Nucleation at quantized vortices and the heterogeneous phase separation in supersaturated superfluid 3He-4He liquid mixtures
aa r X i v : . [ c ond - m a t . o t h e r] M a r Nucleation at quantized vortices and the heterogeneous phase separation insupersaturated superfluid He- He liquid mixtures
S.N. Burmistrov and L.B. Dubovskii
NRC ”Kurchatov Institute”, 123182 Moscow, RussiaandMoscow Institute of Physics and Technology,141700 Dolgoprudnyi, RussiaE-mail: burmistrov [email protected]
Supersaturated superfluid He- He liquid mixture, separating into the He-concentrated c -phaseand He-diluted d -phase, represents a unique possibility for studying macroscopic quantumnucleation and quantum phase-separation kinetics in binary mixtures at low temperatures down toabsolute zero. One of possible heterogeneous mechanisms for the phase separation of supersaturated d -phase is associated with superfluidity of this phase and with a possible existence of quantizedvortices playing a role of nucleation sites for the c -phase of liquid mixture. We analyze the growthdynamics of vortex core filled with the c -phase and determine the temperature behavior of c -phasenucleation rate and the crossover temperature between the classical and quantum nucleationmechanisms.PACS: 67.60.Q- Solutions of He in liquid He;64.60.Q- Nucleation;64.70.Ja Liquid-liquid transitions;
Key words : macroscopic quantum nucleation, supersaturated superfluid He- He liquid mixture,quantized vortex, heterogenous phase separation
A. Introduction
This year E.Ya. Rudavskii celebrates 80. Our meet-ing with Eduard Yakovlevich has taken place about thesame time when the Department of Physics of QuantumFluids and Crystals has started the systematic exper-imental study on the phase separation kinetics of su-persaturated He- He mixtures. This study has laidthe foundations for new field of physics, namely, macro-scopic quantum nucleation or kinetics of first-type phasetransitions in condensed matter at temperatures so closeto absolute zero that the classical thermal-activationphase-transition mechanism becomes completely ineffec-tive. Under the influence of pioneer experiments and per-sonal charm of Eduard Yakovlevich we, keen at that timewith the theory of macroscopic quantum tunneling andthe role of dissipative processes, have turned to the studyof the low-temperature phase-separation kinetics of liquid He- He mixtures and the energy dissipation effects asso-ciated mainly with the diffusion of impurity He atoms.In 1969 during the study of degenerated He- He liquidmixtures there is demonstrated a possibility of preparingthe metastable state of supersaturated superfluid He- He liquid mixture in the lack of free liquid-vapor sur-face [1]. For
T <
70 mK, there are obtained the long-livedsupersaturated liquid mixtures staying in the metastablestate for two and more hours. The experimental stud-ies of phase separation kinetics in liquid He- He mix-tures have been started by Brubaker and Moldover [2]and then continued in 80-ies [3–5]. The experiments wereperformed in the high temperature
T > He- He liquidmixtures in the region of lower temperatures
T <
200 mKwith the aim of detecting the transition from the thermal-activation phase-separation regime to the quantum un-derbarrier one. There have been used new methods ofexperimental study. One method, developed in Kharkovin B.Verkin Institute for Low Temperature Physics andEngineering, is based on a continuous change in the Heconcentration directly in the course of experiment at con-stant pressure and temperature due to varying the os-motic pressure and fountain pressure. The variation ratein the He concentration is usually about 10 − % per sec-ond. The state of the liquid mixture is recorded by twomethods [6–10]. The acoustic method is based on thesound velocity variation in the course of phase separa-tion of liquid mixture and the capacitive one is based onthe measurement of dielectric permittivity. The jump-like reduction in the He concentration, recorded simul-taneously with the sound and capacitive methods, corre-sponds to separating a metastable supersaturated liquidmixture into two phases.As is known, due to existence of the effects of osmoticpressure and fountaining in a superfluid phase of liquid He- He mixture the temperature inhomogeneity leadsreadily to the pressure and concentration inhomogeneity.To avoid the effect of this factor, the process of preparinga supersaturated liquid mixture should occur under in-variable temperature and sufficiently slowly in order notto set the He atoms in motion. Thus, employing thedependence of the separation line on the pressure, theauthors of work [11] have proposed another scheme ofcontinuous pressure variation using a thin capillary (su-perleak) through which the superfluid He componentalone can flow. This results in changing the pressure and He concentration in the experimental cell. In the courseof experiment the total He amount in the cell remainsunvaried. The rate of pressure variation is about 0.25atm/hr, corresponding to the He flow as about a micro-gram per second and as a concentration rate 6 · − % persec. This is by a factor of 10 –10 as compared with thatin the phase separation experiments on liquid mixturesin the vicinity of the tricritical point. The final results ofthe research and their discussion are given in work [12].So far, the theoretical interest [13, 14] has mainly beenfocused on the homogeneous mechanism of phase sepa-ration in the supersaturated d -phase of liquid He- Hemixture. In spite of careful analysis a scepticism hasremained. The possible mechanisms for inhomogeneousphase separation of liquid mixture have not received aproper attention. In particular, this concerns a possi-bility of the presence of remnant quantized vortices andan estimate of the crossover temperature for the thermaland quantum regimes of phase separation.Here we consider one of possible heterogeneous mech-anisms for the phase separation in the supersaturated d -phase of liquid He- He mixture, which is associatedwith superfluidity of d -phase and with possibility of ex-istence of quantized vortices. The vortices, in its turn,can play a role of nucleation sites for the c -phase of liq-uid mixture. The idea goes directly back to works [15–17]where the phenomenon was analyzed of He atom adsorp-tion onto the quantized vortex core in superfluid He-II.Unlike the previous works [18] dealing with the thermal-activation nucleation mechanism alone, we here concen-trate our attention on the growth dynamics of vortex corefilled with the c -phase and determine the nucleation ratein the quantum region as well as the thermal-quantumcrossover temperature. B. The quantized vortex structure in the saturated He- He liquid mixture
Let us consider the structure, core radius and energyof rectilinear vortex with one circulation quantum in thesaturated superfluid d -phase of liquid He- He mixture.It is well known that He atoms tend to be localized andtrapped with the vortex core. Below we analyze a simplemodel with the rigid core for quantized vortex in thesaturated superfluid He- He liquid mixture, assumingthe vortex core to be filled with the c -phase. Beyondthe vortex core of radius R the superfluid velocity V s at distance r from vortex line is subjected to equation: V s ( r ) = ~ /m r as a result of circulation quantization H V s dl = 2 π ~ /m .The condition for equilibrium of d -phase in the bulkrequires the constancy of thermodynamic potentials andtemperatureΦ( P, T, Z, V s ) + V s ≡ Φ( P, T, Z ) | r = ∞ ,Z ( P, T, c, V s ) = const ≡ Z ( P, T, c ) | r = ∞ ,T = const ≡ T | r = ∞ . Here Φ = µ /m , potential Z = µ /m − µ /m is conju-gated to the mass He concentration, and µ , and m , are the chemical potentials and masses of He and Heatoms, respectively.Using the known variations for the thermodynamic po-tentials δ Φ = δPρ − σ δT − c δZ − ρ n ρ δ (cid:18) V s (cid:19) ,δZ = ∂∂c (cid:18) ρ (cid:19) δP − ∂σ∂c δT + ∂Z∂c δc − ρ n ρ δ (cid:18) V s (cid:19) , we find that the variations of concentration c and pres-sure P are determined with the following equations ∂Z∂c ∇ c = − ρ ∂ρ s ∂c ∇ V s , ∇ P = − ρ s ∇ V s . (1)Let us put P , c and T for the magnitudes of pressure,concentration and temperature far from the vortex at theinfinity r = ∞ . Next, while the pressure and concentra-tion variations are small, i.e. δP ≪ P and δc ≪ c , wefind approximately P ( r ) − P ≈ − ρ s V s , (2) c ( r ) − c ≈ − ∂ρ s /∂cρ ∂Z/∂c V s . As is seen from the last equation, the concentrationvariation in the superfluid liquid mixture is completelydue to dependence of superfluid density ρ s upon the He concentration. Since the condition for the ther-modynamic stability of liquid mixture implies always ∂Z/∂c > ∂ρ s /∂c <
0, the He con-centration enhances always in the regions with the largermagnitudes of superfluid velocity. From the physicalpoint of view it is also evident from a possibility to reducethe kinetic energy of the superfluid component ρ s V s / ∂Z∂c ≈ Tm c . This gives the known [15] Boltzmann concentration dis-tribution c ( r ) = c exp( − U eff /T ) with the effective attrac-tive potential U eff ( r ) = ∂ρ s /∂cρ · m V s ( r )2 . This distribution can only be justified for the distancesprovided that the He concentration does not exceed themagnitudes corresponding to the condition of nondegen-eracy T F ∼ ~ m (cid:18) ρcm (cid:19) / ≪ T. For the smaller distances or in the degenerate liquidmixture, the growth of concentration occurs slower andpower-like. For the more realistic estimate of the concen-tration behavior at the close distances to the vortex core,it is necessary to know the detailed behavior of quantities Z = Z ( P, c, V s ) and ρ s = ρ s ( P, c, V s ).Note that the He concentration in the d -phase, in anycase, cannot exceed magnitude c λ corresponding to the λ -line at which the superfluid component density ρ s van-ishes or the magnitude c sp corresponding to the spinodalline at which the derivative ∂Z/∂c = 0 and the d -phasebecomes absolutely unstable. These conditions restrictthe minimum size of vortex core. Emphasize that theconcentration is c λ > c sp at the temperatures below thetricritical point temperature T t and, on the contrary, at T > T t the λ -line lies ahead of the spinodal and thetransition to the c -phase occurs continuously in concen-tration. Accordingly, we can expect the distinctions inthe vortex structure at temperatures above and belowthe tricritical point temperature, namely, the concentra-tion discontinuity at a vortex radius if T < T t and thecontinuous variation with a kink at the vortex radius if T > T t .For the temperature region T > T t , when the phaseseparation of liquid He- He mixture does not take placebut only the d -to- c phase transition occurs at some tem-perature and pressure-dependent concentration c λ , thevortex core radius can grow infinitely as the liquid mix-ture concentration c approaches the c λ one. In fact, forradius R λ where the He impurity concentration reachesthe magnitude c λ , we have c ( R λ ) = c λ ( P ( R λ )). Accord-ing to Eq. (2) c λ (cid:18) P − ρ s V s , V s (cid:19) = c − ∂ρ s /∂cρ ∂Z/∂c V s λ -line dependenceon the pressure and superfluid velocity, we arrive at c λ ( P ) − ρ s dc λ dP V s − ∂ρ s /∂V s ∂ρ s /∂c V s = c − ∂ρ s /∂cρ ∂Z/∂c V s . Hence we find R λ = m (cid:16) − ∂ρ s /∂cρ ∂Z/∂c + ρ s dc λ dP + 2 ∂ρ s /∂V s ∂ρ s /∂c (cid:17) / √ ~ ( c λ − c ) / −→ ∞ as c → c λ .In order to realize the vortex of large core radius belowthe tricritical point temperature T t , one should have theimpurity concentration close to that at the spinodal. Inother words, the liquid He- He should already be super-saturated, i.e. it should be in the metastable region. Aswill be seen below, the presence of a vortex, playing a roleof nucleation site, prevents from achieving the highly su-persaturated state of the d -phase. Thus, the vortex coreradius remains finite and cannot exceed some maximumvalue, starting from which the vortex becomes absolutelyunstable against the core expansion. This entails thephase separation of a liquid mixture.In the d -phase the above distinctions and specific fea-tures in the vortex core behavior as a function of tem-perature and concentration could be observed with theaid of second sound. The second sound absorption isvery sensitive to the presence of the c -phase, allowing usto detect the variation in the total volume of the nor-mal c -phase in vortices under varying the liquid mixtureconcentration. C. Energy of nucleation on a vortex
First, we find the energy of rectilinear vortex in the d -phase close to saturation. As a liquid mixture separates,the vortex core is progressively filling with the normal c -phase playing a role of a nucleus of new phase. We needto know the energy of the system as a function of vortexcore radius R . In equilibrium the thermodynamical po-tentials and temperature of the normal c -phase obey thefollowing requirements: Φ ′ ( P ′ , Z ′ , T ′ ) = const ,Z ′ ( P ′ , c ′ , T ′ ) = const and T ′ = const . This yields the ordinary conditions: c ′ ( r ) = const and P ′ ( r ) = const.The boundary conditions at the vortex core r = R ,usual for the equilibrium between two phase, imply anequality of thermodynamical potentials, temperatures,and pressures involving the Laplace pressure:Φ ′ ( P ′ , Z ′ , T ′ ) = Φ( P, Z, T, V s ) + V s ≡ Φ( P, Z, T, V s = 0) | r = ∞ ,Z ′ ( P ′ , c ′ , T ′ ) = Z ( P, c, T, V s ) ≡ Z ( P, c, T, V s = 0) | r = ∞ ,T ′ = T and P ′ = P ( R ) + α/R. (3)The surface tension coefficient α = α ( P ′ − P, c ′ − c ),in general, depends on a difference of the pressures andconcentrations at the interface. For brevity, we omit thenotation for temperature.Let us denote the impurity concentration far fromthe vortex at r = ∞ as c = c ps ( P ) + ∆ c and, cor-respondingly, impurity concentration inside the core as c ′ = c ′ ps ( P ) + ∆ c ′ . Here c ps ( P ) and c ′ ps ( P ) are the He impurity concentrations at the phase separation line.Assuming ∆ c ≪ c ps ( P ) and ∆ c ′ ≪ c ′ ps ( P ) as well as δP = α/R + P ( r = R ) − P to be small, we expandEqs. (3) in deviationsΦ ′ ( P, c ′ ps ( P )) + 1 + β ′ ρ ′ ∆ P − c ′ ∂Z ′ ∂c ′ ∆ c ′ = Φ( P, c ps ( P )) − c ∂Z∂c ∆ c, and Z ′ ( P, c ′ ps ( P )) − β ′ ρ ′ c ′ ∆ P + ∂Z ′ ∂c ′ ∆ c ′ = Z ( P, c ps ( P )) + ∂Z∂c ∆ c, where coefficients β and β ′ are defined as usual β = cρ ∂ρ∂c and β ′ = c ′ ρ ′ ∂ρ ′ ∂c ′ . Hence we arrive readily at the deviations of concentrationand pressure from their equilibrium magnitudes insidethe vortex ∆ P = ρ ′ (cid:18) c ′ ps ( P ) − c ps ( P ) (cid:19) ∂Z∂c ∆ c, ∆ c ′ = (cid:20) β ′ (cid:18) − c ps ( P ) c ′ ps ( P ) (cid:19)(cid:21) ∂Z/∂c∂Z ′ /∂c ′ ∆ c. Thus, if d -phase is saturated (∆ c = 0), the impurityconcentration inside the core is equilibrium c ′ = c ′ ps ( P ).Introducing the imbalance between the phases∆Φ = [ c ′ ps ( P ) − c ps ( P )] ∂Z∂c ∆ c, we arrive at the equation determining the core radius R :∆ P ≡ α/R + P ( R ) − P = ρ ′ ∆Φ . To find pressure P ( R ), we involve the behavior of su-perfluid density ρ s as a function of P , c , V s and userelations (1). Then we have ∇ ρ s = ∂ρ s ∂P ∇ P + ∂ρ s ∂c ∇ c + ∂ρ s ∂V s ∇ V s = − ρ s B ∇ V s , where B is equal to B = ρ s ρ ( ∂ log ρ s /∂c ) ∂Z/∂c + ∂ρ s ∂P − ∂ log ρ s ∂V s , and the superfluid density is approximately given withthe following equation: ρ s ( r ) = ρ s (cid:18) − B V s ( r )2 (cid:19) . From ∇ P = − ρ s ( r ) ∇ ( V s /
2) we find the pressure P ( r ) = P − ρ s V s ( r )2 + ρ s B V s ( r )8 + . . . Accordingly, core radius R satisfies − ρ ′ ∆Φ + αR − ρ s V s ( R )2 + ρ s B V s ( R )8 + . . . = 0or − ρ ′ ∆Φ + αR − ρ s ~ m R + ρ s B ~ m R + . . . = 0 . Multiplying it with 2 πR and integrating over R , we ob-tain the energy U ( R ) per unit length for a c -phase nu-cleus in the form of quantized vortex U ( R ) = − ρ ′ ∆Φ πR + 2 παR + ρ s π ~ m ln LR + ∂α∂c ∂ρ s /∂cρ∂Z/∂c π ~ m R − π ~ m ρ s B R + . . . (4)Here surface tension α is assumed to be only depen-dent on the He impurity concentration at the c - d inter-face but a possible pressure and core radius dependenceis neglected. The length L is a usual cutoff one for avortex. The origin for the terms in the vortex energy isobvious. Note only that the third and fifth terms arisewholly from the kinetic energy of superfluid component(1 / R ρ s ( r ) V s ( r ) d r . In essence, equation (4) repre-sents an expansion of linear energy of a nucleus in itsinverse radius [19]. D. Thermal and quantum nucleation rate on thevortex. The crossover temperature to the quantumregime
We consider here the quantum nucleation rate of c -phase on the vortex at zero temperature and the thermal-quantum crossover temperature in nucleation. First ofall, it is necessary to understand the behavior of potentialnucleus energy U ( R ). To simplify and to obtain the clearanalytical formulae, we restrict ourselves with the threefirst terms of expansion in equation (4) U ( R ) = − ρ ′ ∆Φ πR + 2 παR + ρ s π ~ m ln LR .
FIG. 1: The behavior of potential energy U as a function of vortex core radius R in the various ranges of imbalance ∆Φ. As is seen from Fig. 1, energy U ( R ) as a function of im-balance ∆Φ has a various behavior and a different num-ber of extrema depending on the vortex core radius. Forunsaturated (∆Φ <
0) liquid mixture, there is a singleextremum and the corresponding vortex state is abso-lutely stable (Fig. 1a). Within the intermediate imbal-ance range 0 < ∆Φ < ∆Φ cr there are two extrema in thepotential energy (Fig. 1b) as a function of core radius at R ± R = 2 p (cid:16) ± p − p (cid:17) ,R = ~ ρ s αm , p = ∆Φ∆Φ cr . (5)Here R is the core radius in the saturated liquid mixture∆Φ = 0. For the imbalance larger than the critical one ρ ′ ∆Φ cr = α R = α m ρ s ~ , (6)there are no extrema (Fig. 1d). This entails an appear-ance of the line of absolute instability. In other words,vortices which sizes exceed R cr = 2 R become unstableagainst vortex core expansion and the phase separation ofa liquid mixture proves to be unavoidable. Hence, onlyfor the imbalance range 0 < ∆Φ < ∆Φ cr correspond-ing to R < R < R , we observe the metastable statewhich can be destabilized as a result of thermal or quan-tum fluctuations depending on the temperature. Thesespecific features are well evident in Fig. 2.Such behavior of nucleus energy as a function of nu-cleus size and imbalance differs in kind from the case ofhomogeneous nucleation of spherical drop. The point isthat there is a competition of two opposite factors in thepresence of a defect similar to vortex. If, for instance, anucleus grows, the contribution associated with the sur-face tension increases and the other due to effect of adefect decreases. For negative and small positive ∆Φ,these two contributions result in the minimum for thepotential energy (Figs. 1a and 1b). On the contrary, ifthe imbalance is large and the critical nucleus is small,the effect of surface tension is small as well. Then, for FIG. 2: The diagram for the equilibrium between vortex lineand superfluid He- He liquid mixture. Here R is the vortexcore radius and ∆Φ is the imbalance of a liquid mixture. ∆Φ > ∆Φ cr , the total energy is determined, in the firstturn, with the terms decreasing gradually and, therefore,has neither minimum nor maximum. This results in theunstable state. All these features, involving the similarphase diagrams R —∆Φ hold for other defects [20], e.g.for a charged ion which influence decays with the distancetogether with electric field.At the first sight, due to existence of critical value∆Φ cr one may expect that the phase separation of a liq-uid mixture with the quantized vortex will occur at thesame magnitude of imbalance in all experiments. How-ever, the process of phase separation can take place inthe metastable region 0 < ∆Φ < ∆Φ cr before the crit-ical imbalance is achieved. Since the phase separationof metastable state is a random process described withsome probability and dependent on imbalance, the ex-perimental magnitudes acquire some dispersion aroundcertain magnitude ∆Φ < ∆Φ cr . This magnitude anddispersion of data depend both on the decay probabilityand on the rate of varying the liquid mixture imbalancein experiment.For the high temperatures, the nucleation rate, deter-mined as a nucleation probability per unit time at onenucleation site, is governed with the usual Arrhenius for-mula for thermal fluctuationsΓ cl = ν exp( − U L /T )where ν is the frequency of attempts. The activationenergy U L for a vortex of length L is expressed as U L = L ∆ U with ∆ U = U ( R + ) − U ( R − ) determined fromthe difference between the maximum and the minimumof energy U ( R ) (Fig. 2b):∆ U παR = 2 p p − p − ln 1 + √ − p − √ − p = (1 − p ) / for p → , p − ln p for p → . It is seen that the nucleation rate enhances drasticallyas p = ∆Φ / ∆Φ cr → q = ν q exp( − A L / ~ ) . Here ν q is the attempt frequency and A L = AL is thedoubled underbarrier action for vortex length L where A is the quantity per unit length of a vortex.To calculate the quantum probability, it is necessaryto estimate the effective mass of the expanding vortexcore. We treat the boundary conditions at the vortexcore surface r = R ( t ) as for the boundary between twodifferent phases. First of all, we have a conservation forthe radial component of mass flow across the boundary − ρ ′ ˙ R = r ( R ) − ρ ˙ R, = ρ n V n + ρ s V s . Here we assume that the c -phase of density ρ ′ in thevortex core is at rest ( V ′ = 0). The second condition isa continuity for the tangential component of momentumflux density tensor Π ik Π ′ θ r − ′ θ ˙ R = Π θ r − θ ˙ R, where for the superfluid phaseΠ ik = ρ n V n i V n k + ρ s i V n i V s k + P δ ik . Accordingly, we have ρ n V n θ ( V n r − ˙ R ) + ρ s V s θ ( V s r − ˙ R ) = 0 . Treating the normal component of superfluid d -phase asa liquid with the properties of an ordinary viscid fluid, we should require an equality of tangential componentsfor the velocities of adjacent fluids. Then we have V n θ ( R ) = V ′ θ ( R ) = 0 . From the above three equations we find the magnitudesof velocities at the boundary r = R ( t ) V s r ( R ) = ˙ R,V n r ( R ) = − ρ ′ − ρ n ρ n ˙ R,V n θ ( R ) = 0 . As is seen, the mass transfer across the surface of the ex-panding vortex core is connected with the normal com-ponent flow alone.On the neglect of the compressibity of a liquid mixturethe distribution for the radial components of superfluidand normal velocities is equal to V s r ( r ) = V s r ( R ) R/r,V n r ( r ) = V n r ( R ) R/r , r > R ( t ) . In the logarithmic quasistationary approximation and onthe analogy with the two-dimensional nucleation [21] wecan estimate the kinetic energy of the expanding vortexcore as 2 πρ eff R ln Λ( R ) R ˙ R M ( R ) ˙ R with the following effective density ρ eff ρ eff = ρ s + ( ρ ′ − ρ n ) ρ n . (7)The cutoff parameter Λ( R ) is of the order of the soundvelocity multiplied with the typical time of the core ex-pansion Λ( R ) ≈ s eff τ ( R ). Since there are two soundvelocities in the superfluid liquid mixture, velocity s eff is some weighted average of first and second sound ve-locities. In other words, length Λ( R ) is an effective sizeof sound propagation region for the time of the under-barrier nucleation evolution, i.e., size of the perturbedmedium surrounding the nucleus. Here we do not con-sider possible energy dissipation effects due to viscosityand diffusion in the process of vortex core growth.So, action A is calculated between the classical turningpoints in the potential ∆ U ( R ) = U ( R ) − U ( R − ) as A = 2 Z R c R − p M ( R )∆ U ( R ) dR (8)where R c is the exit point from the barrier U ( R c ) = U ( R − ). Since the logarithm is a slowly varying function,for our aim it is sufficient to estimate the growth time ofa nucleus as follows: τ ≈ (cid:18) M ( R ) R U ( R ) (cid:19) / R ≈ R c . The analytical expressions for the effective action aresucceeded to obtain for the two limiting cases. For thesmall degree of imbalance p = ∆Φ / ∆Φ cr ≪
1, the con-tribution to the vortex energy, associated with the super-fluid motion, plays a minor role and the effective actionis mainly governed with the magnitude of the surfacetension: A ( T = 0) = 32 √ π p / (cid:18) αρ eff R ln s R αp (cid:19) / ∝ (∆Φ) − / ln / (∆Φ) − . The quantum critical radius R c exceeds significantly thecore radius R in a weakly supersatured liquid mixture R c = 2 αρ ′ ∆Φ = 8 R p ≫ R . The typical time of nucleus growth reads τ ( R c ) ≈ (cid:18) ρ eff R c α (cid:19) / ∝ (∆Φ) − / . Let us turn to the other limit p = ∆Φ / ∆Φ cr → U ( R )2 παR = p − p (cid:18) R − R − R − (cid:19) − (cid:18) R − R − R − (cid:19) . As p →
1, the distance between the points of entrance R − and exit R c reduces to R c − R − = 3 R − p − p but the growth time of a nucleus increases τ ≈ (cid:18) ρ eff R α √ − p (cid:19) / . With the help of Eq. (8) the action per unit length canbe estimated as A = 192 π − p ) / (cid:18) αρ eff R ln 8 s ρ eff R α √ − p (cid:19) / . Thus, effective action vanishes with approaching to theinstability. Accordingly, the probability of quantum nu-cleation grows drastically.The next important point in the low temperature ki-netics is a crossover temperature T q between the quan-tum and classical nucleation regimes. A simple estimatefor T q can be obtained from comparing the classical andquantum actions under assumption ν q ≈ ν . So, T q = ~ U L /A L . Hence we have in the limit of small p = ∆Φ / ∆Φ cr ≪ T q = p / √ π (cid:18) α ~ ρ eff R ln[ s ρ eff R / ( αp )] (cid:19) / . (9)In this region of imbalance the crossover temperaturegrows as the imbalance enhances.On the other hand, as the imbalance tends to the criti-cal value, i.e. 1 − p ≪
1, the crossover temperature startsto reduce according to relation T q = 572 (1 − p ) / (cid:18) α ~ ρ eff R ln[8 s ρ eff R / ( α √ − p )] (cid:19) / . (10) FIG. 3: The various types of the c -phase nucleation ontoa quantized vortex in a supersaturated He- He liquid mix-ture. The solid line shows the thermal-quantum crossovertemperature. The dashed line is the spinodal, separating themetastable states from absolutely unstable ones.
The total behavior for the crossover temperature T q as a function of imbalance is shown in Fig. 3. Notehere that the crossover temperature maximum T q, max isshifted in the direction to the line of absolute instability.Another specific feature, apparently, inherent in all tran-sitions near the spinodal, is that the nucleation mecha-nism becomes again the thermal one instead of quantumas a function of supersaturation ∆Φ or imbalance degree[20] in the intermediate vicinity to the line of absoluteinstability (spinodal). This takes place though the tem-perature is less than T q, max . For a rectilinear vortex, thecrossover temperature T q is naturally independent of itslength.We do not consider here the effect of dissipative phe-nomena on the nucleation rate of c -phase nucleus at thequantized vortex core, which can be associated with vis-cosity, impurity He diffusion, and impossibility of usingthe quasistationary approximation for rectilinear vortex.In principle, one can here distinguish the hydrodynam-ical and ballistic regimes of nucleus growth. However,for the region of supersaturation close to critical value, apossibility of hydrodynamical R c ≫ l ( T ) regime, where l ( T ) is the mean free path of excitations in a liquid mix-ture, is unlikely in the quantum nucleation region sincethe quantum vortex core radius R c in the nucleus growthdoes not exceed several vortex core radii R in the sat-urated liquid mixture (about a few tens of angstrom).For the ballistic R c ≪ l ( T ) regime, one can suppose thatthe friction coefficient is directly proportional to the corearea. Accordingly, the friction coefficient per unit lengthis µ ( R ) ∝ R . In the viscous M ( R ) τ − ( R ) ≪ µ ( R ) limita simple estimate gives T q ≈ α ~ R µ (2 R ) r − ∆Φ∆Φ cr , ∆Φ → ∆Φ cr ; A ( T < T q ) ≈ π (cid:18) − ∆Φ∆Φ cr (cid:19) (2 R ) µ (2 R ) (cid:18) − T T q (cid:19) and in the other ∆Φ ≪ ∆Φ cr limit T q ≈ πα ~ µ ( R c ) R c ∝ (∆Φ) , R c = 2 R , (∆Φ ≪ ∆Φ cr ) ; A ( T < T q ) ≈ µ ( R c ) R c ∝ (∆Φ) − . Comparing these formulae with the previous ones, itis seen that the qualitative character for the behaviorof effective action A ( T ) and crossover temperature T q re-mains unchanged as a function of imbalance ∆Φ. The di-agram T q –∆Φ conserves its shape in kind (Fig. 3) thoughthe quantum nucleation region reduces and the thermalactivation region increases beside the instability line. E. Rapid nucleation line and the numericalestimates.
Below we discuss some consequences from the equa-tions above and perform the numerical estimates con-nected with the c -phase nucleation on a quantized vortexin the supersaturated superfluid He- He liquid mixture.First, we analyze in kind the possible positions of therapid nucleation line in the T –∆Φ diagram of nucleationregimes. The rapid nucleation line exists also as in thecase of homogeneous nucleation of spherical drops [13, 14]due to very drastic dependence of nucleation rate on theimbalance. The rapid nucleation line separates the re-gion where the nucleation rate is practically zero and su-persaturated liquid mixture does not separate infinitelylong on the time scale of experimental period from theregion where the phase separation occurs almost instan-taneously.After preparing the metastable state 0 < ∆Φ < ∆Φ cr at temperature T the liquid mixture separates eventu-ally for the expectation time τ obs . Thus the nucleationprobability is approximately equal to unity W (∆Φ , T, τ obs , N nuc ) ≡ τ obs N nuc Γ(∆Φ , T ) = 1 . (11)Here Γ stands for either Γ cl or Γ q in the correspondencewith the temperature range and N nuc is the number of nucleation sites. Equation (11) determines the rapid nu-cleation line ∆Φ sat ( T ) in diagram T –∆Φ. This corre-sponds to the experimentally achievable supersaturation.For the nucleation probability, we have W = τ obs N nuc νe − A (∆Φ ,T ) / ~ = (cid:26) τ obs N nuc ν cl exp( − U L (∆Φ , T ) if T > T q (∆Φ) ,τ obs N nuc ν q exp( − A L (∆Φ , T ) if T < T q (∆Φ) . Hence one can see that the position of the rapid nucle-ation line depends on the temperature, the number ofnucleation sites, and the rate of sweeping the liquid mix-ture imbalance. Since the shape of potential energy U ( R )depends strongly on the closeness to the instability line,the effect of the sweep rate on the position of the rapidnucleation line here is more essential as compared withthe case of homogeneous nucleation.Depending on the expectation time τ obs and the num-ber of nucleation sites, one can discern two opposite casesin the position of the rapid nucleation line in the T –∆Φdiagram (Fig. 4). The first case is restricted with theinequalityln( ντ obs N nuc ) ≫ √ π (cid:18) αρ eff R L ~ ln s ρ eff R α (cid:19) / (12)and corresponds to the limit of low nucleation rates Γ.This corresponds to the large lifetime of a supersaturatedliquid mixture against the decay channel considered. Inthis case (Fig. 4a) the rapid nucleation line lies far fromthe instability line and ∆Φ sat ≪ ∆Φ cr . Therefore, theexistence of the instability line has no significant effecton the nucleation kinetics. In the classical thermal acti-vation region the attainable supersaturation is stronglytemperature-dependent according to ∆Φ sat ∝ /T . Inthe quantum T < T q region the attainable supersat-uration is almost independent of temperature. Corre-spondingly, the crossover temperature T q , proportionalto (∆Φ sat ) / , is significantly smaller than the maximumcrossover temperature T q, max . FIG. 4: The schematic for the rapid nucleation lines (solidlines): (a) for low nucleation rates Γ and large expectationtime τ obs , (b) for high nucleation rates Γ and small expecta-tion time τ obs . For the opposite case of high nucleation rates when in-equality (12) is invalid, the existence of instability affectsessentially the position of the rapid nucleation line at suf-ficiently low temperatures (Fig. 4b). As the temperaturelowers, the rapid nucleation line should closer approachthe instability line since the smallness of potential barriercan compensate a decrease of temperature in the classi-cal exponent, providing us the high nucleation rates. Asa result, in the thermal activation regime the tempera-ture behavior for the attainable critical supersaturationsshould go over from drastic ∆Φ sat ∝ /T to the smootherone ∆Φ sat = ∆Φ cr (cid:16) − ( T /T ∗ ) / (cid:17) in the low temperature region if T . T q, max . Here T ∗ is some typical temperature which can be determinedfrom Eq. (11) with the classical exponent at p → T < T q, max (Fig. 3). How-ever, as is seen from Fig. 3a and 3b, the observation ofsuch reentrant behavior is impossible under the fixed nu-cleation rate Γ. To do this, it is necessary to vary anyparameter in equation (11), e.g. the number of nucle-ation sites N nuc . In liquid He- He mixture this can bedone with introducing vortices intentionally before thestart of the phase separation process in the d -phase of aliquid mixture.Let us turn to the numerical estimates of the resultsobtained. We start from the calculation of the criti-cal value ∆Φ cr which plays a key role in comprehendingthe phase separation in a superfluid liquid mixture withquantized vortices. As for the surface tension, we take themagnitude for the flat interface between the bulk phasesof liquid mixture at zero pressure α =0.0239 erg/cm [22]. Using simplest estimates (5) and (6), we find ρ ′ ∆Φ cr ≈ R cr =12.7 ˚A. Arelatively large critical vortex core justifies an applicabil-ity of macroscopic approximation to some extent. On ac-count of estimating the derivative ∂Z/∂c ≈ · erg/gfor the limiting critical value of saturation responsiblefor the vortex core instability, we arrive at ∆ x cr =(1.9–2.0)% [19]. Thus the estimate agrees with that obtainedin work [18].Expansion (4) in the inverse core radius involves the in-homogeneous distribution of concentration, pressure andsuperfluid density in the d -phase. However, the effectof these terms proves to be small and counts about 5%.If we neglect the contribution associated with coefficient B in (4), putting B = 0, and expecting the order-of-magnitude estimate of ∂α/∂c ≈ α , we find somewhatsmaller but the close value for the limiting supersatu-ration ∆ x cr =1.80%. This is connected with the rel- atively large critical core radius. Note also that value R cr =12.7 ˚A correlates with the core radius R sp ≈ He concentrationat the core boundary corresponds to the spinodal of thebulk d -phase and equals x sp ∼ T q . To es-timate the latter, it is necessary to know the effectivedensity ρ eff (7) which proves to be approximately equalto ρ eff ≈ . ρ s at the phase separation line and lieswithin the range 0.23 – 0.40 g/cm . The highest possi-ble crossover temperature T q, max can be estimated eitherfrom (9) at p = 1 or from (10) at p = 0 .
5. This results inrelatively small temperatures T T q, max . c -phase nucleationin the supersaturated d -phase when a quantized vor-tex plays a role of nucleation site. Though the maxi-mum possible supersaturation of the d -phase in the pres-ence of vortices proves to be in the almost satisfactoryagreement with the observable magnitudes of supersat-uration, the estimate for the crossover temperature tothe quantum nucleation regime is rather small as com-pared with the temperature at which the transition intemperature behavior is observed for the critical super-saturation ∆ x cr of a liquid mixture. Varying the valuesof physical parameters in order to increase T q, max , wehave an enhancement of the maximum attainable con-centration at which the vortex core instability occurs.Then the agreement with experiment becomes worse inthis parameter. So, the assumption about the heteroge-neous phase-separation mechanism in a supersaturatedsuperfluid He- He liquid mixture with quantized vor-tices as nucleation sites, apparently, cannot describe theexperimentally observed picture of phase separation onthe whole.An obstacle for quantitative comparison arises fromthe exponential behavior of nucleation rate. In such sit-uation from the experimental point of view it would beuseful to study the effect of the number of nucleationsites on the phase separation of a liquid mixture underthe planned and controlled introduction of quantized vor-tices into the d -phase. One of possibilities is an experi-ment in a rotating cryostat and the study of the phaseseparation rate of liquid mixture as a function of rotationvelocity. Since in the limit of small density of vortices thenucleation rate is proportional to the number of vortexlines, the observable nucleation rate should also be pro-portional to the rotation velocity.For the large rotation velocities, especially when thespacing between the vortex lines is comparable with thecore sizes, the critical value of supersaturation ∆Φ cr andthe potential barrier, separating the metastable statefrom unstable, are strongly suppressed. Accordingly, thephase separation rate of a liquid mixture should dras-tically grow in the limit of high rotation velocities. Inaddition, due to difference in the centrifugal energy of He and He atoms the spatial He distribution becomes0inhomogeneous over the bulk of a rotating fluid, facilitat-ing the reduction of the critical supersaturation ∆Φ cr . Inany case it is known that the He impurities strongly af- fect the process of nucleating the quantized vortices insuperfluid He. [1] J. Landau, J.N. Tough, N.R. Brubaker, and D.O. Ed-wards,
Phys. Rev. Lett. , 283 (1969).[2] N.R. Brubaker and M.R. Moldover, Nucleation of phase-separation in He- He. — Proc. 13-th Int. Conf. on LowTemp. Phys., 1972, eds. W.J. O’Sullivan, K.D. Timmer-haus, and E.F. Hammel (Plenum Press, NY, 1973), vol. I ,pp. 612-617.[3] J.K. Hoffer, L.J. Campbell, and R.J. Bartlett, Phys. Rev.Lett. , 912 (1980).[4] P. Alpern, Th. Benda, and P. Leiderer, Phys. Rev. Lett. , 1267 (1982).[5] J. Bodensohn, S. Klesy, and P. Leiderer, Europhys. Lett. , 59 (1989).[6] V.A. Mikheev, E.Ya. Rudavskii, V.A. Chagovets, andG.A. Sheshin, Fiz. Nizk. Temp. , 444 (1991).[7] V.A. Maidanov, V.A. Mikheev, N.P. Mikhin, N.F. Ome-laenko, E.Ya. Rudavskii, V.K. Chagovets, and G.A.Sheshin, Fiz. Nizk. Temp. , 943 (1992).[8] V.A. Mikheev, E.Ya. Rudavskii, V.K. Chagovets, andG.A. Sheshin, Fiz. Nizk. Temp. , 1091 (1992).[9] V.A. Mikheev, E.Ya. Rudavskii, V.K. Chagovets, andG.A. Sheshin, Fiz. Nizk. Temp. , 621 (1994).[10] V.K. Chagovets, V.A. Mikheev, E.Ya. Rudavskii, andG.A. Sheshin, J. Low Temp. Phys. , 827 (1995). [11] T. Satoh, M. Morishita, M. Ogata, and S. Katoh,
Phys.Rev. Lett. , 335.[12] E. Tanaka, K. Hatakeyama, S. Noma, S.N. Burmistrov,and T. Satoh, J. Low Temp. Phys. , 81 (2002).[13] I.M. Lifshitz, V.N. Polesskii, and V.A. Khokhlov,
ZhETF , 268 (1978) [ Sov. Phys. JETP , 137 (1978)].[14] S.N. Burmistrov, L.B. Dubovskii, and V.L. Tsym-balenko, J. Low Temp. Phys. , 363 (1993).[15] L.S. Rent and I.Z. Fisher, ZhETF , 722 (1968) [ Sov.Phys. JETP , 375 (1969)].[16] T. Ohmi, T. Tsuneto, and T. Usui, Progr. Theor. Phys.(Japan) , 1395 (1969).[17] L. Senbetu, J. Low Temp. Phys. , 571 (1978).[18] D.M. Jezek, M. Guilleumas, M. Pi, and M. Barranco, Phys. Rev. B , 11981 (1995).[19] S. Burmistrov, V. Chagovets, L. Dubovskii, E. Rudavskii,T. Satoh, and G. Sheshin, Physica B , 321(2000).[20] S.N. Burmistrov,
Phys. Rev. B , 214501 (2012).[21] S. N. Burmistrov and L.B. Dubovskii, J. Low Temp.Phys. , 131 (1994).[22] L.S. Balfour, J. Landau, S.G. Lipson, and J. Pipman, J.de Physique ,39