Nucleation and the Tunneling Phenomenon

J. C. Martinez

Math-Science Centre, Ngee-Ann Polytechnic, 535 Clementi Road, Singapore 599489

Received February 11, 1998; accepted May 8, 1998

The rate of nucleation of a stable phase in a homogeneousmetastable medium is computed within the context of particletunneling through a modified cubic potential, bypassing the con-ventional thermodynamic approach. Comparison with the stan-dard Becker–Doring results hints at good prospects for the presentprocedure. Some fundamental issues are raised regarding currentapproaches. © 1998 Academic Press

Key Words: homogeneous nucleation; WKB approximation;tunneling; surface tension.

INTRODUCTION

While studies of the nucleation of a stable phase in ahomogeneous metastable medium date back to Wilson’s worka hundred years ago, progress in experimental work has notbeen smooth because of the notoriously explosive character ofthe nucleation rate and the difficulty in isolating extrinsicfactors that tend to mask the effects of homogeneous nucle-ation. Recently, however, it was reported that intrinsic quan-tum nucleation had been observed in supersaturated3He-4Hemixtures at the milliKelvin range (1). In as much as nucleationis an important ingredient in the understanding of first-orderphase transition phenomena, the time is now ripe to work outan accurate quantitative theory that can be compared withexperiment. Naturally, the quantum mechanism—tunnel-ing—is an important feature that should be included in such atheory. Indeed in the past few years a number of approaches totunneling, even for the case of low barriers, have becomeavailable (2).

There are two routes of access for understanding nucleationphenomena. In the classical approach pioneered by Becker andDoring and by Zeldovich, one studies the kinetic evolution ofdrops subject only to an evaporation-condensation mechanismin which a drop of size, sayl , gains or loses one molecule (3).To solve the drop-growth equations they invoked a detailedbalance argument and restricted themselves to a time-indepen-dent solution. A fundamental assumption in this approach isthat clusters of greater than a specified critical size are removedfrom the system, thereby determining the boundary conditionsto be satisfied by the solutions. Over the decades refinements ofthis initial formulation have been made, and we may mentionamong the more recent ones the inclusion of cluster-size de-

pendence of vapor pressure, the appreciation of scaling nearthe critical point, and numerical studies. There is also a hydro-dynamic model which yields results almost identical to Beckerand Doring’s (4). Reviews of these developments can be foundin Refs. (5). Perhaps the major criticisms of this approach areits firm grounding on a steady-state solution of the dynamicalequations obeying detailed balance (which principle may notbe strictly valid for the small number of particles involved) andthe neglect of the tunneling mechanism, which we feel isparamount in quantum nucleation.

The other approach, initiated by Lifshitz and Kagan for thequantum phenomenological model and by Affleck for a field-theoretical version (6), comes from a different direction. Sincenucleation involves the overcoming of a potential barrier inconfiguration space, the focus of this approach is the tunnelingof molecules across the modified cubic potential formed by thesum of the volume and surface free energies of a nucleatingdrop within a supersaturated vapor phase (Fig. 1),

V 5 4pr 2g 2 e4

3pr 3, [1]

whereg is the surface tension,e 5 kTrL ln s (proportional tothe difference in chemical potentials between the phases),s the saturation,T the temperature, andrL the density of theliquid phase. Although we invoke a phenomenological modelrelying on thermodynamic analysis, the qualitative successachieved in the nuclear liquid drop model offers some justifi-cation for considering Eq. [1] as a useful candidate for thequantum tunneling potential. We may rewrite Eq. [1] moreconveniently as

V~r ! 51

2mv2r 2S1 2

r

RcD , [2]

wherem is the molecular mass,v2 5 8pg/m, Rc 5 3g/e. Byassuming that drops accrue mass independently of each other,one avoids having to deal directly with kinetic processes.(Indeed at this level, it would appear that such processes are ofless importance than the tunneling process. Our final resultsseem to bear this out.) The question of reversibility is thussidestepped. The crucial issue in this context is the construction

JOURNAL OF COLLOID AND INTERFACE SCIENCE205,476–481 (1998)ARTICLE NO. CS985655

4760021-9797/98 $25.00Copyright © 1998 by Academic PressAll rights of reproduction in any form reserved.

of a suitable kinetic energy term corresponding to the abovepotential energy. Lifshitz and Kagan address this by treating agrowing drop as an incompressible spherical fluid and associ-ating its overall kinetic energy with the kinetic energy of aparticle having radius-dependent mass; that is, matter outsidethe drop moves with a velocity dependent on the radius of thedrop. Recently, Nakamuraet al. (7) generalized this idea bygranting greater freedom to the radius dependence of the mass.In as much as the accretion of mass is the result of singleparticles tunneling through the potential barrier, one followingthis line of argument would be “overcounting” the mass be-cause the nucleating core is essentially quiescent during thetunneling buildup and only the surface is involved in thisaccretion process. Our treatment below makes use of the singleparticle amplitude and identifies it with the propagation ofparticles just outside the interface between the vapor and liquidphases. The drop, on the other hand, grows via the accumula-tion (propagation) of particles just below this interface. Once adrop has reached a certain size, it will grow continuously. Thisis called the critical drop. There is no unanimity regarding this.In the classical view this corresponds to a drop of radiusr 52Rc/3. However, here we will taker 5 Rc, which is more inline with the tunneling process. Modern quantum tunnelingstudies incorporate dissipative effects as well (8); we considerhere a simple study of the tunneling rate within the context ofpotential [2]. While many works (5) focus on the kinetic-thermodynamic aspects of nucleation, our main interest here ison the importance that the tunneling mechanism may have onhomogeneous nucleation. We see below that it might wellaccount for much of the growth of the critical drop if accurateknowledge of its density profile can be obtained.

It may be pointed out that a proper treatment of the nucle-ation phenomenon should be a field theoretical one. Neverthe-less it is clear that the centerpiece of such an approach is thelifetime of a metastable particle which can be obtained (tosome degree of accuracy) without appealing to field theory.Moreover this quantity—the lifetime—is employed in field-

theoretic approaches ignoring the fact that a drop is really acollection of a number of particles, each of which has its ownparticular decay lifetime. It would also be helpful to gaininsight into the importance of collective modes, excited states,and surface effects. Thus, a serious study of the decay ofmetastable decay without recourse to field theory is justified, ifonly to identify the main mechanisms involved. We will seebelow that this calls for careful scrutiny of the decay lifetime.Although Affleck (6) has pointed out that the tunneling mech-anism is the dominant process for low-temperature nucleationwhile the classical one is mainly responsible for high-temper-ature nucleation, efforts toward understanding the former lagfar behind those already made with the latter.

Within the context of interface science, this work empha-sizes the paramount importance of a careful understanding ofthe dynamics of growth at the interface between the liquid andvapor phases. We believe that the explosive nature of thenucleation phenomenon is directly linked with this. The issueof surface density profile is also noted and should acquiregreater importance in the task of fine-tuning the correspon-dence between theory with observation.

WKB THEORY

We begin by deriving the decay rate of a particle trapped inthe potential well. Although many deviations have been givenbefore, they all consider only a particle in the ground state (9).As we will point out below this could undercount the nucle-ation rate by a very large factor. Moreover a detailed derivationcould allow us to see where possible improvements may bemade. The Schrodinger equation for a spherically symmericpotential V(r ) is (2(\2/ 2m)¹2 1 V)f 5 Ef, and if wesubstitutef 5 (C(r )/r )Ylm(u, w) we obtain, after separation,the one-dimensional radial problem with an effective potentialV 5 V(r ) 1 [ l (l 1 1)\2/ 2mr2]. In the following, we willconcentrate on the corresponding one-dimensional radial prob-lem for C and setl 5 0. We expect thel Þ 0 case to be lessimportant because of the centrifugal barrier (but we can stilllook at excited states withl 5 0). Let us look first in thevicinity of Rc where the one-dimensional Schrodinger equationhas the approximate form

\2

2m

­2

­r 2 C 21

2mv2Rc~r 2 Rc!C 5 En,vC. [3]

We focus on solutions whose low-energy spectrum resemblesthat of the oscillator. Hence we set

En,v 5 \vSn 11

21 vD , [4]

|v| ! n. This is because, in the nucleation problem,\v !barrier height and we expect low-energy particles tunneling

FIG. 1. The potentialV(r) with V(Rc) 5 0. The turning points arer1 andr2 for a particle of energyE.

477NUCLEATION AND THE TUNNELING PHENOMENON

through the potential hump. The solution of Eq. [3] in terms ofAiry functions Ai and Bi is

C~r ! 5 N~Ai ~ x! 1 Bi~ x!!, [5]

wherex 5 (R9c 2 r )x, x3 5 m2v2Rc/\2, R9c 5 Rc 2 En,v/F,

F 5 (1/ 2)mv2Rc, which for r @ Rc, represents a sum ofoutgoing and reflected waves (10)

C~r !3 Np21/ 2@x~R9c 2 r !#21/4Hexp2

3@x~R9c 2 4!#3/ 2

1i

2exp2

2

3@x~R9c 2 r !#3/ 2J ~r 3 `!. [6]

The constantN will be specified below.Near the origin the one-dimensional radial Schrodinger

equation is that of a harmonic oscillator

2\2

2m

­2

­r 2 C 11

2mv2r 2C 5 En,vC, [7]

whose solutions are the parabolic cylinder functions

C 5 Dn1v~6y!, y 5 r /a0, a0 5 Î~\/ 2mv!. [8]

In general theDn1v(6y) are linearly independent except in thecase thatn 1 v is integer, and in this instance they coincide.Wheny is large (but such that the particle is still close to theorigin relative toRc) the following asymptotic expressions (11)are available

Dn1v~ y!3 S r

a0D n1v

e214

r2

a02

Dn1v~2y!3 2Î2p

G~2n 2 v! S2a0

r Dn111v

e14

r2

a02 , [9]

the sign being chosen for later convenience. Whilen is a(radial) quantum number,v is not. This latter is a manifestationof the observation that our problem does not really involve astationary state; we are dealing with a metastable state andv isassociated with this fact (12).

Let us next calculatev, which is proportional to the decayrate, by appealing to a WKB analysis. That is, we connect thesolutions [5] and [8] via an approximate one with the aid of thepotential [2]. We study the scattered wave (r @ Rc) by setting

C~r ! 5 exp21

\$W~r ! 1 \W~1!~r !% [10]

and substituting into the Schrodinger equation with potential[2]. One finds thatW(r ) andW(1)(r ) satisfy

1

2m S­W

­r D2

2 V~r ! 5 0,

1

m

­W~1!

­r

­W

­r2

1

2m

­2W

­r 2 5En,v

\. [11]

The first equation has the solution

W~r ! 5 Er9

r

dr9Î2mV~r 9!, [12]

wherer9 is, in principle, arbitrary. We shall denote the integralon the right-hand side byB. The second equation has thesolution

W~1!~r ! 51

2ln

r

4RcS1 2

r

RcD 1/ 2

2En,v

\lnSF1 2 Î1 2

r

RcGY

F1 1 Î1 2r

RcGD 2 ~n 1 v!ln

4Rc

a0, [13]

where a suitable integration constant has been included. In thelimit r /Rc ! 1, we find that exp(2W/\ 2 W(1)) agrees withthe first line of Eq. [9]. Whenr /Rc ' 1, the second log of Eq.[13] may be ignored, so, effectively,

W~1! 5 lnS1 2r

RcD 1/4 1

161/4 S4Rc

a0D2n2v

, [14]

and, hence, whenr /Rc ' 1,

expS2W

\2 W~1!D < 2~xRc!

1/4S4Rc

a0D n1v

3 e212 B@x~Rc 2 r !#21/4expF2

3x~Rc 2 r !G . [15]

Comparing this with Eq. [6], we obtain an expression for thenormalization factor,

N 5 2p1/ 2~xRc!1/4~4Rc/a0!

n1vexpS21

2BD , [16]

if we ignore the minor difference betweenR9c andRc.We repeat a similar calculation for the reflected wave by

writing (subscript 1 refers to reflection)

478 J. C. MARTINEZ

C1~r ! 5 exp21

\~W1~r ! 1 \W1

~1!~r !!. [17]

TheseW1’s satisfy the same equations [11], but we choose theappropriate boundary condition such thatW1(r ) 5 2W(r ). Acalculation similar to Eq. [13] yields

W1~1!~r ! 5

1

2ln

r

4RcS1 2

r

RcD 1/ 2

1 S1

21 n 1 vD

3 ln SF1 2 Î1 2r

RcGYF1 1 Î1 2

r

RcGD

2 ~1 1 n 1 v! lna0

4Rc2 ln

Î2p

G~2n 2 v!~2!n1v11. [18]

When r /Rc ' 1, we find

exp2 W1~1! < SRc 2 r

16RcD21/4S a0

4RcD 11n1v 2p

G~2n 2 v!~2!n111v ,

@19#

and we now compare this with the second term of Eq. [6]. Withthe aid of Eq. [16] and the relation

v 5 2iGn

2v, [20]

whereGn is the decay rate associated withEn,v, one obtains

Gn <v

Î2pS4Rc

a0D 112n e2B

n!. [21]

The integralB is evaluated between the limitsr 5 0 andr 5Rc. When n 5 0, the standard result emerges (9, 13). If weassume a Boltzmann distribution for the energy levels, weobtain the equilibrium decay rate (b21 5 kT)

G 5On50

` Gnexp~2bEn!

On50` exp~2bEn!

5 ~1 2 e2b\v!G0expFS4Rc

a0D 2

e2b\vG , [22]

which is manifestly wrong because the ratio (Rc/a0) turns outto be very large. This raises the issue of the definition of thelifetime (12) and the validity of the concept of an averagedecay rate. Indeed, we already noted that nucleation has to bea non-equilibrium process. Because the ratio (Rc/a0)2n is typ-ically of the order of 104n, the result [21] predicts extremelylarge contributions even for moderately lown states. It appears

then that a calculation looking only at ground state contribu-tions grossly underestimates the lifetime. On the other hand,intuitively, we do not expect the excited states to have such adominant role, as the above results imply. We resolve this issuebelow.

APPLICATION TO NUCLEATION

The nucleation process is the inverse of the decay process. IfdG (a 3 b) is the differential rate for the reactiona 3 b,unitarity stipulates that (14)

Edbca

dG~a3 b!

db5 Edbcb

dG~b3 a!

da,

wherecbdb gives the number of particle states in the volumerangedb of particle statesb. This allows us to equate thenucleation rate with the decay rate. For a molecule of energyE, the integral in the exponential factor in [21] may bewritten as

B 5 Rc E0

1Î 2m

\2 H27

4L0~ y2 2 y3!Jd y, [23]

wherey 5 r /Rc andL0 5 (16/ 27)pRc2g is the height of the

potential barrier (Fig. 1). We identify Eq. [21] with the mole-cules propagating just outside the interface between the vaporand liquid phases. The growth of the drop, however, will beassociated with the liquid phase. By continuity and assumingquasi-stationary conditions, we have that the interfacial veloc-ities of the particles in the vapor phasevv and in the liquidphasevL are related by

rLvL 5 rvvv, [24]

where ther’s refer to the densities. Hence in order to applythe above to the liquid, we must modify Eq. [23] by ap-pending a factorrv/rL into the exponential. (This could bea signal for important collective modes.) If now the drop isof radius r, there will have to ben#4pr2dr new particlesacquired to increase the drop’s radius bydr. Here n# is thenumber density in the liquid phase (not to be confused withthe radial quantum numbern). In the following, we absorbthe factor (4Rc/a0)

2n in [21] into the exponent, since it arisesfrom the excited states. Then the probability for formationof the critical drop is

479NUCLEATION AND THE TUNNELING PHENOMENON

expF2rv

rLE

0

1

n#Rc34py2dy

3 HE0

y Î 2m

\2 S27

4L0~y92 2 y93!DRcdy9J 1 2n lnS4Rc

a0Î rv

rLDG5 expH23

rv

rLÎ 27m

2\2L0NRcI 1 2n lnS4Rc

a0Î rv

rLDJ , [25]

whereN 5 4pRc3n# 0/3 is the number of particles in a drop of

radiusRc and uniform densityn# 0, and

I 5 E0

1

y2$E0

y

Îx2 2 x3dx%d y < 0.0643. [26]

We have appended a density ratio in the second term of theexponent for a reason we will explain in the next paragraph.Examination of the integral of Eq. [25] reveals a ln25 s depen-dence on the saturation. The exponent of Eq. [25] will bereferred to as “exponent.”

The factor (v/=2p)(4Rc/a0) to the left of the exponentialin Eq. [21] is to be interpreted as a frequency factor. Followingthe argument above leading up to Eq. [24], we must replacevby v (rv/rL) anda0 by a0/=(rv/rL). Thus we have the replace-mentG3 G (rv/rL)3/2. (This applies to Eq. [25] as well.) So wemodify the frequency factor (statistical prefactor) by appendingthis density ratio and write the new frequency factorv# as

v# 5 Srv

rLD 3/ 2 v

Î4pS4Rc

a0D . [27]

The density ratio discussed above has the effect of softeningthe dominant behavior of the (Rc/a0)2n term in the decay rate.The number of droplets formed per second per unit volume isthen the product ofv# , the probability for formation of onedroplet [25] and the number of centers of formation of droplets(which is roughlyp/kT). This is our result for a “rectangulardensity profile” in which the density is uniform up toRc andzero beyond.

We can improve on our results by employing a “trapezoidal”form for the number density (Fig. 2), where we assume that thedensity drops linearly fromn0 to 0 over a surface of thicknessRc/10. The frequency factorv# remains unchanged. However,Imust now be replaced by 0.0512. The number of particles in adroplet is about 84% that ofN above.

In Table 1 we tabulate typical values ofv# and the “expo-nent” for ethanol atT 5 293 K for the two density profiles usedand compare with the Doring–Becker result (15) given in Table2. In our case,m ' 7.7 3 10226 kg, v ' 2.733 1012 rad/s,Rc ' 1.633 1029 (ln s)21 m, rv/rL ' 1.443 1024, the barrierheight is L0 ' 1.13 3 10219/ln2 s (@ \v), and s is the

FIG. 2. Trapezoidal number density profile for a drop.

TABLE 1Parameters in the Tunneling Probability for Nucleation per Volume per Second

s Rc (1029 m) v# (109 s21)

Rectangular density profile Trapezoidal density profile

n “Exponent” n “Exponent”

1.9 2.55 1.2 45 68.8 33 66.92.0 2.36 1.2 29 58.7 20 59.22.1 2.21 1.04 19 50.7 13 48.52.4 1.88 0.88 5 36.2 2 36.2

TABLE 2Parameters in the Tunneling Probability for Nucleationper Volume per Second for the Doring–Becker Theory

s Rc (1029 m) v# (109 s21) “Exponent”

1.9 1.7 0.27 65.22.0 1.54 0.3 55.92.1 1.43 0.32 482.4 1.21 0.36 35.1

480 J. C. MARTINEZ

saturation. Notice thatL0 @ kT ' 14\v. In comparing Tables1 and 2, one should bear in mind the fact that the drop radiusis defined slightly differently in the Becker–Doring theory.Since there are more particles away from the center of the dropand the potential (2) peaks at 2Rc/3, the issue of radius is acrucial one. The “exponent” can be improved upon with betterparametrizations of the density profile. The trend in Table 1points toward promising possibilities.

OUTLOOK

We observe that our calculation, based solely on a phe-nomenological model of the tunneling mechanism, is able togive a reasonable picture of the nucleation rate. There is noreference to the principle of detailed balance at all. The“exponent” depends on the radial quantum numbern, whichis large for low saturation and small for high saturation. Itappears that only when saturation is large enough is theground state sufficient to account for most of the nucleationtaking place.

Future work can go in at least four directions. Since the“exponent” is evidently sensitive to the density profile, moreaccurate studies of the drop surface would be valuable.Along another direction, the frequency factor depends onvwhich in turn depends on the surface tension. As Fig. 1shows, however, the quadratic part of the potential is dom-inant for smallr. In other words surface tension effects aremore important deep in the interior of the drop. This castssome doubt regarding the use of a value ofg obtained fromobservations of surface phenomena. Finally, a field-theo-retic model may also be contemplated in which the lifetimeof a metastable state is correctly defined and incorporatedinto a multi-particle system.

ACKNOWLEDGMENT

The author is grateful for the assistance of the Lee Foundation.

REFERENCES

1. Satoh, T., Morishita, M., Katoh, S., Hatakayama, K., and Takashima, M.,PhysicaB197,397 (1994).

2. Karrlein, R., and Kleinert, H.,Phys. Lett.A187, 133 (1994).3. Frenkel, J., “Kinetic Theory of Liquids,” p. 367 ff. Oxford Univ. Press,

Oxford, 1947.4. Langer, J. S., and Turski, L. A.,Phys. Rev.A8, 3230 (1973).5. Wu, D., in “Solid State Physics” (H. Ehrenreich and F. Spaepen, Eds.),

Vol. 50, p. 38 ff. Academic Press, San Diego, 1995; Gunton, J. D., SanMiguel, M., and Sanhi, P.,in “Phase Transitions and Critical Phenomena”(C. Domb and J. L. Lebowitz, Eds.) p. 267 ff. Academic Press, San Diego,1983; Hanggi, P., Talkner, P., and Borkovec, M.,Rev. Mod. Phys.62,251(1990); Mel’nikov, V. I., Phys. Reports209,1 (1991).

6. Lifshitz, I. M., and Kagan, Yu.,Sov. Phys.-JETP35,206 (1972); Affleck,I., Phys. Rev. Lett.46, 388 (1981).

7. Nakamura, T., Kanno, Y., and Takagi S.,Phys. Rev.B51, 8446 (1995).8. Weiss, U., Grabert, H., Ha¨nggi, P., and Riseborough, P. S.,Phys. Rev.

B35, 9535 (1987).9. Coleman, S.,Phys. Rev.D15, 2992 (1977); Schmidt, A.,Ann. Phys. (NY)

170, 333 (1986); Weiss, U., and Haeffner, W.,Phys. Rev.D27, 2916(1983).

10. Abramowitz, M., and Stegun, I., “Handbook of Mathematical Functions,”formulas 10.4.59, 10.4.63. US Gov. Printing Office, Washington, DC,1968.

11. Morse, P. M., Feshbach, H., “Methods of Theoretical Physics,” p. 1641.McGraw-Hill, New York, 1953.

12. Boyanovsky, D., Willey, R., and Holman, R.,Nucl. Phys.B376, 599(1992).

13. Caldeira, A. O., and Leggett, A. J.,Ann. Phys. (NY)149,374 (1983);153,445 (1984) (E).

14. Weinberg, S., “The Quantum Theory of Fields,” p. 150. Cambridge Univ.Press, Cambridge, 1995.

15. Ulbricht, H., Schmelzer, J., Mahuke, R., and Schweitzer, F., “Thermody-namics of Finite Systems and the Kinetics of First-Order Phase Transi-tions,” p. 17. Tuebner, Leipzig, 1988.

481NUCLEATION AND THE TUNNELING PHENOMENON