u) is even, we take the path cq in the

complex plane passing along the imaginary coordinate axis and closed by the right semicircle of large radius q . The path must bypass the poles u 40 and u4i

k

2(n 11) by the left semicircles of small radius and the poles u 42i

k

2(n 11) by the right ones. For the chosen path we have

1 2 pi_c



q F(u) du K_c qK cQ 1 2Res F( 0 ) 4 1 2 ( 2 n 13)(xX)n (n 11) z2 . Then, using the residue theorem, we find for (B.1)

J 42( 2 n 13)(xX) n 2(n 11) z2 1 1 2(n 11)1z2 _{l 41}

!

2 (21)l 11 aljn(alx) j_{n 11}(al) 3 (B.4) 3[yn(alX) jn 11(al) 2jn(alX) yn 11(al) ] with a14 z, a24

k

2(n 11), 0 GxGXG1.

All other sums over roots of (2.14) are found in the analogous way, for instance

!

j 41 Q 2 j_n( m_{njx) jn}( m_njX) (z2 2 mnj2 ) jn2( mnj)( 2 n 2mnj2 ) 4 (B.5) 4 1 2 n 2z2

!

l 41 2 (21)l 11 aljn(alx) j_{n 21}(al) [yn(alX) jn 21(al) 2jn(alX) yn 21(al) ] ,

where a14 z, a24k2 n and mnj are the roots of the equation mnjjn8 ( mnj) 1 (n11) jn( mnj) 40 , j 41, 2, R .

In contrast to the sums (B.4), (B.5) the so-called Fourier-Bessel sum over roots bnj

of (2.12) is well known [32]:

!

j 41 Q _{2 jn(bnjx) jn(bnjX)} (z2_{2 b}nj2 ) jn 112 (bnj) 4 zjn(zx) jn(z) [ yn(zX) jn(z) 2jn(zX) yn(z) ] . (B.6) R E F E R E N C E S

[1] BRATKO D., BLUM L. and WERTHEIM M. S., J. Chem. Phys., 90 (1989) 2752.

[2] BITSANIS I., MAGDA J. J., TIRRELL M. and DAVIS H. T., J. Chem. Phys., 87 (1987) 1733. [3] HEINBUCH U. and FISHER J., Phys. Rev. A, 40 (1989) 1144.

[4] VANDERLICK T. K., SCRIVEN L. E. and DAVIS H. T., J. Chem. Phys., 90 (1989) 2422. [5] BUG A.L., Int. J. Thermodyn., 10 (1989) 469.

[6] ZHOU Y. and STELLG., Mol. Phys., 68 (1989) 1265.

[7] MORELLI S. and SANTANGELOR., Nuovo Cimento C, 14 (1991) 377. [8] TOUGH R. J. A. and VAN DEN BROECK C., Physica A, 157 (1989) 769. [9] FISHER I. Z., Sov. Phys. JETP, 34 (1972) 878.

[10] IL8INV. V., DAVYDOVA. S. and ANTONCHENKOV. YA., Fundamentals of Physics of Water (Naukova Dumka, Kiev) 1991 (in Russian).

[11] ALDER B. J. and WAINRIGHT T. E., Phys. Rev. A, 1 (1970) 18. [12] ALLEY W. E. and ALDER B. J., Phys. Rev. A, 27 (1983) 3158. [13] RYTOV S. M., Sov. Phys. JETP, 6 (1958) 130.

[14] LANDAU L. D. and LIFSHITZ E. M., Statistical Physics, Vol. 2 (Pergamon, Oxford) 1980. [15] LISY V., Phys. Lett. A, 150 (1990) 105.

[16] LISY V., ZATOVSKY A. V. and ZVELINDOVSKY A. V., Phys. Rev.E, 50 (1994) 3755. [17] LOVESEYS. W. and SCHOFIELDP., J.Phys. C, 9 (1976) 2843.

[18] ZATOVSKII A. V. and ZVELINDOVSKII A. V., Sov. Phys.-Tech. Phys., 35 (1990) 1078. [19] Proceedings of the II International Meeting on Small Particles and Inorganic Clusters,

Lausanne, 1980; Surf. Sci., 106 (1981).

[20] PETROV U. I., Physics of Small Particles (Nauka, Moscow) 1982 (in Russian). [21] MONTROLL E., J. Chem. Phys., 18 (1950) 183.

[22] TAMURA A. and ICHINOKAWA T., J. Phys. C, 16 (1983) 4779.

[23] KRISTENSEND., JENSEN E. J. and COTTERILLR. M. J., J. Chem. Phys., 60 (1974) 4161. [24] BOGOMOLOVV. N. , KLUSHINN. A., OKUNEVAN. M., PLACHENOVAE. L., POGREBNOIV. I. and

CHUDNOVSKII F. A., Sov. Phys. Solid State, 13 (1971) 1256. [25] TAMURA A. and ICHINOKAWA T., Surf. Sci., 136 (1984) 437.

[26] LANDAU L. D. and LIFSHITZE. M., Fluid Mechanics (Pergamon, Oxford) 1982. [27] LANDAU L. D. and LIFSHITZE. M., Statistical Mechanics (Pergamon, New York) 1980. [28] MORSEP. and FESHBACH H., Methods of Theoretical Physics, Vol. 2 (McGraw-Hill, New

York) 1953.

[29] KADANOFFL. P. and MARTIN P. C., Ann. Phys., 24 (1963) 419.

[30] M. ABRAMOWITZand I. A. STIGUN(EDITORS), Handbook of Mathematical Functions (Dover Publ., New York) 1965.

[31] BULAVINL. A., GARAMUSV. M., KARMAZINAT. V. and SHTANKOS. P., Colloid J., 57 (1995) 856. [32] BATEMANH. and ERDE´LYIA., Higher Transcendental Functions (McGraw-Hill Book, New