Analysis of transition state theory for condensation

NOTES1 Reaction coordinate and the activated complex ofAnalysis of transition statethe condensationtheory for condensationAs shown in fig. 1, we consider the condensation ofa vapor molecule. Assuming that in the liquid-vapor ∞-WANG Zunjing, CHEN Min & GUO Zengyuanexistent system, the molecular average distance of theliquid cluster is R, the average distance of the condensingDepartment of Engineering Mechanics, Tsinghua University, Bejjingmolecule from the surrounding molecules is R。. For the100084, ChinaLennard-Jones fluid as shown in fig. 2, a condensing mo-Abstract By statistically analyzing the condensationlecule moves from the vapor into the liquid cluster ac-process and reconsidering the transition state theory focording to the following steps:condensation and evaporation, we first presented a complet-ed theoretical formula of the condensation coefficient.Namely, the unknown parameters contained within the tran-sition state theoretical calculation of the condensation coeffi-D-cient are determined. The condensation coefficients calculat-ed from this formula agree well with those from molcular0080dynamics simulations. With this formula, the classical ex-pression of the condensation flux developed from the gasa)b)(ckinetic theory can be used to predict the condensation flux.Fig. 1. Sketch map of the molecular condensation process. (a) BeforeKeywords: condensation, transition state theory, L-J fluid.the condensing vapor molecule enters the liquid cluster; (b) when thevapor molecule enters the liquid cluster; (C) when the volume of theEvaporation and condensation at the liquid-vaporliquid cluster expands.. ( , The condensing vapor molecule; .C, theinterface has drawn much attention for many years due toliquid molecules, --. the moving direction of molecules.its significant and extensive applications in a variety offields of science and engineering, but many points remainunsolved because of its complexity. According to thekinetic theory, the condensation flux can be expressed byL-J potentialintroducing a condensation coefficient (Hertz _Knudsenequation)"- 41. However, the condensation coefficient wi-thin this classical formula can only be estimated throughexperiments. Moreover, a precise measurement of thecondensation coefficient is very difficult. There existedmany disagreements among the experiments by differentauthors in literature. For example, the differences betweenthe measured condensation coefficients of water at similartemperatures in different experiments may be veryFig.2. Lennard-Jones potential. 1, Before the condensing vapor mo-largel4s. As a statistical approach to reaction dynamics,lecule enters the liquid cluster; 2, when the vapor molecule enters thetransition state theory has provided several statistical ex-liquid cluster; 3, when the volume of the liquid cluster expands.pressions of the condensation coefficient, yet the conden-(1) Before the condensing molecule enters the liquidsation coefficient can not be theoretically predicted. Thecluster, R is initially equal to the average distance of theobstacle lies in the unknown parameters within the statis-liquid molecules, r, and R。is initially equal to the averagetical expressions, such as the partition function of thedistance of the gas molecules, r。. With its distance fromactivated complex (i.e. the transition state), the free volu-the liquid-vapor interface diminishing, the average dis-me of the activated complex, etc.6 I. Recently, with thetance of the condensing molecule from the surroundingmolecular dynamics method, investigators gained somemolecules, R。，decreases, hence its potential first fallsvaluable microscopic information of the liquid-vapordown, and then rises.interfacial properties and interphase transport9 -I21. How-(2) When the condensing molcule enters the liquidever, these numerical studies have not been effectivelycluster, the molecular average distance of the liquid clus-combined with the existing theories of evaporation andter, R, decreases from r to r' , and the average distancecondensation.In this contribution, based on the microscopic under-of th中国煤化工m the surrounding ones,standing of the condensation process obtained from mo-R,the variation of R。lecular dynamics simulations, we statistically analyzed the"1YHCNMHG'condensation process with transition state theory, andlarger unan Tnat ol Kp ana 93) and three gener-from r to r, and the average distance of the condensingalized momentum P1, P2, P3) respectively. A mechanicalmolecule from the surrounding ones, Rg, increases fromstatus of a molecule can be represented with a corre-后tor, so the potential of the condensing molecule fallssponding point in the six-dimensional phase space, andthe hyper volume element in the phase space isdown.dt = dq,dqzdq3dp dprdp3.Taking s= R- R, as the reaction coordinate of theThe classical Hamilton of the molecule iscondensation, the potential energy of the condensing mo-H=H(p, q)= H(q,, 92>93;P1, P2, P3).lecule along the reaction path can be shown in fig. 3,Based on the first assumption of the transition state theory,where ri is the average distance of the liquid molecules, rgthe ratio of the activated complexes across the potentialis the average distance of the gas molecules,△E is thebarrier to all the vapor molcules colliding with the liquid-activated energy for the condensation, U。is the potentialvapor interface isbarrier relative to the zero-point energy (For condensation「'.e-BHdpdp2dpzdqdqzdq3reaction, zero-point energy is taken as the average poten-(3tial of the bulk vapor). The area close to the potential“e-BHdpdpdpsdqjdqpdq3barrier is the saddle point field of the potential surface, δis the width of the saddle point field. According to thwhere r is the phase space of the gas molecules collidingtransition state theory, for condensation, there are threewith the interface, and Ts is the phase space of the activat-assumptions:ed complexes which can get across the potential barrier.Since the vapor phase is assumed to be the ideal gasl4, theu↑Hamilton can be decomposed toH=p 12m+吃12m+防/2m,(4)where H is the Hamilton, and m is the mass of the mole-Saddel pointcule. According to the classical statistical mechanics, the, region。number of all the possible states in the phase space of thecolliding vapor molecules isβ些=」。e P2mdp」e 2mdp2」_ 。e 2mdp;Tgδ0(5Fig.3. Potential energy of the colliding molecules along the reactionin which ["dq["dq2["dg3 is defined as the freepath,(1) Even in the absence of equilibrium between e-volume of the vapor molecules, β =1/kgT, kg is Boltz-actant (vapor) and product (liquid) molecules, the activat-mann' s constant, T is the temperature of the system, r。 ised complexes that are becoming products are distributedthe average distance of the gas molecules. From eq. (5)，among their states according to the Maxwell-Boltz- mannthere islaw.(2) Molcules that have crossed the transition statesf eH"ndndpdqdqrdqs =，2四3. (6)in the direction of products cannot turn around and reformreactants.For「'e-BHdpndndpjdqjdqzd3 in eq. (3), based on the(3) In the transition states, motion along the reactioncoordinate may be separated from the other motions andthird assumption of the transition state theory, we havetreated classically as a translation.H =NE+ps /2m+ p2 /2m+ p3 /2m, .2 Statistical analysis of the condensation for mona-is th中国煤化工y for the condensation, ptomic fluidand.fYHc N M H Glong the reaction path,p2henta in the other two di-Let us consider a system of monatomic fluid in can-mensions. According to the transition state theory for theonical ensemble (i.e. the total number of molecules N, the condensation, if the colliding molecules, which movevolume V, and the temperature T of the system are fixed). .along the positive reaction path, i.e. from reactants toChinese S府亦数letin Vol.47 No. 11 June 2002953NOTESproducts, have enough activated energy, they can cross the1.transition states to transform to the liquid molecules.Therefore, in the colliding phase space, the number ofArgon0.8phase states of the vapor molecules which can cross thetransition states is0.6-f'e-B"dpdrndsxdq,dg,dqs0.4_β丝_β瞪=e-β°AEf°e "2mdpJ_ e' 2mdp2]」_ e 2mdp3一口- MD simulation by Tsuruta et al.-0- MD simulation by the authors(8).0--◆- Transition state theory by the authors° dgax]"dg,8090-100110120130whereds[da2J"dqsis the free volume of theTemperatureKactivated complex, r。is the average distance of the vaporFig. 4. Comparison between the transition state theory analysis and themolecules, 1 is the average distance of the liquid mole-molecular dynamics simulation.cules, and s is the reaction coordinate. Integrating eq. (8),ratio of the condensed molecules to the incident vaporwe haveones into the liquid phase, plays an important role on theestimation of heat and mass transfer rate across the liquid-广e-BH1 dadrdqsdpsvapor interphase. By statistically analyzing the condensa-tion process with transition state theory, we presented a-B 81(20(rg-r). 营(9)completed transition state theoretical expression of the2| βcondensation coefficient. The condensation coefficient of aIntroducing eqs. (6) and (9) into eq. (3), we havemonatomiefluidis 0。=(rg-rj)/ rg"se . It decreasesy )3/2f=eB.NE. 1(2四(g-n)3with the increment of the temperature. The condensation2(βcoefficients calculated in this note are consistent with the(10)results from the molecular dynamics simulations byAccording to the second assumption of the transition statedifferent authors.theory, f in eq. (10) is the ratio of the condensed molecu-Acknowledgements This work was supported by the National Naturalles to the colliding molecules, i.e. condensation coefficientScience Foundation of China (Grant No. 50106004) and the Fundamen-0。From eq. (10), the condensation coefficient of thetal Research Foundation of the Mechanical Engineering School ofmonatomic fluid is .Tsinghua University.Referencese-B AE(11).Schrage, R. w.. A Theoretical Study of Interphase Mass Transfer,in which rg is the average distance of the vapor moleculesNew York Columbia I Jniv. Press. 1953.L abuntsov. D. A An analvsis of intensive(r。=vl/3, Vg is the free volume of the vapor molecules),condensation. Hieh TemperatureEngl.Transl.i.1967 5:579.r is the average distance of the liquid moleculesCammenga. H. K，Evaporation mechanisms of liquids，CurrentTopicsinMaterialsSciences(edKaldis EAmsterrdamNorth-(i=V|/3, V is the free volume of the liquid molecules),Eames, I. W, Marr, N. J.. Sabir, H., The evaporation coefficient△E is the activated energy for the condensation, β = l/kgT,. Steinfeld, J. I, Francisco, J. S., Hase, w. L., Chemical Kineticskp is Boltzmann' s constant, and T is the temperature of theand Dynamics, New Jersey: Prentice Hall, 1989, 308system.Mortensen, E. M.. Eyring, H，Transmission coefficientsevaporation and condensation, J. Phys. Chem, 1960, 64: 846.The condensation coefficients of argon at a series of8. Fujikawa, S., Maerefat, M., A study of the molecular mechanism oftemperatures calculated from eq. (11) and those fromvapour condensation, JSME International Jourmal, 1990, 33(4); 634.molecular dynamics simulations are compared in fig. 4,。Tsuruta, T, Nagayama, G., A molecular dynammics approach towhere the hollow symbols represent separately the mo-interphase mass transfer between liquid and vapor, Heat Transferand Transport Phenomena in Microscale (ed. Celata, G. P.), Newlecular dynamics simulation results by Tsuruta et al.!9] andYork: Begell House, 2000, 432- -439by the authorsl12, and the solid symbols represent the10. Wang, Z. J., Chen, M.. Guo, Z. Y., Molecular dynamics study onresults of the transition state theory by the authors. TheFluid Phase Equilibria, 2001,中国煤化工condensation coefficients from the transition state theore-aoka, Y, Molecular mechanismtical calculation agree well with those from molecular[HC N M H GThermal Sciene & Eninering,dynamics simulations.12. Wang, Z. J., Chen, M.. Guo, Z. Y, Molecular dynamics study on3 Conclusionevaporation and condensation, J. Xi an Jiaotong Univ. (in Chi-nese), 2001, 35(11): 1126.Condensation coefficient, which is defined as the(Received January 8, 2002)954Chinese Science Bulletin Vol. 47 No. 11 June 2002