Numerical calculation of the rate of homogeneous gas-liquid nucleation in a Lennard-Jones system

J. Chem. Phys. 110, 2159 (1999); h ps://doi.o g/10.1063/1.477826 110, 2159 © 1999 Ame ican Ins i u e o Physics. The mally d i en escape o e a ba ie o a bi a y shape Ci e as: J. Chem. Phys. 110, 2159 (1999); h ps://doi.o g/10.1063/1.477826 Submi ed: 13 Ma ch 1998 . Accep ed: 21 Oc obe 1998 . Published Online: 12 Janua y 1999 A. N. D ozdo , and J. J. B ey ARTICLES YOU MAY BE INTERESTED IN Theo y o ac i a ed a e p ocesses: Exac solu ion o he K ame s p oblem The Jou nal o Chemical Physics 85, 1018 (1986); h ps://doi.o g/10.1063/1.451844 Two no el app oaches o he K ame s a e p oblem in he spa ial di usion egime The Jou nal o Chemical Physics 111, 6481 (1999); h ps://doi.o g/10.1063/1.479945 Ac i a ed a e p ocesses: Gene aliza ion o he K ame s–G o e–Hynes and Lange heo ies The Jou nal o Chemical Physics 97, 2422 (1992); h ps://doi.o g/10.1063/1.463081 The mally d i en escape o e a ba ie o a bi a y shape A. N. D ozdo a) and J. J. B ey Fı ´sica Teo ´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, Se illa 41080, Spain ~Recei ed 13 Ma ch 1998; accep ed 21 Oc obe 1998! The K ame s heo y o he he mally ac i a ed a e o escape o a B ownian pa icle om a po en ial well is ex ended o a ba ie o a bi a y shape. The ex ension is based on an app oxima e solu ion o he unde lying Fokke –Planck equa ion in he spa ial di usion egime. Wi h he use o he Mel’niko –Meshko esul o he unde damped B ownian mo ion an o e all a e exp ession is cons uc ed, which in e pola es he co ec limi ing beha io o bo h weak and s ong ic ion. I gene alizes in a na u al way a ious di e en a e exp essions ha a e al eady a ailable in he li e a u e o pa abolic, cusped, and qua ic ba ie s. Applica ions o symme ic pa abolic and cusped double-well po en ials show good ag eemen be ween he heo y and es ima es o he a es om nume ical calcula ions. © 1999 Ame ican Ins i u e o Physics. @S0021-9606~99!01404-X# I. INTRODUCTION E e since he pionee ing con ibu ion o S an e A hen- ius, he p oblem o he mally d i en escape om a me a- s able s a e has become one o he mos undamen al p ob- lems in physics and chemis y.1The mode n heo y o ac i a ed a e p ocesses is essen ially due o K ame s,2who p o ided a dynamical amewo k o he o iginal concep s o A henius. The unde lying idea o he K ame s heo y is o model he escape p ocess by he mo ion o a B ownian pa - icle wi h mass weigh ed coo dina e xin a po en ial o mean o ce V(x). The dynamics is go e ned by he ollowing Fokke –Planck equa ion o he p obabili y densi y P(x, , ) o inding he pa icle a ime a posi ion xwi h eloci y :2 ] P~x, , !5@2 ] x1V8~x! ] 1 g ] ~ 1 b 21 ] !#P~x, , !. ~1.1! He e he p ime deno es he de i a i e wi h espec o x, g is he ic ion coe icien , and b he in e se ene gy a ailable om he he mal ba h, b 215kBT. The po en ial is assumed o ha e a well wi h minimum a xw,0, sepa a ed om he con inuum by a ba ie a x50 o heigh E52V(xw). He eby we se o con enience V(0)50. The quan i y o in e es is he escape a e Go he pa icle om he well. The la e can always be w i en in he o m G5 m GTST ,~1.2! whe e GTST is he ansi ion s a e heo y ~TST! esul GTST5 H A 2 p b E 2` 0dx e2 b V~x! J 21,~1.3! and m is a ansmission coe icien desc ibing he de ia ion o he a e om GTST . K ame s s udied he dependence o he escape a e on he ic ional damping in wo egimes, namely, o small and in e media e o la ge ic ion g . In he o me egime, he coupling be ween he sys em and he ba h is assumed o be anishingly weak so ha he a e limi ing s ep is he ans e o ene gy om he ba h o he pa icle. The ansmission coe icien akes in his case he o m m ~ g →0!5D~ g →0!52 gb E xp 0dx A 22V~x!,~1.4! whe e Dis he dimensionless loss o ene gy pe oscilla ion o a pa icle wi h ene gy close o he ba ie heigh , and xp he le -hand side u ning poin o he asymp o ic unde damped ajec o y, V(xp)50. In he in e media e o la ge ic ion e- gime, when he ans e o ene gy becomes as enough o main ain he mal equilib ium o escaping pa icles, he a e limi ing s ep is spa ial di usion ac oss he ba ie egion. One o he basic assump ions o he K ame s heo y2in his egime is a pa abolic ba ie app oxima ion. I consis s in di iding he ull po en ial in o a pa abolic ba ie pa U~x!52 1 2 2x2,~1.5! wi h 252V9(0), and an anha monic co ec ion eading V~x!5U~x!1O~x3!.~1.6! In he immedia e icini y o he ba ie op which domina es he dynamics, he nonlinea i y o V(x) anishes as e han he pa abolic pa 21 2 2x2and he e o e can be neglec ed. This yields he ollowing exp ession o he ansmission co- e icien : m pb5 A 11 g 2 4 22 g 2 .~1.7! I should be no ed ha Eq. ~1.7!is alid o g * /(2 p b E). Consequen ly, in he ex eme high ba ie ~low empe a u e!limi , b E→`, one will ul ima ely almos always be in he spa ial di usion egime. K ame s’ model, al hough simple, is o wide- anging signi icance o a de ailed unde s anding and e alua ing he in luence o he medium on eac ion a es. I has ound a i- ous gene aliza ions o he ull ic ion ange,3non-Ma ko ian ac i a ed a e p ocesses,4,5 mul idimensional sys ems,6and cases wi hou de ailed balance7~ o a e iew see Re . 1!.In a! Pe manen add ess: Ins i u e o High Tempe a u es, 13/19 Izho skaya S ee , 127412 Moscow, Russia. JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 4 22 JANUARY 1999 21590021-9606/99/110(4)/2159/5/$15.00 © 1999 Ame ican Ins i u e o Physics all hese in es iga ions he ba ie is assumed o be pa abolic, hough his assump ion is no always me in eal physical and chemical ba ie c ossing p ocesses. Fo example, he ba ie o cha ge ans e eac ions is o en o a cusp-shaped o m.8 K ame s also de i ed he ansmission coe icien o a cusp- shaped ba ie ,2U(x)52a u x u , m cusp~ g →`!5a g A 1 2 p b ,~1.8! bu his exp ession is alid only in he asymp o ic limi o la ge ic ion whe e Eq. ~1.1!can be app oxima ed by a Smoluchowski equa ion. The e a e a ious a emp s in he li e a u e o b idge he s ong ic ion limi esul o a cusp- shaped ba ie wi h he TST alue, m TST51, a ze o damping.9,10 An analogous in e pola ing o mula is known o a qua ic ba ie .1Only e y ecen ly, Be ezhko skii e al.11 ha e ex ended his o mula o an a bi a y nonpa a- bolic ba ie o he o m U~x!52~a/ a ! u x u a .~1.9! Thei gene aliza ion eads11 m a 5 H E 2` `dx exp@ b U~x!# J 21 3 E 2` `dx exp H b F U~x!21 2 g 2x2 G J .~1.10! The e is, howe e , a ce ain i ony he e; he abo e o mula ag ees wi h he known escape a es o nonpa abolic ba ie s, bu ails o ep oduce he exac esul o a pa abolic ba ie . In he la e case, i yields ins ead o Eq. ~1.7!an app oxi- ma e exp ession m 25~11 g 2/ 2!21/2.~1.11! The aim o his pape is wo old. Fi s , we wan o p esen an app oxima e a e o mula, which indeed is alid o a bi a ily shaped ba ie s and in e pola es be ween he limi s o small and la ge ic ion. And second, we wish o compa e his o mula wi h exac nume ical a es in di e en ypes o po en ials. II. INTERPOLATING FORMULA To begin wi h we conside he spa ial di usion egime. Ou pu pose is o de i e an app oxima e solu ion o he Fokke –Planck equa ion which would allow one o eco e Eqs. ~1.7!and ~1.10!. This goal can be achie ed in many di e en ways.12 He e we employ he lux o e popula ion me hod de eloped by K ame s.2Wi hin i s scope, he escape a e is de ined as he a io o a s a iona y di usion cu en a he op o he ba ie o he popula ion o he well. Acco d- ingly, we ha e o look o a cu en ca ying s a iona y p ob- abili y densi y P(x, ), ha smoo hly ma ches he equilib- ium dis ibu ion Peq~x, !5exp@2 b V~x!21 2 b 2#~2.1! in he well and anishes beyond he ba ie . The wo s a ion- a y densi ies a e ela ed by a o m unc ion j (x, ), P~x, !5 j ~x, !Peq~x, !,~2.2! which is de e mined om $ 2 ] x1@V8~x!2 g # ] 1 gb 21 ] 2 % j ~x, !50. ~2.3! Once he o m unc ion is known, he eac i e lux o mula yields o he ansmission coe icien m 5 b E 2` `d j ~ 0, !exp S 21 2 b 2 D .~2.4! Following K ame s, we app oxima e he po en ial V(x) en e ing Eq. ~2.3!by i s ba ie pa U(x). The la e is no necessa ily pa abolic, i may be a sum o a bi a y ~pa abolic and nonpa abolic! e ms U~x!52 1 2 2x22a a u x u a 2¯.~2.5! Mo eo e , we assume ha j (x, ) is a unc ion o some lin- ea combina ion o xand , j ~x, !5 j ~%!,%5cx1b .~2.6! Then, i is no di icul o check by di ec subs i u ion ha in leading o de in %and ( b E)21an app oxima e solu ion o Eq. ~2.3! eads j ~x, !5Z21 E % `dy e b U~y!,~2.7! wi h %5 A /~ gm pb!@x2~ m pb / ! #.~2.8! In he abo e m pb is gi en by Eq. ~1.7!, while he no maliza- ion cons an Zis de ined by he equi emen ha he o m unc ion j (x, ) app oaches uni y in he ini ial well and ze o in he p oduc side. This immedia ely yields Z5 E 2` `dy e b U~y!.~2.9! I will be ecalled he e ha he ba ie ~ empe a u e!is as- sumed o be high ~low!enough so ha he po en ial can be well app oxima ed by i s local beha io in he icini y o he ba ie op. O he wise one can use in Eqs. ~2.7!and ~2.9! ins ead o he ba ie pa U(x) he ull po en ial V(x) i sel . In such a case, he in eg a ion has o be es ic ed o he ba ie egion wi h a lowe limi a , say, xwand he uppe limi a a alue beyond he ba ie om whe e he ec ossing p obabili y o a pa icle wi h ze o ini ial eloci y can sa ely be neglec ed. Inse ing Eq. ~2.7!in o Eq. ~2.4!, we ob ain he ollow- ing exp ession o he ansmission coe icien : m ab5Z21 E 2` `dx exp H b F U~x!21 2~ g / m pb!x2 G J . ~2.10! I is a simple ma e o check ha o a pa abolic ba ie he abo e o mula coincides wi h he exac K ame s esul , Eq. ~1.7!, while o a pu ely nonpa abolic ba ie ( 50) i e- p oduces Eq. ~1.10!. One may also no e ha i ag ees in he limi ing case o high ic ion wi h he ansmission ac o o an a bi a ily shaped ba ie ollowing om he co espond- ing Smoluchowski equa ion13 2160 J. Chem. Phys., Vol. 110, No. 4, 22 Janua y 1999 A. N. D ozdo and J. J. B ey m ~ g →`!5 H g A b 2 p E 2` `dx e b U~x! J 21 ,~2.11! and educes o uni y a ze o damping. A a e exp ession alid in he ull damping ange can be ob ained by making use o an elegan app oach de eloped by Mel’niko and Meshko .3This gi es in a s aigh o wa d way m 5 m abA~D!,~2.12! wi h A~D!5exp S 1 p E 0 `dx ln $ 12exp@2D~x211 4!# % x211 4 D , ~2.13! whe e Dis gi en by Eq. ~1.4!. I should be no ed ha he ansa z o w i ing a uni o m o mula o nonpa abolic ba ie s as a p oduc o a spa ial di usion exp ession and he depopu- la ion ac o Ais ad hoc. I ollows nei he om Mel’niko and Meshko no om Pollak, G abe , and Ha ¨nggi u no e heo ies. I is ou aim he e o p o e he u ili y o Eq. ~2.12! by compa ing wi h exac nume ical a es. The la e is no so ob ious as one migh hink. Speci ically, Mel’niko and Meshko de i ed he depopula ion ac o ~2.13!unde he assump ion ha he escape dynamics can be desc ibed by a p obabilis ic in eg al equa ion in ene gy-ac ion a iables, whose G een unc ion co esponds o he ba ie ajec o y. Fo a smoo h po en ial he ajec o y ha lea es he ba ie wi h he en i e ene gy close o ze o e u ns o i a e ime T→`. This in ini e ime, howe e , is no longe ue o a cusped ba ie whe e he ime is o he o de o he pe iod o pa icle oscilla ion in he well. Thus he in e es ing issue we shall add ess in ou nume ical applica ions is as ollows: Does he ini e pe iod o he ba ie ajec o y spoil he ap- plicabili y o Eq. ~2.12!? III. NUMERICAL RESULTS The aim o his sec ion is o p esen exac nume ical a es o di e en ypes o po en ial ba ie s ha would allow one o es analy ical p edic ions. One migh , a i s , belie e ha his issue should ha e been se led long ago, mainly because o i s con inuous impo ance in many p oblems o chemical physics. To he bes o ou knowledge, howe e , he e a e no nume ical solu ions o such a ype, o he han hose ob ained in Re s. 10 and 11 unde he assump ion ha he po en ial consis s only o a ba ie pa . This assump ion esul s in a mono onic dependence o he ansmission coe - icien on g ; he coe icien inc eases wi h dec easing g and eaches i s maximal alue a ze o damping, when he e is no coupling be ween he sys em and he ba h. I is clea ha he da a so ob ained a e no sui ed o es ing analy ical p edic- ions in he mos p oblema ic in e media e and weak damp- ing egimes. He e we deal wi h ac i a ed a e p ocesses in a symme - ic double-well po en ial o he o m V~x!5E 112a@x424a u x u 22~12a!x2#,a.2 1 2. ~3.1! I s ba ie pa a ies wi h he pa ame e a om a pu ely pa abolic (a50) o a pu ely cusped (a>1) ba ie , see Fig. 1. Acco dingly, he equency en e ing ou a e exp ession eads 25 H 4~12a!E/~112a!21 2,a<1, 0a.1. ~3.2! The me hod used o nume ically sol e Eq. ~1.1!will be de- sc ibed elsewhe e.14,15 Table I shows a lis o he i s non- ze o eigen alue in he conside ed po en ial o b E510 and a50, 0.5, and 1. The calcula ion is pe o med o e a la ge ange o g which co e s all egimes o chemical in e es , om he unde damped B ownian mo ion o he spa ial di u- sion egime. Be o e es ing he alidi y o he p esen a e exp ession, we no e ha Eq. ~2.12!gi es he ansmission coe icien o he escape om a me as able s a e. Using he app oach sug- ges ed by Mel’niko and Meshko ,3 he coe icien o a symme ic double well can be w i en as m 5 m abA2~D!/A~2D!.~3.3! FIG. 1. Di e en shapes o he po en ial V(x), Eq. ~3.1!, o a50~ he dashed line!and a51~ he solid line!. TABLE I. Fi s nonze o eigen alue o symme ic double-well po en ials, Eq. ~3.1!wi h b E510 and a50, 0.5, and 1. Exponen ial no a ion 2k means ha he numbe p eceding is o be mul iplied by 102k. g a50a50.5 a51 0.05 0.17124 0.14624 0.14424 0.1 0.30424 0.25924 0.24724 0.25 0.59324 0.49424 0.45624 0.5 0.86824 0.71324 0.64024 1 0.10623 0.88924 0.79124 2 0.10623 0.95224 0.85624 5 0.85824 0.91424 0.85024 10 0.60724 0.78624 0.77024 20 0.36124 0.55724 0.56424 50 0.15424 0.26824 0.28624 100 0.78025 0.13324 0.14524 1000 0.78326a0.13425a0.14725a aExac es ima e o he eigen alue calcula ed om he espec i e Smolu- chowski equa ion. 2161J. Chem. Phys., Vol. 110, No. 4, 22 Janua y 1999 A. N. D ozdo and J. J. B ey The leas non anishing eigen alue o he co esponding Fokke –Planck ope a o is hen gi en by wice he a e de- ined by Eq. ~1.2!. The nume ical alues o he ansmission coe icien ex ac ed in his way a e exhibi ed in Fig. 2, o- ge he wi h he analy ical p edic ions ob ained in e ms o Eq. ~3.3!. As e idenced by Fig. 2, he app oxima e a e ex- p ession gi es an uppe bound o he exac esul o he a e in he pa abolic double-well po en ial. Fo he cusped po en- ials he heo y o e es ima es he a e in bo h limi s o weak and s ong ic ion and unde es ima es i in he in e media e ic ion egion. I is also seen ha o all alues o a he bes ag eemen is achie ed in he s ong damping limi ( g *100). Wi h dec easing g he e o made by he ansa z ~3.3! inc eases and eaches maximal alues in he weak damping egion ( g &0.1). The heo e ical exp ession o e es ima es he a e in his egion by 14% o a pa abolic ba ie (a 50) and by 18% o a pu ely cusped ba ie . I should be poin ed ou ha he same is ue o he u no e heo y o Pollak, G abe , and Ha ¨nggi.5As we ha e shown in ecen pape s,15,18 hei heo y also conside ably o e es ima es he a e in he weak ic ion egime. Finally, o conclude his sec ion we no e ha he ba ie equency appea ing in Eq. ~2.10!may s ill be le e en i he ba ie is pu ely nonpa abolic. In such a case, i should be ea ed as a a ia ional pa ame e .16 Ye ano he way o im- p o e he a e o mula is o ake in o accoun ini e-ba ie co ec ions. These a e ob ainable sys ema ically in bo h e- gimes o weak17 and in e media e o s ong ic ion.12 A u - he imp o emen o he o e all a e exp ession can be achie ed by using in Eq. ~2.10!a p ope ly de e mined ene gy loss o he de e minis ic pa icle dynamics. In con as o he weak ic ion exp ession o Dp oposed by Mel’niko and Meshko , Eq. ~1.4!,3as well ha sugges ed by Pollak, G ab- e , and Ha ¨nggi5in hei u no e heo y, he de e minis ic app oach o his quan i y yields an app oxima ion which e- mains co ec in he ull damping ange, ega dless o he pa icula shape o he po en ial ba ie .15,18 IV. CONCLUDING REMARKS In his pape , an app oxima e o mula o he a e o escape o e an a bi a ily shaped ba ie has been con- s uc ed by means o he lux o e popula ion me hod and he app oach by Mel’niko and Meshko . The esul ing exp es- sion ag ees in he limi ing case o high ic ion wi h he a e ollowing om he co esponding Smoluchowski equa ion and, in he ex emely unde damped egime wi h he a e ob- ained by K ame s om a di usion equa ion in ene gy ~ac- ion! a iables. I gene alizes in a na u al way he known a e o mulas o pa abolic and nonpa abolic ba ie s. Besides, we ha e p esen ed o he i s ime nume ically exac a e cons an s o po en ials wi h di e en ba ie shapes in all egimes o chemical in e es , om unde - damped o o e damped B ownian mo ion. These esul s p o- FIG. 2. T ansmission coe icien and pe cen age e o , 1003~app oxima e2exac !/exac , made in m by using Eq. ~3.3!. Exac nume ical esul s a e shown by ci cles. ~a!a50; ~b!a50.5; ~c!a51. 2162 J. Chem. Phys., Vol. 110, No. 4, 22 Janua y 1999 A. N. D ozdo and J. J. B ey ides he necessa y ounda ion o es ing a ious di e en a e exp essions ha al eady exis in he li e a u e. Compa i- son wi h he nume ical da a shows ha he p esen o e all a e exp ession is a he accu a e in he s ong damping limi , unde es ima es he a e by ;0%–18% in he in e media e ic ion egion and o e es ima es he a e by ;14%–23% in he weak damping egime. ACKNOWLEDGMENTS One o us ~A.N.D.!is g a e ul o A. M. Be ezhko skii, P. Talkne , and V. Yu. Zi se man o many help ul discus- sions. 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