Temperature and pressure constraints near the freezing point

PHYSICAL REVIEW 8VOLUME 43, NUMBER 41FEBRUARY 1991 Tempe a u e and p essu e cons ain s nea he eezing poin Juan J.Mo ales and Ma iaaia J.Nue o Depa amen o de FIsica, Facul ad de Ciencias, Uni e sidad de Ex emadu a, 06011Badajoz, Spain Luis F.Rull Depa amen o de Fssica Teo ica, Uni e sidad de Se illa, 41080Se illa, Spain (Recei ed 26 Feb ua y 1990; e ised manusc ip ecei ed 3May 1990) The iso he mal-isoba ic ensemble molecula -dynamics me hod MD(T,p, N) p oposed by Nose and Hoo e is used o s udy he Auc ua ions in a wo-dimensional Lenna d-Jones Auid, close o he eezing poin . The Tand pcons ain s in his me hod do no a ec he dynamical beha io o he sys em, since spon aneous Auc ua ions in he densi y allow he sys em o eeze and mel jus as do he Tand pAuc ua ions in he mic ocanonical ensemble MD(E, V, N) close o he mel ing zone. I. INTRODUCTION The dynamics o a omic sys ems wi h con inuous in- e ac ions be ween he a oms equi es nume ical in eg a- ion o Hamil on's equa ion o mo ion, ' dH dpi d ; =- '= =P d 'I dp; d PI= dU( ) d , whe e M=T(p)+ U( ) is he Hamil onian o he X-body sys em, T(p )and U( )being he kine ic and po en ial ene gies, espec i ely; New onian mechanics implies ha he ene gy and momen um a e he conse ed a iables o he mo ion. In adi ional molecula -dynamics (MD) ex- pe imen s, he o al ene gy E, he numbe o pa icles N, and he olume Va e conse ed as he dynamics o he sys ems e ol es in ime. The ime a e age o any p ope - y is an app oxima e measu e o he mic ocanonical en- semble MD(E, V, X). Fo ce ain applica ions, i may be desi able o pe o m dynamical simula ions a cons an empe a u e and/o p essu e leading o he canonical en- semble MD( T, V, N) and he iso he mal-isoba ic ensemble MD(T,p, X) Howe e so. me o hese echniques need he in oduc ion o a i icial p ocesses in o de o make his p ac ical. Such schemes may be classi ied as non- New onian MD, since hey a e no based on Hamil oni- ans o mo ion o eal sys ems, bu use a i ices o scaling ha esul in ic i ious o ces in o de o sa is y some adop ed de ini ion o cons an Tand p. Howe e , i is possible o show igo ously ha ce ain o hese me hods gi e p ope equilib ium ensemble a e ages. Se e al me hods ha e been de eloped o simula e he canonical ensemble, including he "s ochas ic" me hod in which collision wi h an imagina y hea -ba h pa icle is aken in o accoun , he "ex ended-sys em" me hod" in which adeg ee o eedom is included which ep esen s he ese oi in con ac wi h he sys em, and he "con- s ain s" me hod in which he eloci ies a e escaled a each ime s ep by some ac o in o de o ix he kine ic ene gy o he sys em. Al hough o he me hods do no gene a e s a es in he canonical ensemble, hey seem e y use ul o s udying he changing s a e and sys ems each- ing equilib ium a adi e en empe a u e. Fo he cons an -p essu e MD simula ion, in all app oaches i is ine i able ha he sys em box mus change olume as in cons an -p essu e Mon e Ca lo (MC) simula ions. The nomencla u e is he same as o he MD(T, V,N). The s ochas ic me hod is aMC calcula ion in which he Me opolis echnique is applied o he andomly changed olume AV. The ex ended me hod in ol es coupling he sys em o an ex e nal a iable V, he olume o he sys em, whe e he coupling mimics he ac ion o a pis on on a eal sys em. The cons ain s me hod makes he ins an aneous p essu e acons an o he mo ion. In wo o he pa icula me hods, one allows he simula ion box o change shape as well as size, and in he o he he app op ia e ensemble is no well iden i ied. In a o me s udy' we p esen ed some MD simula ions o asys em e y close o he mel ing zone whe e Auc ua- ions play an impo an ole in he dynamic p ope ies o he sys em. The simula ions we e pe o med using he adi ional mic ocanonical ensemble, i.e.,wi h no scaling p ocedu e, and we ound luc ua ions o empe a u e and p essu e which spon aneously make he sys em oscil- la e be ween wo poin s o s a e. We conclude ha simu- la ions wi h any cons ain s me hod, such as he cons an - empe a u e and/o cons an -p essu e me hods, a e inco ec nea he mel ing zone because he e a e p oblems ela ed o luc ua ions wi h espec o space and ime and he cons ain s can diminish he luc ua ions o no allow hem a all. Ne e heless, we now see his con- clusion o be no comple ely gene al, because he sys em beha io depends on he me hod used and consequen ly on he way in which he cons ain s a e applied. Among he echniques e iewed he e we chose he combina ion o he ex ended sys em me hod due o Nose and he cons ain s me hod due o Hoo e , "leading o Nose-Hoo e (NH) o mula ion. 'The e we e wo easons o he choice: i s , because he NH scaling p o- cedu e ep oduces bo h he canonical and he iso he mal-isoba ic p obabili y densi y in he phase space 43 3514 QC1991 The Ame ican Physical Socie y 43 TEMPERATURE AND PRESSURE CONSTRAINTS NEAR THE. . . 3515 II. SIMULATIONS AND RESULTS The equa ions o mo ion o he MD(T,p, N) ensemble in he NH o mula ion a e ' dx; p; m "D (3) dp) = ;— (g — E)p;, (4) whe e x;= , /V' a e he educed coo dina es o he pa icle i, Dis he dimension o he space, , is he o ce, and gand e=V/DV a e he iso he mal and isoba ic ic- ion coe icien s which couple he sys em o a ese oi a T=T„and p=p„, espec i ely. The dynamical equa ions o he ic ion coe icien s a e N2 DNk T— d ,.&I(DNkT )(5) o aclassical mechanical sys em, and second, because his me hod had been used p e iously o show ha he dynamical beha io o he NH equa ions does no des oy he long- ange bond co ela ion in he Auid nea he eezing poin . 'In he ollowing we shall s udy he be- ha io o he empe a u e, p essu e, and densi y Auc ua- ions in he MD( T,p, N) ensemble in he wo-dimensional (2D) sys em nea he eezing poin and he esul s will be compa ed wi h hose ob ained by MD(E, V, N) nea he mel ing poin . ' and =(p — p,„)V/(kT ), d (6) whe e ~z and ~a e he iso he mal and isoba ic elaxa- ion imes o he sys em espec i ely which gi e he cou- pling s eng hs. Bo h elaxa ion imes ha e aGaussian dis ibu ion in he NH heo y and hey a e ela ed o he mean collision ime. '' The NH iso he mal-isoba ic simula ions we e pe - o med o a20 Lenna d-Jones sys em o X=400 pa i- cles wi h acu oA dis ance o , =2.5o..The s a e poin was se up a kT/e=0. 7and po =0.83. This poin is a Auid e y close o he eezing poin ' wi h aco espond- ing p essu e o pe /E =2 64. .These alues o empe a- u e and p essu e we e chosen as he ixed ex e nal Tand p,„o he ese oi in Eqs. (5) and (6). The o he inpu alues we e an=0. 005 and *= /&N =0'.3.'To in- eg a e he equa ions o mo ion he Tox ae d algo i hm' was used wi h a ime s ep h=0.005(mo /E)', and he sys em was allowed o e ol e in ime abou 120000 h ak- ing a e ages alues e e y 800 h. In Fig. 1 he Auc ua ion o empe a u e is plo ed agains he numbe o ime s eps o he MD(Tp, N) en- semble. The luc ua ions a e e y small h oughou he dynamic e olu ion o he sys em, in spi e o he icini y o he eezing zone whe e he Auc ua ions a e la ge. This means ha he iso he mal ic ion coe icien wo ks well a in e changing ene gy be ween sys em and ese - oi h ough Eq. (5), diminishing he luc ua ions a ound he ixed empe a u e as much as possible, as was o be expec ed in p inciple. The inal mean alue o he em- pe a u e was kT/c, =0.7000+0.0007. These Auc ua ions 0.75 4J CC l— UJ CL LU LU 0.65 50 100 x800 NUMBER 0F TINE STEPS FIG. 1. Reduced- empe a u e luc ua ions s numbe o ime s eps o a wo-dimensional luid close o he eezing zone. 3516 JUAN J.MORALES, MARIA J.NUEVO, AND LUIS F.RULL 43 2.70, 2.64 2.58 100 x800 NUNBER OF TINE STEPS FIG. 2. As Fig. 1, bu o he educed p essu e. a e abou 14 imes smalle han hose o he co espond- ing MD(E, V,X) simula ion in he mel ing zone. ' Simila beha io is ound o he p essu e Auc ua ions whe e he dynamical equa ion o he isoba ic ic ion coe icien Eq. (6) main ains he sys em a cons an p es- su e while allowing i o change i s olume. These Auc- ua ions look mo e egula ha he empe a u e luc ua- ions and g ea e by abou 3 imes. The inal mean alue o he p essu e was po. / =2. 641+0.002, wi h Iuc ua- ions 40 imes smalle han he p essu e Auc ua ion o he MD(E, V, N), as can be seen by compa ison o he s anda d de ia ion and Fig. 2wi h he esul s gi en in Re . 18. The ime e olu ion o he densi y is e y di6'e en , as Fig. 3shows. The luc ua ion is i egula and la ge due o he absence o cons ain s. On wo occasions, a e l— LLJ 0.87 0,82 0.77 50 j.00 x800 NUNBER OF TINE STEPS FICx. 3. As Fig. 1, bu o he educed densi y. 43 TEMPERATURE AND PRESSURE CONSTRAINTS NEAR THE. ..3517 abou 44000 and 104000 ime s eps, luc ua ions appea spon aneously which allow he sys em o each densi ies wi hin he eezing zone. 'The inal mean alue o he densi y was po. =0.827+0.010, his s anda d de ia ion being he same as ha co esponding o he empe a u e in he MD(E, V, N) ensemble a mel ing. These la ge spon aneous luc ua ions disappea when he sys em is a away om he eezing zone: simula ion in he liquid sys- em a po. =0.7606 showed as anda d de ia ion o +0.0027, luc ua ions nea ly ou imes smalle han be- o e. III. DISCUSSION AND CONCLUSIONS I we compa e hese esul s MD(T,p, N) nea he eez- ing poin wi h hose MD(E, V, N) co esponding o he solid nea he mel ing zone, 'we can see ha he wo sys- ems beha e in he same way physically. The mic o- canonical ensemble allows empe a u e and p essu e o Auc u a e in he same way as he densi y in he iso he mal-isoba ic ensemble. Nea aphase ansi ion, bo h ensembles mus unde go spon aneous luc ua ions ha d i e he sys em back and o h be ween wo poin s o s a e. The p oblem o he non-appea ance o he Auc- ua ions a ises when he me hod used o simula e he MD(T,p, N) is inco ec , e en hough i yields ime a e - ages o he modynamically in e es ing a iables o wi hin he s a is ical accu acy o mos expe imen al da a. A e y in e es ing discussion can be ound in Re s. 7and 16, whe e he alidi y o he s ochas ic me hod used o he olume change in Re . 7 o in es iga e he mel ing ansi ion is e u ed in Re . 16, since he e o ob- ained wi h ha me hod no only changes he co ec lo- ca ion o he ansi ion bu also mis ea s any densi y luc ua ions in i s icini y. We can he e o e conclude ha he cons ain s o empe a u e and p essu e in he ex ended Nose and Ho- o e me hod no only do no al e he na u al e olu ion o he sys em, bu also gi e he co ec dynamical beha - io including he zone o la ge luc ua ions such as he eezing ansi ion. Du ing he p epa a ion o he p esen pape me hods o implemen ing empe a u e and p essu e con ols a equilib ium and nonequilib ium in he NH dynamics ha e been published. 'These me hods inc ease he e iciency o la ge-seal: simula ions bu he de ailed e- sul s a e no ye a ailable. ACKNOWLEDGMENT The coope a ion o he Uni e si y o Ex emadu a Compu e Cen e is g a e ully acknowledged. See, o example, H. Golds ein, Classical Mechanics (Addison- Wesley, Reading MA, 1980), o M. P. Allen and D. J.Tildes- ley, Compu e Simula ion o Liquids (Cla endon, Ox o d, 1989}. F.F.Ab aham, Ad . Phys. 35, 1(1985). H. C. Ande sen, J.Chem. Phys. 72, 2384 (1980). 4S. Nose, Mol. Phys. 52, 255 (1984). 5D. J. E ans and G. P. Mo is, Compu . Phys. 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