Physica A 164 (1990) 245-264 North-Holland

ADIABATIC HEATING EFFECT NEAR THE GAS-LIQUID CRITICAL POINT

Akira ONUKI Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan

Richard A. FERRELL Department of Physics and Astronomy, University of Maryland, College Park, MD 20742, USA

Received 7 December 1989

A new feature of macroscopic thermal relaxation processes is investigated in pure fluids near the gas-liquid critical point. If the total volume of the fluid is fixed, an adiabatic heating process can be the dominant temperature equilibration mechanism near criticality. That is, if the boundary temperature is changed, the resultant boundary diffusion layer acts as an efficient piston instantaneously producing an adiabatic volume change in the interior region. As a result, inhomogeneities of the fluid temperature are greatly reduced on a time scale rapidly going to zero at the critical point. On the other hand, when the two phases coexist below 1",, the adiabatic process induces a flow of latent heat through the interface. This flow persists for a long time and slows down the equilibration.

I. Introduction

Thermal equilibration processes in fluids near the gas-liquid critical point are known to be complicated and puzzling [1-3], mainly because of the following reasons: (i) Severe density stratification is produced on earth due to the strong divergence of the isothermal compressibility K r [4]. (ii) In many cases the processes are not described by the linearized hydrodynamic equations due to a number of critical anomalies, such as the divergence of the thermal conductivity A and those of the specific heats, Cp and C v [3]. (iii) The analysis is very difficult in the presence of the two phases [1, 2]. (iv) Pressure perturbations propagate as sound waves and change rapidly in time, while the entropy (per unit mass) changes diffusively and undergoes critical slowing- down. This leads to an adiabatic heating effect if the total volume of the fluid is fixed [5, 6]. For example, if a sample is warmed from the boundary, a thermal expansion near the boundary instantaneously causes a nearly homogeneous

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246 a . Onuki attd R .A . Ferr,,ll / Adiabatic h,'atbrg near g,m-liquid critical point

pressure increase Bp throughout the sample. Then the sample is adiabatically warmed by the amount (BT),, = (OT/ipp), Bp even far from the boundary.

The aim of this paper is to illustrate the essence of this adiabatic effect. which seems to have not been fully discussed in the literature, in the simplest manner. To this end we will mostly neglect the gravity, but we will briefly discuss as to how it blurs the adiabatic cffcct in subsection 2.6. Our analysis, even in its present form, can explain a seemingly puzzling observation in space by Nitsche and Straub [7]. We will also discuss the effect when the two phases coexist. Following a lecture by one of the present authors 15l, two groups have performed numerical analysis of the effect: Boukari ct al [8] have prcsentcd an illustrative example of the effect near the critical point, and Zappoli et al. 191 have examined initial pressure and temperature changes in supcrcritical Iluids. These two papers supplement our previous letter 161 and thc present paper. because our papers mainly consist of analytic results.

2. Thermal relaxation in the one-phase region at fixed volume

2.1. Linear hydrodynamics

We use the linearized hydrodynamic equations to describe small perturba- tions in a homogeneous, quiescent, pure Iluid. We further neglect the spatial dependence of the pressure deviation Bp, which is then a fimction of lime t only. Namely, we are interested in slow processes with time scales much longer than the acoustic time scale l . /c. L being the cell length and c being the sound velocity. Using the relation

= , + Ill

the thermal diffusion equation can be written t*s

d ~ B T ' = ( ~ ) , - ~ t ~ I ' + t • . (2)

Here Bs and 8T are the deviation of the entropy per unit mass and that of the temperature and D = A IpCl, is the thermal diffusion constant, p being the mass density.

The +s can also be written in terms of the density deviations 8p and +p as

Bs = ( ~ ) , , 8p + ( ~ ) , Bp. (3)

We consider the ease in which the total volume V of the sample is fixed as in

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 247

the case of C v measurements. Then, we have to first order in the deviation

~V =1 [ drSv 1 [ drBp O, (4) v J p J

where v = 1/p is the volume per unit mass and the integral is within the cell. We hereafter denote the spatial average by ( . . . ),

xf ('-')=~ d r ( . . . ) .

Relation (3) yields

(5)

aT aT

where use has been made of the relation

= ( a T ] (ap~ ( a p ) ( a p ) Cv 1

Hereafter we use the notation

q, 3,~- ~--~v > 1. (9)

It follows that 0 < a < 1. The spatial average of (7) leads to a natural relation for the case of a fixed volume,

( ~ T ) = ( T / C v ) ( B s ) = Q(t)/pVC v . (10)

It goes without saying that, if p is fixed, Cp appears in (10) instead of C v. Note that the second term of (7) is independent of space and represents the adiabatic temperature change that is present even if 8s is localized only near the boundary. We also notice that the second term of (7) can be easily much

(7)

(8)

() Bp= _~s (Bs) = ~s p ( p T V ) - ' Q ( t ) ' (6)

where Q(t) is the total heat supply from the outside to the sample through the boundary. The temperature deviation can then be written in terms of 5s and (~s) as

248 A. Onuki am/R.A. Ferrell I Adiabatic" heating near gas-liquid ('ritical point

greater than the first term near the critical point since "y---, ~ at criticality. Thus the adiabatic heating effect becomes increasingly important on approaching criticality.

If use is made of (6) and (8), the modified heat conduction equation (2) reads

( ] ) /J B T = a ST +DT' -BT (11) fit

The spatial integral of (1 1) over the cell becomes the equation of the total energy conservation as ought to be the case,

dt Q ( t ) = p V C v B'l" =A d ~ r n . ~ ' . B T . (12)

where J'do" ( . . . ) is the surface integral and n is the outward normal unit vector at the boundary.

2.2. Quick thermal equilibration

The thermal equilibration can be much quickened in the case (~ ~ 1. To show this let us change the boundary temperature by an infinitesimal amount T~ = Ej T~. at t = 0. We shall see that the thermal boundary layer has the usual diffusion length (Dr) ~ "- in an early stage. Then IV~TI-T~/(Dt) ~~- at the boundary and (12) yields

iJ (87") A "!', lit L p C v (Dt) I.. , (13)

where L is the characteristic linear size of the cell. This is integrated to give

i T-~ ~ 7(Dt ) ' "-. (14)

Owing to the large factor ~,, the right-hand side of (14) quickly reaches ~'( 1 ) fi)r t ~ g , where

t t = L'- /~, ' -D ~ L"~. : . 7 , (15)

being the thermal correlation length. Note the relations A x ~, C), ~ ~ ' ", and C v ~ ~"';', and q ~ L~-e 1"7 with e = ( T - T,.)/T,. on the critical isochore. Hereaf- ter ~,. b,/3, and & are the usual critical exponents. Strikingly the time q rapidly goes to zero near the critical point, while the usual relaxation time L21D (for

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 249

systems at constant pressure) grows as is well known. At t ~ t t the boundary diffusion layer has a thickness of order (D t~ )1~z~T-~L and is still much thinner than the cell lenght L. We may prove that the thickness of the boundary layer remains of order (Dt) ~/2 until the usual entropy diffusion time t ~ L2/D = y2t I.

Nitsche and Straub measured the center temperature of a cylinder filled with SF 6 as response to the linearly increased wall temperature in space [7]. They found that the difference of the two temperatures stayed unexpectedly small. We believe that our theoretical results provide an explanation for their finding.

To examine the relaxation process more in detail, let us solve (11) in a simple one-dimensional case, 0 < x < L, under the following boundary condi- tions:

8T(x,t)--->O ast---*0, 8 T ( O , t ) = S T ( L , t ) = T , f o r t > 0 . (16)

We assume that T~ is sufficiently small and the linear scheme is applicable. The details of the calculations will be deferred to the appendix and we here summarize the main features. The average temperature deviation behaves as

( S T ) / T I ~ 4 ( t ] ~ l l ) 112 for t ~ < t I , (17)

1 - ( S T ) l T , ~ 2 ( t , l ' t r t ) tl2 for t, ~ t < ~ y 2 t ~ . (18)

The difference between the average change (ST} and the change [ST]i . in the interior region is very small for t .~ L2/D = y2t~. Let us define a normalized temperature deviation,

G(x, t) = [ T, - ST(x , t)] l[ T, - (ST)i . ] , (19)

which is 0 at x = 0 and tends to 1 in the interior region. We may define the effective thickness x e , ( t ) of the boundary layer by ( G ) = 1 - 2 x e , / L . Then we shall see in the appendix

2xe,tl L = [ (ST) - (ST) i . ] / [ T 1 - ( S T ) , . ] - ( Dt)I'21L . (20)

In the boundary layer near x = 0, the temperature profile is calculated as [6]

[" erf(.~) for t .~ t I , (21a)

G(x ' t) - t l - ei2 + ~ ~ for tl ,~ t ,~ T2tl , (21b)

where

.~ _ x / (4Dt ) l /2 (22)

2511 A. Om&i and R.A. Ferrell I Adiabatic heating near gas-fiquid critical point

The third term on the right-hand side of (21b) is dominant near the boundary where .~ ~< (t~/t) ~': or x ~ LI'y. Therefore. the heat flux A dT/dx at the boun-

3.'2 dary is equal to ½A(T t - [ B T l i , ) L / D t >-t in the time region t t ~ t ~ 7-t I in accord with (12). On the other hand, for t ~ 7 " - t ~. we have the following exponential decay:

I - BT'(x, ( ) I T , ~ (2/"I)11 - cos(2rrxlL)l e x p ( - 4 w ' - D t l L ' - ) , (23)

which is the first term, n = I, of (A.13) i,I the limit y----, >-. The temperature firstly rises as l ' - ' for t<~tt , secondly undergoes the

polynomial relaxation (>.t t:_,) in the wide region, t, <~t~7"-t~. and linally shows the exponential relaxation for t~7"-t~ with the decay rate 4w'-D/L". which is fot=r times larger than that of the usual case of lixed pressure. We expcct that this is a general aspect in the linear regime, independent of the geometry of the cell, if use is made of a suitably chosen characteristic length L. The temperature profile in the time region t ~ 7 " - t , is characterized by two distinctly different spatial regions, the boundary layer with thickness (Dt ) r'-

and the plateau-like interior regio,1. Zappoli et al. 191 aptly described the rolc of the boundary layer as a piston compressing or expanding the interior region adiabatically.

2.3. Density change

The density change is also of great interest. Using (6) ~ (10) we obtain the following two expressions:

/ \ t ip Bp = 124

= - (25)

In the time region 0 < !.¢ t,. (ST) is considerably smaller than T, and (24) indicates that the density change at the boundary is of thc I'ollowi,lg order:

(",'/ ( ~ p ) h ~ \ i t 7 , 1 v T I for t ~ t I . (26)

Hereafter [ - - - It, and [- .. ]in denote the value at the boundary and that in the interior region, respectively. In particular, if the fluid was at the critical density p =p,. before the temperature change, we have ,1 h ---(,Sp)b/& ~ e-~'et (where e, = T , / T ~ ) in the linear regime. As a result the boundary layer can stay within the critical isochore only when 7". is very small,

I , , I = I T,/T,I ~ ~'~' ~ (27)

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 251

where/3 is also the usual critical exponent nearly equal to 1/3. In this linear case, (25) shows that the density change in the interior region is much smaller as

(00) (SO)in ~ ~ s 5p = - - ? p C v ( S T ) i n . (28)

Since (Op/Os)~, is nearly constant in the critical region, we find the ratio in the form

1(Sp),o/(Sp),l-~-'(sr)/r,~" fort~t~. (29)

In terms of G(x, t) defined in (19), the density profile can be written as

( O p ) [ T , - ( S T ) i . I ( I - G ) . (30) 5 O-(Sp)in-- ~'~ p

In the time region t, <-t<~y2t,, from (18) and (21b) we find

5p - (Sp), . ~ 2 (~°P) ,, T,[l_Y:(t,/t),,2l(t,l~O,,2 e _ e . (31)

.~ being defined by (22).

2.4. Nonlinear case

If (27) does not hold as in the case of e , - e , we should have (ap/ aT) , 0cA~ t~ '~ near the boundary from the scaling consideration, so that

A~ = (Sp)h/p= -I~,1 ~'~+~' for I~, I ~> ~+~ (32)

In this nonlinear regime the boundary layer is brought into an off-critical region. There, the linear relations so far cannot be used (at least in their present forms) until A b decreases down to the order of e 6. However, we may assume that the interior region is still linearly perturbed. This is satisfied for l(sp),.I ~p~, so that a t [ S T ] i n ~ T, (28) yields

I~,1 ~ ~+~. (33)

Then we can make some estimations in the early stage. Note the relation

1 5 0 = 1 ( 0 0 ) 5S+ 1 (~-p;) p p ~s p p , S P ' (34)

252 A. Onuki and R.A. Ferrell I Adiabatic heath~g near gas-liquid criti('al point

where (ap/Os)p =-p"(aT/ap):~ is virtually a constant in the critical region, ISTI ~ T= and 15p[ ,~ p,.. Integrating (34) over the space we obtain

(' ~V=(p"T) ' ~ pQ(t)+ p .,. V 3 p = O . (35)

Note that (Op/i~p)., ~ C v. In the early stage, in which the boundary layer is much thinner than L, we may replace the average (p 'l(8pli~p),) by the value

- I . F) in the interior region [p (0p/~p).,]~, to obtain

BP ~ [ ( ~s )~,]m (p T V ) ~Q(t) .

The temperature deviation in the interior region is then

( ~ T ) . , ~ (1_7_) ( p V ) I Q ( t ) . ~¢V" in

(36)

Next we calculate the counterpart of the characteristic time t 1, (15). in the nonlinear regime and denote it by t,, t. We assume that the heat flux at the boundary is given by AI',/(Dt)' "- in the early stage t ~< t.i whcre A and D are the wdues at the boundary. It decreases as the density deviates from the critical isochore. Then. from (37). the countcrpart of (13) becomes

,,,~ ( ~ ' / ' ) ' - ~ - h ,,,

Integrating (38) and setting (~7"),,,

t.i = [ L~-/( D )h][( C,.),,,/( C/, ),, 1-'

= tl(,~ c, , ) , , , / ( a c , , ) , , .

7", , we lind

(39a)

(39b)

T h e second line holds when the interior region has only slightly deviated from the initial thermodynamic state. It" further use is made of (32), we obtai,1

t.i ~ I t ( e , / e / ~ " "~)<;' ' "~'~/:~ "";') ( 4 0 )

Hence t.i > 11 and, if e I = ~', it follows

t.i ~ t, ~" /' 7. L'-e I" , ( 41 )

where p = (~ + ~, - i)(/, + ~ ) / ( ~ + j,) ~ 0.71. At t = ix. A,,(t)/AI,(0) decays to

(37)

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 253

1/2, say. Note that we have not yet examined the subsequent relaxation to the linear regime, in which ½ > Ab(t)/Ab(0 ) > e ~.

We notice that the time t.~ generally goes to zero as e---> 0 under (33) and is of course much shorter than the entropy diffusion time DIL 2 = y2t~. There- fore, we may expect that the exponential relaxation still occurs on the time scale of yzt I even in the nonlinear case, (32) and (33), despite the initial nonlinear relaxation in the boundary layer.

2.5. Adiabatic energy change

It is also of great interest to examine the amount of the internal energy increase in the interior region, which is nothing but the energy transported by acoustic modes from the boundary into the interior region. Far from the boundary the internal energy e per unit mass changes as

B e = - p B o = p p~-~ppj Bp=p s(Bs), (42)

where Bs = 0 and use has been made of (6). The total energy change (BE)i . in the interior region is written as

(BE), . = V~,p Be = ( V J V ) a , Q , (43)

where V~, is the volume of the interior region and is nearly equal to the total volume V in the early stage since 1 - VJV~(Dt)I~21L. Note that (43) holds even in the nonlinear case (32) from (36). The coefficient a, is defined by

p (O_~pT) -I) P ( 0 ~ ) (44)

where use has been made of (#TlOp)., = a(OT/Op)p from (8). As is well known (0pl#T)p tends to a constant at the critical point. The value of a s at the critical point is 1/6.02 for Xe, 1/6.98 for CO z, 1/3.92 for 4He according to ref. [10], and 1/4 from the van der Waals equation of state. We have hence a very simple result,

(BE)i . ~ a,Q, (45)

in the early stage in which V - V~, <~ V. From energy conservation the internal energy change in the boundary region is

(SE)b = Q - (BE)i . ~ (1 - as)Q. (46)

254 A. Om~M and R.A. Ferrell ~ .4diabatic heating near ga.~'-Iiquid critical point

The a., turns out to be the fraction of energy adiabatically transported into the interior region. Relations (45) and (46) hold cvcn far from the critical point in the early stage, V - I,~, ,~ V. under a fixed total volume.

2.6. Gravity effect

We linally consider the adiabatic effect in the presence of gravity. "I'o simplify the problem we neglect both rapid variations occurring on the acoustic time scale L/c and collective motions triggered by the hydrodynamic in- stabilities. The geometry is assumed to be one-dimensional in the region. 0 < x < L, with the x-axis in the vertical direction under a lixcd total wflume V. The vclocity field v(.r. t) is supposed to be present only along the x-axis and is then longitudinal, related to the temporal variation of the density by at,/ ax = - t J 'ap/at . In such quasi-stationary situations, the velocity field should be very small and the pressure gradient should be locally balanced with the density wtriation by

a a.--r p(x, t) = gp(x. t ) , i47l

g being the gravitational acceleration. In the case of a fixed vohunc the pressure is composed of thrcc parts.

p(x, t) = p,,,l(.r) + ~p,(t) + ~pl(.r. t ) . (4S)

The first part p~,l(.r) is the equilibrium pressure related to tile equilibrium

density p~.q(x) by 14, 101 a

ax P~'l(r) = g#"l(r) (4t~a)

or a

a.~ /X~'l(X) = g" (49b)

where /x~q is the chemical potential per unit mass ill equilibrium. The sccond part ~po(t) is the homogeneous pressure change tinder consideration so far, determined by the fixed volume condition (4). The third part is a new part induced by the density deviation 5p(x. t )= p(x. t ) - p,,q(x) and is determined by (47). Integration of (47) yields

~p~(x, t) = g f dx' ~p(.r', t ) . II

(5O)

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 255

so that ~p~(0, t )=ap~(L, t ) = 0 from (4). This part changes rapidly in the boundary layer and slowly in the interior region.

Since Cp and A can strongly depend on the height x in gravity near the critical point, the heat conduction equation for a small temperature deviation ST(x, t) should be written as

O aT=(O~) 0 1 O AO_ST (51) Ot • ~ ~ip + p---~p O-x 0x "

Here (OT/Op).,. is nearly a constant, ap =3p~+apl is given by (48), and Cp ~ ~'~ and A ~ ~ with ~: being the height-dependent correlation length. We have now a closed set of equations describing the temperature relaxation, although numerical analysis seems to be required in general.

We consider the thermal relaxation after a change of the boundary tempera- ture as in the previous subsections. Notice that, even under gravity, the initial stage of the relaxation should be still characterized by the presence of the boundary layer acting as a piston and the interior region where as ~0 . As a simplifying result, furthermore, we can show that the pressure and temperature changes in the interior region are nearly homogeneous in most realistic conditions. To show this we first estimate ~p,. Its maximum in space just after the boundary temperature change is estimated as

laP,l g( ) r(Ss) L . (52)

If lap, l,~ lap,,I is assumed, (35) still holds and we have

Q(t)=(as)~((°:p) aPt,. (53)

Thus,

lap,/ap,,l gL/ ( c-') , (54)

where (c") is the spatial average of the square of the sound velocity c-' = (Opl Op)~. We assume that the right-hand side of (54) is much smaller than 1. The adiabatic temperature change in the interior region is then given by

(st),.----(°Tpr) apo - - 1 1 (Cv) pV Q(t), (55)

which is a generalization of (10). The flow rate OQlOt is determined by A and D at the boundary if the diffusion layer is sufficiently thin (compared to the

256 A. Onuki and R . A . Ferrell / Adiabatic heating m, ar gas-liquhl crith'al point

inverse of gO In ~ l a x ] at the boundary, which characterizes the inhomogencity due to gravity). Then we have an equation for a ( ~ T ) ~ . , l a t in the same form as (13) or (38), in which (Cv) appears instead of Cv or ( C v ) , , . Similarly to (15) or (39) we find the corresponding relaxation time,

t~ ~ I L- / (O)dl ( C ,. ) / ( % )d - . (56)

The interior region has now been characterized as a medium whose specilic heat is effectively given by the averagc { C t,).

3. Thermal relaxation in the two-phase region at fixed volume

So far we have treated a pure fluid in the one-phase region. On the other hand, when two phases coexist, we may expect further interesting effects. Let us suppose a macroscopic interface separating the gas phase and the liquid phase at the center of the cell. The adiabatic process after warming or cooling at the boundary shifts the thermodynamic states of the two phases in the interior region upward for warming (or ~p > 0) and downward for cooling (or ~ p < 0 ) in the p - T phase diagram (or more conveniently in the s - T phase diagram because s is unchanged). The subsequent phcnomena are very differ- ent depending on the magnitude of $p. if 8p is positive and large, the two phases will be shifted into the one-phase region and the interface will vanish eventually. If ~p is negative and its magnitude is large, we should observe emergence of small domains via nucleation or spinodal decomposition through- out the sample. As the simplest case, if I pl is sufficiently small, the two phases will not exhibit any instability and return to thermodynamic states on the coexistence curve characterized by a new temperature. For simplicity we will focus our attention on the last case assuming that the linear treatments are allowable.

First notice that there should arise a Ilux of latent heat through the interface. This can be readily understood as follows. The entropy difference between the two phases is given in the initial equilibrium state (t <0) by

(AS) in t = m s ( T c - - ] ' ) / ~ (57)

where A, is a constant. At t = 0 the boundary temperature is changed to T + T t and in the ultimate final state, t--->~, the entropy difference becomes

(As)ri, , = Aa,.(T ¢ - T - T,) ti

(As)i,t - ~ A , ( T ~ - "1") iJ ' T t . (58)

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 257

We have assumed J TjJ ,~ T , - T in the second line. Denoting the volume fraction of the minority phase by 4,, the total amount of latent heat, which flows through the interface, is estimated as

(59)

This should be compared with the externally given heat (10). The latter will be written as Q.xt. By setting (ST) = T~ (because we are considering the final state), we find

Q . , J Q , x t ~ q b ~ L A , ( T ~ - T ) ~ - ' l C v ~ ~bY ~'2 , (60)

where use has been made of/3 - 1 + & = ½ (& - ~,). The above ratio can well exceed 1 even for small ~b.

We expect that the current through the interface persists for a long time of the order of the diffusion time L2/D. Then, how is the internal current induced? Recently one of the authors [11] has shown that the gas phase becomes slightly warmer than the liquid phase against a small pressure increase. This is because the thermodynamic derivative (0 TlOp), takes slightly different values in the two phases,

= 1 + + . . . s cri - - " ~ '

(61)

where ( O T / O p ) c r i is the critical-point value, the plus sign is for the gas phase, and the minus sign for the liquid phase. The coefficient a c is a universal number of order 1 [11]. Therefore, for a pressure change 8p, the resultant temperature difference between the two phases is

( 0 T ) a~

~ 2 a~ (ST),. (62) v~

where (ST)i . = (0 T/Op) , 8 T is the adiabatic temperature change in the interior region. Recall the relation (18) for t t ~<t~ < y2tl, which indicates that T I - (ST)~n decays below (AT)g_ e for t ~< ts_ e. By equating the right-hand sides of (18) and (62), the crossover time ts_ e is calculated as

ts_ e = y t I = L 2 / y D o~ L2~: -°'7 . (63)

For t >~ tg_ t the main inhomogeneity in the system turns out to exist near the

258 A. Onuki and R . A . 1"2,rrell I Adiabatk" heathzg near gas-liquhl crithal poh~t

interface of the two phases. The diffusion length at t ~ t ~ . , is given by - ~ L / y and is still thinner than the cell size. Notice that t~, goes to ( D t g _ t ) l ~" ':~-

zero on approaching the critical point. In the Ct. measurements by Brown and Meyer Ill and by Dahl and Moldover

[2], they observed rapid thermal relaxation times in the one-phase case even close to the critical point. We believe that the adiabatic heating effect should have been crucial in their experiments. On the other hand, in two-phase states, they observed markedly different, long relaxation times, which apparently suggested anomalously slow motion of an interface [2, 31. However, in the light of the adiabatic effect, the slow relaxation can bc explained differently: their measured relaxation times are much longer than tg , and the two groups should have measured relaxation times of latent heat exchange between gas and liquid regions.

4. Concluding remarks

1) The critical speeding up at lixed volume is surprising in view of the well-known critical slowing-down at fixed pressure.

2) "rile adiabatic effect in tile two-phase case is more surprising. The two phases undergo slightly different, :ldiabatic temperature changes due to rela- tion (61). The resultant slow process of latent heat exchange seems to explain the puzzling findings reported in refs. [I, 21 in the two-phase case. Further experimental investigations on this point should be very informative. It is worth noting that the adiabatic heating process also gives rise Io ;.111 interesting effect of large enhancement of the sound attenuation when fluids are undergo- ing phase separation i l l I. This is because pressure variations associated with ultrasound induce oscillatory heat Ilows between the two phases and large dissipation occurs at low frequencies.

3) The space experiment by Nitsche and Straub [7 i called for the examina- tion of the adiabatic effect I51. However, it should be stressed that the effect itself exists even on earth as shown in subsection 2.6. it has not bccn apparent on earth and has remained unnoticed.

4) In binary mixtures spinodal decomposition experiments have been per- formed by adiabatically changing the pressure [12. 13 I. In pure fluids spinodal decomposition and nucleation can be instantaneously induced ahnost in the same manner by lowering of the boundary temperature.

5) In this paper we have neglected rapid spatio-temporal wlriations of the pressure. Such variations have been numerically cxamincd in rcf. 19[ on the acoustic time scale, which justifies our assumption that the pressure w~riation is a function of time only.

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 259

Acknowledgment

A.O. would like to thank Dr. M.R. Moldover for bringing this problem to his attention.

Appendix

To solve (11) under (16) we define dimensionless quantities by

X = x l L , z=DL-"t , F(X,r)=~T(x,t)IT I . (A.I)

The Laplace transformation of F(X, ~') will be denoted by

F(X, p) = f dr F(X, I") exp(-pr) . (A.2) 0

Then, (11) becomes

0 2 pP= ap( P) + ~ F. (A.3)

Using P = 1 Ip at X = 0 and 1, we may solve (A.3) in the form

F = a ( P ) + ( 1 -a (P) )cos[k (2X- l ) ] c o s k ' (A.4)

where

i pt:2 (A.5) k = ~ or p = - 4 k 2.

The spatial average of (A.4) yields

p ( P ) = + (7 - (A.6)

where

~b(k) = tan(k) Ik = tanh(x/"~/2)/(x/-p/2). (A.7)

Substituting (A.6) into (A.4) we find

e AP-- 1 - pP= ( 1 - cos[k(2X-cos k 1)l ) / t l + ( 7 - 1),b(k)]

= [ 1 - cos(2kX) - tan(k) sin(2kX)]/[1 + (7 - 1)4~(k)]. (A.8)

260 A. Onuki and R .A . Ferrell I Adiabatic heating near gas-liquid critical point

Hereafter AF is the Laplace transform of AF---- 1 - F. the amount by which F(X, ~') has not yet reached its full equilibrium value of 1. Of particular interest is the value of AF in the interior of the fluid, far from the diffusion layers, in the early time region in which the two characteristic spatial regions are well defined. In the interior region the oscillating terms in the numerator of (A.8) disappear leaving the X-independent Laplace transform

I

aP~, , (p) = P 1 + ( 3 , - l)q~(k) " (A.9)

Deforming the contour of integration in the complex-p plane to pass around the poles of A ~ , ( p ) that lie along the non-positive real axis of p gives

,X/'].(r) ~ -4,~,... = e K,,. (A. 10) n ~ I

with

2(r - i) R,, 3'(~/- I) + k;; (A.II )

The contribution from the pole p = 0 is given by 7 ~ and has been neglected. The poles in the complex-p plane are located, according to (A.5) and (A.7). at

"1 p,, = - 4 k 7 , . where

k l !

tan(k,, ) - (A. 12) 3 ' - I "

The values of k,, may be taken to be positive and we find n - ts < k,,/'tr< n. More generally AF(X , z) may be expressed as

AF(X, r) = R,[I - cos(2k,,X) - tan(k,, ) sin(2k,X)] e .4,/,, n - I

(A.13)

For further manipulation of (A.10) and (A.13) we change the summation variable to

k l ! u, , - ( A . 14)

3 / -1

and the time variable to

"7 = ~'/r I . (A. 15)

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 261

where

1 ¢ ' , - , , 2 • ( A . 1 6 )

4(y I )

In the case y -> 1, by virtue of (A.12) and (A.14), we have a dense distribution for the summation variable u, . The spacing of consecutive discrete values is

kn+ ! - k,, Au,, =- u,,+l - u,, = (A.17)

~ , - 1

To a first approximation, from (A.13) in the limit y--~ ~, we have k,,+i - k,, = ~r, so that

Au,, -- ,~ 1 (A. 18) y - 1 "

The density of terms in the summation is therefore given by

1 3, - 1 ( A . 1 9 ) p ( . ) = ,,,,,,----~- I .

Substitution of this expression into (A.11) gives

AFi, ~ 2 f du _ e_,? ~ = e ~ erfc(V~') T~u-"

0

(A.20)

where erfc = 1 - erf is the complementary error function. For very early times, and for times later than ~'~, (A.20) simplifies to

I 1 - 2 ( ' 7 / ' a ' ) I t 2

av,.O-) ~ L (z m~) ''~

for 0 ~< 7,~ 1, (A.21a)

for 1 ,~ 7,~ 7 2 . (A.21b)

Similarly, for 7<~ 7 2, AF(X , r) is approximated by

AF( X , ~') ~ - "7-------w2 i du -=~[1 cos(uP() u sin(u.~')] ~ , l + . e - _ ,

0

(A.22)

where use has been made of (A.12) and

9~ = 2(3, - 1 )X. (A.23)

262 A. Onuki aml R.A. Fen'ell i Adiahath" heating ,ear gas-liquhl crith'al i,,,i,t

The intcgrat ion with respect to u in (A.22) is exact ly pe r f lmned to give

AF(X. r ) / ~ F , , ( r ) "~ I - cxp(.l~") e r f c (V~ ~ .~"/2%'@)/erfc(V~). (A.24)

This yields (21a) and (21b). The space d e p e n d e n c e of the t e ,npcra tu re prolile near the boundar ies will be

exhibi ted explicitly e lsewhere 161. Here we will limit ourselves to those salient features of the profile that c a , be inferred from the space average of A/- '= I ; p _ ~b namely

- ,l,(k ) _ t

l'(a/3> = I + (,/- l)4,(k) 3, ---Z-i + P "~/~;'' " (A.25)

Thus . the amoun t by which the average falls below the Ilat. plateau-l ike. inter ior value is

a~at:')-=<.x,~'>-a~;,- ' (a/-;. ') 3 , - I - p "

so that the inverse Laplace t ransform yields

I ~ ( a F ) = < a f ) - a ~ , , - ( a ~ i , - l )

" y - I

(A.26)

(A.271

We can define an effect ive thickness of the bounda ry diffusion l aye r s . .V . . , by

(2X~,n,.)(AP].) = - N a F ) . (A.28)

f rom ( a F } = ( I - 2X..n.) ~/:],,. so

X , . , = 2(3," I) - I . (A.29)

Subst i tut ion of (A.21a, b) gives

| j , , (÷/~r) - = (4r/~r) ~-' . 0 ~ r "~ r I , (A.3lla)

2 -~ :2 = (.n.r)t _,. 7", '~ 7" '¢ 1 . (A.30b)

A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid critical point 263

The early time development expressed by (A.30a) is nothing other than ordinary heat diffusion without any adiabatic effect. The later behavior shown by (A.30b) differs only by a numerical factor indicating that entropy diffusion continues, at these later times, with its penetration depth still of order ~.~2

For relatively very long times ~" of order 1, (A.30b) indicates that the entropy diffusion penetration depth has grown to be of the same order of magnitude as the total thickness of the fluid sample. For such times the recasting of AFi,0" ) in the form of an integral becomes inaccurate, and (A.20), (A.21b) and (A.30b) are no longer valid. In fact, as there is no longer any flat, plateau-like interior region, AFt, loses its unique significance and becomes ill-defined. We therefore concentrate our attention on (AF) and return to its representation in terms of a sum similar to (A.10),

1 (AF) = 2 7 ~ , e

,.°1 y (y - 1) + kT, (A.31)

For ~" of order 1, only a few terms of the series contribute, and, for these, we have k] .~ y(y - 1) and can make the additional approximation k , , = ~ n . Thus,

2 ( a F ) - - y - I ,,1Z= e - 4 ~ 2 r n 2 . (A.32)

It is worthwhile to verify that (A.32) joins on smoothly to the results obtained earlier for (AF) in the intermediate regime (y - 1) -I .~z,~ 1. If 1-,~ 1, (A.32) becomes

(AF) ~ 2 f y-----~ d n e -4"2r'2 ~ (1/~r'7) t'2 , (A.33)

0

which is in complete agreement with (A.21b) for y >> 1.

Note added in proof

Recently Behringer and Meyer [14] have reexamined data of thermal relaxation times in 31-le near the gas-liquid critical point [15, 16] in the light of the adiabatic effect. These experiments were performed at a fixed volume, but the boundary conditions were different from those in this paper. They have found that the adiabatic effect resolves a previously unexplained difference between the data and the calculated relaxation times in the fixed pressure condition.

264 A. Onuki and R.A. Ferrell I Adiabatic heating near gas-liquid crith'al point

References

Ill G.R. Brown and H. Meyer, Phys. Rev. A 6 (1972) 364. [2J D. Dahl and M.R. Moldover, Phys. Rev. A 6 (1972) 1915. [3J M. Gitterman, Rev. Mod. Phys. 50 (1978) 85. [4] M.R. Moldovcr. J.V. Sengers, R.W. Gammon and R.J. Hocken, Rcv. Mod. Phys. 51 (1979)

79. [5] A. Onuki. A contribution to: Thermal Equilibration Near the Critical Point, a NASA-

sponsored workshop held at the National Institute of Standards and Technology (Gaith- ersburg, MD. 16-17 March 1989, unpublished).

[61 A. Onuki, H. Hao and R.A. Ferrell, Phys. Rev. A 41 (1990) 2256. [7l K. Nitsche and J. Staub, Proc. 6th European Symp. on Material Science under Microgravity

Conditions (Bordeaux, France, 2-5 December 1986). [81 H. Boukari, J.N. Shaumeyer, M.E. Briggs and H. Ga,mnon, Phys. Rev. A 41 (19~)) 22~). [9l B. Zappoli. 1). Bailly, Y. Garrabo.,,. B. I.e Neindre. P. Guenoun :rod D. Beysens. Phys. Rev.

A 41 (1990) 2264. [l()] P.C. tlohcnberg and M. Barmatz, Phys. Rcv. A 6 (1~J72) 2,~. I I I I A . Onuki, prcprint. [121 N.('. Wong and C.M. Knoblcr, J. Chem. Phys. (~ (1~77) 47(J7. [13[ E.A. Clark :rod J.V. Sengers. Physica A 118 (1983) 3(~). [141 R.B. Behringer and H. Meyer, private communication. [15l C.E. Pittman. L.H. Cohen and FI. Meyer. J. Low. Temp. Phys. 4¢~ (1~821 115. [161 L.H. Cohen, M.L. Dingus and 1t. Meyer, J. I.ow. Temp. Phys. 61 (19~51 7~.