The thermodynamic principle determining the interface temperature during phase change
TTHE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACETEMPERATURE DURING PHASE CHANGE
TOM Y. ZHAO † AND NEELESH A. PATANKAR † , ∗ Abstract.
What is the interface temperature during phase transition (for instance, fromliquid to vapor)? This question remains fundamentally unresolved. In the modeling of heattransfer problems with no phase change, the temperature and heat flux continuity conditionslead to the interface temperature. However, in problems with phase change, the heat fluxcondition is used to determine the amount of mass changing phase. This makes the interfacetemperature indeterminate unless an additional condition is imposed. A common approachin the modeling of boiling is to assume that the interface attains the saturation temperatureaccording some measure of pressure at the interface. This assumption is usually appliedeven under highly non-equilibrium scenarios where significant temperature gradients andmass transport occur across the interface. In this work, an ab-initio thermodynamic prin-ciple is introduced based on the entropy production at the interface that fully specifies theassociated temperature under non-equilibrium scenarios. Physically, the thermodynamicprinciple provides a theoretical limit on the space of possible phase change rates that canoccur by associating the mass flux with a corresponding interfacial entropy production rate;a stronger statement is made that a system with sufficient degrees of freedom selects themaximum entropy production, giving the observed phase change rate and associated inter-face properties. This entropic principle captures experimental and computational values ofthe interface temperature that can deviate by over from the assumed saturation values.It also accounts for temperature jumps (discontinuities) at the interface whose difference canexceed ◦ C . This thermodynamic principle is found to appropriately complete the phasechange problem. 1. Introduction
In phase transition (e.g. liquid to vapor), the fundamental principle that dictates the temperature at theinterface between the two phases has been debated and it remains an open question.The interface temperature is presumably intrinsically connected to the rate of phase change as well as theinterface velocity. This is critical information to understand and model phase change heat transfer in varietyof heat exchange applications. This information is also crucial in designing water purification processeslike membrane distillation, energy storage systems using latent heat batteries, , additive manufacturingtechniques involving molten metal jets , and phase change memory technologies for nonvolatile solid statestorage. ( † ) Northwestern University, Department of Mechanical Engineering: 2145 Sheridan Road, Evanston, Illi-nois 60208, USA ( ∗ ) E-mail: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b THE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE
Theoretical and computational models typically assume that the interface between the two phases at-tains the saturation temperature.
Experimental work using thermocouples with thicknesses on the orderof microns have resolved interface temperatures that are found to deviate significantly from the satu-ration assumption. Theoretical attempts to find a different interface condition, to replace the saturationtemperature condition, include the kinetic theory and the statistical rate theory. Kinetic theory expresses the entropy generation at the interface using a constitutive relationship with theparameter φ representing the kinetic mobility, or the relative strength of molecular attachment to a surface.However, the evaluation of φ requires an empirical evaporation coefficient α , which is difficult to measureand can deviate by over three orders of magnitude from the theoretical value of unity. The kinetic theoryalso underestimates the temperature jump measured in experiment by 3 to 4 orders of magnitude. The statistical rate theory uses quantum mechanics to describe a relationship between the rate of phasechange and the change in entropy associated with a molecule transferring from the liquid to the vaporphase. After measuring the interface properties (including temperatures) of the liquid and vapor sidefrom experiment, the mass flux from phase change can be calculated based upon the material properties ofthe fluid, the molecular vibrational frequencies and the partition function for the fluid molecule. From acomputational standpoint or generally in scenarios where the interface temperatures and properties are notknown a priori, the rate of phase change cannot be obtained via this method and vice versa.In this work, we determine the thermodynamic relationship between the temperatures of both phases atthe interface and the rate of interfacial entropy production σ . This mapping provides a theoretical limit onthe space of possible interface temperatures and phase change rates; there exists a maximum rate of entropyproduction due to the competition or inverse relationship between the entropy jump from phase change andthe heat flux carried away from the interface. This space of possible σ is bounded from below by the secondlaw of thermodynamics σ ≥ .Finally, we propose a stronger thermodynamic principle that fully determines the interface temperaturesduring the time evolution of a phase change system. It is found that the interface temperatures whichmaximize the entropy production rate σ capture the full range of both experimental and computational dataon interfacial properties of different fluids and solids under evaporation, condensation, and freezing processes.This thermodynamic principle prefaced on the maximum rate of entropy production also determines therate of phase change as a function of material properties and temperature boundary conditions in the farfield; properties and field variables at the interface are not known or fixed a priori.The proposed entropy condition closes the formerly incomplete problem of phase change under nonequi-librium scenarios. 2. The missing interface condition
The complete problem without phase change.
We first consider a well-posed problem comprisinga two component system in which no phase change can occur (for instance, with water and oil). Figure 1shows a finite, one-dimensional system where the number of equations and boundary conditions can be easilycounted. The governing equations for the incompressible species A and B across the two phase interface are ∂u c ∂x = 0 , (1) ρ c ∂u c ∂t = − ∂p c ∂x , (2) ρ c c p,c ∂T c ∂t + ρ c c p,c u c ∂T c ∂x = k c ∂ T c ∂x , (3)where the subscript c ∈ [ A, B ] . The velocities, pressures and temperatures in each phase are denoted u c , p c and T c respectively. Similarly, k c , ρ c and c p,c refer to the thermal conductivity, density and specific heat HE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE 3 (cid:5) (cid:3) (cid:1) (cid:2) (cid:4)
Figure 1.
Finite, one-dimensional system with two fluid components A and B . The inter-face between the two phases is located at x = d ( t ) . A motionless wall bounds the domainat x = 0 , and a moveable piston at x = L ( t ) + d ( t ) controls the pressure in the system. Thewall is held at constant temperature T W and the piston at temperature T P . The radius ofcurvature of the interface R can be nonzero.capacity at constant pressure. The boundary conditions at the motionless wall areat x = 0 , u A = 0 , (4) T A = T W . (5)The boundary conditions at the piston areat x = L ( t ) , p B = p P , (6) T B = T P . (7)The interface conditions areat x = d ( t ) , ρ A (cid:0) u A ( d ) − u S (cid:1) = ρ B (cid:0) u B ( d ) − u S (cid:1) , (8) ρ A (cid:0) u A ( d ) − u S (cid:1) = 0 , (9) [ Q ] = − k B ∂T B ∂x + k A ∂T A ∂x = 0 , (10) T A ( d ) = T SA = T SB = T B ( d ) , (11) p A = p B + γκ, (12)where u S is the interface velocity, T W is the wall temperature, T P is the piston temperature, p P is the pistonpressure, and [ Q ] is the jump in heat flux across the interface, γ is the surface tension between the twospecies and κ is the interface curvature ( κ = 1 /R in one dimension). The temperatures T SA , T SB representthe respective values for each species A, B at the two-phase interface. Note that for simplicity, we have madethe assumptions that the interface is massless, surface tension is constant and there is no temperature jumpacross the interface .There are 6 unknown field variables ( u A , T A , p A , u B , T B , p B ) and six sets of conservation equations(eqn. 1, 2, 3) for components A and B. The four mass and momentum conservation equations (eqns. 1and 2 for components A and B) are first order differential equations that each require a single boundarycondition (eqn. 4, 6, 8, 12). The two energy equations (eqn. 5 for components A and B) are second orderdifferential equations that each require two boundary conditions (eqn. 5, 7, 10, 11). Without phase change,the mass continuity condition at the interface (eqn. 9) specifies the interface velocity u S = u A ( d ) = u B ( d ) and completes the problem. THE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE
The ill-posed problem with phase change.
Consider a scenario where phase change occurs betweenthe two species (for instance, with liquid water and water vapor). The governing equations are the same,but the interface conditions change . Conservation of mass at the interface statesat x = d ( t ) , ρ A (cid:0) u A ( d ) − u S (cid:1) = ρ B (cid:0) u B ( d ) − u S (cid:1) . (13)From energy balance at the interface, the mass flux due to phase change is given as u B − u A = (cid:0) /ρ B − /ρ A (cid:1) [ Q ]∆ H = (cid:18) − k B ∂T B ∂x + k A ∂T A ∂x (cid:19) (cid:18) /ρ B − /ρ A ∆ H (cid:19) . (14)For simplicity, the temperatures of the two phases at the interface are often assumed to be continuous, suchthat T A ( d ) = T SA = T SB = T B ( d ) . (15)Momentum conservation at the interface gives p A = p B + γκ + (1 /ρ A − /ρ B )( [ Q ]∆ H ) , (16)where viscous terms are neglected in the momentum conservation eqn. 16 in the 1D limit. Here, ∆ H = h A − h B is the latent heat of phase change expressed as the difference between the enthalpy of phases A andB, h A and h B respectively.As the mass conservation eqn. 13 is no longer identically equal to zero due to the possible phase changebetween species A and B, the interface velocity u S becomes unspecified. The energy conservation conditioneqn. 14 at the interface can be borrowed to fix the value of u S , since the mass flux is determined by thethermal energy diffused to the interface relative to the latent heat of phase change.However, once the heat flux interface condition is used to determine the rate of phase change, the interfacetemperature T S becomes unspecified. Thus, the problem becomes ill-posed due to either the missing interfacevelocity u S or temperature T S .3. The entropy condition at the two-phase interface
The entropy condition at the two phase interface has been explored in the literature , but the resultingstatement on the interfacial entropy production rate σ (in units of energy per unit time and per unit area)is weak when referencing the second law of thermodynamics in or near equilibrium: σ ≥ . We refer tothis inequality as weak in that it is not sufficient to specify a particular interface temperature or velocity.Additionally, the statement of the second law leaves the entropy production rate unbounded in a semi-infiniterange.In this section, we will first present without reference to proposed laws or lemmas in nonequilibriumthermodynamics the entropy production rate across a two phase interface. Then for the simplified case ofphase change across an interface without a temperature jump, it can be physically shown that the entropyproduction term is bounded from above by a maximum value. The range for σ becomes finite, leading to astronger statement on the possible macrostates accessible to the system. HE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE 5
The rate of entropy production σ at the two phase interface is given in general by ( T S ) σ = ˙ m A (cid:18) ( T SA − T S ) s A + g A − g S + 0 . v A − v A · v S + v S ) (cid:19) + ˙ m B (cid:18) ( T SB − T S ) s B + g B − g S + 0 . v B − v B · v S + v S ) (cid:19) + q A · ˆ n A (1 − T S /T SA ) + q B · ˆ n B (1 − T S /T SB ) + T S q S · ∇ S (1 /T S ) − ˙ m A ρ A ( τ A · ˆ n A ) · ˆ n A − ˙ m B ρ B ( τ B · ˆ n B ) · ˆ n B , (17)where T SA and T SB are the absolute temperatures of the respective phases at the interface, and T S is theabsolute interface temperature. Similarly, the variables g A,B,S are the free enthalpies, s A,B are the entropiesper unit mass and q A,B,S are the heat fluxes. The unit normal vectors ˆ n A and ˆ n B are directed towards theinterface; this convention agrees with the concept of a local entropy source at the interface . The stresstensors in each species are given by τ A and τ B , while the surface gradient is represented by ∇ S . Finally, themass fluxes across the interface are denoted as ˙ m A = ρ A ( v A − v S ) · ˆ n A and ˙ m B = ρ B ( v B − v S ) · ˆ n B . Eqn.17 for the entropy source term comes from combining the evolution equations for mass, momentum, energyand entropy at the interface .Next, we can find a simplified expression for the entropy production rate in a 1D system across a massless,infinitesimally thin interface (Fig. 1). Let c ∈ [ A, B ] . By definition, the sum of the free enthalpy g c and theproduct of temperature with the entropy density T Sc s c of each phase is simply the saturation enthalpy, sincethe temperature and pressure dependencies of the two terms cancel to give g c + T Sc s c = g c, sat + T Sc s c, sat ( T Sc ) = h c, sat . (18)The mass flux across the interface due to phase change ˙ m is specified by the energy balance equation at theinterface (eqn. 14) with ˙ m = 1 h A, sat − h B, sat ( q A · ˆ n A + q B · ˆ n B ) = − ˙ m A = ˙ m B . (19)We can thus simplify the full expression for the entropy production rate at the two phase interface to T S σ M = − ˙ m (cid:18) − T S s A (cid:19) + ˙ m (cid:18) − T S s B (cid:19) + q A ( − T S /T SA ) − q B ( − T S /T SB ) − ˙ m (cid:18) ρ A − ρ B (cid:19) . (20)In this 1D expression for the entropy production rate σ M , the dependence of the heat fluxes q c and massflux ˙ m on the interfacial temperatures T S , T SA , T SB can be estimated from the boundary conditions in thefar field and the assumption of linear temperature profiles in each phase. On the other hand, the variationof each species’ entropy density s c with interface temperature can only be specified after more informationis known about the composition of each phase.For the particular case of phase change between vapor (species V) and liquid (species L), the entropydensities can be expressed in terms of the pressure and temperature in each phase as ( T S ) s V = ( T S ) s V, sat − RT S ln (cid:18) p V p sat ( T SV ) (cid:19) , (21) ( T S ) s L = ( T S ) s L, sat − ρ L ( p L − p sat ( T SL )) , (22)where ρ L is the density of the liquid phase, R is the specific gas constant and p sat is the saturation pressureassociated with the interface temperature T SV or T SL of each phase. THE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE
Following the 1D formulation of σ M (eqn. 20), the entropy production rate at the interface of a vapor-liquid system becomes σ LV ( T SV , T SL ) = ˙ m (cid:18) s V, sat − s L, sat − R ln (cid:18) p V p sat ( T SV ) (cid:19) + 1 T SL ρ L ( p L − p sat ( T SL )) (cid:19) + q V ( − /T SV ) − q L ( − /T SL ) − ˙ m T S (cid:18) ρ V − ρ L (cid:19) . (23)This poses the question: what is a reasonable choice for T S ? For simplicity, we take T S ≈ T SL in this workon the basis that the Knudsen layer in the liquid phase is significantly smaller than that in the vapor .3.1. Physical insight into the existence of a maximum entropy production rate.
The entropyproduction rate σ at the two phase interface is bounded below by the second law of thermodynamics. In asimplified 1D system, we now show that σ is bounded above by a maximum, finite value.Consider the setup introduced in Fig. 1, where component A is water vapor (V) and component B is liquidwater (L). The wall at x = 0 is superheated to temperature T W , while the piston at x = L is maintained atthe saturation temperature T P = T sat corresponding to the applied piston pressure p = p P .We make further assumptions to simplify the analysis and provide physical intuition into the competingeffects that drive the entropy production rate to achieve a finite maximum value. First, let us suppose thatthe temperature in each region is diffusion dominated, such that the profile for T is linear in the vapor andliquid phases. Next, we take T S = T SV = T SL , acknowledging that this should only hold in special instancessuch as when the system is in equilibrium. With this, the mass flux at the interface (eqn. 24) becomes ˙ m = ρ L ( u S − u L ) = 1∆ H (cid:18) k L T P − T S L − k V T S − T W d (cid:19) . (24)Meanwhile, the entropy production rate at the interface (eqn. 23) is simplified to σ s LV ( T S ) = ˙ m (cid:18) s V − s L (cid:19) − ˙ m ∆ HT S , (25)where we have neglected the cubic term in mass flux, since ˙ m << is typically a good approximation. If thelengths d , L , the temperatures T P and T W as well as the piston pressure p P in the far field are both fixed,then σ is only a function of T S .As T S increases, the magnitude of ˙ m decreases along with the net heat transfer to the interface. Meanwhile,the difference s V − s L decreases as well (eqn. 21 and 22) with larger T S . Thus the first term in the simplifiedinterfacial entropy evolution σ s LV is inversely proportional to the interface temperature.The second term - ˙ m ∆ HT S is directly proportional to T S . As the interface temperature increases, ˙ m , ∆ H and T S all decrease, such that the negative of their product increases. Due to the competition between theentropy flux due to phase change and the heat flux from the interface in eqn. 25, σ s LV reaches a maximumvalue with respect to T S , analogous to how the change in Gibbs free energy goes through a minimum inthe classical heterogeneous nucleation theory due to the competition between surface tension and volumetricfree energy as the radius of the nucleus varies .Fig. 2 shows that σ sLV achieves a finite maximum value at an interface temperature T ∗ S satisfying T sat Figure 2. The entropy evolution at the interface (eqn. 25) normalized by the maximumvalue ( σ (cid:48) sLV = σ sLV max TS ( σ sLV ) ) as a function of the normalized interface temperature T (cid:48) S . Theentropy production rate σ (cid:48) C reaches a maximum between T (cid:48) S = 1 (the wall temperature) and T (cid:48) S = 0 (the saturation temperature) due to opposing dependencies between the mass fluxand the difference in phase free entropies on the interface temperature. The temperaturecorresponding to the maximum σ (cid:48) sLS drops towards the saturation value as the distance d between the wall and interface increases to macroscale lengths. The vapor and liquidproperties of water (thermal conductivities, latent heat of phase change, etc.) as a functionof temperature were referenced from NIST . The boundary conditions and location wereset to L = 0 . m, T wall = 550 K, T sat = 373 . K, p P = 1 atm. Linear temperature profilesare assumed in the liquid and vapor domains.3.2. The maximum entropy principle at the two-phase interface. Having demonstrated that theinterfacial entropy production rate achieves a maximum, finite value for phase change systems, we nowpropose an entropic condition to determine the exact temperatures at the interface.In nonequilibrium thermodynamics, it has been proposed that a process follows the path along which theentropy produced in the system at each step is maximized, subject to conservation laws as well as externalconstraints such as prescribed thermodynamic forces or fluxes .The maximum entropy production principle (MEPP) therefore seeks to generalize the inequality formu-lation of the second law of thermodynamics, which only states that the entropy production is either positivefor irreversible processes or zero for reversible ones ; alone, the second law gives a possible range of discon-tinuous interface temperatures T SA and T SB that satisfy σ ≥ , but does not pinpoint an actual value for T SA , T SB .MEPP has been explored with series of proofs in the literature ranging from variational analyses tostatistical mechanics considerations . Functionally, MEPP can be used as a variational principle tosolve the Boltzman equation ; in climate models to predict surface temperatures and cloud coverage ; insolid state physics to predict dendritic structure and growth rates . THE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE Here, we propose that the MEPP closes the phase change problem at the two phase interface, in that themaximum entropy production rate can be used to exactly specify the interface temperatures. In general,suppose that the discontinuous temperatures T ∗ SA and T ∗ SB give an optimal solution to max( σ ( T SA , T SB ) )while satisfying the imposed constraints on thermodynamic fluxes or forces and conservation laws, representedby the series of conditions F i ( T SA , T SB )=0 for i ∈ [1 , , ..., N ] ; in general, the dependence of σ ( T SA , T SB ) onthe interface temperatures is given by eqn. 17. Then T ∗ SA and T ∗ SB are the temperatures observed at the twophase interface when the diffusive time scale τ D of the system is smaller than the evolutionary time scale τ E .Here, we take τ D = d c /α c to be the thermal diffusion time scale such that α c is the thermal diffusivityof phase c ∈ [ A, B ] at the associated interface temperature and d c is the relevant length scale occupied byphase c . The evolutionary time scale τ E = d c /u S is the length scale divided by the interface speed u S ,which likewise is a function of the interface temperature. We find that for systems that satisfy τ D < τ E , theinterface temperatures and phase change rates predicted by MEPP capture the corresponding data obtainedfrom experiment.The physical intuition behind this closure condition is that the trajectory of states corresponding to themaximum entropy production rate at each step reflects the most probable path observed in the system underthe constraints imposed by fixed thermodynamic fluxes or forces as well as conservation laws . Endresshowed that for the Schlögl model of a first order phase transition with noise, the probability of observinga particular trajectory at nonequilibrium steady state increases exponentially with the entropy productionrate . Specifically, the most probable trajectory of states is the one that maximizes the entropy productionrate, while minimizing the classical and stochastic action along that path. The latter condition representsthat the governing equations describing the system are satisfied.4. Results The interfacial temperatures that maximize the entropy production rate can be used to describe phasechange features in both simulation and experiment. Only properties in the far field need to be know a priori inorder to predict temperatures and mass fluxes at the interface. The efficacy of the proposed thermodynamicclosure principle prompts its use in experiment and continuum simulation to capture the evolution of theinterface under nonequilibrium behavior.4.1. Liquid-vapor interface temperature at nanoscale. In this work, we use molecular dynamics (MD)simulations to explore nonequilibrium phase change across the liquid-vapor interface of water within nanome-ters of the superheated wall. The simulation setup in Fig. 1 is established across a distance L + d of nm,with phase A adjacent to the superheated wall referring to water vapor and phase B adjacent to the pistonreferring to liquid water; further details are provided in the Methods section below.Figure 3 demonstrates that the liquid and vapor side temperatures ( T ∗ SL and T ∗ SV ) associated with themaximum entropy production rate at the two phase interface (eqn. 23) well captures the interfacial tem-peratures T SL, MD and T SV, MD measured using molecular dynamics as the system evolves during the phasechange process. The constant saturation temperature T sat = 373 . K corresponding to the pressure p P = 1 bar applied at the piston completely fails to model the non-constant, discontinuous dynamics of the interfacetemperatures.In the window of time presented in which the vapor film thickness d increases from . to nm, the twophase interface shifts away from the superheated wall mainly due to expansion of the superheated vapor. Infact, the average mass flux measured via MD when the interface position . < d < nm is ˙ m = − . kgm s ,which means that condensation occurs at the interface. The mass flux that maximizes the entropy productionrate averaged over . < d < nm is ˙ m ∗ = − . kgm s . Thus, this entropic interface principle accuratelydescribes the physically unintuitive mode of condensation at the interface near a superheated vapor andestimates the correct order of magnitude of the phase change rate of a nonequilibrium, nanoscale process. HE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE 9 Interface Position (nm) I n t e r f a c e T e m pe r a t u r e ( K ) Figure 3. The evolution of liquid and vapor side interface temperatures as a function ofthe interface distance d from the superheated wall, predicted by the maximum entropy rateprinciple proposed in this work and measured via MD. The temperatures as averaged over6 molecular dynamics simulations from independent initial conditions are well captured bymaximizing the interfacial entropy production rate at each time step. The vapor and liquidproperties of water (thermal conductivities, latent heat of phase change, etc.) as a functionof temperature were referenced from NIST . The boundary conditions were T W = 575 K, T P = 373 . K, p P = 1 bar. Linear temperature profiles were assumed in the liquid andvapor domains.Possible sources of error in this analysis include the deviation of associated fluid properties such as thermalconductivity or enthalpy from bulk values for the SPC/E water model compared to real water as well as forthe nanoscale film compared to the bulk phase . Additionally, it was assumed that the temperature profilesin both the liquid and vapor follow linear regimes.In this system, as in all MD simulations and experimental outcomes included in this work, the thermaldiffusion time scale τ D is smaller than the evolutionary time scale of the interface τ E . Here, τ DL ≈ − s inthe liquid phase and τ DV ≈ − s in the vapor phase are one and three orders of magnitude smaller than τ E , respectively. Intuitively, this means that the system has time to resolve interfacial temperatures thatmaximize the entropy production rate before the interface shifts to a new location, such that the interfacecan be considered stationary with respect to the entropy production rate.4.2. Ice-liquid interface temperature at nanoscale. The agreement between the proposed entropyproduction interface principle and molecular dynamics simulation holds for phase change between solidand liquid water as well. Figure 4 shows that the ice and liquid side interface temperatures T ∗ SI and T ∗ SL corresponding the maximum interfacial entropy production rate well describe the the highly non-equilibriuminterface temperatures measured by MD during the freezing process.Note that the entropy production rate σ IL at the interface between the ice and liquid phases can bederived from the general 1D expression (eqn. 20) by taking the pressure dependence of the ice phase entropyto be ( T S ) s I = ( T S ) s I, sat − ρ I ( p I − p sat ( T SI )) . (26) The interfacial entropy production rate σ IL can then be expressed as σ LV ( T SI , T SL ) = ˙ m (cid:18) s L, sat − s I, sat − T SL ρ L ( p L − p sat ( T SL )) + 1 T SI ρ I ( p I − p sat ( T SI )) (cid:19) + q L ( − /T SL ) − q I ( − /T SI ) − ˙ m T S (cid:18) ρ L − ρ I (cid:19) , (27)where the difference in liquid and ice phase entropies s L, sat − s I, sat for the mW water model was given byHolten et al .The interface temperatures appear approximately continuous ( T SI, MD ≈ T SL, MD ≈ T S, MD ≈ K), butnonetheless form non-monotonic profiles with the boundary conditions in the liquid and ice domains T BC ≈ K. Thus, the simplest assumption of a constant temperature distribution in both regions T S, MD ≈ T BC fails. The freezing point of the mW water model T F = 274 . K exceeds T S, MD by around K and is thusalso a poor predictor.The best estimate for the interface temperature is obtained by maximizing the entropy produced at theice-liquid interface while assuming linear temperature profiles in both the liquid and vapor domain. Theinterfacial velocity associated with this maximum entropy rate principle u ∗ S = 4 . m/s also agrees with theinterface velocity measured via molecular dynamics u S, MD = 4 . m/s.4.3. Experiments at mesoscale. The stochastic rate theory (SRT) suggested by Ward et al. advancedmeticulous experiments to measure the interfacial temperature jump between two phases . Asnoted, the SRT gives reasonable estimates for the temperature jump if the interface temperature on eitherthe liquid ( T SL ) or vapor ( T SV ) side as well as the mass flux across the interface is measured first.Fig. 5 shows that the experimental interface temperatures measured via micro-thermocouples are wellcaptured by the vapor and liquid interface temperatures T ∗ SV , T ∗ SL that maximize σ LV (eqn. 23). Therefore inapplications where interface properties are not available a priori, the maximum entropy production conditionmay be used to pinpoint the absolute temperatures of both phases at the interface and the mass flux due tophase change. Only far field properties such as temperature and pressure must be input into this analysis.The bulk temperature profiles in Fig. 5 A are non-monotonic due to the interfacial temperature jump,and the liquid side interface temperature T SL is not bounded by the temperature conditions in the far field.This escapes a straightforward description from existing theory and heretofore falls under the umbrella of’nonlinear, transient evolution’. However, the nonequilibrium thermodynamic mechanism proposed in thiswork suggests that the interfacial liquid and vapor temperatures are selected for by maximizing the rateof entropy produced due to phase change; this non-monotonic and "unbounded" behavior of the interfacetemperatures is therefore deterministic, rather than stochastic.Another set of experimental results are visualized in Fig. 6. The temperatures of each phaseat the interface in Fig. 6 A and mass fluxes due to phase change in Fig. 6 B are well described by thethermodynamic principle proposed in this work. This agreement holds for evaporation and condensation ofwater under laminar and turbulent conditions, as well as for evaporation of octane. Although temperaturejumps in these sets of experiments are typically smaller, the nonequilibrium phase change processes examinednonetheless exhibit the distinct unbounded characteristic wherein the interface temperatures can lie outsidethe range of the far field temperature conditions. In all cases, only far field pressures and temperatures wereused as input into the maximum entropy rate principle; all properties on the interface were determined bymaximizing the entropy produced σ LV (eqn. 23).5. Discussion To gain a physical understanding of nonequilibrium phase change in a liquid-vapor system, we observe theinterfacial temperature jump, mass flux and entropy production rate associated with the average pressure HE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE 11 Position (nm) T ( K ) Ice Liquid water 240 260 280 300240 Figure 4. A ) The temperature profile of an ice-liquid system as measured via moleculardynamics by Wang et al using the mW water model and as calculated by the maximumentropy production interface condition proposed in this work. Note that molecular dynamicsplaces the interface temperature T SL, MD ≈ T SI, MD ≈ K to be K greater than theboundary temperatures T BC ≈ K but around K less than the freezing point of mWwater T F = 274 . K. The temperature distribution is non-monotonic and yet cannot beapproximated accurately by the saturation temperature assumption. B ) The interfacialentropy production rate exhibits a maximum value at T ∗ SL = T ∗ SI = 263 . K. The ice andliquid properties of water (thermal conductivities, latent heat of phase change, etc.) weredrawn from the mW water properties reported by Wang et al. . Linear temperature profileswere assumed in the liquid and ice domains.and temperature at the interface (Fig. 7). For this example, the widths of both the vapor and liquid domainsare fixed at mm. The piston cooling the liquid reservoir is held at a constant temperature of K, whilethe wall temperature is varied for each value of pressure imposed on the system (Fig. 7 D ). Given theseboundary conditions, the maximum entropy production rate is used to determine the interface temperatures T ∗ SL and T ∗ SV , the average of which is plotted on the x axis. Similarly, the average pressure at the two phaseinterface is tabulated on the y axis.Fig. 7 A overlays the interfacial temperature jump ∆ T S = T ∗ SV − T ∗ SL on the average pressure-temperaturediagram at the interface. It is notable that interfacial temperature continuity is a special condition confinedto a single contour, whereas the majority of the phase space is dominated by the existence of a temperaturediscontinuity. The sign of this jump, whether T ∗ SV > T ∗ SL or vice versa, cannot in general be predicted bythe equilibrium binodal curve. The assumption of temperature continuity is not generally reliable when thelocal interface rests in equilibrium.Another way to characterize the interface is to look at the mass flux due to phase change (Fig. 7 B ).The contour along which no phase change occurs is likewise ill predicted by the binodal in general; all threecurves (binodal, ˙ m = 0 , and ∆ T S = 0 ) only intersect at the point T W = T ∗ SV = T ∗ SL = T P , which reflectsconstant temperature profiles in both phases. This corresponds to the bulk system being in equilibrium (Fig.7 C ). -600 -400 -200 0 200 400 600298.75298.8298.85298.9298.95299299.05 298 300 302 304 306 308 310298300302304306308310 A B Liquid Vapor Figure 5. A ) The experimental temperature profile of a two phase water system is welldescribed by the predicted interface temperatures for the liquid ( T SLM ) and vapor ( T SV M )side using the maximum entropy principle. The interface is located at x = 0 micron, where apronounced temperature jump creates a non-monotonic temperature distribution such that T SL is not bounded by the liquid or vapor temperatures in the far field. B ) The interfacialtemperature jump is well described by the maximum entropy principle. Only data sets thatprovided all necessary information such as boundary conditions, distances to the interface,etc. were included in the plot to avoid using any unknown properties to ’fit’ the data. Lineartemperature profiles were assumed in the liquid and vapor domains. The vapor and liquidproperties of water and octane (thermal conductivities, latent heat of phase change, etc.)as a function of temperature were referenced from the IAPWS formulation .The two contours ˙ m = 0 and ∆ T S = 0 separate the interfacial phase space into four regions. In thetop left and bottom right sectors, phase change conforms to our physical intuition around the binodal.That is, as temperature increases or pressure decreases past the coexistence curve, vapor becomes the bulkstable phase and vice versa. The top right and bottom left sectors characterize the metastable phases thatare involved in processes like capillary condensation below and capillary evaporation above the binodal.Thus this entropic interface condition gives a complete description of possible phase change processes innonequilibrium scenarios.Another issue of note is that Fig. 7 C displays agreement with the minimum entropy production rateprinciple . This principle says that over the relaxation time scale of a stationary nonequilibrium systemwith some thermodynamic forces fixed and others free, the thermodynamic fluxes in the system conjugate tothose unfixed forces will disappear. This drives the system toward the minimum of the entropy productionrate, which occurs at equilibrium. Indeed, the global minimum in the entropy production rate is associatedwith the equilibrium interfacial temperature and pressure, at the intersection of the two contours ˙ m = 0 and ∆ T S = 0 . If the wall temperature is allowed to evolve over time from the initial condition (unfixed) ratherthan be held to a constant value, the system would eventually relax to this equilibrium state in which thetemperature profiles in both phases are constant and equal.On the shorter time scale, or if all thermodynamic forces are held constant, the associated thermodynamicfluxes adjust in order for the system to achieve the maximum entropy production rate for each specific, HE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE 13 260 270 280 290 300255260265270275280285290295300 0 2 4 6 8 10 12-0.500.511.522.533.54 Figure 6. A ) The experimentally measured temperature jump as well as the distinct liq-uid and vapor side temperatures at the two-phase interface are captured by the maximumentropy principle. This general agreement between experiment and theory holds for evapo-ration (Exp 1, 3, 7, 8, 9) and condensation (Exp 4, 5) of water; evaporationof octane (Exp ; evaporation of water under turbulent conditions (Exp ; andevaporation of water heated on the vapor side (Exp 10, 11, 12) . B ) The average phasechange rate at the interface in units of mass per area, per unit time drawn from the sameexperiments are also captured by the maximum entropy principle. Only data sets thatprovided all necessary information such as boundary conditions, distances to the interface,etc. were included in the plot to avoid using any unknown properties to ’fit’ the data;data points that overlapped significantly in the plot were also excluded for clear visualiza-tion. The vapor and liquid properties of water and octane (thermal conductivities, latentheat of phase change, etc.) as a function of temperature were referenced from the IAPWSformulation .average interfacial pressure and temperature plotted in Fig. 7 C . The minimum entropy production rateprinciple suggests that a stationary nonequilibrium system with sufficient degrees of freedom will tend towardminimum value of the entropy production rate over the relaxation time scale, whereas the maximum entropyproduction rate principle tells us that a nonequilibrium system on a shorter time period or under constantthermodynamic forcing will find the state corresponding to the maximum in the entropy production rateas the fluxes vary. Indeed, both principles can apply simultaneously, in that a stationary nonequilibriumsystem may approach the state associated with the minimum entropy production rate on a longer time scaleby taking individual steps over a short time scale that maximize the entropy production rate at each step,while satisfying thermodynamic constraints and governing laws.Thus, the maximum entropy production rate allows us to accurately pinpoint the interfacial propertiesof a nonequilibrium system undergoing phase change in a thermodynamically consistent manner. It is themissing condition needed to close the two phase problem when phase change occurs across the interface. Inaddition, the minimum entropy production rate informs the trajectory of a stationary phase change systemover longer time scales, within the permissible phase space set by the presence of fixed thermodynamic forcesor fluxes. 300 350 400 45010 -7-6-5-4-3-2-10123 300 350 400 45010 300 350 400 45010 -0.02-0.015-0.01-0.00500.0050.010.015 300 350 400 45010 -100-50050100150200250300350 A) Δ" = " − " B) ̇) EvaporationCondensation Δ" > 0Δ" < 0Evap.Δ" < 0 Cond.Δ" > 0Cond.Δ" < 0 Evap.Δ" > 0 C) log( ) D) " Figure 7. The nonequilibrium properties at the two phase interface mapped onto theaverage interfacial pressure and temperature. The distance of the interface from the wallis 3 mm, corresponding to the width of the vapor domain. The distance of the interfaceto the piston is likewise 3 mm, corresponding to the width of the liquid domain. Thepiston cooling the liquid reservoir is held at a constant temperature of K, while the walltemperature is varied. A ) The temperature jump at the liquid vapor interface is generallynonzero. The interface temperature is only continuous along the black contour. B ) Themass flux due to phase change across the interface is positive for evaporation and negativefor condensation. No phase change occurs along the white contour. C ) The logarithmof the entropy production rate at each interface pressure and temperature shows that theglobal minimum is located at equilibrium on the binodal, where the continuous interfacetemperature and zero mass flux contours intersect. The point of intersection corresponds toa constant temperature profile equal to the far field piston temperature K in the liquiddomain. D ) The wall temperature T W is varied from K to K at different pressuresto obtain the phase diagrams of the interfacial temperature jump, mass flux and entropyproduction rate. The vapor and liquid properties of water (thermal conductivities, latentheat of phase change, etc.) as a function of temperature were referenced from the IAPWSformulation . HE THERMODYNAMIC PRINCIPLE DETERMINING THE INTERFACE TEMPERATURE DURING PHASE CHANGE 15 Conclusion The maximum entropy production rate at the interface closes the phase change problem by determiningthe interface temperature and velocity. The predictions from the proposed entropic interface condition wellcapture nanoscale temperatures and mass fluxes for liquid, vapor and solid phase change at the nanoscale.The condition also accurately predicts experimental data on interface temperature jumps and mass fluxesfor different fluids under both turbulent and laminar flows at mesoscale. This agreement suggests thatat most length and time scales, the interface properties are dominated by a deterministic thermodynamicprinciple (that of entropy production maximization) rather than stochastic or transient behavior which mustbe modeled probabilistically.The maximum entropy principle can be used directly to design phase change systems to achieve desiredmass fluxes or interface properties for the applications discussed prior. It can also be used to model nanoscaleand mesoscale effects in continuum level simulations of multiphase flows, where the saturation temperaturehas been the standard approximation. 7. Methods To gauge interface properties under nanoscale evaporation conditions as set up in Fig. 1, moleculardynamics simulations were carried out with LAMMPS software. A total of 32085 SPC/E molecules ofliquid water were equilibrated at saturation temperature T = 373 . K in the canonical ensemble withconstant pressure (1 atm) imposed by a piston constrained to move only in the direction orthogonal to thebottom surface . The solid surface and piston were constructed using two graphene sheets with armchairlattice orientation, and the interaction between these planes and the SPC/E water molecules was governedby the 6-12 Lennard Jones pair potential with the depth of the potential well fixed at 0.05 kcal/mole . Afterthis equilibration step, the liquid water adjacent to the bottom surface was heated to a target temperatureof T = 575 K, whereas the liquid adjacent to the piston was held at constant, saturation temperature tosimulate nonequilibrium heat transfer conditions . The lateral simulation box size in the plane parallel tothe surface and piston was 8 nm by 8 nm. The perpendicular dimension varied as the vapor film thicknessevolved in time. 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