Vibrational contribution to the surface specific heat of a fluid

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Vibrational contribution to the surface specific heat of a fluid

B. Djafari-Rouhani, L. Dobrzynski

To cite this version:

B. Djafari-Rouhani, L. Dobrzynski. Vibrational contribution to the surface specific heat of a fluid.

Journal de Physique, 1982, 43 (3), pp.523-529. �10.1051/jphys:01982004303052300�. �jpa-00209422�

Résumé.

Nous donnons des résultats analytiques pour la contribution vibrationnelle à la chaleur spécifique 0394CV(T) ^{de basse} température ^d’un liquide

visqueux. ^{Il est} montré par

méthode de fonctions de Green

qu’en présence ^d’ondes capillaires ^{de surface}

0394CV(T)

^{MT4/3 + NT2 + PT8/3 +} QT3 ⁺ O(T10/3).

Les coefficients M, N, ^{P et} Q ^{sont donnés} analytiquement

^fonction de la densité du liquide, de la tension super- ficielle et de la vitesse longitudinale ^du

dans le volume du liquide.

Abstract

We report analytical results for the vibrational contributions to the low temperature ^surface specific

heat 0394CV(T) ^of

^{non viscous} liquid It is shown by

Green’s function method that when capillary

^exist

on

the surface of such

liquid

0394CV(T)

^{MT 4/3 + NT2 + PT8/3 +} QT3 ⁺ O(T10/3).

The coefficients M, N, ^{P and} Q

given ⁱⁿ closed form

functions of the density ^{of the} liquid, the surface tension and the velocity ^{of the} longitudinal ^bulk

Vibrational contribution to the surface specific ^{heat of a fluid}

B. Djafari-Rouhani ^{and L.} Dobrzynski

Laboratoire des Surfaces et Interfaces (*), I.S.E.N., 3,

F.-Baës, 59046 Lille Cedex, ^France (Reçu ^le 23 juillet 1981, accepté le 24 novembre 1981 )

Classification Physics ^Abstracts

62.60

65.20

1. Introduction.

The surface contribution to the

specific ^{heat of a finite} crystal ^has been studied exten-

sively [1] and its low temperature T2 ^{behaviour is} well established both theoretically ^and experimen- tally. Although ^Atkins [2] has shown that a T4/3 contribution to the low temperature ^surface specific heat ACv(7) ^{of a} liquid may be due to hydrodynamic capillary ^waves, he established this result by taking

into account only the existence of these capillary

waves, without considering ^the corresponding per- turbation of the bulk density ^{of states.} Recently

losilevskii [3] reported ^that LBCy(1) behaves like T2

at low temperature, ^without any mention of the T 4/3 behaviour predicted by ^Atkins [2].

Therefore a thorough examination of this question

was needed We report ^{here such a} study ^{that takes}

into account the effect of the capillary ^{waves as well as}

the perturbation of the bulk waves due to the existence of the surface. We use for this _purpose a Green’s function method [4]. ^Different formulations [3, 5-8]

of this method were already used for the study ^{of other}

vibrational properties ^of liquid ^surfaces, namely

surface modes and Brillouin scattering.

In section 2 we outline the derivation of the Green’s function for a semi-infmite liquid The ‘fluid will be assumed compressible and viscous. The general ^for-

mulation of this problem is similar to that of an iso-

tropic ^solid [9]. Going ^{then to} the limit of an inviscid

liquid, ^we give ^the contribution of the surface to the

density ^{of states and to the} specific ^heat (Section 3).

-This last quantity ^{contains terms} with different powers of the temperature T, coming from different contri- butions to the density ^{of states. We} give ^{here all these}

terms up through ^{the one in T3 which has the same}

power ^as the contribution of the bulk waves to the

specific ^heat.

2. Formulation of the Green’s function.

The motion of the bulk fluid is governed by ^{the Navier-}

Stokes equation

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004303052300

524

where v is the velocity, p the density, p ^and y’ the coeffi- cients of shear and dilatation viscosity ^and p the pressure. For small velocities, ^the equation (1) ^{can be}

linearized by neglecting ^{the term} (v.V)v. Assuming

that the fluid excitations execute harmonic motion with a time dependence e-irot, ^the velocity ^{v is related}

to the displacement ^u by

Finally, ^{we assume} that the pressure ^term in equa- tion (1) ^{is due} only ^{to the} density fluctuations giving

rise to dilatation and compression of the fluid That is,

we neglect the contribution due to the thermal fluc- tuations. Then

where A is the compressibility of the fluid which is related to the (longitudinal) ^sound velocity ^C

With the above assumptions, ^the stress tensor in the

liquid ^can be written as

with _a, fl

1, 2, ^3.

This form of the stress tensor is completely ^{analo- .}

gous ^to that obtained in elasticity theory ^{for an isotro-} pic solid. The following correspondances ^{should be}

made

where C1 ^and Ct ^are the velocities of longitudinal ^and

transverse sound waves in the solid

The derivation of the bulk Green’s function can be done step by step along ^{the same lines as for an isotro-} pic ^solid [9]. ^{Let us recall the} equation ^satisfied by ^this

Green’s function, ^after making ^{a Fourier} analysis parallel ^{to the} plane ^{of the surface} (plane X3

0)

and a rotation of the _X,l X2 ^axes which brings ^the xl axis along ^the wavevector kll parallel ^to the surface

where gij stands for gi,(k 11 W’ I X3, x3). ^{Let us} also recall that equation (7) ^{leads to} g12 =g21 =g23 =g32 =0,

that is to a decoupling ^between sagittal ^{and transverse} polarization. ^The solution of equation (7) for the bulk fluid is exactly ^{the same as for an} isotropic ^solid [9],

when we make use of the relations (6). ^In the presence of a free surface, equation (7) ^{has to be solved} subject

to the boundary conditions expressing ^{the absence of}

the stresses at the surface. These boundary equations

can be written in correspondence ^{with those for the}

solid [9], but it is _necessary to take into account the effect of the surface tension and the possibility ^of

surface viscosity [10]. ^The general contributions of these two effects to the surface stresses are of the

form, [10]

where _yo is the surface tension of the unstretched surface, ^s = 07 (where ^{S is the} liquid ^surface

O(Log S)

area) give the variation of the surface tension under

stretching, ^and p,, ,us

^{take account} of the surface

viscosity. Using the notation

the boundary conditions at the free surface can be

written

Let us define, ^{as for the} solid, ^the quantities

and separate in the Green’s function the bulk term

g(o) from the surface contribution g(S)

The solution for g(S) ^{can be written in the} following

form [9]

Insertion of these solutions into the boundary ^condi-

tions (10) yields ^the unknown coefficients A,, A,, A’, A;,

B. The full expressions for these coefficients are given

in the Appendix.

Let us just mention that the frequencies ^{of the}

surface waves are given by ^the poles ^{of this Green’s} function, ^{and are then solutions of the} equation

In the simpler ^{case of a non viscous} liquid ^{with s}

^0,

we obtain the usual capillary ^{wave solution}

3. Surface density ^{of states and} specific ^{heat of an}

inviscid liquid.

^The above Green’s function will be used in this section to study ^the surface contribution

to the density ^{of states and then to the low} temperature specific ^{heat of a} liquid ^{We limit ourselves to a} simpler

case than that of section 2 by assuming ^s

^{0 and} by neglecting viscosity, ^but by keeping the surface tension yo and the liquid compressibility.

The variation due to the surface of the bulk density

of states follows from the imaginary part ^{of the trace} of the surface contribution to the Green’s function

Since the dependence ^{of the Green’s} function elements

on X3 and x; is of the form e-(aiX3+ajX3), ^where i, j

^t

or 1 (see ^the Appendix), ^the integration ⁱⁿ (15) ^is straightforward ^{Then we} go ^to the limit of E

0 and ^zero viscosity which leads to the following simple

result

We have to point ^{out that on} taking ^the imaginary part ⁱⁿ equation (16), ^the quantity ^{W2 should be} replaced by W2 + itl where ’1 ^{is an} infinitesimal

positive number. We then get ^three different contri- butions to An

An(l) results from the first term in (16)

It represents ^a discontinuity ^{in the} density ^{of states}

at the threshold of the bulk bander

C, k 11 ^{; a similar}

result was obtained [11] ^{at the} longitudinal ^threshold

in isotropic ^solids.

The second term in equation (16) gives ^{rise to the two}

other contributions to An. The first of these contri-

butions is

where mo(k jj) ^{is the frequency} of the surface capillary

wave, the solution of the equation (14&), ^that appears

as a pole ^of the second term in equation (16). ^{Let us} point ^{out that} rearranging equation (14b) by ^means

of the definition of _ai (Eq. (11)) ^{leads to the} following

third order polynomial ⁱⁿ kj

Standard resolution of this equation ^shows [12] ^that

it has only ^{one real root}

526

Fig. ^1.

^Surface density ^{of states} (in arbitrary units) ^for

given wavevector kil. ^{The two vertical straight lines}

respond ^{to two} delta functions of weights ¹ and - 4 respec-

tively. ^{The first}

gives ^the position of the surface capillary

wave, the second

is situated at the threshold of the bulk band. The figure is drawn for the parameters corresponding

to liquid mercury (neglecting viscosity) ^{and for} k 11 1-- 10’ ^{m - ’.}

with

and

For a given k the ^{frequency mo of this} surface mode is below the threshold of the bulk band C, k I I*

Finally ^{there is a variation of the} density ^{of states}

inside the bulk band that for m > C, k is

An example ^{of this} density ^{of states is} given ⁱⁿ figure ^{1. Now, we can} evaluate the change ^{in the} specific ^{heat due to the} surface, calculating separately

the contribution of each part ^{of the} density ^{of states}

Let us recall that the surface contribution to the free energy is given by

where S is the surface _area, and

Then, by using ^{the relation}

we obtain the following ^formal expression ^{for the surface} specific ^heat

The contribution of ån(l) to the specific ^{heat results} directly ^from equations (27) ^and (18)

where we have defined

where C(u) is the Riemann zeta function.

The second contribution to AC«n ^{comes from} ån(2), ^{that is from the surface} _capillary ^wave

In the equation (30) mo is a function of k ^through

equations (20)-(22), ^which give ^the dispersion ^{of the}

surface wave.

Since the low temperature specific ^{heat is} governed by ^{the low energy} excitations of the system, ^we develop equation (21) ⁱⁿ powers of (oo ^{to obtain}

Inserting this result into equation (30) ^{one finds} (with ^{the aid of} the relation k dkjj ^{II -L} d(k 2))

The term T4/3 in (32) ^{is the one obtained} previously [2]

for the contribution of the capillary ^{wave to the} specific heat while further corrections are of the form

T2, T8/3, T10/3, T14/3, T16/3,... ^{Let us note that the}

AC ^{term is} partly compensated by ^{the T2 term} appearing ⁱⁿ AC.

The last contribution to åCy(T) results from ån(3).

In the denominator of An (Eq. (23)), ^{let us rewrite}

the term k’ ’ ^{in the new form}

pointing ^{out that the} region of interest for k II is 0 k ^COICI. As the low lying excitations are the most important ^{for our} calculation, ^{we can} keep ⁱⁿ

the preceding expression ^the p2 W4/ YÕ ^{as the} leading part, the remainder being considered in a Taylor

series expansion

Then we get immediately

Finally the surface specific ^{heat of a non viscous}

fluid can be obtained by adding the contributions of

equations (28), (32) ^and (33). ^{Let us} write it in the

following ^form

where the coefficients are

The function I(u) ^{was defined} by equation (29) ^and

the arbitrary temperature To ^was introduced just ⁱⁿ

order to get ^for M, P, ^{N and} Q quantities ^{with the}

same dimension (m- 2). The contribution in T 4/3 is the same as the one predicted by ^Atkins [2]. ^Iosi-

levskii’s [3] result for the specific ^{heat of a free} liquid

surface is the same as our contribution AC(v) (Eq. (28)) coming ^{from the threshold at} Cl. ^{He did not take into}

account any localized surface mode below the bulk bands and obtained finally that the creation of a free surface decreases the density ^of vibrational states.

In principle the creation of a free surface produces ^the opposite effect, ^{as in our} present calculation.

The last term in equation (34) is of the same order in T as the contribution of the bulk waves to the

specific ^{heat, that is} [13]

where V is the volume of the liquid

528

Let us illustrate the above calculation by ^the example

of liquid helium, comparing ^the magnitude ^{of the}

different coefficients M, N, P, Q. ^{For the} liquid 4He,

the velocity of sound and the density ^are [3]

and the surface tension is [14J yo = 3.5 x 10 - ’ J. m - ’.

Taking To

^{1 K one} finds in units of 101’ m-2

We can also _compare the AC( 3) contribution, ^which behaves like T , ^{to the bulk} _specific ^heat

The above quantities ^{enable us to find for} liquid

helium the relative importance of the surface specific

heat for a given ratio of the surface to the volume.

Finally, ^{a different} example ^is given by liquid ^mer-

cury for which [7]

It is more convenient to take To

^{300 K in this case.}

Then in units of 102° m-2 M = 14.8, ^{N =} 2, ^{P -} 5, Q = - ^{5.3 and}

The higher ^terms play a ^more important ^{role here}

than in the case of liquid ^helium.

4. Discussion.

We present in this section a

summary and discussion of the results obtained above.

The Green’s function of ^a liquid ^{with a} free surface has been calculated taking ^{into account} the surface tension and the viscosity. ^{Various forms of} this Green’s function were recently ^{used to discuss the} intensity ^of

the light ^{scattered from a} liquid ^surface [6-8]. ^The expressions derived here are shown to be formally

identical with those for an isotropic ^solid. Therefore, they put ^{the solid and the} liquid ^{on the same mathe-}

matical footing. Also, they ^can easily ^be generalized ^to describe interfaces between liquids ^{or between solids}

and liquids.

The contribution of the surface to the density ^of

states and to the specific ^{heat are} then derived for an

inviscid fluid For a given kil ^{the density of states is} composed ^{of three} components : ^a delta function

(of weight one) ^{at the} frequency We of the surface

capillary ^wave, another delta function (of weight - 4) yielding ^{a loss of} states at the threshold of the bulk

band, ^and finally ^a continuous variation inside the bulk band. Adding ^the viscosity ^{to the} problem ^enlar-

ges the two delta peaks ^{which are the more} important component of the surface density ^{of states. In} parti- cular, ^at the limit of low viscosity, the delta function

corresponding ^{to the} capillary ^wave yields ^{a lorentzian}

centred at Wc of width Ao) -- I p --I’ - ^{P for typical values}

of k ⁱⁿ scattering experiment, ^{let us} say 101_107 m-1,

We is given ^{to a} good approximation by (oc - Yo p k 3/2 11 ^P

i.e. the result for an incompressible ^fluid (C - oo).

As the relative width Aw/we of this lorentzian is an

increasing ^{function of} kjj, ^the viscosity plays ^{a more} important ^{role for} higher wavevectors.

The surface specific ^{heat of an} inviscid fluid AC,,

was calculated as a series expansion ^{in the} temperature

T up through ^{the term in T 3. The} capillary ^wave gives

the main contribution to ACv which is in T4/3 and then correction terms of type T2, T8/3,

T10/3, ^etc...

The term in T2 is partially compensated by ^the

contribution of the delta peak ^{in the} density ^{of states}

situated at the bulk threshold Finally the variation of the density ^{of states} inside the bulk band only

contributes to ACv by ^{a term in T 3 which has the same}

power dependence ^{as the bulk} specific ^heat.

Obviously ACv ^{as a whole is} proportional ^{to the}

surface area while the bulk specific ^{heat is} proportional

to the volume. The examples given ^{above show that}

even for droplets of micron size the determination of ACv ^needs good precision (about 10-4) ^{as well as}

low temperatures. Finally ^{we should stress that for}

very small droplets (1) the surface area is not planar, (ii) ^the wavevectors are limited by ^{a minimum cut-off}

of the order of the inverse of the droplet ^radius, and (iii)

the viscosity may play a ^{role at these} wavevectors.

Then further studies of the properties reported ^here including viscosity and different geometries ^{are needed}

Acknowledgments.

^We would like to thank Pro- fessors A. A. Maradudin and R. F. Wallis and Dr.

V. Velasco for their interest in this work.

APPENDIX

Expression of the Green’s function for the semi- infinite liquid.

^{We do not} give ^{here the bulk}

Green’s function of the liquid which have exactly ^the

same expression ^{as for an} isotropic ^solid [9], ^{once the}

values of C, ^and C, given by equation (6) ^{are used.}

The coefficients At, A,, At, A’, B appearing ^{in the}

surface part (Eqs. (12)) ^are given by

with

and

References

[1] ^{See for} references, MARADUDIN, ^A. A., WALLIS, ^{R. F.}

and DOBRZYNSKI, L., ⁱⁿ Surface phonons ^and pola- ritons, ^Handbook of Surfaces and Interfaces,

Vol. 3, ^{Ed. L.} Dobrzynski (Garland Press, ^New York) ^1980.

[2] ATKINS, ^K. R., 1953 in Statistical Physics ;

LANDAU, ^{L. and} LIFSHITZ, E. M., (Pergamon Press) 1959, p. 460.

[3] IOSILEVSKII, ^YA. A., Phys. ^{Rev. B 20} (1979) ^1473.

[4] DJAFARI-ROUHANI, B., Thesis, Université Paris-Sud

(1978).

[5] GARCIA-MOLINER, F., ^Ann. Phys. (Paris) ² (1977) ^179.

[6] PLATERO, G., VELASCO, ^{V. R. and} GARCIA-MOLINER, F., Phys. ^Scr. 23 (1981) ^1108.

[7] LOUDON, R., Proc. R. Soc. London A 372 (1980) ^275.

[8] BOUCHIAT, ^{M. A. and} MEUNIER, J., J. Physique Colloq.

33 (1972) C1-141.

LANGEVIN, ^{D. and} BOUCHIAT, ^M. A., J. Physique ³³ (1972) ^101.

[9] MARADUDIN, ^{A. A. and} MILLS, ^D. L., ^Ann. Phys. ¹⁰⁰ (1976) ^262.

[10] HANSEN, ^R. A., AHMAD, ^J. A., Prog. Surf. Member Sci.

4 (1971) 1.

[11] BURT, ^M. G., ^J. Phys. ^{C 6} (1973) ^855.

[12] Handbook of Mathematical functions, Eds. M. Abra-

mowitz, ^{I. A.} Stegun (Dover Publications, ^I.N.C.

New York) p. 17.

[13] See reference [2], p. 200.

[14] American Institute of Physics ^{Handbook, 3rd Edition,}

Ed. D. E. Gray (Mc ^Graw Hill) 1972, p. 2-207.