Collective excitations in liquid methanol: A comparison of molecular, lattice-dynamics, and neutron-scattering results

J. Chem. Phys. 96, 7696 (1992); h ps://doi.o g/10.1063/1.462370 96, 7696 © 1992 Ame ican Ins i u e o Physics. Collec i e exci a ions in liquid me hanol: A compa ison o molecula , la ice- dynamics, and neu on-sca e ing esul s Ci e as: J. Chem. Phys. 96, 7696 (1992); h ps://doi.o g/10.1063/1.462370 Submi ed: 03 July 1991 . Accep ed: 14 Feb ua y 1992 . Published Online: 31 Augus 1998 J. Alonso, F. J. Be mejo, M. Ga cía-He nández, J. L. Ma ínez, W. S. Howells, and A. C iado ARTICLES YOU MAY BE INTERESTED IN Molecula -dynamics simula ions o dime hylsul oxide-me hanol mix u es The Jou nal o Chemical Physics 123, 154507 (2005); h ps://doi.o g/10.1063/1.2085052 Raman Spec al S udies o Wa e S uc u e The Jou nal o Chemical Physics 40, 3249 (1964); h ps://doi.o g/10.1063/1.1724992 In es iga ion o he empe a u e dependence o dielec ic elaxa ion in liquid wa e by THz e lec ion spec oscopy and molecula dynamics simula ion The Jou nal o Chemical Physics 107, 5319 (1997); h ps://doi.o g/10.1063/1.474242 Collec i e exci a ions in liquid me hanol: A compa ison o molecula , la ice- dynamics, and neu on-sca e ing esul s J. Alonsoa) Ru he o d Apple on Labo a o y, Chil on, Didco , Oxon OXII OQX, Uni ed Kingdom F. J. Be mejo Ins i u o de Es uc u a de la Ma e ia, Se ano 123, E-28006 Mad id, Spain M. Ga c a-He nandeza) Ru he o d Apple on Labo a o y: Chil on, Didco , Oxon OXII OQX, Uni ed Kingdom J. L. Ma inez Ins i u Laue Lange in, 156X, F-38042 G enoble Cedex, F ance W.S. Howells Ru he o d Apple on Labo a o y, Chil on, Didco , Oxon OXII OQx, Uni ed Kingdom A. C iado Depa amen o de F'isica Ma e ia Condensada, Uni e sidad de Se illa, Apa ado 1065, E-4080 Se illa, Spain (Recei ed 3 July 1991; accep ed 14 Feb ua y 1992) The collec i e dynamics o liquid me hanol-d4 is s udied by means o molecula -dynamics simula ion. The model po en ial is alida ed by means o la ice ene gy calcula ions and shows a e y good ag eemen wi h he expe imen ally ob ained c ys al s uc u e. Cen e -o -mass densi y and momen um luc ua ions a e in es iga ed in he (Q,w) egion which is also accessible o inelas ic neu on-sca e ing (INS) echniques. A simple iscoelas ic model p e iously used o he analysis o INS da a is es ed agains he dynamic s uc u e ac o compu ed om he simula ion. A di ec compa ison wi h he INS esul s hemsel es is also ~ade and quali a i e ag eemen is ound. Also, a en a i e assignmen o he peaks appea ing m he cu en -cu en co ela ions is made on he basis o la ice-dynamics calcula ions o he polyc ys alline low- empe a u e a phase. I. INTRODUCTION The collec i e dynamical esponse o s ongly associa - ed liquids has a ac ed a subs an ial e o om heo e ical, compu e simula ion, and expe imen al app oaches (see, o ins ance, Re . 1). Con a y o wha was obse ed in molecu- la liquids composed o pa icles in e ac ing ia nea ly pu e Lenna d-Jones po en ials such as liquid ni ogen/ he p es- ence o s ong in e molecula co ela ions induced by he hyd ogen bond was expec ed o ende hese sys ems amena- ble o expe imen al measu emen s using inelas ic neu on sca e ing. As a ma e o ac , he possibili y o obse ing sho -wa eleng h collec i e exci a ions in he kinema ic ange accessible o he mal neu ons was poin ed ou some ime ago by Impey, Madden, and McDonald3 on he basis o molecula -dynamics (MD) simula ion esul s o liquid wa- e . A subsequen expe imen on inelas ic neu on sca e ing (INS)4 opened up a deba e conce ning he na u e o he obse ed exci a ion. On he o he hand, he in e p e a ion o he expe imen al esul s was pu in o ques ion by he simula- ion esul s o Wojcik and Clemen i,5 bu a numbe o pape s on a simpli ied e sion o he p ojec ion ope a o echnique, as well as compu e simula ions, seemed o ein o ce he cu - en in e p e a ion o he neu on-sca e ing esul s.6 A s udy o he cohe en inelas ic esponse o liquid a) Pennanen add ess: Ins i u o de Es uc u ade 1a Ma e ia, Se ano 119, E- 28006 Mad id, Spain. me hanol was pe o med la e 7 and p o ided addi ional suppo o he hesis which assigned an acous ic na u e o he obse ed modes, in opposi ion o hose pos ula ing a " as -kine ic-mode" na u e o he exci a ion. The easons o he choice o his ma e ial we e i s ela i e simplici y com- pa ed o wa e and he ac ha i s low mel ing poin enabled he s udy o he collec i e e ec s wi h a e y small con ibu- ion om single-pa icle ( ansla ion and o a ion) modes.8 The main di icul y a ising om he analysis o INS spec a was he need o de elop a simple analy ic model use- ul o da a analysis pu poses which encompasses mos o he ele an collec i e dynamical pa ame e s. A ela i ely sim- ple exp ession was ound7 which, based upon a iscoelas ic ea men ,9 was able o ep oduce he mos p ominen spec- al ea u es, and he e o e was used o in e he dispe sion beha io om he obse ed inelas ic in ensi ies. The p esen s udy has se e al aims. Fi s o all, i is in- ended o ob ain p ecise in o ma ion ega ding he collec i e dynamics o his sys em (a p elimina y accoun o mos ly single-pa icle esul s has al eady been gi en 10) which is suppossed o be simple han wa e since mos o he hyd o- gen-bond ne wo k will be cons i u ed by long winding chains 10 ins ead o he complica ed a angemen s ound in he liquid and solid phases o wa e . On he o he hand, he p esen exe cise will se e o es he alidi y o he app oxi- ma ion used o analyze he neu on da a by pe o ming a simila analysis o he simula ed dynamical s uc u e ac- o s. Finally, i will y o shed some ligh in o he o igin o 7696 J. Che n. Phys. 96 (10), 15 May 1992 0021-9606/92/107696-14$06.00 @ 1992 Ame ican Ins i u e o PhySics Alonso e a/.: Collec i e exci a ions in liquid me hanol 7697 collec i e modes o he han he no mal sound whose phys- ical o igin is s ill an open con o e sy.4,5 The ou line o he pape is as ollows. Some ele an o mulas conce ning densi y and pa icle cu en luc u- a ions a e e iewed in he nex sec ion. In Sec. III we sum- ma ize he main cha ac e is ics o he molecula -dynamics simula ions as well as he way in which he in e molecula po en ial has been alida ed, and we in oduce some esul s on mos ly s a ic p ope ies. Sec ion IV is de o ed o he s udy o he spec al ea u es o he au oco ela ion unc- ions o densi y and pa icle cu en luc ua ions. A com- pa ison wi h he ele an unc ions ob ained om a la ice- dynamical model o he polyc ys al will also be made and he o igin o high- equency modes which a e shown o be common o he liquid and polyc ys alline phases will be discussed. The modeling o S( Q,liJ) and he compa ison wi h inelas ic neu on-sca e ing esul s a e he subjec o Sec. V. Finally, ou conclusions and summa y a e p esen ed in he las sec ion. De ails abou he la ice-dynamics calcula ions a e gi en in he Appendix. II. THEORETICAL BACKGROUND Dynamic p ocesses in molecula liquids ha e a g ea e complexi y han in mona omic liquids due o he exis ence o addi ional o a ional and ib a ional deg ees o eedom. As a i s app oxima ion, i is cus oma y o analyze he dynam- ics o molecula cen e s o mass in e ms o he o mulas de eloped o mona omic liquids. I 1,12 In his s udy we will be mainly in e es ed in he collec- i e beha io as cha ac e ized by he luc ua ions in he den- si y and pa icle cu en s. As usual, his will be done h ough hei ime-au oco ela ion unc ions (ACFs), F(Q, ) =J... <PQ( +s)p_Q(s», N J/(Q, ) = ~ ([q'jQ( +s)] [q'LQ(s)]>, J/(Q, ) =J... ([b'jQ( +s)][b'LQ(s)]>, (1) N known as he in e media e sca e ing unc ion and co ela- ion unc ions o he longi udinal and ans e se compo- nen s o he pa icle cu en , espec i ely. In he p eceding equa ions N is he numbe o molecules; he a e age is aken o e ini ial condi ions s and wa e ec o s Q o magni ude Q; q and b a e uni ec o s pa allel and pe pendicula , espec- i ely, o he wa e ec o Q; inally, he spa ial Fou ie com- ponen s o he mic oscopic numbe densi y and pa icle cu - en a e de ined by N pQ( ) = L exp[ -iQ·Ra( )], a=1 N jQ (I) = L Va ( ) exp[ -iQ·Ra ( )], a=l (2) whe e Ra ( ) and Va ( ) a e he posi ion and eloci y ec o s o he cen e o mass o molecule a a ime . The powe spec um o he in e media e sca e ing unc ion is known as he dynamic s uc u e ac o and is gi en by I l~ S(Q,liJ) = - d F(Q, ) cos(liJ ) , 1T 0 (3) wi h simila exp essions o he spec a o he cu en au o- co ela ion unc ions (CACFs) J[(Q, ) and J (Q, ). F om heo e ical and compu a ional s andpoin s i is ad an ageous o s udy he sho - ime beha io o he in e - media e sca e ing unc ion, which p o ides in o ma ion abou he equency momen s o he dynamic s uc u e ac- o . Expanding F( Q, ) in a Taylo se ies, we ob ain F(Q, ) = Scm (Q) I - liJ~ -+ liJ~liJ7 - - '" , ( 2 4 ) 2! 4! (4) whe e (5) and liJi a e he educed second and ou h equency mo- men s o S(Q,liJ), espec i ely. The e a e analogous exp es- sions o he longi udinal and ans e se componen s o he pa icle CACF; and hei educed second momen s, liJ7(Q) and liJ; (Q), a e ela ed o he Q-dependen elas ic cons an s el1 (Q) and G~ (Q), espec i ely. 11 F om he de ini ions o Eqs. (I) and (2) and exploi ing he s a iona y p ope y o ime ACFs i ollows ha F( Q, ) and J[ ( Q, ) a e no independen . 3 In ac , hei powe spec- a a e ela ed h ough II liJ 2S(Q,liJ) = Q2J[(Q,liJ). (6) III. COMPUTATIONAL DETAILS AND STATIC PROPERTIES A. The po en ial model A subs an ial numbe o a emp s ha e been egis e ed whe e model po en ials we e de eloped in o de o ep o- duce he mal and s uc u al p ope ies o he liquid phase as well as single-pa icle ime-co ela ions unc ions and he dynamics o bond b eaking and o ming in he hyd ogen- bond (HB) ne wo k. 14 - 18 Howe e , o he ex en o ou knowledge no se ious a emp o in es iga e he collec i e dynamics in me hanol om a molecula -dynamics poin o iew has been made so a . Fo his kind o compu a ion e y long MD uns a e needed in o de o p o ide accep able s a is ics; he e o e, lexible molecula models become imp ac ical since hey usually equi e ime s eps a leas I o de o magni ude smalle han he igid models. Among he la e ones, wo ha e been a o ed by mos esea che s. 14,15 As he eliabili y o bo h models is compa able,15,19 he choice be ween hem becomes a ma e o p e e ence. Fo his wo k we ha e cho- sen he model p oposed by Haughney, Fe a io, and Mc- Donald ls which is a si e-si e model, each si e-si e in e ac- ion being a Lenna d-Jones plus a Coulombic e m. B. La ice ene gy minimiza ion The adop ed model o he in e molecula in e ac ions has been also applied o he low- empe a u e c ys al phase 20 o me hanol-d4. The eqUilib ium c ys al s uc u e co e- sponding o his po en ial model has been ob ained as he minimum-ene gy con igu a ion. Fo his pu pose, a New- J. Chem. Phys., Vol. 96, No.1 0, 15 May 1992 7698 Alonso e al.: Collec i e exci a ions in liquid me hanol on-Raphson minimiza ion p ocess wi h he p og am 21 WMIN has been ca ied ou s a ing a he expe imen al c ys- al s uc u e, using he cell pa ame e s and molecula o a- ions and ansla ions as a iables. The Ewald me hod 22 has been used o deal wi h he Coulombic sums, and he cu o dis ance o 12 A has been adop ed o he Lenna d-Jones in e ac ions. The changes occu ing in he p ocess by he la ice pa- ame e s a, b, and ca e 2.9%,3.8%, and 1.6%, espec i ely, whe eas he maximum shi in he a omic coo dina es is 0.3 A. This o de o disc epancy be ween expe imen al and cal- cula ed c ys al s uc u es is usual in c ys al packing calcula- ions using a om-a om po en ial models; he e o e, we can assume ha he p oposed model ep oduces ai ly well he expe imen al c ys al s uc u e. C. The MD algo i hm Simula ions o a 256 CD 3 OD molecule sys em we e ca - ied ou using he compu e p og am desc ibed in Re . 10, which was w i en ollowing he hin s gi en by Allen and Tildesley.23 The p og am compu es he ajec o y o he sys- em subjec ed o cubic pe iodic bounda y condi ions using New onian classical mechanics (NVE-P ensemble). The molecules we e ea ed as igid bodies composed o six mass poin s (modeling he a oms in he molecule) and h ee in e - ac ion si es loca ed a he oxygen, ca bon, and hyd oxylic deu e ium posi ions. The Ca esian equa ions o mo ion we e in eg a ed using he eloci y e sion o he Ve le algo- i hm 23 wi h a ime s ep o 10 - 14 s, and he RA TILE algo- i hm 24 was used o implemen he holonomic cons ain s equi ed o keep all in amolecula dis ances ixed. The mo- lecula geome y and pa ame e s de ining he in e ac ion po en ial we e hose p oposed by Haughney, Fe a io, and McDonald. 15 Only wo mino modi ica ions on he po en- ial model we e in oduced. Fi s , he masses o he hyd o- gen a oms ha e been subs i u ed by ha o deu e ium in o de o simula e he ully deu e a ed compound (which is he one s udied by neu on sca e ing); and, second, he long- ange in e ac ions we e unca ed by using a swi ch unc ion 23 ,25 o um o smoo hly all he in e ac ions be- ween pai s o molecules whose cen e s-o -mass sepa a ion was g ea e han he cu o dis ance. Following he in oduc ion o he swi ch unc ion he in e ac ion ene gy be ween wo molecules, a and/3, becomes Uap( ;a, jp ) = S(R ~p)U~p( ;a, jp), (7) whe e RaP is he dis ance be ween cen e s o mass o bo h molecules, and Sex) is he unique i h-o de polynomial sa is ying {I o x<R L Sex) = 2 o o x>R u, (8) and ha ing con inuous i s and second de i a i es a he end poin s o he in e a1. 25 In he p esen calcula ions we ook RL = 12.25 A and Ru = 12.65 A o bo h uns. No e ha due o he la ge cu o adius and he elec oneu ali y o me hanol molecules he leading e m in he unca ed po en- ial co esponds o he dipole-dipole in e ac ion and be- ha es as R;;/. The unc ion U~( ;a, jp) is he po en ial unc ion e e ed o as model HI by Haughney, Fe a io, and McDonald I 5 and has he o m o a sum o si e-si e in e - ac ions. D. Accu acy o he esul s The use o pe iodic bounda y condi ions imposes some es ic ions upon he leng h and ime scales o he phenome- na ha can be s udied wi h con en ional molecula dynam- ics. F om a dynamical poin o iew he quan i y o in e es is he ecu ence ime (de ined as 'T ee = LboJc, whe e c is he eloci y o sound) ,26 which is abou 1.4 ps o he simula- ions epo ed he e. In o de o ensu e ha no measu able a i ac s we e in oduced due o ecu ence e ec s, he F( Q, ) o he lowes Q alue analyzed (0.25 A - I) was ex- amined in de ail and no spu ious ea u es ha could be a - ibu ed o ecu ence e ec s we e ound o imes smalle han 4 ps. Besides, co ela ion unc ions compu ed om i- ni e ime-leng h MD ajec o ies a e subjec o nume ical and s a is ical e o s;27,23 he e o e, he measu able ACF is gi en by MD( ) = ( ) + €( ), whe e ( ) is he ue ACF and €( ) ep esen s he e o e m. Taking his in o accoun , we see ha he spec um o a MD ACF is gi en by 1 L"" MD(W) = - d MD( )W( ) cos(w ) 1T 0 = [ ew) + €(w)] ® W(w), (9) whe e w( ) is some window unc ion and W(w) i s Fou ie ans o m. The e o e, he a ainable spec um is a con olu- ion o he ue spec um wi h he window unc ion plus a noise e m. Besides, due o he possible exis ence o ecu - ence e ec s as men ioned ea lie , he window should decay o ze o o imes o he o de o 'T ee o a oid he con amina- ion o he spec a by ecu ence e ec s. Howe e , his wo - sens he esolu ion in he equency domain and a low Q, whe e he spec al ea u es o in e es a e na ow and loca ed a low equency, i seems ha he e is no choice bu o use a wide window hough i could esul in an inc eased spec al noise le el. The in e media e sca e ing unc ion and pa icle cu - en au oco ela ion unc ions ha e been compu ed using he as Fou ie ans o m (FFT) me hod 23 o bo h he - modynamic s a es, and o many alues o Q spanning om 0.25 o 5 A - I. In all cases an a e aging o e he whole se o allowed wa e ec o s (compa ible wi h he pe iodic bound- a y condi ions) was pe o med. In o de o ob ain an es ima ion o he quali y o ou esul s he 200 K un was subdi ided in o eigh sub uns o 20 ps leng h. Au oco ela ion unc ions we e compu ed o hese sub uns and la e a e aged and used o es ima e he s anda d de ia ion o he mean. Fo Q = 0.25 A -I ( ha wi h only h ee independen wa e ec o s could be consid- e ed a un a o able case) he e o in F(Q, ) was a ound 10% du ing he i s 5 ps. Fou ie ans o ming he eigh sub uns sepa a ely we ound he e o in S( Q,w )/S( Q) o be again 10% and cons an in he equency ange s udied. As he numbe o allowed wa e ec o s inc eases wi h Q he e o s diminish acco dingly down o 3 % o Q> 2 A - I. The as decay o cu en au oco ela ions unc ions wi h ime J. Che n. Phys., Vol. 96, No. 10, 15 May 1992 Alonso e al.: Collec i e exci a ions in liquid me hanol 7699 enables us o employ he e o analysis o Zwanzig and Ailawadi,27 inding a maximum e o o ± 0.05 o he no - malized CACFs a low Q which educes o ± 0.02 as Q inc eases. E. Some esul s on s a ic p ope ies Two o he h ee he modynamic s a es s udied in Re . 10 a e eanalyzed in his wo k, wi h special emphasis on he collec i e, ime-au oco ela ion unc ions. The a e age al- ues o simple he modynamic p ope ies we e compu ed along wi h he ajec o ies, and a e shown oge he wi h den- si ies and un leng hs in Table I. The un leng hs we e chosen long enough so ha we can a e age o e he longes pe iod luc ua ions o he he modynamics p ope ies obse ed in he sys em. The s a ic s uc u e ac o as well as he pa ial pai - co ela ion unc ions ha e been analyzed in a p e ious pa- pe . 10 The educed second and ou h momen s o S( Q,{i) , {i)o, and (i)[o ha e been p esen ed in Fig. 1 oge he wi h hei espec i e ideal-gas limi s. Ou esul s a e e y simila o hose o Wojcik and Clemen i o wa e S hough he e- quencies in ol ed a e somewha smalle o me hanol. The second momen shows a s ong dependence on he s a ic s uc u e ac o Q < 2.5 A -I as expec ed om Eq. (5), dis- playing a p onounced dip a Qp [i.e., he posi ion o he main maximum o Scm (Q) ], and emaining close o i s ideal alue om he e on. The ou h momen does no app oach i s ideal limi so quickly hough i oscilla es a ound i s sel - al- ue om ela i ely low Q. Besides, he dimensionless quo ien o (i)/ and i s single-pa icle coun e pa , namely (i) 'l (Q) = (n~ + 3Q2kB TIM) 112, (10) exhibi s li le o no dependence on he he modynamic s a e, n~ being he "Eins ein equency" (see Table I). On he o he hand, he oscilla ions a ound he sel - alue decay e y slowly, being s ill no iceable a Q~ 5 A -I hough hey lose ampli ude o Q> 2.5 A-I, which demons a es ha collec- i e e ec s a e impo an down o leng h scales o jus a ew angs oms (i.e., nea es -neighbo dis ances). Resul s o he second equency momen o he spec a o he ans e se CACFs also show independence o he he modynamic s a e (a leas o he wo s a es s udied he e) when exp essed in dimensionless o m: {i) (Q)/{i),;,,' ( Q), whe e (i)~el (Q) = (n~ + Q2kBTIM) 112. (11) Collec i e e ec s seem o be impo an only in he e- gion Q < I A - I; o la ge momen um ans e s he CACF beha es mos ly like i s single-pa icle coun e pa . The in i- ni e- equency mac oscopic shea modulus, Goo' can be es i- ma ed om he low-Q alues o (i) (Q). App oxima ing Goo by Goo (Q = 0.25 A - I) we ge alues (Table I) ha a e abou hal hose ound o liquid wa e .s IV. DENSITY AND CURRENT FLUCTUATIONS A. Densi y luc ua ions The dynamic s uc u e ac o S( Q,{i) has been plo ed in Figs. 2 and 3 as a unc ion o he angula equency (i) o he smalles accessible alues o he momen um ans e . A 200 K he wo lowes -Q spec a (0.25 and 0.35 A - I) exhibi dis inc i e B illouin side peaks a equencies (i) B o 0.44 and 0.56 X 10 13 adls, espec i ely. Then he peak becomes o e - damped as he momen um ans e is inc eased, hough i is s ill no iceable as a weak shoulde in he spec a o momen- um ans e s below 0.6 A - I. Ano he in e es ing ea u e o S( Q,{i) is he inc ease o he in ensi y in he equency e- gion a ound 2X 1013 ad/s, which a Q = 0.86 A -I mani- es s i sel clea ly as a shoulde in he spec um. The p es- ence o his shoulde sugges s he exis ence o a second exci a ion in he liquid coexis ing wi h he lowe - equency mode. A 300 K he o e all si ua ion is e y simila , al- hough he B illouin peaks al eady appea as o e damped a he lowes alue o he momen um ans e accessible o ou MD expe imen (see Fig. 3). F om Fig. 2 i is clea ha he peak maxima whene e isible show a no iceable dispe sion (a leas up o 0.4 A-I). TABLE I. A e age alues o simple he modynamic p ope ies o simula ed liquid me hanol·d4. Run numbe s a e as in Re . 10. Quan i y Uni s Run 1 Run 3 Densi y kglm 3 943.0 885.0 L"' n nm 2.5333 2.5875 Elapsed ime ps 177.00 100.10 Tempe a u e" K 202.3 ± 0.3 297.7 ± 1.0 Uln e b kl/mol -39.88 -34.79 P essu e" MPa 52.0 105.3 n.3 ps -2 463.7 402.2 Scm (0) 0.029 ± 0.009 0.038 ± 0.003 XT 10- '0 Pa-' 6.5 ± 1.9 6.3 ± 0.5 cT mls 1290 ± 190 1340 ±60 CII (Q = 0.25 A. -') 10· J/m 3 15.77 13.16 G~ (Q=0.25A.··') 10· J/m 3 6.72 5.96 "The s anda d de ia ion o he a e age empe a u e has been es ima ed by aking in o accoun he s a is ical ine iciency o he da a (Re . 23). "No long· ange co ec ions ha e been applied. J. Che n. Phys., Vol. 96, No.1 0, 15 May 1992 7700 Alonso e al. : Collec i e exci a ions in liquid me hanol 2.5- ..--... n 2.0- --- '"Cl o I-< M 1.5- .- 0 ,........ '--" --- 1.0- :3 0.5- c» * * * • wz(Q) • ..... ....... 4>+' ~ .•.•• 4> #, + ~ ... ..-" ... ++ .. , + .. , .' ...... . ....... -........ e ......... . ..... - ... . ::::::::::~:::::::::~ ........... . .' * • .. * * • • • * • - .... ~ o 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Q / A-I FIG. 1. "Dispe sion ela ions" o he educed second and ou h momen s o he dynamic s uc u e ac o , S( Q,(() a 200 K. Thei ideal-gas (high-Q) limi s, which a e gi en by Q(kBT 1M) 1/2 and Q(3k BT 1M) 112, espec i ely, a e shown as s aigh do ed lines. The eloci y o p opaga ion o his exci a ion can be es i- ma ed om he equency o he side maxima o S( Q,w), W B (Q), acco ding 0 28 (12) which in he hyd odynamic limi educes o he adiaba ic sound eloci y. Subs i u ing he abo e-men ioned equen- cies, we ob ain sound eloci ies o 1760 and 1600 m/s o Q = 0.25 and 0.35 A-I, espec i ely. Ou simula ion esul s seem o ag ee easonably well wi h ecen measu emen s o he hype sonic eloci y.29 In pa icula , he adiaba ic sound eloci ies epo ed a e 1475 and 1080 m/s o liquid me ha- nol-d4 a 200 and 300 K, espec i ely. The highe alue ob- ained om ou 200 K simula ion can be in e p e ed as a posi i e sound dispe sion in he simula ed sys em. B. Longi udinal cu en luc ua ions In o de o in es iga e ho oughly he dispe sion o he B illouin mode and o cla i y whe he a second mode exis s, we ha e unde aken he calcula ion o he spec um o longi- udinal cu en luc ua ions which is ela ed o S( Q,w) h ough Eq. (6). Some well-known bene i s o s udying he spec a o he cu en ACF a e (i) J1 (Q, ) is oscilla o y and decays as e han F( Q, ); he e o e, he spec a a e less a - ec ed by unca ion o windowing e ec s; (ii) he e is no cen al peak and consequen ly no o e lapping wi h he quasielas ic componen ; (iii) he spec a always exhibi , a leas , a pai o peaks (S okes and an i-S okes); and (i ) he high- equency pa o he spec a is enhanced, allowing he s udy o weak high- equency exci a ions. Cons an -Q sec ions o he spec um o he longi udinal cu en luc ua ions a e shown in Fig. 4. A i s sigh , he mos s iking ea u e is undoub edly he appea ance o a sec- ond peak a equencies a ound 2.25 X 1013 ad/s, which is he mos in ense in he in e al 0.70 < Q < 1.25 A-I. The Q dependence o he equency o hese wo maxima has been plo ed in Fig. 5. The low- equency mode is acous ic in na- u e and i is ela ed o he peak appea ing in S( Q,w) a low Q, hough i now appea s a somewha highe equencies (see Fig. 5) due o he w2 ac o in Eq. (6); i s "dispe sion ela ion" shows no signi ican quali a i e di e ence om wha has been ound in simple liquids. 28 Namely, i is s ongly a ec ed by s uc u al e ec s a low Q [showing a p onounced dip in he neighbo hood o Qp ], bu beyond 2.5 A-I kine ic e ec s become p edominan , as can be seen om he compa ison wi h he ideal-gas-limi ing alue w:::(Q) = Q(2k B T/M) 112. The second mode exhibi s a mo e complex beha io . To begin wi h, i canno be unambiguously dis inguished as a peak in he whole Q ange examined in his s udy, bu usual- ly a shoulde can be obse ed a equencies a ound 2 X 10 13 ad/s in all he spec a no showing a peak. Second, because o he s ong o e lap be ween he wo modes i is di icul o asce ain whe he his highe - equency mode shows any measu able dispe sion. In addi ion, we ound in ou p e ious single-pa icle s udy 10 ha he cen e -o -mass eloci y au o- co ela ion unc ion (VACF) has a seconda y peak a 2.04 X 1013 ad/s which ag ees app oxima ely wi h he posi- ion obse ed he e. The in ensi y o bo h modes exhibi s im- po an oscilla ions a low Q ha a e p og essi ely damped J. Chem. Phys., Vol. 96, No.1 0, 15 May 1992 Alonso e al.: Collec i e exci a ions in liquid me hanol 7701 0.50 A-I 0.55 A-I 0.5 1.0 1.5 2.0 W / ( 10 13 ad/s ) 0.61 A-I 0.70 A-I 0.74 A-I o78A-I 0.82 A-I 0.86 A-I 2 3 w / ( 1013 ad/s ) FIG. 2. Cen e s-o -mass dynamic s uc u e ac o o liquid me hanol a 200 K. S( Q, iJ) is shown as a unc ion o he angula equency, iJ, o he lowes II alues o he momen um ans e . No e he di e en equency anges used in bo h ames. The o dina e uni s a e (10 12 ad/s) -1. M o ...... x o 0.5 0.24 A-I 0.34 A-I 0.42 A-I 0.49 A-I 0.54 A-I 1.0 1.5 2.0 w / ( 10 13 ad/s ) FIG. 3. Cen e s-o -mass dynamic s uc u e ac o o liquid me hanol a oom empe a u e. S( Q, iJ) is shown as a unc ion o he angula equency, IJ, o he lowes i e alues o he momen um ans e . The o dina e uni s a e (10 12 ad/s) - I. 0'0.15 II ..., ~ ...;- _0.10 ~ 2: ...;- 0.05 (a) 0.08 0.07 0'0.06 II do.0 5 ...;- 0.04 - '3 dO.0 3 ...;- 0.02 0.01 (b) 0.25 A-I 0.35 A-I 0.43 A-I 0.55 A-I 0 2 3 w / ( 1013 ad/s ) 0.61 A-I 0.70 A-I 0.74 A-I 0.78 A-I 0.86 A-I ° 234 5 W / ( 1013 ad/s ) FIG. 4. Cons an -Q sec ions o he spec a o he no malized longi udinal cu en au oco ela ion a 200 K as a unc ion o he equency. No e he di e en equency anges used in bo h ames. The o dina e uni s a e (10 12 ad/s) -1. as Q inc eases; howe e , hey oscilla e in opposi e di ec ions. The in ensi y o he high- equency mode is ound o be oughly p opo ional o OJ m (Q) while he acous ic mode ol- lows a end p opo ional o i s in e se, which is in ag ee- men wi h i s asymp o ic beha io a high Q (i.e., ideal gas). A new sound eloci y c, (Q) can be de ined simila ly o cB(Q) bu using he equencies om he maxima o he CACF spec a ins ead o hose o S( Q,OJ). The high- and low- equency limi s o he sound eloci y a e deno ed by COO (Q) and CO (Q), espec i ely, and a e gi en by COO (Q) = OJ,CQ)/Q, CO CQ) = YcT(Q), C 13) whe e is he quo ien o he speci ic hea s and cT(Q) = OJ o (Q)/Q is a wa e- ec o -dependen gene aliza- ion o he iso he mal sound eloci y. Finally, in Fig. 6 we ha e collec ed all ou da a conce ning sound eloci ies. No e ha we ha e plo ed CT (Q) ins ead o he low- equency J. Chem. Phys., Vol. 96, No. 10,15 May 1992 7702 Alonso e a/.: Collec i e exci a ions in liquid me hanol "' - 2.~ 11 2.0- .... .,., '0 ..... 1.~ - 0' 1.0- 1 o.~ I~I }~'! i+ dl~~H H i · o Q j A-I FIG. 5. F equency o he maxima o he longi udinal cu en spec a o me hanol a 200 K as a unc ion o he momen um ans e . Open ci cles ep esen he acous ic mode while as e isks gi e he loca ion o he second exci a ion. The maxima obse ed in S( Q,Cll) a low Q ha e been indica ed as squa es. Finally, he s aigh do ed line ep esen s he ideal-gas asymp o ic limi . E o ba s ha e been es ima ed as he hal wid h a hal maximum (HWHM) o he window unc ion used o Fou ie ans o ming he cu - en ACF. To aid compa ison he igu e has been plo ed on he same scale as Fig. I. sound eloci y due o ou lack o eliable in o ma ion on o he simula ed sys em. The disag eemen be ween C B (Q), C I (Q), and C T (Q) a low Q is a clea symp om o he s ong sound dispe sion [no e ha in o de o econcile he alues o c B (Q) and Co (Q) we would ha e o assume ~2]. A 200 Ke en C B (Q) and C I (Q) do no coincide, indica ing ha he B illouin peaks a e b oad and o e lap subs an ially wi h he cen al peak. On he o he hand, he low-Q beha io o c, (Q) expe iences an impo an change a 300 K, eaching a maximum alue and, possibly, beginning o con e ge owa ds Co (Q) jus beyond he lowes Q we a e able o in es- iga e in ou simula ion (0.25 A-I). 4. 300 K 3. 200 K ---. en 3. - S ~ 2.~ -2. C; '(:)1. 1. o. o 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 QjA-I Q/A-l FIG. 6. Sound eloci ies cT(Q), c~ (Q), c,(Q), and cB (Q) as a unc ion o he momen um ans e (see ex o de ini ions). The open ci cles co e- spond o c,(Q), he squa es o cB(Q), and he solid lines o cT(Q) and c ~ (Q), as indica ed. c. T ans e se cu en luc ua ions We ha e unde aken he calcula ion o ans e se CACF in o de o ge some insigh in o a pa o he collec- i e dynamics no easily amenable o expe imen a ion. Ex- pe imen al echniques such as neu on sca e ing do no cou- ple easily o ans e se modes (unless some kind o di use "umklapp" p ocess is pos ula ed). In ac , o da e molecula dynamics is he mos eliable sou ce o in o ma ion abou ans e se exci a ions in dense luids o he kinema ic ange o ou in e es . The equency o he maximum o he ans e se CACF spec a has been plo ed in Fig. 7 as a unc ion o Q. A i o he low-Q da a o he "dispe sion ela ion" o a s aigh line gi es an es ima e o he eloci y o p opaga ion which is 940 ± 20 mls a 200 K. Co esponding spec a and dispe - sion ela ion a 300 K look quali a i ely he same. Howe e , a his empe a u e he e is a subs an ial o e lap o he S okes and an i-S okes peaks o he lowes Q which dis o s o some ex en he dispe sion ela ion; o example, a he lowes Q he alueo J, (Q,liJ = 0) is abou 1/2 o he alue a he maximum, while a 200 K i is 1/20 o i . In addi ion, he eloci y o p opaga ion has now educed o 750 ± 25 m/s. The li e ime o he exci a ion 30 can be es ima ed om he ecip ocal o he hal wid h a hal maximum o he peaks. A 200 K we ind li e imes o 1.06, 0.71, 0.42, and 0.35 ps a Q = 0.25, 0.35, 0.43, and 0.50 A - I, espec i ely, while a 300 K he li e imes a e 0.64,0.49,0.35, and 0.30 ps o simi- la alues o momen um ans e . An ex apola ion o he dispe sion ela ion o lowe Q, a 300 K, sugges s ha he peaks a ± liJ m (Q) will coalesce in o a single one o Q < 0.1 A-I, bu he e is no indica ion o ha happening o any ini e Qa 200 K. Undoub edly, he eason why me hanol seems o be able o suppo eely p opaga ing shea wa es wi h wa eleng hs as la ge as 50-100 A is he exis ence o a e y s able hyd ogen-bond ne wo k, which is i sel connec - ed o he s ong di ec ionali y o he hyd ogen-bond in e ac- ion. ---.0. en - ~ ~ o. o ..... c '1 o. o 0.5 1.0 1.5 2.0 Q / A-I FIG. 7. F equency o he maximum o he ans e se cu en au oco ela- ion spec a o me hanol a 200 K as a unc ion o he momen um ans e . E o ba s as in Fig. 5. J. Chem. Phys., Vol. 96, No.1 0, 15 May 1992 Alonso e al.: Collec i e exci a ions in liquid me hanol 7703 Fo momen um ans e s g ea e han 1.0 A-I he ans e se CACF beha es mos ly as a single-pa icle coo di- na e. Finally, o end he analysis o he spec a o he ans- e se cu en we jus men ion ha a conspicuous seconda y maxima can be obse ed a he same equency as he hi d ( e y weak) maxima in he VACFspec a, 10 i.e., 3.57X 1013 ad/s, in almos all he spec a wi h Q be ween 0.5 and 1.5 A-I, hough i is ex emely weak. In he low-Q egion he spec a con ain some ipples ha p eclude he obse a ion o weak signals as his seconda y peak. D. la ice-dynamics calcula ions In o de o explo e he physical o igin o he modes ap- pea ing in he simula ion as well as o ha e a e e ence o compa e wi h, we ha e unde aken a la ice-dynamics (LD) calcula ion o a polyc ys al sample ollowing he p ocedu e desc ibed in he Appendix. F om he calcula ed dispe sion ela ions o he 24 modes co esponding o o a ions and ansla ions o he ou molecules wi hin he uni cell, a quan i y di ec ly com- pa able wi h he cu en co ela ion unc ions has been de- i ed. Se e al plo s co esponding o di e en alues o he momen um ans e a e shown in Fig. 8. F om inspec ion o he igu e he ollowing commen s a e in o de : (a) The mani old o acous ic peaks appea s as a well-de ined en i y up o Q alues o abou 0.5 A, showing no iceable dispe sion; and (b) wo se ies o peaks o a mixed ( o a ional and ans- la ional) cha ac e a e clea ly appa en om 0.5 A onwa ds, cen e ed a abou 2.2 and 3.0X 1013 ad/s. Such equencies can be easily co ela ed wi h p ominen ea u es appea ing in he calcula ed ib a ional densi y o s a es (DOS) which is displayed in Fig. 9. A close inspec ion o he igu e whe e a compa ison o a om-a om DOS is made o he c ys al and liquid phases e eals he ollowing ea u es: • The pu ely acous ic modes a e appa en in he polyc ys al as a Debye con ibu ion (Le., wi h a dependence upon (2) up o equencies o 5 X 10 12 ad/s. The i s wo in ense peaks cen e ed a ound lOX 10 12 ad/s a e o mixed ( o a- ional and ansla ional) cha ac e as has been e idenced om he analysis o he mode eigen ec o s. • The polyc ys alline DOS up o w = 15X 10 12 ad/s be- comes in he liquid phase a b oad, s uc u eless o a ional con ibu ion cen e ed a abou he same equency as in he solid. The sound mode ( ansla ional o cen e -o -mass) con ibu ion appea s in he liquid as a b oad dis ibu ion cen e ed a abou 5 X 10 12 ad/s, which will gi e ise a low-Q alues o a ini e- equency esponse in he S( Q,w). A no- iceable amoun o ansla ion- o a ion coupling is appa - en , which e idences he ac ha no pu e ansla ional o o a ional modes a e o be expec ed o occu in a molecula ma e ial wi hin he kinema ic ange accessible o MD o INS s udies. • A se ies o in ense peaks cen e ed a 22, 30, and 38 X 10 12 ad/s in he polyc ys al appea as a b oad s uc u e be ween w = 15x 10 12 and w = 30X 10 12 ad/s in he liquid in a e- quency egion showing also a no iceable amoun o o a ion- ansla ion coupling. The in ense peak which appea s in he solid a abou 50 X 10 12 ad/s also shows up in he liquid, al hough he modes con ibu ing o i a e now so ened, which leads o a educ ion in he peak equency o abou 37 X 10 12 ad/s. F om he analysis o some o he mode eigen ec o s he componen s o he mode pola iza ions can be ob ained. As an example, i was ound ha he wo low- equency peaks co espond o modes wi h an a e age cha ac e o Tx(y,z) = 0.20 (0.23, 0.36) and RX(Y,z) = 0.74 (0.47, 0.11) o he low- equency peak, and Tx(y,z) = 0.21 (0.22,0.67) and Rx(y,z) = 0.24 (0.62,0.14) o he highe - equency one. He e Tx(y,z) and Rx(y,z) deno e he a e aged ampli Udes o he ansla ional and o a ional molecula deg ees o ee- dom exp essed in he p incipal axis o he ine ia molecula ame (in inc easing ine ia momen o de ). The e o e, i seems clea ha he compa ison o he cu - en -cu en co ela ions as well as he DOS o bo h poly- c ys al and liquid phases e eals ha mos o he obse ed ea u es in he luid phase can be assigned by aking he LD esul s as a e e ence. Fu he discussion on his will be de- e ed o he Discussion sec ion. V.INELASTIC NEUTRON SCATTERING AND MODELING OF S{Q, o) Recen ly, Be mejo e aU ha e measu ed he inelas ic neu on-sca e ing spec a o liquid me hanol-d4, I( Q,w), which is ela ed o he a om-a om dynamic s uc u e ac o s no U no no I(Q,w) ex: I ~S~l (Q,W) + I I bibjSi/Q,w) , i=1 41T i=lj=1 (14) whe e na is he numbe o a oms pe molecule, (J inc and bi a e he incohe en sca e ing c oss sec ion and he cohe en sca e ing leng h o neu ons o a om ype i, and S ij ( Q,w) is he a om-a om dynamic s uc u e ac o o a oms i and j ( he supe sc ip sel makes e e ence o i s sel -pa ). How- e e igo ous, he p eceding equa ion would no be e y use- ul as a model due o i s la ge numbe o ee pa ame e s and a simpli ied one based in a well-known iscoelas ic app oxi- ma ion o simple liquids has been p oposed7 I model ( Q,w) ex: Scm (Q) exp( -p,Q 2) X {Rqe (Q,W) + P6 (pi - P6 )7 } [W7(W2 -PF)] 2 + (W 2 -P6)2 ® O(w), (15) whe e Scm (Q) deno es he s uc u e ac o o molecula cen e s, he exponen ial is a Debye-Walle e m, Rqe (Q,w) ep esen s he cen al quasi elas ic componen , he second e m inside he cu ly b acke s is he usual iscoelas ic sca - e ing law o simple liquids,9 O(w) is he expe imen al es- olu ion unc ion; and ® deno es a con olu ion ope a ion. To keep he numbe o ee pa ame e s o a minimum he Max- wellian elaxa ion ime 7 is es ima ed using he p esc ip ion due o Lo esey,9 7 -1=2[(PF-P6)!1 ]1I2. (16) As usual, he Q dependence o he educed momen s om J. Chem. Phys., Vol. 96, No.1 0, 15 May 1992