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Dynamical Growth Rate of a Diffuse Interface

arXiv:hep-ph/9307348v1 27 Jul 1993TPI–MINN–93–34/TNUC–MINN–93–17/TJune 1993Dynamical Growth Rate of a Diffuse Interfacein First Order Phase TransitionsRaju VenugopalanTheoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455Axel P. VischerPhysics Department, Oregan State University, Corvallis, OR 97331andSchool of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455AbstractWe compute the dynamical prefactor in the nucleation rate of bubbles or dropletsin first order phase transitions for the case where both viscous damping and ther-mal dissipation are significant.This result, which generalizes previous work onnucleation, may be applied to study the growth of bubbles or droplets in condensedmatter systems as well as in heavy ion collisions and in the expansion of the earlyuniverse.PACS indices: 64.60 Qb, 44.30.+v, 82.60.Nh, 68.10.-m1

Near the critical temperature for a first order phase transition, matter in acertain phase is metastable and small fluctuations in the state variables may acti-vate the nucleation of bubbles (a term we use below to represent either bubbles ordroplets) of a more stable phase of matter. If these bubbles have a radius larger thansome critical value, they begin to grow exponentially The nucleation and growthof these critical bubbles has historically been of great interest in the physics ofliquid–gas phase transitions and in condensed matter physics [1].

More recently,they have been studied in the context of first order phase transitions in the earlyuniverse [2, 3, 4] and in high energy heavy ion collisions [5].A general kinetic theory of homogeneous nucleation was developed by Langer [6].In the neighbourhood of a first order phase transition, when the critical radii of thebubbles exceed the correlation length, a reduced description of nucleation in termsof a coarse–grained free energy is appropriate. Langer and Turski [7] used such aphenomenological approach to show that the nucleation rate of bubbles could bewritten as a product of three termsI = κ2πΩ0 exp(−∆F/T) .

(1)The dynamical prefactor κ determines the exponential rate of growth of bubbles ofa critical radius, beyond which size the bubbles are stable. For the bubbles to growbeyond the critical radius, latent heat must be carried away from the surface ofthe bubble.

This is achieved through thermal dissipation and/or viscous damping.Kawasaki [8] and Turski and Langer [9], neglecting viscous damping, show that κ forthe condensation of a supersaturated vapour is linearly proportional to the thermalconductivity of the vapour. The statistical prefactor Ω0 is a measure of the phasespace volume of the saddle point region of the free energy functional and ∆F is thechange in the free energy required to activate the formation of a critical bubble.Recently, the theory of Langer and Turski has been used by Csernai and Ka-pusta to study nucleation in relativistic first–order phase transitions [3].

The baryon2

density in the systems studied is negligible. In the absence of a net conserved charge,the thermal conductivity vanishes and the expression of Langer and Turski for thedynamical prefactor is no longer applicable.

It was shown in Ref. [3] that, for thesystems studied, the latent heat could be transported from the growing bubble byviscous damping instead of thermal dissipation; the new expression for the dynam-ical prefactor depends linearly on the shear viscosity of the surrounding medium.The resulting expression for the pre–exponential factor differs significantly from ear-lier estimates where, on dimensional grounds, the prefactor was taken to be T 4 orT 4c .

Here T is the temperature and Tc is the critical temperature of the first orderphase transition.In this work, we derive a general expression for the dynamical prefactor in thenucleation rate of critical sized bubbles in first order phase transitions. This formulamay be used to study nucleation in liquids and gases and in condensed mattersystems where both the viscosity and thermal conductivity are significant.

It mayalso be used to estimate the probability of formation of superheated quark–gluondroplets or supercooled hadronic bubbles in the baryon–rich matter produced in highenergy heavy ion collisions [10]. Further applications also include the expansion ofthe early universe and the formation of neutron stars.Consider two phases of matter, phase A and phase B, where phase A is ametastable state which decays into the more stable phase B.

Phase A, for instance,may be a supersaturated vapour which when supercooled, nucleates bubbles of aliquid phase B. When the radius R∗of the critical bubble of phase B is greater thanthe correlation length ξ in phase A, the behaviour of the system can be described interms of a coarse–grained energy functional F. This functional depends on the fluiddensity n(⃗r), the local fluid velocity U(⃗r) and the temperature T(⃗r).FollowingRef.

[7], we make the ansatz that F = FK + FI where FK is the kinetic energy.The interaction term FI is the sum of the Helmholtz free energy and the van derWaals–Cahn–Hilliard gradient energy [11]. The formalism for the coarse–grained3

free energy functional is also valid for relativistic systems where the pressure iscomparable to the energy density and the fluid velocity U(⃗r) in the local rest frameis small compared to the speed of light.In addition to the homogeneous phases A and B, the free energy is also station-ary for a configuration whose solution is a generalization of the van der Waals soliton,which has the hyperbolic tangent–like density profile. To determine the expansionof the bubble about this stationary configuration, we linearize the hydrodynamicequations around the stationary configuration: n(⃗r) = ¯n(⃗r) + ν(⃗r), ⃗U(⃗r) = ⃗0+ ⃗U(⃗r)and T(⃗r) = T0 + θ(⃗r), where the quantities ν, ⃗U and θ correspond to small devi-ations in the density, velocity and temperature, respectively, from their stationaryvalues.

They approach constant values away from the interface. We derive below ageneral expression for κ which does not depend on any specific parametrization ofthe free energy.In Ref.

[9] relations were derived between the velocity potential and the densityand temperature functions on either side of a diffuse interface. These relations, theKotchine conditions [12], are generalizations of the well–known Rankine–Hugoniotdiscontinuity conditions for shocks.

For instance, in the former case, the velocityof matter diffusing through the interface is a function of position and falls offawayfrom the interface. In the later case, the velocity of the matter is a constant.

Itwas shown in Ref. [9] that these Kotchine conditions give the correct dispersionspectrum for capillary waves.We now use the Kotchine conditions for a spherically growing bubble to derivean expression for the dynamical prefactor κ.Our derivation is similar to thatof Turski and Langer but differs from theirs in some key aspects.

The Kotchineconditions for a spherical bubble are[nUR] = [n]dRdt ,(2)[P] = −2σR ,(3)4

[µ] = 0 ,(4)lnanb[UR][n]= −λ(∇T)R −43η + ζURdUdrR. (5)In the above, the brackets denote the difference in the bracketed quantity across theinterface.

For instance, [n] ≡nb −na = ∆n, where the subscripts denote phase Band phase A, respectively. Also, UR is the velocity of matter through the interface,dR/dt is the velocity of the bubble wall, P is the pressure and µ the chemicalpotential.

The latent heat per particle is given by l, λ is the thermal conductivityand η and ζ are the shear and bulk viscosities respectively.The first Kotchine condition, Eq. (2), is the matter continuity relation acrossthe interface.

The second Kotchine condition is the well–known Laplace formula forthe surface tension. The third denotes the continuity of chemical potentials betweenthe two phases at Tc.

The final Kotchine condition equates the latent heat producedper unit area per unit time at the interface to the energy dissipated per unit areaper unit time.Combining Eq. (2) and Eq.

(5), the total energy flux transported outwards isgiven by∆wdRdt = −λdTdr −43η + ζURdURdr . (6)Here ∆w is the difference in the enthalpy densities of the two phases.

From thecontinuity relation, ∂tν = −⃗∇· (¯nUR), one may show (see the discussion precedingEq. (77) in Ref.

[3]) on very general grounds that the radial dependence of the veloc-ity at the interface UR ∝1/r2. Hence, dU(r)/dr|r=R = −2UR(r)/R.

Substitutingthis relation in the above equation, we obtain∆wdRdt = −λdTdr + 243η + ζ U 2RR . (7)We wish to obtain a similarly simple expression for the gradient in the temper-ature dT/dr.

For the systems we consider, we may assume that the temperature5

varies slowly across the bubble wall. If we represent the temperatures in the twophases by Ta and Tb, we can define an average temperature T and the variation θ0by Ta = T + θ0 and Tb = T −θ0.

In the quasi–stationary approximation ∇2θ ≈0.The solution to the Laplacian is then [13]θb=θ0∀r ≤R ,θa=θ0Rr∀r > R ,(8)where θ0 is a constant. Hence,dTdr |r=R = −θ0R .

(9)To determine θ0, we use the continuity of chemical potentials µa = µb across theinterface of the critical bubble (the third Kotchine condition).Then, using thefirst law of thermodynamics and assuming a large latent heat (strong first ordertransition), we arrive at the relationPbnb−Pana≈−lθ0T . (10)Now from the second Kotchine condition, Eq.

(3),Pb = Pa + 2σR . (11)Substituting this equation in Eq.

(10), we obtain, after a little algebraPa = na∆n2σR + nblθ0T. (12)The temperature difference between the two phases, 2θ0, is due to the dissipationof latent heat.

For the critical bubble, θ0 = 0, which implies thatPa = 2naσ∆nR∗. (13)Replacing Pa in Eq.

(12) with the above expression, we obtain finally for θ0 therelationθ0 = 2σT∆wR 1R∗−1R. (14)6

Substituting this result for θ0 in Eq. (9), we havedTdr |R = −2σT∆wR 1R∗−1R.

(15)We have one further unknown–the velocity UR(r) of matter diffusing throughthe surface of the growing bubble. If there exists a net momentum flux through theinterface, then from Laplace’s formula,∆w U 2R = 2σ 1R∗−1R.

(16)We have omitted the shear term in the above equation since it represents a higherorder contribution to the linearized hydrodynamic equations.Combining our results in Eq. (15) and Eq.

(16) with Eq. (7), we obtain thefollowing expressiondRdt =2σ(∆w)2R 1R∗−1R λT + 243η + ζ.

(17)If R−R∗∝exp(κt), we obtain finally our general result for the dynamical prefactorκ =2σ(∆w)2R3∗λT + 243η + ζ. (18)In the limit of zero baryon number, λ →0, and we obtain the result of Csernai andKapusta.

If the matter is baryon–rich but viscous damping is negligible, η, ζ →0,we obtain the result of Kawasaki, and Turski and Langer. Since the results forthe dynamical prefactor were known in the two limits, and since only terms linearin the transport coefficients are retained, the above result might easily have beenintuited.

However, many of the ingredients in our calculation are generally valid,and hence the dynamical prefactor could also be computed in a similar manner formore complicated systems.We should point out that there are several assumptions that have been madein our derivation of the dynamical prefactor.Our result is strictly valid whennon–linear effects can be ignored and the linearized hydrodynamic equations are7

applicable. Further, for the coarse–graining description to hold, the radii of thebubbles must be larger than the correlation length.

We have also assumed thatheating due to dissipation is slow, causing the temperature to vary slowly acrossthe bubble wall. Finally, we have assumed in our derivation that the phase transitionis strongly first order–releasing considerable latent heat.To summarize, we have derived above an expression for the dynamical prefactorwhich governs the growth of critically sized bubbles nucleated in first order phasetransitions.

Our results are applicable to the wide range of phenomena where bothviscous damping and thermal dissipation effects are important. In a following pa-per [10] we will discuss one such application–the nucleation of quark–gluon dropletsin baryon–rich hadronic matter created in high energy heavy ion collisions.AcknowledgementsWe would like to thank J. I. Kapusta for inspiration and many useful comments.We also thank L. P.Csernai for a careful reading of the manuscript.

One of us(R.V.) would like to thank J. S. Langer and J. P. Donley for a prompt response toa query.

This work was supported by the U. S. Department of Energy under grantDOE/DE–FG02–87ER40328 and by the Gesellschaft F¨ur Schwerionen ForschungmbH, under their program in support of university research.8

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