A first-order phase-transition, a super-cooled fluid, and a glass in a twodimensional. and A. Baram (2)

Transcription

1 A first-order phase-trasitio, a super-ooled fluid, ad a glass i a twodimesioal lattie gas model E. Eiseberg ( ) ad A. Baram (2). Shool of Physis ad Astroomy, Raymod ad Beverly Sakler Faulty of Exat Siees, Tel Aviv Uiversity, Tel Aviv 69978, Israel 2. Soreq NRC, Yave 8800, Israel Abstrat. Studyig the series expasio of the thermodyami potetial for the hard ore N 3 lattie gas model, we provide evidee for a first order phase trasitio with a fiite jump i desity ad etropy, i agreemet with umerial trasfer matrix alulatios. The solid brah termiates at the ritial ativity, but the fluid brah otiues beyod, desribig a super-ooled fluid. It termiates with / p ~ 0.85 ad fiite etropy per lattie site. This termiatio desity is lose to the radom-losest-pakig desity of the glassy state obtaied for ifiitely-fast oolig. The model thus exhibits a thermodyami meta-stable glassy phase with fiite Edwards' ompativity. PACS: F, C, Pf

2 The lattie gas model with earest-eighbors hard ore exlusio is log kow to udergo a otiuous phase trasitio from a low desity disordered fluid state to a ordered solid phase [-6]. O the other had studies of the otiuum model of three dimesioal hard spheres (whih are the limit of lattie models with a exteded exlusio) idiate a first order trasitio with oexistee of solid ad super-ooled fluid phases [7,8]. The ature of the trasitio i the hard disks ase is ot lear yet [9]. It is therefore expeted that the ature of the phase trasitio depeds o the rage of the hard ore iteratio. Bellemas ad Orba [0], ad later Orba ad Va Belle [], have studied umerially the lattie gas model of hard partiles o the two dimesioal square lattie, with iteratio rage that exteds up to the third shell of earest eighbors (N 3 model). This model is idetial to the model of hard-ore symmetri rossshaped petamers o the square lattie. Usig the matrix method of Kramers ad Waier [3] they studied rigs of ifiite legth ad fiite width M, for M=5, 0, 5 [0], ad 20 []. The symmetry of the model requires that the width M is a multiple of 5, otherwise the system aot reah the ofiguratio of losest pakig, whose desity is 5. Aalysis of the umerial results suggests that the N 3 model p exhibits a first order phase trasitio. This model is believed to be the simplest model that exhibits a first order trasitio. Furthermore, we have reetly show that this is also the simplest model that exhibits a glass trasitio, resultig from a radom fillig proess otrolled by diffusio (RSAD) [2]. This proess termiates i a stable radom pakig state, at the desity (3) (here ad i the rp

3 followig, the umber i paretheses is the uertaity i the last digit). Therefore, it is worthwhile to study the properties of this exlusio model, fousig i partiular o the relatios betwee the ritial values: exp( ), ), ) (the f ( s ( ritial ativity, the desity of the fluid at the trasitio, ad that of the solid at the trasitio, respetively) ad the termiatio values of the fluid ad the solid brahes, lookig for sigatures of the glassy behavior i the thermodyami properties. I this letter we study the ature of this phase trasitio through the aalyti properties of the low-desity ad high-desity expasios of the thermodyami futios as a futio of the ativity. Lookig at the aalyti properties of these series we show the existee of a first-order trasitio ad the oexistee of the solid phase with a super-ooled phase, extedig beyod the ritial ativity. We support our results by extedig the umerial matrix-method alulatios to M=25. The termiatio desity of the super-ooled fluid brah, as derived from extrapolatig the low-desity series, turs out to be very similar to rp, the limitig desity of the glassy phase obtaied for ifiitely fast oolig through the RSAD proess [2]. This last result suggests that the super-ooled phase might have the harateristis of a glassy state for fiite large ativities beyod the trasitio. If this is the ase, this model may serve as a miimal model for studyig the properties of strutural glasses.

4 We start by desribig the umerial results obtaied usig the matrix method, whih we exted here up to M=25. The approximats for the desity M ) M d(l d ( max as well as their derivatives are pratially M-idepedet outside of the ritial regio. However, for , their M-depedee is profoud. I ) partiular, the ompressibility d K has a sharp peak, whose height diverges d 2 like M, ad width dereases like M 2. This is iterpreted as a sigature of a first order phase trasitio, where a otiuous trasitio exhibits oly a weaker, logarithmi, divergee of this peak [3]. Table presets the ompressibility maximum (M ), the pressure at exp( ), the ompressibility maximum ad the desity at the maximum for the differet values of M, as well as the desities of the left (fluid) ad right (solid) ifletio poits of the peak. M Table p( ) ( ( M )) K max f s () (2) The ritial hemial potetial overges like ~ 4 M towards it s asymptoti value, leadig to (), or (4). The pressure at the ritial ativity overges like 2 M towards its asymptoti value p (2). Oe thus

5 sees that the matrix method alulatios provide quite aurate estimatios for the ritial parameters, ad the ature of the global miimum of the free eergy. However they do ot give a aurate estimatio of the desity gap at the trasitio, ad do ot provide ay iformatio o the possibility of meta-stable solid or fluid phases. We ow tur to study aalytially the asymptoti expasio of the thermodyami potetial. The thermodyami futios of the solid brah a be expaded aroud the losest-pakig ofiguratio, i terms of u=/, the ativity of holes. The pressure ad desity are give by: ' p( ) {l( ) b u } (a) 5 ' ( ) { b u } (b) 5 We derive the first six oeffiiets ' b of the high-desity series by the eumeratio method of ref. [2]. The iteger oeffiiets ' b are preseted i table 2: (the first four oeffiiets were give i ref [], with a mior error i the fourth oe). Table ' b A ratio-method type aalysis leads to the followig ubiased fit for the oeffiiets: b ' ( ) (2)

6 The maximum deviatio is of.5% for =2, ad the higher order oeffiiets are reprodued with relative auray better tha 0-4. The value of the leadig expoetial term beig so lose to (as estimated from the umerial matrix method results) strogly suggest that is ideed the radius of overgee of the high-desity series. I additio, we verified that Levi [3] ad Pade approximats of the series for diverge for ' b u. We therefore fix the value of the expoet, ad fit the oeffiiets by the two-parameter biased expressio: b ' ( ) (3) Assumig this form holds for the higher-order oeffiiets as well, we may sum the series for the pressure ad its derivatives at the trasitio poit. We fid the d ompressibility ~ to diverge at du like ( ) with the ritial expoet 0.0(), ad the ritial desity ad pressure are give by: s ( ) { (.905 ) O( )} 0.90() (4a) ps ( ) { (2.905) O( )} (2) 2 (4b) 5 where (x) is the Riema eta futio. Both ritial parameters are i a good agreemet with Levi ad Pade approximats. Note that the value of the ritial pressure obtaied here from the series expasio is idetial to the umerial value obtaied by the matrix method (table ). The fluid phase is desribed by the low-desity Mayer luster expasio i powers of the ativity. The overgee radius of this series is determied by a o-physial

7 sigularity at a poit o the egative axis 0, whih is typially very lose to the origi ompared to. Usig the M=25 matrix, we derived the 2 th ad 3 th luster oeffiiets (the first oeffiiets were already foud i []). Utiliig the 3 available luster oeffiiets ad the uiversal asymptoti form of the luster oeffiiets of two dimesioal repulsive models [4,5]: 5 ( ) 6 6 b ( ) ( ) (5) 0 we estimate the loatio of the leadig sigularity by the Levi aeleratio method[3] to be ( 0 ). The ratio is very big ad it seems, aively, that the series aot be extrapolated to the trasitio regio ad beyod. However, oe a use the well-defied patter of the luster oeffiiets of lassial repulsive systems to overome this obstale. Takig ito aout the fat that the etire spetrum of the tri-diagoal matrix represetatios (Yag-Lee eroes) of suh systems lies o the real axis [6], it is possible to trasform the series to tratable forms i spite of the sreeig effet of the leadig sigularity. The result of this approah were also ofirmed by Levi summatio of the ativity series applyig the trasformatio wexp(aw). Table R R

8 Followig the matrix represetatio otatio of [6], table 3 presets the values of the matrix elemets of the symmetri tri-diagoal matrix R, defied by: ( ) ( ) b R, where b are the Mayer expasio oeffiiets. The matrix elemets are obtaied from the first 3 luster oeffiiets. They rapidly overge to ostat asymptoti values alog the mai ad sub-diagoal, as expeted for repulsive models [6]. Thus, the matrix R beomes a tri-diagoal Toeplit-like matrix, whose asymptoti ostat values B (diagoal) ad A (off diagoal) are related to the two brah poits of the fluid thermodyami futios at 0 ad t by: 0 2A B t 2A B (6) where t is the termiatio poit of the fluid phase. Extrapolatig the values give i table 3, oe fids (see figure 2) that 2A+B=2.4907(3), i exellet agreemet with 0 the above diret estimate of the o-physial sigularity loatio 2.495(5), while 2A-B extrapolates to (3), leadig to 80(25). It thus follows that oe a ertaily exlude the possibility of the fluid brah termiatig at the trasitio ativity, leadig to the olusio that a super-ooled fluid exists for the t ativity rage. t Usig the asymptoti Toeplit form for the R matrix, ad employig the extrapolated values of the matrix elemets, we omputed a approximat for the desity (). The pressure of the fluid phase at the ritial ativity is estimated to be p ( T ) (4), i agreemet with the matrix method estimate T

9 p (2). The differee is greater tha the ombied uertaities, but still very small. The desity at the ritial ativity is estimated to be ( ) (0). Thus, the desity gap at the trasitio is (2 ) (or T 0.65() i uits of the losest-pakig desity), ad the etropy gap at the trasitio s is l( ) 0.23(7) k (per site). At the termiatio poit of the super-ooled fluid, the desity is ( s T t ) 0.75() ad the etropy is 0.08( ). Notably, this k termiatio desity is remarkably lose to the radom losest pakig desity obtaied for this model upo ifiitely fast oolig [2]. It is thus temptig to suggest that as the ativity ireases, the super-ooled fluid brah freees ito a glass, with limitig desity beig the radom losest pakig desity, ad fiite Edwards' ompativity [7]. I summary, we have studied the aalyti properties of the high-desity ad lowdesity ativity series of the thermodyami futios for the N 3 lattie-gas model. We fid that the solid brah termiates at fiite ativity, suggestig the existee of a phase trasitio. The termiatio ativity ad pressure are i exellet agreemet with those determied by the umerial trasfer matrix method, as well as with the pressure as alulated from the fluid brah at the ritial ativity, supportig the validity of the aalyti expasio. The fluid brah does ot termiate at the ritial ativity but rather extrapolates to desribe a meta-stable super-ooled fluid phase, whih termiates with fiite etropy per site (Edwards' ompativity [7]) ad desity very lose to that of a glass obtaied by ifiitely fast oolig of the system. We thus

10 suggest that the simple N 3 lattie gas model a serve as a miimal model for the study of strutural glasses. Referees. R.J. Baxter, J. Phys. A, 3, L6 (980). 2. D.S. Gaut ad M.E. Fisher, J. Chem. Phys., 43, 3233 (965). 3. F.H. Ree ad D.A. Chestut, J. Chem. Phys., 45, 3983 (966). 4. D.S. Gaut, J. Chem. Phys., 46, 3237 (967). 5. R.J. Baxter, I.G. Etig ad S.R. Tsag, J. Stat. Phys., 22, 465 (980). 6. A. Baram ad M. Fixma, J. Chem. Phys., 0, 372 (994). 7. B.J. Alder ad T.E. Waiwright, J. Chem. Phys., 27, 208 (957). 8. W.G. Hoover ad F.H. Ree, J. Chem. Phys., 49, 3609 (968). 9. K. Bider, S. Segupta ad P. Nielaba, J. Phys. Cod. Matter, 4, 2323 (2002). 0. A. Bellemas ad J. Orba, Phys. Rev. Lett., 7, 908 (966).. J. Orba ad D. Va Belle, J. Phys. A, 5, L50 (982). 2. E. Eiseberg ad A. Baram, J. Phys. A, 33, 729 (2000). 3. S.N. Lai ad M.E. Fisher, J. Chem. Phys., 03, 844 (995). 4. D. Levi, It. J. Comp. Math., 3, 37 (973). 5. A. Baram ad M. Luba, Phys. Rev. A, 36, 760 (987). 6. A. Baram ad J.S. Rowliso, J. Phys. A, 23, L399 (990). 7. S. F. Edwards ad R.B.S. Oakeshott, Physia A, 57, 080 (989).

11 Figures Figure : The desity (relative to the losest pakig desity p =/5) as a futio of the ativity. The thik lies desribe the thermodyamially stable fluid ad solid phases, with a fiite jump i desity for the ritial ativity (4), ad the log-dashed lie desribes the meta-stable super-ooled fluid. Matrix method results for M=20 ad M=25 are show as well, iterpolatig betwee the fluid ad the solid. The iset presets the ompressibility data as alulated by the matrix method, for M=20 (dashed) ad 25 (solid), exhibitig a peak at the ritial ativity.

12 Figure 2: The diagoal R (irles) ad twie the off-diagoal 2R,+ (squares) elemets of the tri-diagoal matrix R as a futio of, ad their extrapolatio for >>, as a futio of / 2.