Kinetics of the initial stage of isothermal gas phase formation D. Kashchiev and A. Firoozabadi Reservoir Engineering Research Institute, Palo Alto, California 94304

(Received 10 September 1992; accepted 3 December 1992)

A theoretical description is proposed of the kinetics of the initial stage of gas phase formation in supersaturated liquids at constant temperature. General expressions are given for the time dependence of the number N of bubbles nucleated, of the probability P of appearance of at least one nucleus bubble and of the total volume V, of the newly formed gas phase in the case of progressive or instantaneous bubble nucleation. Different definitions of the critical supersaturation for gas phase formation are considered and their experimental applicability is discussed. The general results are applied to the particular cases of gas phase formation at constant and steadily declining pressure and the corresponding time dependence of X P, and Vg is obtained. Expressions for the critical supersaturation are also derived and it is found that this quantity can be rather sensitive to the rate of pressure decline when gas phase for- mation occurs via instantaneous bubble nucleation.

I. lNTRODUCTlON

Gas phase formation in supersaturated liquids is a first-order phase transition which occurs by bubble nucle- ation, growth, and coalescence. Initially, the bubbles are small and far from each other and coalescence is of no importance. The kinetics of the initial stage of the process are therefore describable only in terms of the rates of bub- ble nucleation and growth. A description leading to better understanding of the overall process of gas phase forma- tion is important not only scientifically, but also for the numerous applications of this process in technology. In petroleum reservoir engineering, for example, there is a need to know how gas builds up in oil reservoirs, since the factors controlling the process are directly related to the efficiency of oil recovery.ld

So far, considerable attention has been paid to the study of the rates of bubble nucleation7-l2 and gro~h.‘0>‘3-‘8 In contrast, relatively little is known about the role played by these quantities in the overall process of gas phase formation. This role is well understood and suc- cessfully analyzed in other cases of new phase formation, e.g., crystallization of melts and precipitation in solu- tions.‘g-21 It may be expected that the results known from the analysis of these cases can be a good basis for describ- ing also the kinetics of gas phase formation.

The aim of this paper is (i) to develop a theoretical framework for description of the initial stage of the overall process of gas phase formation; (ii) to give a general ex- pression for the time dependence of the total volume of the newly formed gas phase; and (iii) to apply the general results for determination of the critical supersaturation in the particular cases of isothermal gas phase formation at constant and steadily declining pressure.

II. PHYSICAL MODEL

We consider a liquid phase of volume V. at a constant absolute temperature T. At the initial moment t=O the liquid is put under pressure p<pa p,(T) being the equilib- rium pressure, i.e., the pressure at which the liquid and the

corresponding gas phase can coexist. As time goes on, p may either remain constant or vary with time, but under the condition that it is always below pc In other words, the liquid is always supersaturated, the supersaturation s being defined as

s=p,-p. (1)

Thermodynamically, s is related to the driving force APE,q(p) -,uJp) for gas phase formation by the approx- imate formula A,u=kT ln(p,J’) in which k is the Boltz- mann constant, and pl and pg are the chemical potentials of the bulk liquid and gas phase, respectively. At p=pe there is no driving force for gas phase formation (s=O, Ap =O) and it is said that the liquid is saturated. Gas phase forma- tion is motivated energetically only for p <pe when the liquid is supersaturated, i.e., when s> 0 so that Ap > 0. Under such conditions the build-up of the gas phase occurs by nucleation, growth and coalescence of bubbles into the bulk of the liquid and/or on the surface of the solid or liquid “walls” confining the liquid. At the beginning of the process coalescence is of no importance, since the bubbles are relatively small and sufficiently far from each other. The initial stage of the overall process of gas phase forma- tion is, therefore, describable only in terms of the nucle- ation rate J( m-’ s-i > and the growth rate Gsdr/dt of the bubbles, Y being the bubble radius.

Ill. RATE OF BUBBLE NUCLEATION

A. Constant supersaturation

When the liquid is kept at a constant supersaturation s, the rate J of bubble nucleation may be time-independent. Then it is reliably expressed by the stationary rate Jo of bubble nucleation, which fairly generally may be written down as7-i2

Jo=A exp( - M*“k/kT) =A exp( - B/2). (2) This formula applies to both single- and multicomponent liquids provided the pressure inside the nucleus bubble is practically equal to pe. The quantities A( mm3 s-l) and

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B(Pa2) are kinetic and thermodynamic parameters, re- spectively, related to the elementary act of nucleus growth and the nucleation work wh. They are given by

A=epZ, (3)

B= 16?ro$/3kT, (4)

where ~~0.01-0.1 is the so-called Zeldovich factor, p( s-t) is the frequency of increasing the nucleus volume by the volume u, of a single molecule, Z(mw3) is the number density of points in space on which a bubble can be nucleated, and a&J rnp2) is an effective specific surface energy.

Different physical conditions lead to different expres- sions and values for p, Z, and a,f (and, thereby, for A and B), 7-12 but an important point is that A and B may be considered as practically independent of s. In the case of homogeneous nucleation of spherical bubbles oef equals the specific surface energy o of the gas/liquid interface and Z= l/u,= 1O28 m-‘. In this case, for evaporation- mediated increase of the volume of the nucleus bubble f* is given by the Knudsen-type formula9 f*=4?rF2p$ (2rmkT) 1’2 in which m is the molecular mass and p r 10 nm is the radius of the nucleus bubble. Thus, at T = 300 K, with values of e=O.Ol, pe= 103-10’ Pa and a,f=o=3-70 mJ m-‘, from Eqs. (3) and (4) it follows that A= 1036- 1040 mm3 s-r and B= 1O’4-1O’8 Pa2 for homogeneous bub- ble nucleation.

Another case to note is the heterogeneous nucleation of bubbles, for instance, in a porous medium. If the me- dium is regarded as constituted of monodisperse grains (or droplets) of radius R and number density N,(md3) and if the nucleus bubbles are formed on the surface of the grains, as is known7-l2 to be given by o,f=41’3a. The correction factor 4 is defined by # = V&/Y* and is a number be- tween 0 and 1, since the volume V& of the heterogeneous nucleus bubble is always smaller than the volume v* = (4/ 3)7rf13 of the corresponding homogeneous nucleus bubble. This factor is a function of the grain radius R and of the liquid/grain wetting angle 8 which is expressed by the Young formula

cos e= (q--a,)/0

in terms of o and the specific surface energies o1 and ag of the liquid/grain and gas/grain interfaces, respectively. If the grains are large enough (R>fi), 4 is the following function of e only7-l2

4(e) = (i/4) (2+3 cos e--c0s3 e),

which shows that 4= 1, l/2, and 0 at complete (e=O>, “half” incomplete (8=?r/2) and no (e=n> wetting, re- spectively. This means that heterogeneous bubble nucle- ation is thermodynamically favored only when the liquid does not completely wet the grains, as only then is C$ < 1.

In the case of heterogeneous nucleation, a decrease in p is also possible because of the reduction of the area of the gas/liquid interface of the nucleus bubble. Also, Z will be given by Z=4rR2ZfiP where ZS(mm2) is the number density of these points on the grain surface on which het-

erogeneous bubbles can be nucleated. For densely packed grains Ng= 1/(4/3)7rR3, so that Z=Z,/R. In the absence of nucleation-active sites on the surface of the grains Z, = l/sar 10” mm2 (s, is the atomic area). This means that Z= 102’ m-‘, e.g., for grains of radius R = 1 ,um. This is less than Z for homogeneous nucleation and can be further reduced to Z=NgrRm3 if every grain provides one nucleation-active site only (then 4z-R2ZS= 1) . The general conclusion is, therefore, that for heterogeneous bubble nu- cleation the values of both A and B are smaller than those for homogeneous nucleation. For bubbles nucleated heter- ogeneously in a porous medium, A may increase with de- creasing grain radius R to a power from 1 to 3.

B. Variable supersaturation

When nucleation takes place at variable supersatura- tion, in general, J is a complicated function of time t (Refs. 22-24). For slow enough changes of the supersaturation, however, J can be approximated by the quasistationary nucleation rate corresponding to the instantaneous values of the supersaturation. For bubble nucleation at time- dependent supersaturation s(t) we shall, therefore, have J(t) = Jo[s(t)] so that in view of Eq. (2),

J(t) =A exp[ -B/$(t) 1. (5) According to Refs. 23 and 24, this formula is valid when the rate ds/dt of supersaturation change is low enough to satisfy the condition

~dAj../dt~<kT/n*[s(t)]~,[s(t)]

which with A,x=kT ln(pdp) takes the form

~ds/dt~<p/n*[s(t)]~,[s(t)l. (6) Here n* is the number of molecules in the nucleus bubble, and r,=4/r38p (Refs. 24 and 25) is the nucleation time-lag. For p > 0. lp,, with values of pe= lo3 Pa, n* = 100 and r,= 1 ,us it thus follows from inequality (6) that Eq. (5) is a good approximation if

I ds/dt I Q lo6 Pa/s, a requirement which is usually met in practice.

(7)

IV. RATE OF BUBBLE GROWTH

After a bubble has surpassed the nucleus size (i.e., when Y > ?), it can grow irreversibly with a certain growth rate G. It is generally accepted109’3-‘5 that for an isolated supernucleus bubble G is controlled by mass, momentum, and/or heat transfer across the bubble/liquid interface. Mass transfer occurs via evaporation and condensation at this interface and/or diffusion in the liquid when this is multicomponent, momentum transfer is governed by hy- drodynamic forces connected with the bubble capillary pressure and the liquid inertia and viscosity, and heat transfer takes place by flow of heat from the liquid towards the bubble. It has been found’0*‘3-18 that the different re- gimes of growth result in different r(t) dependencies which, in general, are rather complicated. However, when the effects of the bubble/liquid interface curvature on

D. Kashchiev and A. Firoozabadi: Isothermal gas phase formation 4691

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4692 D. Kashchiev and A. Firoozabadi: Isothermal gas phase formation

chemical potential are neglected, at constant supersatura- tion many of these dependencies show exact or approxi- mate proportionality of Y to some power of t. This finding is supported also by available experimental y(t) data. For example, the rate of bubble growth controlled only by the inertia of the liquid has a constant value Gc given by the formula

Go = (2v,&3m) 1’2. (8) following from the Rayleigh equation.16 Hence, in this re- gime the bubble radius obeys a linear growth law as long as s remains time-independent. Neglecting the bubble initial radius r(0) which is just above the nucleus radius p, we have

r(t) = (2u&3m) 1’2t. (9)

For instance, with m/v,= 1 g/cm3, Eq. (8) yields Go=8 m /s at s=105 Pa. In the same regime, in the important case of supersaturating the liquid at constant rate c( Pa/s), i.e., when

s(t) =pe-p=ct, (10)

Eq. (8) predicts the following time-dependent growth rate if c is small enough

G(t) = (2~ c/3m) 1’2t1’2. m

Accordingly, for the bubble growth law we find

r(t) = (8v,c/27m) “‘X2

provided r(0) SO.

(11)

(12)

provided r(0 j is negligible. Here D is the diffusion coeffi- cient of dissolved gas molecules, and KS is the solubility constant in the supposed Henry-type proportionality be- tween pressure and gas concentration in the liquid. If, e.g., T=300 K, D= 10m9 m2/s and KS= lOi Pa-’ m-‘, at s/p, =O.l and t= 1 s, Eq. (17) predicts G= 1 ,um/s. When the supersaturation increases steadily but slowly enough, in view of Eq. (lo), as long as ct<pe, the growth rate from Eq. (17) will change to a time-independent one given by

Go= (DK&Tc/p,) 112, (19)

and the growth law Eq. (18) will become linear [r(O) is again neglected],

r(t) = (DK&Tc/p,) 1’2t. (20)

Having in m ind the above results, it seems reasonable to assume that at both constant and linearly increasing supersaturation the rate of bubble growth may be repre- sented (accurately or approximately) in the following fairly general form:

G(t) =vKs”tv-l (21)

for constant supersaturation and

G(t)=(u+v)Kc”t”+“-’ (22)

for steadily increasing supersaturation. These growth rates correspond to power-type growth laws given by

Another case characterized by power dependencies of G and Y on t is the case of bubble growth in the regime of mass transfer. For instance, for bubble growth controlled by evaporation and condensation, at constant s the growth rate is also constant and when s<pe, it is given by9

Go= (kT?/2rmpz)“2. (13) Accordingly, the bubble radius obeys a linear growth law

r ( t) = ( kTs/2rmpz ) 1’2t (14) provided r( 0) ~0. For example, with m = 1O-22 g, eat T=300 K, and s/p,=O.l, Eq. (13) yields Go=8 m /s. Equations ( 13) and ( 14) change in the case of increasing supersaturation at constant, sufficiently low rate c. Using Eq. (10) in Eq. ( 13) results in a time-dependent growth rate of the form (ctgp,)

G(t) = ( kTc2/2nmpz) 1’2t (15) which for r(0) r0 corresponds to the following growth law:

r(t) =Ks”t”, (23)

- r(t) =Kc”tO+” (24)

for constant and steadily increasing supersaturation, re- spectively. Here the powers v > 0 and o > 0 and the growth constant K have to be specified for each particular regime of growth. As before, Eqs. (21)-(24) imply that the su- persaturating rate c is low enough and that the bubble initial radius r(0) is vanishingly small. Also, the v= 1 and w + v= 1 cases correspond to bubbles growing linearly with time at a constant growth rate G,=Ks@ or Kc”.

V. GENERAL FORMULATIONS

Knowing the bubble nucleation and growth rates Jand G makes it possible to determine theoretically the following important characteristics of the overall process of gas phase formation:

The number N of bubbles nucleated in the liquid until time t,

J-

* N(t) = v, J( t’)dt’.

0 r(t) = (kTc2/8Pmpi)“2?. (16)

Again in the regime of mass transfer, when growth is diffusion-controlled and dr/dt= DK&Ts/p,r, at constant supersaturation s<p, known dependencies arer4,i8

G(t) = ( D&kTs/2p,) 1’2,- 1’2, (17)

r(t) = (2DK&Ts/p,) 1’2r1’2, (18)

J. Chem. Phys., Vol. 98, No. 6, 15 March 1993

(25)

This formula is valid until the initial volume V. of the liquid is not significantly exhausted and neglects the com- pressibility of the liquid at time-dependent s. These two restrictions apply also to Eqs. (26)-(28) and (30) below and to all results following from them.

The probability P to form at least one nucleus bubble until time t,

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P(t)=l-exp[--N(t)]=l-exp -V. [ lJ(t’)dt’].

(26)

This formula was derived by Toschev et a126 by using the fact that the nucleation events are random in time and obey the Poisson distribution law. With the help of Eq. (26), the mean time r for the appearance of at least one nucleus bubble can be found,

7= I,” tdP(t)= Jom exp[ -VoJofJ(t’)dt’]dt. (27)

The total volume Vg of the gas phase at time t,

s

t V,(t) = vo J(P)q)(t,t’)dt’. (28)

0

Here vb(t,t’) is the volume at time t of a bubble nucleated at time t’<t and is given by

t 3 vb(t,t’) =a?(t-t’) =a

[I G( t” - t’)dt” 1 (29)

t’

provided the initial volume of the bubble is vanishigly small [a is a time-independent shape factor, e.g., a=47r# ( f3)/3 for bubbles of equilibrium shape, and r( t -t’) is the effective radius at t of a bubble formed at t’=g]. Equation (28) corresponds to the case of the so-called progressive nucleation (PN) when in the liquid new bub- bles are continuously nucleated among the growing ones. A limiting case of IQ. (28) is the case called instantaneous nucleation (IN) when bubble nuclei of number density No (mm3) are formed at once at t’ =0, so that later they only grow. In this case nucleation and growth are decoupled in time and, since J(f) is formally expressed by the Dirac S-function as J( t’ ) =N,6 ( t’ ) , Eq. (28 ) becomes

V,(t) = v&o%w). Here, according to Eq. (29),

vb(t,O) =a?(t) =a [ Jot G(t”)dtrj3

(30)

(31)

when the initial volume of the nucleus bubble is negligible. The above formulations are general enough to describe

the initial stage of the overall process of gas phase forma- tion in many cases of physical significance. In every con- crete case the dependencies of J and G on time t, supersat- uration s, and the material parameters of the studied system have to be specified in order to obtain the corre- sponding dependencies of N, P, r, and Vr An important case to note is the case of gas phase formation at time- independent rates Jo and Go of nucleation and growth. Such a case may be realized when the liquid is kept under constant pressure p<p, so that the supersaturation s does not change with time. In this case Eqs. (25)-( 31) lead to the following simple kinetic dependencies:

Since the bubbles do not nucleate and grow infinitely fast, some time always elapses from the moment of super- saturating the liquid until the appearance of the gas phase. This so-called induction time (or period) depends on the supersaturation and on the experimental technique used for detecting the first portion of the newly formed gas phase. That is why, given the experimental technique, a critical supersaturation s,=p,-p, exists at which the in- duction time equals a previously specified arbitrarily cho- sen value (p, is the corresponding critical supersaturation pressure). Physically, this means that for supersaturations between s=O and s=s, the liquid remains effectively in a metastable state. It must be stressed, however, that the critical supersaturation is not a universal quantity, since it depends on how it is theoretically defined and experimen- tally measured. We shall now consider three possible def- initions of the critical supersaturation, which are relevant to different experimental conditions and techniques.

(a) The critical supersaturation to obtain one nucleus bubble.

This definition originates from Volmer7 and reads N( tl) = 1, where tl is the moment of appearance of the first nucleus bubble, i.e., the induction time. In view of Eq. (25), it may be represented in the form

l=V, s

tl J(t’)dt’.

0 (37)

w t> = VoJot,

P(t) = 1 -exp( - Vdot>,

I-= l/V&,

As seen, knowing the time dependencies of J and s allows determination of the so-defined critical supersaturation s, for an arbitrarily chosen value of tl.

(b) The critical supersaturation to form on the average at least one nucleus bubble. (32)

This definition is in fact given by Eq. (27)) (33)

(34) T= Jo- exp[ - Vol J(t’)dt’]dt.

J. Chem. Phys., Vol. 98, No. 6, 15 March 1993

V,(t) = (a/4) VdoGit4, (35)

V,(t) =aV&VoGi$. (36)

These equations are well known in other cases of new phase formation. 19-21126p27 Equations (35) and (36) show that separate information about the kinetics of bubble nu- cleation and growth cannot be obtained by analyzing the V,(t) dependence only, since Jo and Go or No and Go ap- pear combined into a single kinetic parameter governing the gas phase build-up in the case of PN or IN, respec- tively. We emphasize, however, that in the latter case a complete determination of Go is possible if No is indepen- dently known, e.g., if it equals the number density of suf- ficiently small particles (seeds) introduced deliberately into the liquid as nucleation-active sites generating one nu- cleus each. A similar process of crystal phase formation in solutions is known as seeded precipitation and has been recently used for determination of crystal growth rates.28-30

VI. CRITICAL SUPERSATURATION FOR GAS PHASE FORMATION

(38)

D. Kashchiev and A. Firoozabadi: Isothermal gas phase formation 4693

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4694 D. Kashchiev and A. Firoozabadi: Isothermal gas phase formation

Again, choosing arbitrarily a value for the mean time r for appearance of at least one nucleus bubble (which is now the induction time) makes it possible to determine the so- defined critical supersaturation s,

(c) The critical supersaturation to form a detectable gas phase volume V, which is a given fraction a! = V,/V, of the initial volume of the liquid.

This definition reads Vg( t,) = V,, tu being the moment at which the total volume of the gas phase is V, and having the physical significance of induction time. Using Eqs. (28) and (30) leads to the following formulas in the cases of PN and IN, respectively,

s tu

a= J(t’)v&,t’)dt’, (39) 0

a=Novb( t,,O>. (40) As before, determination of the so-defined critical super- sgturation s, is possible upon assigning an arbitrary value to tu. However, in the PN case it is necessary to know not only the time dependencies of J and s, but also of ub (i.e., of G). In the case of IN, only knowledge of ub and s as functions of t is required.

Each of the above three definitions is physically accept- able, but has a lim ited experimental application. The use of definitions (a) and (b) is restricted to experimentation with techniques which are so sensitive that the detection of the very first nucleus bubble is possible. Using these defi- nitions is also correct in the case of gas phase formation by the so-called mononuclear mechanism (e.g., Ref. 27) which is operative for such small V. and J and/or large G that after the appearance of the first nucleus bubble virtu- ally no time is lost for the formation of the detectable volume V, of the gas phase. For time-independent nucle- ation and growth rates the condition for the operativeness of the mononuclear mechanism is27

(JdGob 1’sG’3<l. (41) Physically, definitions (a) and (b) are closely related and become equivalent in the case of stationary nucleation at constant rate Jo, as then Eqs. (37) and (38) lead to iden- tical expressions for t, and T and, accordingly, to the same critical supersaturation.

Definition (c) has a much less restricted experimental usage, since it is relevant to techniques detecting also a macroscopic volume V, of the newborn gas phase. The condition to apply it is the gas phase to be formed by the so-called polynuclear mechanism (e.g. Ref. 27). This mechanism is almost always effective in practice, since it corresponds to such large V. and J and/or small G that many bubbles nucleate and grow in the liquid until the detection of V,. In the case of time-independent nucleation and growth rates the condition for the operativeness of the polynuclear mechanism is the opposite to Eq. (41) (Ref. 27),

(Jo/G& 1’s~o’3~l. (42)

An important point concerning the above definitions is that at a given choice of J, G, and V. they may predict

substantially different values of s,. This means that uncrit- ical employment of any of these definitions for analysis of experimental data may result in m isleading conclusions about the mechanism of bubble nucleation and growth and/or very erroneous values for the parameters charac- terizing these processes.

VII. PARTICULAR CASES

A. Constant supersaturation

1. Kinetic dependencies

We shall now apply the general results from Sec. V to the case of isothermal gas phase formation at constant su- persaturation s. This case is realized when at t=O the liq- uid is put under pressure p <pe which is afterwards kept constant so that s> 0 for t)O. This condition allows sta- tionary bubble nucleation to take place at time- independent rate Jo given by Eq. (2). Assuming that the bubble growth rate and radius can be expressed by Eqs. (21) and (23), from Eqs. (25)-(31) we find explicitly the dependencies of N, P, and Vg on supersaturation and time and of r on supersaturation,

N(t)=V& exp(-B/?)t, (43)

P(t)=l-exp[-Vdexp(-B/sZ)t], (4.4)

r= (l/Vd)exp(B/s2), (45)

Vg(t)=[a/(3v+l)]V,,AK3s3mexp(-B/?)$”+’, (46)

Vs( t) = a V&&3~0~y. (47) Equations (46) and (47) pass into Eqs. (35) and

(36)) respectively, when the bubble growth rate is constant (i.e., at v= 1). Also, they show that the total gas volume Vg depends more strongly on t and especially on s in the PN than in the IN case.

2. Critical supersaturation

We can now apply definitions (a)-(c) from Sec. VI to the above equations in order to determine the critical su- persaturation for gas phase formation. Since Jo is time- independent, tl =r so that definitions (a) and (b) are equivalent and in conjunction with Eqs. (43) and (45) yield the same result,

tl=(1/Vgi)exp(B/s2),

s,= [ 16?r/3kT ln( V&tI) ] 1’2c$.

(48)

Here s, is the critical supersaturation for the appearance of the first nucleus bubble and, as seen, it decreases with de- creasing aef and/or increasing T, but depends relatively weakly on the nucleation kinetic factor A, the initial vol- ume V. of the liquid and the choice of value for tl. Equa- tion (48) is known from the classical nucleation theory7*’ and has been found to be in conformity with experi- ,,,t~7,9,11,‘2

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D. Kashchiev and A. Firoozabadi: Isothermal gas phase formation 4695

~o~,‘.,:““:““:“‘,:““I 0.5 1.0 1.5 2.0 2.5 3.0

a,, (mJ/m”)

FIG. 1. Dependence of the critical supersaturation on the effective spe- cific surface energy for progressive nucleation. Dotted curve, F!q. (48); dashed and solid curves, Eq. (49) with diffusion growth constant K= lo-’ and lo-’ rn/sln Pa’“, respectively.

According to definition (c), from Eqs. (46) and (47) we find that the induction time tu and the critical supersat- uration s, to form a detectable volume V,= a V. of the gas phase are given by

t,=[a(3Y+l)/aAK3s30]1’(3v+1)exp[B/(3~+l)s2], (49)

in the case of PN and by

to= ( a/aNs3) 1’3vs-“‘v,

s,= (a/aN,-JC3t~‘) 1’3o (50)

in the case of IN. Although Eq. (49) is implicit in s,, it shows that the dependence of the critical supersaturation on a,, and T is essentially the same as that from Eq. (48)) since the logarithmic factor changes relatively little with s, Numerically, however, s, from Eq. (49) may be consider- ably higher than s, from Eq. (48) especially for smaller K-values when the bubble growth is relatively slow. This is seen in Fig. 1, where s, from Eqs. (48) and (49) is plotted as a function of oef=#‘30 for heterogeneous nucleation and parabolic growth (Y= l/2) of bubbles with w= l/2 [e.g., in the regime of mass transfer by diffusion according to Eq. (18)]. The other parameters used are 4= l/2 (hemispherical bubbles, a=2?r/3), a=O.OOl, Vo= 10B3 m3 A= 1O25 m-’ s-l, tl= to= 100 s, T=300 K, and K is g&en two values, 10s7 and lo-’ m /s1’2 Pa*‘2 (as indicated in Fig. 1 ), to illustrate the effect of the growth rate on s, Figure 1 shows the decrease of s, from both Eqs. (48) and (49) with decreasing aef and the shift of s, from Eq. (49) towards s, from Eq. (48) with increasing K (i.e., at faster bubble growth). The latter is understandable, as in accor- dance with Eqs. (41) and (42), at higher growth rates the polynuclear mechanism [resulting in s, from Eq. (49)] is bound to be replaced by the mononuclear one [resulting in s, from Eq. (48)]. A g eneral consideration of the inter-

relation between the critical supersaturations for the mono- nuclear and the polynuclear mechanisms is given else- where.27

Finally, we note that in contrast to s, from Eq. (49), Eq. (50) shows that when gas phase formation occurs by IN, s, is determined only by the parameters of bubble growth provided No is s-independent, as it is for instance in the case of seeded new phase formation.28-30 Concerning the induction time tu, we note that in the case of IN it is a much weaker (only power) function of s than in the case of PN. In its turn, the t,(s) dependence for PN is weaker than the t,(s) dependence because of the 3v+ 1 factor in the exponential function.

B. Steadily increasing supersaturation

1. Kinetic dependencies

The general results from Sec. V are relatively easily applicable to the important case of gas phase formation when the supersaturation increases with time at a constant rate c. According to Eq. (lo), this is the case, for instance, when the pressure in the liquid decreases steadily while pe remains practically unchanged. For this case, with the help of Eqs. (5), (22), and (24), from Eqs. (25)-(31) we find the dependencies of N, P, and V, on t and c and of r on c in the following form:

N(t) = V&{exp( -B/c2?)

- [ (3-B) ‘12/ct] erfc( B1’2/ct)},

P(t)=l-exp[ -N(t)],

7= s

m exp[ -N(t)]dt, 0

(51)

(52)

(53)

V,(t) ,aV&3c-(3Y+1) s Ct (ct-s’)3(0+Y) 0

Xexp( - B/f2)ds’, (54)

Vg( t) =a VdyoK3c30t3(0+y). (55) In Eqs. (52) and (53) N(t) is given by Eq. (51) in which erfc(x) is the complementary error function defined by3i

co erfc(x) = (2/~-i/~)

s expt -v2>&. x The above N(t) dependence has also been found in Ref. 3.

Since the integration in Eq. (53) cannot be carried out analytically, in combination with Eq. (5 1) this equation gives only implicitly 7 as a function of the supersaturating rate c. Analytical integration is also impossible in Eq. (54) at arbitrary (w + v) -value and for this reason, in the case of PN, the general dependence of Vg on t and c is not obtain- able explicitly. In contrast, in the case of IN, Vg is an explicit function of t and c. According to Eq. (55), Vg increases with increasing t and c following a power law.

The complicated Eq. (5 1) simplifies considerably for times close enough to the beginning of the process. When

t< (2 B/3c2) 1’2, (56)

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D. Kashchiev and A. Firoozabadi: isothermal gas phase formation

-7 I jE I I I I j! I I I I I I ,

l- 0.4 0.8 1.2 1.6

t xl 0s (s)

FIG. 2. Time dependence of the number N of bubbles nucleated and of the probability P to form at least one nucleus bubble. Solid, dashed, and dotted curves, Eqs. (57) and (58) with supersaturating rate c= 1, 5, and 25 Pa/s, respectively.

N(t) from Eq. (51) becomes

N(t) = ( Vdc2/2B>t3 exp( - B/c2?) (57)

which follows from the series expansion of the erfc- function for large B”2/ct values.31 Accordingly, if Eq. (56) is satisfied, the P(t) function from Eq. (52) has the form

P(t)=1-exp[-(V&2/2B)~exp(-B/c2t2)]. (58)

With a relatively low B= 1014 Pa2, from Eq. (56) it follows that the N(t) and P(t) dependencies are described reliably by Eqs. (57) and (58) as long as ctg8X lo6 Pa.

In contrast to the linear N(t) dependence, Eq. (43) at constant supersaturation, Eq. (57) shows that N increases sharply with t after a certain time delay which, physically, plays the role of induction time. This threshold rise of the number of bubble nuclei is due to both the intuitively ex- pected exponential term and the extra $-factor in Eq. (57).

Figure 2 depicts the time dependencies of N and P from Eqs. (57) and (58) at three supersaturating rates c=l, 5, and 25 Pa/s (as indicated) and Vo= 10m3 m3, A= 102* rn-‘s-l, and B= 1014 Pa2 (this B-value corre- sponds to aef= 3 mJ/m2). It is seen that N and P vary very strongly with time and that, also, they are very sensitive to the supersaturating rate c.

We shall now analyze the t,c-dependence, Eq. (54), of the total volume V’ of the gas phase in the case of PN. The integration in Eq. (54) can be carried out analytically for (o+v)-values satisfying the condition 3(w+v) = 1,2,3,... . The o+v= 1 case corresponds to bubble growth with time-independent rate Go exemplified by Eq. (19) when growth is diffusion-controlled. In this case the integral in Eq. (54) can be expressed in a closed form and the final result is

Vg( t) = (a/4) V&C3c30-4 [ ( c4t4- 7 Bc2?) exp ( - B/c2$)

-4(~B)“~(c~f3--2Bct)erfc(B”~/ct)

+(6Bc2t2-B2)E1(B/c2t2)], (59)

where El(x) is the exponential integral defined by31

El(x) = J-

m y-‘e-Ydy. x

We also note the expressions for V,(t) resulting from Eq. (54) upon integration at w + v= l/3 and 2/3, respectively,

Vg( t) = (a/2) V,@3c30-2 [ c?? exp ( - B/c2?)

-2(~B)“~cterfc(B”~/ct)+BE~(B/c?)], (60)

V,(t) = (a/3) V&C3c30-3[ (c3e--2Bct)exp( - B/c2?)

- (rrB)“2(3c2?-2B)erfc( B112/ct)

+ 3 BctEl ( B/c2t2) 1. (61)

Despite the complexity of the above formulas for Vg( t), in the lim it of small t-values they take a simple and physically more transparent form. Indeed, using the known asymptotic expansions of erfc(x) and E,(x) (Ref. 31), from the above three equations we find that (I? is the complete gamma-function)

X (c2/2B) 3(o+v)+lt9(~+d+3eXp( -B/c2&

(62) provided that t satisfies practically the same condition as Eq. (56). Equation (62) is valid not only for 3 (w +v) = 1,2, and 3, but also at w+v=O, which means that it can be used as a reliable approximation to the general V,(t) dependence, Eq. (54), for arbitrary positive 3(w +v)#O,1,2 ,... . In the w +v= 1 case Eq. (62) reads

V (t)=(3~V&C3c3ti+8/8ti)t12exp(-B/c2~). g (63)

Comparison of Eq. (62) with Eq. (46) shows that the V,(t) dependence has a much less pronounced threshold character at constant than at steadily increasing supersat- uration. The delay time, which has the physical signifi- cance of induction time, is more obvious in the latter case because of both the expected exponential term and the extra tgw+6”+2-factor in Eq. (62). Figure 3 illustrates the dependence of Vg on t from Eq. (63) at three supersatu- rating rates c (as indicated). The values of c, A, and B are those already used for the N(t) plots in Fig. 2, and the other parameters are a = 21r/3 (hemispherical bubbles), w= l/2 [e.g., diffusion-controlled bubble growth after Eq. (19)], and K= lo-’ m /s 1’2 Pa”2. As seen from Fig. 3, the Vg( t) dependence is similar to that of N( t) in Fig. 2. V, is strongly affected by changes in c and is considerably re- tarded, since initially s=O.

In the case of gas phase formation by IN, the character of, the Vg( t) dependence is the same at both constant and linearly increasing supersaturation. Inspection of Eqs. (47) and (55) shows that in both cases Vg is a power function of

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D. Kashchiev and A. Firoozabadi: Isothermal gas phase formation 4697

0.10

0.08

> 0.06 . 9

0.04

0.02

0.00 0.0 I

25

J-4-=

-i- I I

I I I

I I I

I I I I I

I 0 I

L

5 1

0.2 0.4 0.6 0.6 1.0 1.2 1.4 1.6 1.8

tx lo-ys)

FIG. 3. Time dependence of the total volume of the gas phase in the case of progressive nucleation. Solid, dashed, and dotted curves, Eq. (63) with supersaturating rate c= 1, 5, and 25 Pa/s, respectively.

t, somewhat stronger in the latter case, as w>O. Also, in the latter case Yg is a power function of the supersaturating rate c.

2. Critical supersaturation

We now have the results needed for determination of the critical supersaturation s, in the case of gas phase for- mation at constant supersaturating rate. We shall only use definitions (a) and (c) from Sec. VI, because definition (b) requires analyzing the mathematically complex Eq. (53). Moreover, since r and ti are numerically close to each other (see Fig. 2 in which r is roughly the moment at which P has an inflection), definition (b) may be expected to yield results practically equivalent to those based on definition (a).

Definition (a) has to be used in conjunction with N(t) from Eq. (51). However, since the moment tl of appear- ance of the first nucleus bubble satisfies virtually always inequality (56), the use of Eq. (57) is also legitimate. This leads to the following implicit dependence of ti (i.e., of the induction time) on the supersaturating rate c,

t,=[16~/3kTln(3V~kTc2t~/32?ro$)]”2(o~~2/~). (64)

As expected, increasing rate c and temperature T and/or decreasing effective surface energy oef shorten the time needed for the birth of the first nucleus bubble. At the moment tl the supersaturation in the liquid assumes its critical value s, defined by

s,=ct1 . (65) Using this relation allows an equivalent, but more conve- nient presentation of Eq. (64) in the form

c= ( Y&/2B)sz exp( -B/g). (66)

This equation gives implicitly the sought dependence of the critical supersaturation s, on the supersaturating rate c in the scope of definition (a), Equation (66) applies to

2.7

0.8-1 : i ; I ;

0 20 40 60 80 100

c (Pa/s)

FIG. 4. Dependence of the critical supersaturation on the supersaturating rate for progressive nucleation. Solid and dash-dotted curves, Eq. (66) with effective specific surface energy u,r=2.7 and 2.5 m.J/m’, respectively; dashed and dotted curves, Eq. (72) with u,r=2.7 and 2.5 mJ/m’, respec- tively.

gas phase formation by the mononuclear mechanism and predicts a linear dependence if experimental sc( c) data are plotted in ln(c/$) vs l/g coordinates,

ln(c/& =ln(3VdkT/32rgzf) - ( 16?razf/3kT) (l/z). (67)

From the intercept and the slope, the kinetic parameter A of bubble nucleation and the effective specific surface en- ergy of the nucleus bubble can be calculated, respectively.

Figure 4 depicts the s,(c) dependence from Eq. (66) at two different values, 2.5 and 2.7 mJ/m2, of aef (as indi- cated) when Vo= 10s3 m3, A= 1O25 m-’ s-l, and T=300 K. As seen, at a given supersaturating rate, s, is smaller when aef is lower (approximately, s~-c$~). Also, s, is much less sensitive to changes in the rate of pressure de- cline than in oef.

Following the above procedure, let us now use defini- tion (c) to determine the critical supersaturation .s, corre- sponding to the moment tv at which the total volume of the gas phase becomes equal to Vd=aVo. Since t, satisfies practically always the condition given by inequality (56), it is justified to use Eq. (62) instead of the complicated Eqs. (59)-( 61). Thus, we find implicitly the dependence of t, (i.e., of the induction time) on c in the form

d2B) 3(~+~)+1]}1/2~--1~ (68)

In the case of O+Y= 1 (time-independent growth rate) this equation becomes

t,= [ 16r/3kT ln(;laAK3k4T4c3”+8tt2/~~~~)]1’2(a~~2/~), (69)

where n=(2/?r4)(3/16)5=5~10-6. Comparison of the above equations with Eq. (64)

leads to the conclusion that c, aef, and T have essentially the same effect on both t, and tl because of the relative constancy of the logarithmic terms. Defining the critical supersaturation s, by

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4690 D. Kashchiev and A. Firoozabadi: isothermal gas phase formation

s,=ct U) (70) we can represent Eq. (68) by the following more conve- nient formula:

XP+y)+3 exp( -B/s) which in the w +Y= 1 case reads

(71)

c4-‘@= (3aAK3/8a&)$ exp( -B/z). (72)

Equations (7 1) and (72) represent the sought depen-

of the three specific surface energies (i.e., the wetting angle 0) is changed in such a way that the shape factor a dimin- ishes. For instance, if the bubbles are of equilibrium shape, a=47+(6)/3 so that s~-$-~‘~(@+“). This is in contrast with the case of PN in which s, increases with 4 approxi- mately as .~,-o$~-@‘~ [see Eqs. (71) and (72) and Fig. 41. It is important to note as well that using Eq. (75) to analyze experimental s,(c) data (e.g., for gas phase forma- tion by seeded bubble nucleation) allows a separate deter- m ination of the bubble growth parameter K if No is inde- pendently known.

dence of the critical supersaturation on the supersaturating rate in the scope of definition (c) . They give only implicitly s, as a function of c and are applicable to gas phase forma- tion by the polynuclear mechanism in the case of PN. When W+Y= 1, according to Eq. (72), a suitable repre- sentation of experimental s,(c) data are a plot of In ( c4-30/ $“) vs l/s. The resulting linear dependence reads

VIII. CONCLUSIONS

- (16&/3kT) (l/z) (73) and has a slope from which aef can be calculated. The intercept, however, does not allow a separate determina- tion of the nucleation and growth kinetic parameters A and K; as already noted, it yields only the product AK3. From the quality of the linear regression, a best-fit value of w may be determined which could give information about the bubble growth mechanism.

The present study leads to the following conclusions concerning gas phase formation by progressive nucleation or instantaneous nucleation in the cases of constant and steadily increasing supersaturation.

Constant supersaturation. ( 1) The volume of the newly formed gas phase is a

stronger function of time and especially of supersaturation for progressive than for instantaneous nucleation.

(2) The induction time is weakly dependent on super- saturation for instantaneous nucleation, while it is a strong function of supersaturation for progressive nucleation.

Steadily increasing supersaturation. ( 1) In gas phase formation by progressive nucleation

the number of bubbles nucleated increases sharply with time after a certain time delay.

Comparison of Eqs. (66) and (71) shows that the two s,(c) dependencies are similar. This is also seen in Fig. 4 which displays s, from Eq. (72) as a function of c at the two a,fvalues used to calculate s, from Eq. (66). The val- ues of A and Tare also the same, and a = 2~/3 (hemispher- ical bubbles), a=O.OOl, w= l/2 (e.g., diffusion-controlled bubble growth) and K= lo-’ m /s1’2 Pa”2. As seen from Fig. 4, .s, from Eq. (72) is considerably higher than s, from Eq. (66) and, as before, it decreases with decreasing c and oef. Again, s, is much less sensitive to changes in c than in aef.

(2) The volume of the newly formed gas phase in- creases sharply with time for progressive nucleation, while for instantaneous nucleation the volume increase is grad- ual.

(3) The volume of the gas phase also shows higher sensitivity to the supersaturating rate in the case of pro- gressive nucleation than in the case of instantaneous nu- cleation.

(4) For progressive nucleation the critical supersatu- ration is highly sensitive to the effective surface tension and weakly dependent on the supersaturating rate.

Finally, we can also determine the critical supersatu- ration s, for gas phase formation by the polynuclear mech- anism in the case of IN. Applying definition (c) to Eq. ( 55) and using Eq. (70)) we find explicitly the correspond- ing dependencies of tu and s, on the supersaturating rate c,

t”(c) = (a/aNdY3)I/3(0+Y)C-O/(O+Y), (74) Qc) = (cr/aNoK3)1/3(w+Y)CY/(O+Y). (75) Comparison of these equations with Eqs. (68) and

(71) reveals that while in this case t, changes with c es- sentially as in the case of PN, the dependence of s, on c is much stronger than for PN. For example, for difIusion- controlled growth with Y= l/2 and w= l/2, it follows from Eq. (75) that s~-c*‘~, i.e. a 100 times higher rate c should result in a tenfold increase in s,. Similar strong effect of c on s, has been observed in experiments on bubble formation in porous media.ls5 Equation (75) shows also that in the case of heterogeneous IN, s, increases when any

(5) In contrast to progressive nucleation, the critical supersaturation for gas phase formation by instantaneous ’ nucleation shows sensitivity to the supersaturating rate. Also, unlike in the case of progressive nucleation, the crit- ical supersaturation is not strongly affected by the effective surface tension when the gas phase forms by instantaneous nucleation.

ACKNOWLEDGMENTS

This research project was supported by Amoco Pro- duction Co., BP Exploration Co., Elf Aquitaine, Japan Na- tional Oil Corp., Marathon Oil Co., Maersk Oil and Gas A/S, Mobil R&D Corp., Fina Exploration Norway, Phill- ips Petroleum Co., Saudi Aramco, Texaco Inc., and the U.S. Dept. of Energy Contract No. DE-AC22-91 BC14835. Their support is appreciated. We thank Tore Markeset for his help in the preparation of the figures. D. Kashchiev was on leave from the Institute of Physical

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D. Kashchiev and A. Firoozabadi: Isothermal gas phase formation 4699

Chemistry, Bulgarian Academy of Sciences, Sofia, Bul- garia, while he was a visiting scientist at the Reservoir Engineering Research Institute.

‘A. Firoozabadi, B. Ottesen, and M. Mikklesen, SPE Formation Evalu- ation 7, 337 (1992).

*T. F. M. Kortekaas and F. V. Poelgeest, SPE Reservoir Eng. 6, 329 (Aug. 1991).

3Y. C. Yortsos and M. Parlar, Paper SPE 19697, (1989). ‘X. Li and Y. C. Yortsos, Paper SPE 22589, (1991). ‘J. C. Moulu, J. Pet. Sci. Eng. 2, 379 (1989). 6A. El Yousfi, C. Zarcone, S. Bories, and R. Lenormand, C.R. Acad. Sci.

(Paris) 313,7 (1991). ‘M. Volmer, Kinetik der Phasenbildung (Steinkopff, Dresden, 1939). *R. Kaischew and B. Mutaftschiew, Comm. Inst. Phys. Chem. Bulg. Acad. Sci. 2, 75 (1962).

’ 3. P. Hirth and G. M. Pound, Condensation and Evaporation (Perga- mon, Oxford, 1963).

“E. I. Nesis, Kipenie Zhidkostey (Nauka, Moscow, 1973). “M. Blander and J. L. Katz, AIChE J. 21, 833 (1975). I’M. Blander, Adv. Colloid Interface Sci. 10, 1 (1979). “T. Theofanous, L. Biasi, H. S. Isbin, and H. Fauske, Chem. Eng. Sci.

24, 885 (1969). 14J. Szekely and G. P. Martins, Chem. Eng. Sci. 26, 147 (1971). “J. Szekely and S. D. Fang, Chem. Eng. Sci. 28, 2127 (1973).

16M. S. Plesset and S. A. Zwick, J. Appl. Phys. 25, 493 (1954). “D. E. Rosner and M. Epstein, Chem. Eng. Sci. 27, 69 (1972). ‘*K. E. Perepelkin and V. S. Matveev, Gazovye Emulsi i (Khimiya, Len-

ingrad, 1979). “A. N. Kolmogorov, Bull. Acad. Sci. URSS (Sci. Math. Nat.) 3, 355

(1937). “M. Avrami, J. Chem. Phys. 7, 1103 (1939); 8, 212 (1940); 9, 177

(1941). *’ A. E. Nielsen, Kinetics of Precipitation (Pergamon, Oxford, 1964). 22D. Kashchiev, Surf. Sci. 18, 293 (1969). 23 D. Kashchiev, Surf. Sci. 22, 3 19 (1970). 24D. Kashchiev, Cryst. Res. Technol. 19, 1413 (1984). 25D. Kashchiev, Surf. Sci. 14, 209 (1969). 26S. Toschev, A. Milchev, and S. Stoyanov, J. Cryst. Growth 13/14, 123

(1972). “D. Kashchiev, D. Verdoes, and G. M. van Rosmalen, J. Cryst. Growth

110, 373 (1991). **M. C. van der Leeden, D. Verdoes, D. Kashchiev, and G. M. van

Rosmalen, in Advances in Industrial Cvstallization, edited by J. Gar- side, R. J. Davey, and A. G. Jones (Butterworth, London, 1991), p. 31.

“D. Verdoes, D. Kashchiev, and G. M. van Rosmalen, J. Cryst. Growth, 118, 401 (1992).

“M. C. van der Leeden, D. Kashchiev, and G. M. van Rosmalen, J. Colloid Interface Sci. 152, 338 (1992).

3’ Handbook of Mathematical Functions, edited by M. Abramowitz and I. E. Stegun (Dover, New York, 1965).

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