Simulating fluid-crystalline solid equilibria with the Gibbs ensemble

Transcription

1 Simulating luid-rytalline olid equilibria with the Gibb enemble M.B. Sweatman and. Quirke Department o Chemitry, Imperial College, South Kenington, London, UK. SW7 AY. Abtrat We employ the Gibb enemble to imulate ga-liquid-olid equilibrium at the triplepoint o a hited-ore Lennard-Jone ytem. Thi i made poible by generating an aurate anonial Helmholtz ree-energy model o the (deet-ree) olid phae. Thi reeenergy model i generated by a ingle iothermal-iobari imulation at a preure not too ar rom oexitene or whih the hemial potential i known. We alo diu the merit o our method, ompared to other in the literature, or imulating rytal in onined geometrie. Introdution There are everal way to determine phae oexitene or a bulk ytem uing Monte-Carlo imulation. A popular method alulate a ree-energy denity model or eah phae at the ame temperature along both phae branhe in the viinity o the tranition. Coexitene i then eaily determined by inding the point at whih the branhe ro. A problem with thi method i the eort required to alulate the ree-energy denity model, ine many imulation and thermodynami integration may be required. Alternatively, one ould employ a non-boltzmann ampling heme,, or a phae with approah 3, 4, that alulate the ree-energy dierene between the two ompeting phae under given ondition. However, thee approahe an alo require oniderable eort beaue it i uually neeary to repeat thee imulation everal time in order to interpolate the loation where the branhe ro. In the ae o a non-boltzmann ampling heme oniderable eort might alo be required to onnet eah phae with a ontinuou path in order-parameter pae. Alternatively, a muh larger anonial ytem that inorporate a phae interae an be imulated 5, 6. Hene the popularity o the Gibb enemble 7, 8. It main advantage are that oexiting tate are generated pontaneouly during a ingle imulation (given reaonable initial ondition) and that relatively mall ytem an be imulated ine a phae interae i not required. ow onider thee method in the ontext o phae oexitene in onined geometrie (in lit pore or example). For onined ytem it an be oniderably more diiult to deine a uitable thermodynami integration path that doe not ro any irtorder phae boundarie than or the bulk ae beaue o potential urae indued phae tranition. Two approahe an be adopted in the ae o a non-boltzmann ampling method; either imulate in the grand anonial enemble or imulate in the iothermaliobari enemble and employ a uitable method or alulating the hemial potential o that a phae diagram an be drawn (hemial potential i idential in bulk and onined phae at equilibrium). We are limited to the iothermal-iobari enemble i uing a phaewith method. The Gibb enemble ha alo been applied 8, 9 to onined ytem. ow onider luid-rytalline olid oexitene in onined geometrie. Speial tehnique, 0, are needed to alulate the ree-energy denity model o the rytal phae, and it an be an even greater problem to deine the thermodynami integration path. The grand-anonial non-boltmann ampling approah annot be ued beaue o the problem with deet (diued later), and the iothermal-iobari non-boltzmann and phae with method an only be ued i the hemial potential o the luid phae an be alulated ine there are urrently no diret method (that avoid thermodynami

2 integration) or alulating the hemial potential o rytalline olid. The onventional Gibb enemble method ail or the ame reaon a the grand-anonial non-boltzmann method, i.e. partile reation/deletion in the rytal phae lead to deet. The problem with grand anonial enemble imulation o rytalline olid deerve ome diuion. Radhakrihnan and olleague, 3 have employed umbrella ampling (a.k.a. a Landau ree-energy method) in the grand anonial enemble extenively to determine luid-olid oexitene or Lennard-Jone ytem in lit pore. They how how reezing depend on pore width and luid-olid interation trength. However, let u onider a rytalline tate with l lattie poition and d deet (either intertitial or vaanie) in a grand anonial enemble. Partile lutuation an our only via the inertion or deletion o partile. Sine the lattie i ixed in thi imulation at ontant volume, V, only lutuation in d an our. Sine the hemial potential i the dierene in anonial Helmholtz ree-energy between tate that dier by one partile at ontant volume, µ = F(+,V,T) F(V,T), and only deet lutuation an our in the imulation, then the hemial potential in their imulation i the hemial potential o deet not the true hemial potential whih ha a ontribution rom lattie ite lutuation a well. So the work o Radhakrihnan and olleague i lawed in thi repet, i.e. the hemial potential in their work i F( l, d +,V,T) F( l, d,v,t) wherea it hould be F( l +, d,v,t) + F( l, d +,V,T) F( l, d,v,t). The miing ontribution arie rom partile lutuation due to phonon, a problem that i not auaged by analying inite ize eet. The everity o thi problem depend on the relative magnitude o thee ontribution. So, to determine luid olid rytal oexitene in the bulk or in onined geometrie the mot enible option appear to be umbrella or phae with ampling in the iothermal-iobari enemble, provided the hemial potential o the luid phae i eaily alulated. Here, we how how the Gibb enemble an be revived to treat thi problem a well, at leat or pure rytal o the type deribed above. Three-phae oexitene and the Gibb enemble Our method depend on generation o an aurate anonial Helmholtz ree energy model or the olid phae. Thi approah i imilar to that uggeted previouly by Mehta and Koke 4 exept they propoed uing a Helmholtz ree-energy denity model rather than a anonial Helmholtz ree-energy model. ote that the dierene between thee two approahe onern inite-ize eet. Alo, they provided reult or ga-liquid equilibrium only wherea we alulate three-phae equilibrium at the triple-point o the hited-ore Lennard-Jone ytem 5. That i, we ue the Gibb enemble to imulate liquid-ga, liquid-olid and ga-olid phae oexitene where eah ga, liquid and olid phae i idential (within tatitial error) and the olid phae i peret, i.e. ha no deet. A A (VT) Gibb enemble average or a pure ytem i written V ( V ) = Φ( V ) dv exp( β ( F(, V ) + F(, V ))) A(, V ) () = 0 0 where β = / k B T i the invere temperature, = -, V = V V, F ( V ) i the anonial Helmholtz ree-energy or partile in volume V

3 3 ( H ( )) = k T ln( Λ Z( V )) 3 dr F( V ) = k BT ln Λ exp β r B V! () where r indiate the poitional and orientational degree o reedom or partile, A(V) i a anonial enemble average dr! ( H ( r ) = A ( V ) Z ( V ) A( r )exp ) V β (3) where H i the ytem onigurational Hamiltonian, and Λ, the thermal de Broglie wavelength, i an irrelevant ontant in thi tudy. The partition um V Φ( V ) = dv exp( β ( F(, V ) + F(, V ))) (4) = 0 0 normalie the probability o ourrene or eah miro-tate. We ugget that or the olid phae F ( V ) and A(V) ould be alulated a priori or eah relevant anonial enemble, i.e. eah and the Gibb enemble average written A ( βh ( )) V dr x ( V ) = Φ( V ) dv exp( F (, V )) exp r V = 0 0! β A(, V ) (5) where x an be either olid (ubript ) or luid (ubript ). The Gibb enemble imulation then proeed by random intra-box diplaement o luid partile, random inertion and deletion move in the luid phae box (that are equivalent to inter-box traner in a onventional Gibb imulation) and random volume exhange move. Intrabox move are hoen and aepted a per the uual eletion rule. I volume exhange move are hoen o that an amount o volume i tranerred rom one box to the other at random with uniorm probability on the range V max to +V max then they are aepted with probability min {,exp( F β H + log(( V + V ) / V ))} β (6) where V i the hange in volume o the luid box and F = F, V V ) F (, V ) (7) H = H ( ( r ( V + V )) H ( r ( V )) where the partile entre-o-ma poition oordinate are aumed to ale linearly with volume in the uual way. Fluid-phae partile are inerted and deleted aording to the aeptane rule min {,exp( F β H + log( V ) log(( + )!/!))} β (8) where i the hange in the number o partile in the luid phae and x x

4 F = F V ) F (, V ) (9) H = H ( ( r + ) H ( r ) Clearly, i at equilibrium β F + β H i rarely uiiently mall then partile inertion or deletion in the luid phae box will be unlikely and the method will ail. Thi i the ae or pure hard-phere where the hemial potential at olid-luid equilibrium i large and poitive (etimated rom reult in reerene 6 and any equation o tate or the hardphere luid). It i oneivable that a avity bia 7 or taged inertion 8 method might lead to ueul inertion/deletion aeptane rate or thi ytem. However, or the (hited-ore) Lennard-Jone ytem at it triple-point the hemial potential i negative and o partile inertion and deletion hould be aepted at a ueul rate. The method o Mehta and Koke replae F (V) in (5) with (/V)V, where i the Helmholtz ree-energy denity o the olid phae. However, both method onverge to the exat reult in the thermodynami limit. Away rom thi limit thee method will yield dierent reult ine they predit dierent preure and hemial potential or the ame and V. To leading order the dierene in the hemial potential predited by thee method an be hown to be 9 µ ex = P ρ β β ρ ρ P P / ρ ( ) P / ρ (0) Thi dierene inreae with the invere o the ompreibility and o will be more pronouned or imulation where a olid phae, rather than a ga phae, i repreented by a ree-energy model. Sine in their work Mehta and Koke employed a ree-energy denity model or the ga phae thi dierene ould be inigniiant. In thi work we employ reeenergy model or the olid phae and o the dierene might be igniiant or relatively mall. Calulating the anonial Helmholtz ree-energy model Our Gibb enemble method require F ( V ) and A(V) to be alulated or eah relevant anonial enemble o the olid phae. Sine in moleular imulation we are ored to et periodi boundarie aording to the imulation box hape, the required anonial enemble with ixed V annot be imulated diretly whilt maintaining peret rytal truture uing the uual periodi boundary heme. However, an approximate method or alulating the anonial Helmholtz ree-energy o eah required tate o a pure ytem i a ollow. By making ue o the extenive approximation ( () A ex V ) Aex (, V ) or extenive quantitie A ex, and the intenive approximation Ain (, V ) A (, V ) () or intenive quantitie A in, every (V) tate an be related to another tate with a ixed number o partile,, in a volume V /. Thi approximation beome exat in the

5 thermodynami limit. Away rom thi limit thee approximation introdue inite-ize error beaue they pertain to the anonial enemble and beaue o periodi boundary ondition. For intenive quantitie, uh a the hemial potential, the leading order orretion to () i o order (/) (ee equation (0) or example). That i, to redue initeize error () hould be replaed by 0 A (, ) (, ) + ( / )( in V Ain V Ain V ) (3) 0 In general A in i dierent or eah denity, ρ = / V, and an be ound by obtaining A in or two or more value o at the ame denity. But or mall variation o rom thee orretion are alo mall. In thi work we neglet uh orretion on thi bai. In the ae o a olid phae an be made equal to m 0 where m i the number o unit ell and 0 i the number o partile per unit ell. In thi work we make ue o the probability ditribution or tate with volume in the ininiteimal range {V,V + dv} in an - partile PT imulation ( β ( F( V PV + ))dv p ( V, V + dv ) = exp ) µ (4) where µ i the hemial potential. Thi tate that i the hemial potential orreponding to the impoed preure i known then F ( V ) an be alulated abolutely rom the volume probability ditribution o an PT imulation. O oure, the problem i that the hemial potential o a rytalline olid orreponding to a preure P loe to oexitene i generally not known in advane or an arbitrary ytem, and it alulation via thermodynami integration i lengthy. We will diu thi problem in more detail later. The triple-point o the hited-ore Lennard-Jone luid We tet the above method on the hited ore Lennard-Jone ytem beaue; ) it i a imple, well-known model, ) the hemial potential at the triple-point i known and i negative, 3) we an avoid any potential inauray ariing rom long-range interation, and 4) reerene reult have already been alulated 5. The hited-ore potential i given by φ φ r = φ r φ r r r ) LJ ( ) LJ ( ) LJ ( ) ( (5) r r where the ull LJ potential i 6 σ σ φ LJ ( r) = 4ε (6) r r and ε and σ are the energy and length-ale parameter repetively. We hooe the triple-point temperature determined by Errington et. al. 5, T * = k B T / ε = 0.56, and perorm a liquid-ga Gibb enemble imulation, noting the denity, preure and hemial potential o eah phae. Preure i alulated rom the virial relation while the aurate method o Smit and Frenkel 0 i ued to alulate the hemial potential. We then perorm a olid-phae rytal PT imulation at the reulting ga-phae preure.

6 The anonial Helmholtz ree-energy model i generated by re-arranging (4), hooing a uitably mall interval o volume, dv = 0. σ 3, and uing (). We it a ubi urve to our data. The reult o thi PT imulation and the itted anonial Helmholtz ree-energy model are hown in Figure. Alo hown i the (anonial) variation o preure with denity alulated uing two dierent route, i.e. the virial relation and the dierential o -F with repet to V. Good agreement i obtained. Thi ree-energy model i then ued in the ga-olid and liquid-olid Gibb enemble imulation that are perormed aording to the eletion rule in (6) to (9) β F ' βp ( σ -3 ) Denity (σ -3 ) Figure. Reult o the olid-phae PT imulation or a hited-ore Lennard-Jone ytem at P = 0.008ε /σ 3 and βε = The irle are Monte-Carlo reult or the onigurational anonial Helmholtz ree-energy (F = F β - ln(λ 3 )) and the orreponding line i the ubi polynomial it, i.e. the model. The triangle how the preure alulated via the virial relation while the orreponding line i the ame alulated by dierentiating the anonial Helmholtz ree-energy model (with repet to -V). The reult in Table how good agreement with the reerene reult 5 and good oniteny with eah other. The number o partile in the olid phae o eah luid-olid Gibb enemble imulation rarely deviate by more than 00 rom it initial value o 56. When ombined with (0), where the irt and eond dierential o preure with repet to denity are etimated rom the olid phae PT imulation (ee Figure ), we obtain an etimate or the error in β µ ex o about 0.0. Thi i about the ame ize a the tatitial error in our alulation o the hemial potential at ga-liquid oexitene rom the galiquid Gibb enemble imulation (ee Table ). So we onider the variation in to be uiiently mall that () and () are valid, at leat or our anonial Helmholtz reeenergy model. O oure, the irt and eond dierential o the preure with repet to

7 denity in (0) reer to the ininite-ize ytem, but our anonial etimate baed on the PT imulation o the olid phae hould be uiiently aurate to etimate the magnitude o inite-ize eet. The large error orreponding to the liquid-olid Gibb imulation reult rom large and low lutuation in the liquid-olid imulation. So thi type o imulation mut be perormed or a uiiently large number o trial move. Diuion We have hown that luid-olid equilibrium an be imulated in the Gibb enemble provided an aurate anonial Helmholtz ree-energy model o the olid phae an be generated. I the hemial potential o the olid phae orreponding to a given preure i known, or an be eaily alulated, then thi model i eaily generated (with auray dependent on inite-ize eet only) or deet-ree olid by a ingle PT imulation. For greater auray two or more PT imulation an be ued to generate leading order ontribution to inite-ize orretion. In priniple our approah an alo be ued or more omplex ytem uh a interation-ite model o alkane, water and o on in onined geometrie. However, approximation () and () are ueul only or pure ytem. Further modiiation are alo neeary i rytal with deet are to be imulated. Clearly, it i important to be able to eiiently alulate the hemial potential o olid phae. Unortunately, urrent method or alulating the hemial potential o rytalline olid are umberome. We oon hope to preent a new, eiient and diret method or alulating the hemial potential o rytalline olid. Thi alulate the dierene in Gibb ree energy between one ytem and another ytem that i larger by an integral number o rytal unit ell. Thi i ahieved in what we all a dual-enemble by allowing MC move that with between thee ytem and by making ue o non-boltzmann ampling. Further, we hope to extend Tilwani ingeniou retiling algorithm to arbitrary 3D rytal and apply it to the dual-enemble to minimie thi dierene in ytem ize (to rytal unit ell or bulk rytal). Thi would then provide an eiient and diret method or meauring the ree energy and hemial potential o rytalline olid. Table. Reult o the ga-liquid (GL), ga-olid (GS) and liquid-olid (LS) Gibb enemble imulation o thi work and the reult in reerene 5 or the propertie o eah phae at the triple-point (βε = 0.56) o a hited-ore Lennard-Jone model ytem. The braketed number at the end o eah reult or denity, preure and hemial potential i the error to.d., while the braketed number at the end o eah entry or # Move i the number o rejeted trial move at the tart o eah imulation. µ = µ β - ln(λ 3 ) i the onigurational hemial potential. Phae Denity (σ -3 ) P / ε (σ -3 ) µ' / ε # Move Ga (GL) (9) 0.008(6) -3.3() 390(5) Liquid (GL) 0.85(5) 0.00() -3.() 390(5) Ga (GS) (5) () -3.() 4() Solid (GS) 0.936(5) 4() Liquid (LS) 0.80(7) -0.05(9) -3.5(7) 495(5) Solid (LS) 0.933(8) 495(5) Ga Liquid Solid

8 Reerene. B. Smit and D. Frenkel, Undertanding moleular imulation: rom algorithm to appliation (Aademi, ew York, 996).. R. M. Lynden-Bell, J. S. van Duijneveldt, and D. Frenkel, Mol. Phy. 80, 80 (993). 3. A. D. Brue,. B. Wilding, and G. J. Akland, Phy. Rev. Lett. 79, 300 (997). 4.. B. Wilding and A. D. Brue, Phy. Rev. Lett. 85, 538 (000). 5. L. D. Gelb and E. A. Muller, Fluid Phae Equ. 03, (00). 6. B. Chen, J. I. Siepmann, and M. L. Klein, J. Phy. Chem. B 05, 9840 (00). 7. A. Z. Panagiotopoulo, Mol. Phy. 6, 83 (987). 8. A. Z. Panagiotopoulo, Mol. Sim. 9, (99). 9. A. Z. Panagiotopoulo, Mol. Phy. 6, 70 (987). 0. J. Chang and S. I. Sandler, J. Chem. Phy. 8, 8390 (003).. D. Frenkel and A. J. C. Ladd, J. Chem. Phy. 8, 388 (984).. R. Radhakrihnan and K. E. Gubbin, Mol. Phy. 96, 49 (999). 3. R. Radhakrihnan, K. E. Gubbin, and M. Sliwinka-Bartkowiak, J. Chem. Phy. 6, 47 (00). 4. M. Mehta and D. A. Koke, Mol. Phy. 79, 39 (993). 5. J. R. Errington, P. G. Debenedetti, and S. Torquato, J. Chem. Phy. 8, 56 (003). 6. Y. Roeneld, M. Shmidt, H. Lowen, et al., J. Phy.: Conden. Matter 8, L577 (996). 7. M. Mezei, Mol. Phy. 6, 565 (987). 8. D. A. Koke and P. T. Cumming, Mol. Phy. 9, 973 (997). 9. J. I. Siepmann, I. R. MDonald, and D. Frenkel, J. Phy.: Conden. Matter 4, 679 (99). 0. B. Smit and D. Frenkel, Mol. Phy. 68, 95 (989).. P. Tilwani, in Diret Simulation o Phae Coexitene in Solid uing the Gibb Enemble: Coniguration Annealing Monte Carlo (Mater Thei: Colorado Shool o Mine, Golden, 000).