FLUCTUATING HYDRODYNAMICS AS A TOOL TO INVESTIGATE NUCLEATION OF CAVITATION BUBBLES

Transcription

1 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) FLUCTUATING HYDRODYNAMICS AS A TOOL TO INVESTIGATE NUCLEATION OF CAVITATION BUBBLES MIRKO GALLO, FRANCESCO MAGALETTI & CARLO MASSIMO CASCIOLA Department of Mechanical and Aerospace Engineering, Sapienza Università di Roma, Rome, Italy. ABSTRACT Vapor bbbles can be formed in liqids by increasing the temperatre over the boiling threshold (evaporation) or by redcing the pressre below its vapor pressre threshold (cavitation). The liqid can be held in these tensile conditions (metastable states) for a long time withot any bbble formation. The bbble ncleation is indeed an activated process, in fact a given amont of energy is needed to bring the liqid from that local stable condition into a more stable one, where a vapor bbble is formed. Crcial qestion in this field is how to correctly estimate the bbble ncleation rate, i.e. the amont of vapor bbbles formed in a given time and in a given volme of liqid, in different thermodynamic conditions. Several theoretical models have been proposed, ranging from classical ncleation theory, to density fnctional theory. These theories can give good estimate of the energy barriers bt lack of a precise estimate of the ncleation rate, especially in complex systems. Moleclar dynamics simlations can give more precise reslts, bt the comptational cost of this techniqe makes it nfeasible to be applied on systems larger than few tenth of nanometers. In this work the approach of flctating hydrodynamics has been embedded into a continm diffse interface modeling of the two-phase flid. The reslting model provides a complete description of both the thermodynamic and flid dynamic fields enabling the description of vapor-liqid phase change throgh stochastic flctations. The continm model has been exploited to investigate the bbble ncleation rate in different metastable conditions. Sch an approach has a hge impact since it redces the comptational cost and allows to investigate longer time scales and larger spatial scales with respect to more conventional techniqes. Keywords: bbble, diffse interface, flctating hydrodynamics, ncleation, thermal flctations. 1 INTRODUCTION At the moleclar scale, even in conditions of thermodynamic eqilibrim, the flids do not exhibit a deterministic behavior. In fact, going down below the micrometer scale, the effects of thermal flctations play a dominant role in the dynamics of the system. A sitable description of the flid dynamic phenomena at mesoscopic scale is then necessary to inclde the effects of thermal flctations. Since the pioneering work of Landa and Lifshitz (1958, 1959) [1] several models, designed to the description of the hydrodynamic flctations, have been developed in the context of continm mechanics [] contribting to the growing field of flctating hydrodynamics (FH). In recent years there has been an exponential increase of nmerical methods for modeling these effects [3, 4]. These models not only play an important role in flid dynamics, bt a deep nderstanding of these phenomena is necessary for the progress of some of the latest nanotechnologies. For instance the modeling of thermal flctations is crcial in the design of flow microdevices, in the stdy of biological systems, sch as lipid membranes [5], in the theory of Brownian engines and in the development of artificial moleclar motors prototypes [6]. Another problem whose theoretical and technological importance today is widely recognized in the scientific commnity is the problem of ncleation as a precrsor of the phase change in metastable systems, where the rare events in the thermal flctations intensity, cold lead to overcome the energy barriers for phase transitions [7, 8]. This problem is intimately connected to the phenomenon of cavitation [9]. Nowadays the stdy of the above problems, is almost niqely represented by moleclar dynamics (MD) simlations [1], which for a large part of the real systems are often 18 WIT Press, ISSN: (paper format), ISSN: (online), DOI: 1.495/CMEM-V6-N

2 346 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) comptationally too expensive and therefore limited to very small systems, often far from the technological reality. We aim to stdy ncleation processes by means of FH, in order to nderstand the role of thermal flctations in cavitation. Before discssing the mathematical model, it is worth to remember the main featres of ncleation in liqid system. A liqid held at atmospheric pressre can be heated p to a temperatre far beyond its boiling point. In this condition the liqid is called sperheated, or more generally metastable. Metastability can be obtained analogosly by decreasing isothermally the pressre nder its satration vale. At low enogh temperatre at ambient temperatre in the case of water, e.g. the liqid can be stretched down to negative pressre, the so called tensile condition. When the liqid is in metastable conditions a vapor bbble can ncleate with a probability related to the level or sperheating or stretching and we refer to the ncleation event as boiling or cavitation, respectively [9]. Bbble ncleation is an activated process, since an amont of energy is needed to overcome the activation barrier. The presence of imprity or dissolved gas nclei strongly lowers the energy barrier and simplifies the bbble formation. This is the reason why it is extremely easy to experience a cavitation event in water at non-extremely negative pressres even if it has been proven [11] that ltra-pre water can sstain 1 kbar tensions. Several theoretical models have been proposed in order to estimate the energy barrier and the ncleation rate, both in homogeneos and heterogeneos (near bondaries) conditions. The classical ncleation theory (CNT) [1], poses the basis for the nderstanding of the phenomena. More sophisticated theories like density fnctional theory (DFT) [13] or MD simlations can give more precise estimates of the barriers and can correct some mis-prediction of the CNT. Both the methods are extremely powerfl in stationary conditions and need to be copled to specialized techniqes, like the string method [14], to stdy the ncleation events and the transition path [15]. Another promising approach is to se a phase field model where the order parameter is the mass density itself. In stationary conditions it recovers the DFT description with a sqared-gradient approximation of the excess energy [16]. The phase field model has the advantage of being easily extended to nsteady sitations, enabling the fll description of both the thermodynamic and the flid dynamics fields [17]. The model, in its original form, is deterministic and cannot captre spontaneos ncleation originated by thermal flctations, in absence of external forcing. To this prpose, the theory of FH [18] represents the natral framework to embed thermal flctations inside the phase field description. This approach has been sccessflly exploited in [19] to follow the spinodal decomposition in a liqid vapor system. Aim of this work is to stdy the homogeneos ncleation of vapor bbbles in a metastable liqid, by the means of diffse interface approach with the addition of thermal noise. HYDRODYNAMIC FLUCTUATIONS IN EQUILIBRIUM STATE In order to achieve a sitable description of flids at mesoscopic scale the effects of thermal flctation have to be inclded in the classical hydrodynamic eqations. Originally based on phenomenological argments, the theory of FH has been developed by Landa and Lifshitz (1958, 1959) [1], and sbseqently it was framed in a more general contest of stochastic processes theory []. The main idea of Lifshitz and Landa theory is to treat the thermodynamic flxes as stochastic processes. As prescribed by the thermodynamics of irreversible processes at macroscopic level, thermodynamic flxes are the expression of microscopic moleclar degrees of freedom of the thermodynamic system. Under this respect dissipation in flids can be seen as macroscopic manifestation of the energy transfer arising from random moleclar collisions. Ths at mesoscopic scale, thermodynamic flxes have to be modeled as

3 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) 347 stochastic tensor fields, whose statistical properties can be inferred by enforcing the flctation-dissipation-balance (FDB). Before discssing the FDB, we aim to discss the statistical properties of the flctations in an eqilibrim flid. The static correlation fnction of a thermodynamic system in eqilibrim can be evalated from the entropy deviation S from its eqilibrim vale S. For single component systems S can be expressed as a fnctional of flctating fields of mass density, dr(x; t), velocity, dv(x; t) and temperatre, dq(x; t) S = S S = S[ δρ, δv, δθ ] = s( x, t) sdv, (1) V where the integration is over the system volme V, s (X; t) is the entropy density per nit volme and s is its eqilibrim vale (i.e. S is the entropy maximm). Under these assmptions the probability distribtion fnctional for the flctating fields =(dr, dv, dq) is (Einstein s assmption) P eq 1 S [ ] = exp. Z k B Ths the static correlation fnctions can be evalated by solving the path integral () C Z D D (x) = 1 1 = dr dvddq exp s(, ) dv. k V dr dv, dq (3) For small flctations, the entropy fnctional can be expended in a Taylor series arond the eqilibrim vale. A convenient starting point is the expression of the entropy in terms of its natral variables, namely internal energy and density r, S s dv dv s s s s = = + (, ρ) + V δ V + ρ δρ 1 δ δ 1 δ δρ + (4) ρ Where the constraints of energy and mass conservation, 1 δdv = dv( ρδv δv+ δρδv δv + λ δρ δρ), δρ dv =, (5) V V V mst be enforced. The internal energy, whose flctation is in the constraints given above, can be derived from the Helmholtz free energy fnctional. Here we se the famos Van der Waals sqare gradient approximation in order to describe the two-phase liqid-vapor system B 1 F[ ρθ, ] = dv f ( ρθ, ) + λ ρ ρ, V (6) where f is the classical blk free energy and l is the capillary coefficient, related both to the srface tension and to the interface thickness. All terms appearing in the right hand side of eqn. (4) can be expressed in terms of sitable thermodynamic coefficients and of the flctations of density, temperatre and velocity, e.g. d q d m q dr d r dq m q d 1 c s s c dr dr dm ct r dr 1 p =, = + c + c, = + r q s r dq, (7)

4 348 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) where c v is the specific heat at constant specific volme, c T the isothermal speed of sond, p the pressre and µ c = df[r,q]/ dr = f / r l r the (generalized) chemical potential. Assming that the flid is very close to eqilibrim and the flctations are small with respect to the mean vale, the entropy fnctional can be approximated by a qadratic form in the flctating fields, S S dv c 1 c T = V + qr dr l q dr dr r ( ) q d v d v+ rq dq, (8) where S denotes the qadratic approximation arond the maximm. Thanks to this approximation, the generic correlation fnction (3) can be evalated in closed form by integrating Gassian path integrals. We skip here the details of the calclation, bt we refer to [6, ] for an in-depth stdy. As a reslt of the integration, the entire correlation tensor C = reads, C = k Bq c T 4pl exp r r r r rl kbq Id( r r ) r kb q d( r r ) rc (9) We can dedce that, in the Gassian approximation, the eqilibrim correlations for velocity and temperatre are short ranged (delta correlated in space, actally) and the cross correlation of the flctating fields are zero. An important qantity in the theory of liqids is the Forier transform of the correlation tensor, the so called static strctre factor. In particlar the density strctre factor, defined as S(k) = dr(k) dr*(k) (1) assmes a key role, both from an experimental [1] and a nmerical [3] point of view. In eqn. (1) dr(k) is the Forier transform of the density flctation dr(k) = dxe -ik x (r(x) r ) (11) with r being the blk mean density and the symbol * denoting the complex conjgate. For a single component flid embedded with capillarity, the Forier transform of the density component of eqn. (9) reads ρ k Bθ * S(k) = δρ( k) δρ ( k) =. c + ρ λk k (1) T 3 LIFSHITZ-LANDAU-NAVIER-STOKES EQUATIONS WITH CAPILLARITY The dynamics of the mesoscopic system of or interest is governed by a system of eqation expressing mass, momentm and energy conservation, with the addition of stochastic contribtions (Lifshitz-Landa-Navier-Stokes [LLNS] eqations with capillarity):

5 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) 349 ρ + ( ρ) = t ρ + ( ρ ) = p + Σ+ δσ t E + ( E) = ( p+ Σ q) + ( δσ δq), t where S and q are the classical deterministic stress tensor and energy flx, respectively, and where the terms with the prefix d are the stochastic parts, whose statistical properties will be inferred from the flctation-dissipation theorem (FDT). For a simple one-component Newtonian flid embedded with capillarity (the free energy fnctional is expressed as in eqn. 6) the stress tensor S and the energy flx q can be easily dedced by standard non-eqilibrim thermodynamic methods []: (13) = λ T ρ + ρ ( λ ρ) I λ ρ ρ + µ ( + ) I, (14) Flctation-Dissipation-Balance q = λρ ρ k θ. (15) To correctly model the stochastic stress tensor ds and heat flx dq we need to recall the FDT. Here for the sake of brevity we report the fll calclation for the 1D case. Generalization to the 3D case is straightforward and the reslts will be postponed to the next section. The system of eqations (eqn. 13) in the 1D case is rewritten as r+ ( r) = t x l t( r) + x( r) = xp+ m xx+ x + xr xr rl xx r + sw + = p + + rc ( q q) q k q m( ) s W + s W, q t x x xx x q q x where, for the ease of calclation, the energy eqation is here expressed in terms of temperatre. In the eqations µ is the dynamic viscosity, k is the thermal condctivity and the terms s v W v and s q W q represent the stochastic forcing. W is a standard Wiener process and s v/q two sitable operators that will be later identified by means of the FDB. The above system of eqations can be linearized arond the mean soltion {r,, q }. Sch linearization provides a set of stochastic partial differential eqations, whose eqilibrim (statistically stationary) soltion is a Gassian field. The linearized system can be formally expressed in the form t = L + f, where L is the linearized Navier-Stokes operator with capillarity which reads r x ct m 1 L = + l q p r r r x xxx xx x q r c k q p x r c xx (16). (17)

6 35 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) f(x,t) is a Gassian vector process (with three components, in this case) whose correlation is f( xs, ) f ( xq ˆ, ) = Q( xx, ˆ) d ( s q), with Q( xx, ˆ) a matrix depending on x and ˆx Note that delta correlation in time is explicitly assmed. The stochastic forcing f is related to the standard Wiener process Wdt = db by the linear relationship f = KW, where W = {W r,w,w q } T is a Gassian delta correlated process characterized by the correlation W( ys, ) W( yq ˆ, ) = Id( y yˆ) d( s q), and s K = r s q r c is a linear operator acting on the noise. The soltion of the linear system is formally expressed as (18) t L( t s) Lt ( xt, ) = e f( s) ds + e, (19) where the last term which keeps memory of the initial conditions vanishes for large times. Conseqently the eqilibrim correlation is t L( t s) L ( t s) ( xt,) ( xt ˆ,) e Q e ds. = () The above integral can be performed in closed form assming the existence of a Hermitian non-singlar operator E -1 sch that the operator Q can be decomposed as Q= LE 1 E 1 L (1) With this position the integrand appearing in eqn. () is the exact derivative with respect to the delay time s of e L(t s) E 1 e L (t s). Hence eqn. () leads to lim = E 1 = C, () t hence the operator E 1 exists indeed and coincides with the correlation matrix C, see eqn. (9). Given the expression for Q, eqn. (1), and the identity E 1 = C it follows Q = (LC + C L )=(M + M )= k B O (3) where M = -LC and O is called the Onsager matrix. Relationship (3) is the form the celebrated FDB takes for the present system. The nknown operators s /q can finally be identified by enforcing the FDB eqn. (3), Q ˆ K K O (4) ( x, x) d( s q) = WW = kb d( s q), KK =k B O= (LC +C L )=M+M (5) The explicit calclation of the right hand side of this eqation is performed sing the known expression for L and C (eqns.17 and 9) and by taking the transpose of the real matrix M after

7 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) 351 considering that even differential operators are self-adjoint xx = xx anti-adjoint = x x : while odd ones are mk q B ^ M+ M = KK = xx[ d( x x)], (6) r kk Bq ^ d xx[ ( x x)] r c providing the explicit expressions s W = mk q W, s W = kk q W (7) B x q q B x q 3. General form of the stochastic components in a 3D system As shown in the previos sections, for a system in thermodynamic eqilibrim, it is possible to infer the statistical properties of the stochastic flxes by enforcing the FDT, eqn. 4. Hence the covariance of the stochastic process can be written as ˆ Σ dσ( xt ˆ, ) dσ ( xt, ) = Q d( xˆ x ) d( tˆ t ), (8) With Q Σ abnh = k B qm dandbh + dahdbn dabdnh 3, concerning the viscos stress tensor, and ˆ q dqxt ( ˆ, ) dq( xt, ) = Q d( xˆ x ) d( tˆ t ), (9) with Q q αβ = k B θ k δαβ, concerning the heat flx. Moreover it is worth noting that the correlation between thermodynamic force of different tensor rank has to be zero de to the Crie-Prigogine principle i.e. ( dq ( xt, ) dσ ( xt ˆ, ˆ) = ). Ths in the theory of FH, the effect of thermal flctation appears directly in the Navier-Stokes eqations as an external force arising from the flctating part of the thermodynamic flxes. From an operative point of view it is more convenient to re-express the stochastic contribtions in terms of the standard Wiener process as dσ= m k q 1 m q 3 k B W B Tr( W ) I, (3) dq = kk B q W E. (31) W = ( W + ( W ) T )/ is a stochastic symmetric tensor field, and W E is a stochastic vector, with the following statistical properties ^ ^ W ( x, tw ) ( xt, ^ ^ ) = d d d( x x ) d( t t ), (3) ab gd ag bd E ˆ ˆ E W ( xtw, ) ( xt, ) = d d( xˆ x ) d( tˆ t ). (33) a b ab

8 35 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) 4 NUMERICAL SCHEME LLNS eqations have been discretized in the spirit of the method of lines, which consists of two different stages: the first stage concerns the spatial discretization, the second one is focsed on the temporal integrator. Concerning the spatial discretization it is worth stressing that the different physical phenomena described by the LLNS system ask for specialized nmerical techniqes. A crcial point to be addressed is the correct reprodction of the system statistical properties, in particlar the adopted nmerical scheme need to be consistent with the FDB. A necessary condition for the latter restriction is that the mathematical properties of the relevant continm differential operators are conserved in the discrete formlation [3]. Eqation (13) has been discretized on an eqi-spaced staggered grid, following [3]. De to staggering, scalar fields, like density, for example, are located at the cell center while components of vector fields in a given direction are located at the center of the perpendiclar face. The nmerical scheme has been validated by comparing the nmerical eqilibrim static correlations with the theoretical ones in the discretized eqations. Here we report the comparison of the density static strctre factor. In the discrete limit, the theoretical static strctre factor (see eqn. 1) reads k d r kbq St ( kd ) =, c + r lk k T sin( k x kd = + sin( k y x / y / sin( kz z/ + x / y / z / d d (34) (35) is the discrete version of sqare norm of k, arising from the discrete laplacian operator L in Forier space. The nmerical vale of the density strctre factor is calclated, following its definition, as S f (k d ) = dr(k d ) dr*(k d ) (36) As shown in Fig. 1, nmerical reslts are in very good agreement with the theoretical prediction. This ensres that the nmerical scheme is able to reprodce the statistical properties of the system, i.e. the FDB is preserved in the discretized eqations. 5 BUBBLE-NUCLEATION SIMULATIONS The homogeneos vapor bbble ncleation is here stdied by means of the LLNS system of eqations presented in section 3. In particlar bbble ncleation is investigated in a metastable liqid enclosed in a cbic box with periodic bondary conditions. The flid is characterized by an eqation of state that recover the properties of a Lennard-Jones flid [4]. By introdcing the following reference qantities s= m as length = J as energy, m = kg as mass, U r = ( /m) 1/ as velocity, T r =s/u r as time, q r = /k B as temperatre, μ r = m / s as shear viscosity, c r =mk B as specific heat at constant volme and k r = μ r c r as thermal condctivity; the dimensionless fields are defined as r*= r/r r, q*= q/ q r, *=/U r Hence the dimensionless flxes (eqns. 14, 15, 3, 31) read C * = * * + ( * ) * r r C r I C r r + T* m* ( * * + ) 3 * * I, = r r q

9 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) 353 Figre 1: Static strctre factor for a capillary flid in a 3D system. On the first row the qalitative comparison between the theoretical prediction (left) and the nmerical reslt (right). On the second row a more qantitative comparison of the strctre factor at different fixed wave nmbers k y, k z. 3 q* = Cr* * r * k * * q*, dσ* = m q V* t* * W 1 3 m* q* * q * ( ), d * V* t* Tr W I k q = V* t* W * E* where C = lr r /s U r is a capillary nmber, fixed in or simlations as C = 5:44, to reprodce the exact vale of srface tension expected for a Lennard-Jones flid [4]. The system volme V* = (6) 3 has been discretized on an eqi-spaced grid, containing 5 cells in each directions. Several metastable conditions have been investigated and here we report the reslts of for different simlations at fixed temperatre q*= q* eq = 1.5 and different blk densities, exploring the whole range of metastable conditions at that temperatre. In this case the metastable * * * * * range of densities is ρl ρsat, ρ spin = [ 44.,.51 ], where ρ sat and ρ spin are the dimensionless satration and spinodal densities, respectively. Only 1 rns for each simlation have been carried ot in order to perform statistical averages of the reslts since the macroscopical observables, like the ncleation rates, have demonstrated to be statistically robst. A few snapshots of the system evoltion in the different metastable conditions are shown in Fig.. Starting from a homogeneos liqid phase, the hydrodynamic flctations lead the

10 354 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) Figre : Snapshots dring the ncleation process in two different thermodynamic conditions: on the left r*=.46 q*= 1.5, snapshots taken at time t* = 1, t* = 7, t* = 1, t* = 7; on the right r* =.48 q*= 1.5, taken at time t* = 1, t* = 1, t* = 5, t* = 8. system to spontaneosly decompose in two different phases. The vapor nclei starts forming with a complex shape, far from a spherical one, as observed in other works [1]. After reaching the critical size, they start expanding p to a stable eqilibrim state. This new thermodynamic state is characterized by the presence of several stable vapor bbbles in eqilibrim with the srronding liqid. The nmber and the dimension of the bbbles in the latter stage is strictly connected with the initial metastable condition, as evident in Fig.. In Fig. 3 we report a qantitative analysis of the time evoltion of the nmber of bbble nclei that exceed the critical size. The same two different metastable conditions of the snapshots reported in Fig. are here analyzed. As apparent, the dynamics of the system can be divided in two main stages: dring the first one the nmber of bbble increases almost linearly with time (with a constant rate); when the system is poplated enogh, the second stage consists in the expansion-coalescence dynamics when the nclei increase their size p to the eqilibrim radis and some of them coalesce with neighboring bbbles. As shown in Fig. 3, the metastable condition at r L =.46, which, among those we have considered, is the closest one to the spinodal hence the one with lower energy barrier to be overcome to ncleate bbbles leads to a more poplated system and clearly shows more freqent coalescence events. At r L =.48, instead, only the expansion of the bbbles is observed after reaching an almost stable nmber of bbbles. Figre 3: Nmber of bbbles having a size greater than the critical one vs. time, for the thermodynamic conditions r*=.46 q*= 1.5 on the right and r*=.46 q*= 1.5 on the left.

11 M. Gallo, et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 6, No. (18) 355 Figre 4: Ncleation rate comparison between flctating hydrodynamics nmerical reslts (red sqares) and CNT (green circles) at different metastable conditions. In the inset we report the comparison with other athors. The analysis of the first stage of the dynamics, when the nmber of bbble increases, gives access to another crcial observable, the ncleation rate J*, representing the nmber of bbbles formed per nit time and per nit volme. From an operative point of view, the ncleation rate is here calclated as the slope of the linear fit of the initial part of the crves in Fig. 3 [1]. The calclated ncleation rates at different metastable conditions are compared in Fig. 4 with the estimate given by the CNT, formlated here in the context of closed systems, and with some nmerical reslts of other athors [1]. As well-known CNT leads to a theoretical prediction for the energy barrier E and the ncleation rate JCNT = nl g/ mpexp( - E / kbq), where n L is the nmber density of the liqid phase, and the srface tension. Despite the strong assmptions in CNT, the rates calclated with or nmerical simlations are only slightly smaller than predicted by classical theory. An opposite behavior is observed far from the spinodal conditions where CNT over-estimates the ncleation rate, as confirmed by moleclar dynamic simlations in the microcanonical ensemble (NVE) [1]. 6 DISCUSSIONS AND CONCLUSIONS In this work the problem of homogeneos ncleation of vapor bbbles has been addressed by the means of FH, in particlar, a phase field description spplemented by stochastic flxes is exploited here to reprodce the main featres of ncleation process. In this context, starting from the Einstein s theory of eqilibrim field flctations, the intensity of the thermal noise is fixed to reprodce the statistical properties of the hydrodynamic fields. The evoltion of the system is governed by a set of stochastic processes reprodcing the Einstein-Boltzmann probability distribtion for the fields whose deterministic part is represented by the capillary Navier-Stokes eqations. Nmerical reslts show how in a homogeneos metastable liqid, the hydrodynamic flctations lead the system to spontaneosly decompose in two different phases. The ncleation rate strongly depends on the initial thermodynamic conditions, and a good agreement with CNT has been fond. ACKNOWLEDGEMENT The research leading to these reslts has received fnding from the Eropean Research Concil nder the Eropean Union s Seventh Framework Programme (FP7/ 7-13)/ERC Grant agreement no. [339446].

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