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epl draft Phase diagram of the penetrable square well-model Riccardo Fantoni1, Alexandr Malijevsky´2, Andrés Santos3 and Achille Giacometti4 1 National Institute for Theoretical Physics (NITheP) and Institute of Theoretical Physics, Stellenbosch 7600, South Africa 2 E. Ha´la Laboratory of Thermodynamics, Instituteof Chemical Process Fundamentals of the ASCR, and Department of Physical Chemistry, Institute of Chemical Technology, Prague, 166 28 Praha 6, Czech Republic 13 Departamento de F´ısica, Universidad de Extremadura, E-06071 Badajoz, Spain 1 4 Dipartimento di Chimica Fisica, Universita` Ca’ Foscari Venezia, S. Marta DD2137, I-30123 Venezia, Italy 0 2 n a PACS 64.60.De– J PACS 64.60.Ej– 4 PACS 64.70.D-– PACS 64.70.Hz– ] t PACS 64.70.F-– f o Abstract – We study a system formed by soft colloidal spheres attracting each other via a s . square-well potential, using extensive Monte Carlo simulations of various nature. The softness t a is implemented through a reduction of the infinite part of the repulsive potential to a finite one. m For sufficiently low values of the penetrability parameter we find the system to be Ruelle stable - with square-well like behavior. For high values of thepenetrability the system is thermodynami- d cally unstable and collapses into an isolated blob formed by a few clusters each containing many n overlapping particles. For intermediate values of the penetrability the system has a rich phase o diagram with a partial lack of thermodynamic consistency. c [ 1 v 8 0 Pair effective interactions in soft condensed matter σ is the width of the repulsive barrier. For ǫ one r 7physicscanbeofvariousnatureandonecanoftenfindreal recovers the SW model, while for ∆ = 0 or ǫ→=∞0 one a 0 systemswhoseinteractionisboundedatsmallseparations recovers the PS model. . 1as, for instance, in the case of star and chain polymers For finite ǫ , ǫ /ǫ is a measure of the penetrability of a a r 0 [1]. Inthiscase,paradigmaticmodels,suchassquare-well the barrierandwe shallrefer to ǫ /ǫ asthe penetrability 1 a r (SW)fluids,thathavebeenrathersuccessfulinpredicting ratio. PSWpairpotentialscanbeobtainedaseffectivepo- 1 :thermo-physicalpropertiesofsimpleliquids,arenolonger tentialsforinstanceinpolymermixtures[10,11]. Whilein v useful. Instead, different minimal models accounting for themajorityofthecasesthewelldepthǫ ismuchsmaller i a Xthe boundness of the potential have to be considered, the than the repulsive barrier ǫ (low penetrability limit) this r rGaussian core model [2] and the penetrable-sphere (PS) mesoscopic objects are highly sensitive to external condi- amodel [3–5] being well studied examples. More recently, tions(e.g. qualityofthesolvent)andmaythusinprinciple thepenetrablesquare-well(PSW)fluidhasbeenaddedto exhibit higher values of the penetrability ratio ǫ /ǫ . a r thiscategory[6–9]withthe aimofincluding the existence It is well known that three-dimensional SW fluids ex- of attractive effective potentials. The PSW model is ob- hibit a fluid-fluid phase transition for any width of the tainedfromtheSWpotentialbyreducingtoafinite value attractive square well [12–16], the liquid phase becoming the infinite repulsion at short range, metastable against the formation of the solid for a sufficiently narrow well [15]. It is also well established that ǫ , r σ , r  ≤ in the PS fluid (that lacking an attractive component in φPSW(r)= −ǫa , σ <r ≤σ+∆ , (1) the pair potential cannot have a fluid-fluid transition) an 0 , r >σ+∆ , increase of the density leads to the formation of clusters  where ǫ and ǫ are two positive energies accounting for ofoverlappingparticlesarrangedinanorderedcrystalline r a the repulsiveandattractivepartsofthe potential,respec- phase [3,17–19]. tively, ∆ is the width of the attractive square well, and While the novel features appearing even in the one- p-1 Riccardo Fantoni1 Alexandr Malijevsky´2 Andrés Santos3 Achille Giacometti4 dimensional case have been studied in some details [6–9], for any given width ∆/σ < 1 of the well, there is a limit no analysis regarding the influence of penetrability and valueofthepenetrabilityratioǫ /ǫ abovewhichnofluid- a r attractiveness on the phase behavior of PSW fluids have fluid phase transitionis observed. This is depicted in Fig. been reported, so far, in three dimensions. The present 1 where it can be observed that (for ∆/σ < 1) this line paper aims to fill this vacancy. lies outside the Ruelle stable region ǫa/ǫr <1/f∆. A system is defined to be Ruelle stable when the total potential energy U for a system of N particles sat- N 0000....33330000 isfies the condition U NB where B is a finite pos- N ≥ − itive constant [20,21]. In Ref. [8] we proved that in the 0000....22225555 nnnnoooo ttttrrrraaaannnnssssiiiittttiiiioooonnnn one-dimensional case the PSW model is Ruelle stable if ǫ /ǫ < 1/2(ℓ+1), where ℓ is the integer part of ∆/σ. 0000....22220000 a r This result can be extended to any dimensionality d by the following arguments. The configuration which mini- εεεεεεεε////arararar 0000....11115555 ttttrrrraaaannnnssssiiiittttiiiioooonnnn mizes the energy of N particles interacting via the PSW 0000....11110000 potential is realized when M closed packed clusters, each consisting of s = N/M particles collapsed into one point, 0000....00005555 are distributed such that the distance between centers of RRRRuuuueeeelllllllleeee ssssttttaaaabbbblllleeee two neighbor clusters is σ. In such a configuration, all 0000....00000000 0000....0000 0000....2222 0000....4444 0000....6666 0000....8888 1111....0000 the particles of the same cluster interact repulsively, so ∆∆∆∆////σσσσ the repulsive contribution to the total potential energy is ǫ Ms(s 1)/2. In addition, the particles of a given r − cluster interact attractively with all the particles of those Fig. 1: Plot of the penetrability ratio ǫa/ǫr as a function of f∆ clusters within a distance smaller that σ+∆. In the ∆/σ. Thedisplayedlineseparatestheparameterregionwhere two-dimensionalcase,f∆ =6and12if∆/σ <√3 1and thePSWmodeladmitsafluid-fluidphasetransitionfromthat − √3 1<∆/σ <1,respectively. Ford=3,thecaseweare where it does not. The highlighted region (ǫa/ǫr 1/12 for i√nt2e−res1te<d∆in/,σon<eh√a3sf∆1,=an12d,√183,an1d<42∆if/∆σ/<σ1<, r√es2p−ec1-, ∆an/dσǫ<a/ǫ√r2−11/,42ǫa/foǫrr √≤31/118 <for∆√/2σ−<11<) s∆ho/wσs≤<wh√er3e−th1e, ≤ − − − − modelisexpectedtobethermodynamicallystableinthesense tively. The attractive contribution to the total potential energy is thus −ǫa(M/2)[f∆−b∆(M)]s2, where b∆(M) oofnRthueelhleorfiozronatnaylatxhiesrm(ǫao/dǫyrnam0ic)astnadtei.tsTfluhied-SflWuidmtoradnelsiftaiollns accounts for a reduction of the actual number of clus- → is expected to be metastable against the freezing transition tersinteractingattractively,duetoboundaryeffects. This for ∆/σ . 0.25 [15]. The circles are the points chosen for quantityhasthepropertiesb∆(M)<f∆,b∆(1)=f∆,and the calculation of the coexistence lines (see Figs. 2 and 5). limM→∞b∆(M)=0. Forinstance,inthetwo-dimensional Thecrosses are thepointschosen for thedetermination of the casewith∆/σ <√3 1onehasb∆(M)=2(4√M 1)/M. boundarybetweenextensiveandnon-extensivephases(seeFig. − − Therefore, the total potential energy is 3). U (M) 1 N N = + F(M), (2) It is instructive to analyze the detailed form of the Nǫ −2 2M r coexistence curves below (but close to) the limit line of where F(M) (ǫa/ǫr)b∆(M)+(1 f∆ǫa/ǫr). If ǫa/ǫr < Fig. 1. This is presented in Fig. 2. We have explicitly ≡ − 1/f∆, F(M)ispositivedefinite, soUN(M)/N hasalower checked that our code reproduces completely the results bound and the system is stable in the thermodynamic of Vega et al. [12] for the SW model. Following stan- limit. On the other hand, If ǫa/ǫr > 1/f∆, one has dard procedures [12] we fitted the GEMC points, near F(1) = 1 but limM→∞F(M) = (f∆ǫa/ǫr 1) < 0. In the critical point using the law of rectilinear diameters − − that case, there must exist a certain finite value M =M0 (ρl+ρg)/2=ρc+A(Tc T) where ρl (ρg) is the density − such that F(M) < 0 for M > M0. As a consequence, of the liquid (gas) phase, ρc is the critical density, and in those configurations with M > M0, UN(M)/N has no Tc is the critical temperature. Furthermore,the tempera- lowerboundinthelimitN andthusthesystemmay turedependence ofthe densitydifferenceofthecoexisting be unstable. →∞ phases is fitted to the scaling law ρ ρ = B(T T)β l g c − − We have performed an extensive analysis of the vapor- where β = 0.32 is the critical exponent for the three- liquid phase transition of the system using Gibbs Ensem- dimensional Ising model. The amplitudes A and B where ble Monte Carlo (GEMC) simulations [22–26], starting determined from the fit. In the state points above the from the corresponding SW fluid condition and gradually limitlineofFig. 1wehaveconsideredtemperaturesbelow increasingthepenetrabilityratioǫ /ǫ untilthetransition the critical temperature of the correspondingSW system. a r disappears. AtotalnumberofN =512particleswith2N The disappearance of the fluid-fluid transition is signaled particle randomdisplacement, N/10volume changes,and bytheevolutiontowardsanemptygasboxandaclustered N particle swap moves between the gas and the liquid phase in the liquid box. box, on average per cycle, were considered. We find that As discussed, the PSW fluid is thermodynamically Ru- p-2 Phase diagram of the penetrable square well-model 00..7788 suchthatthe systemis metastableifT >Tinst andunsta- ble if T < Tinst. We determined the metastable/unstable 00..7766 crossover by performing NVT Monte Carlo simulations with N = 512 particles initially uniformly distributed 00..7744 within the simulation box. Figure 3 reports the results εεT/T/aa 00..7722 in the reduced temperature-density plane for ∆/σ = 0.5 BB kk 00..7700 and for two selected penetrability ratios ǫa/ǫr = 1/7 and 1/4, the first one lying exactly just above the limit line of 00..6688 SW Fig. 1 while the second deep in the non-transition region. εa/εr=1/6 We workedwithconstantsizemoves(insteadoffixingthe 00..6666 acceptanceratios)duringthesimulationrun,choosingthe 00..00 00..11 00..22 00..33 00..44 00..55 00..66 00..77 00..88 00..99 movesizeof0.15inunitsofthethesimulationboxside. A ρρσσ33 11..2255 4.0 3.5 11..2200 3.0 eexxtteennssiivvee 11..1155 2.5 εεkT/kT/BaBa 11..1100 ε /ε=S1W/8 εkT/Ba 12..50 a r 11..0055 1.0 0.5 nnoonn--eexxtteennssiivveeε /ε=1/7 11..0000 0.0 εaa/εrr=1/4 0.0 0.2 0.4 0.6 0.8 1.0 00..00 00..11 00..22 00..33 00..44 00..55 00..66 00..77 ρσ3 ρρσσ33 22..8800 22..7755 Fig. 3: Regions of the phase diagram where the PSW fluid, 22..7700 with ∆/σ = 0.5 and two different values of ǫa/ǫr, is found to beextensiveornon-extensive(hereweusedN =512particles). 22..6655 The left shows representative snapshots in the two regions. aa 22..6600 εεT/T/ The instability line corresponding to the higher penetrability kkBB 22..5555 case (dashed line) lies above the one corresponding to lower 22..5500 penetrability (continuousline). 22..4455 SW 22..4400 εa/εr=1/11 crucial point in the above numerical analysis is the iden- 22..3355 tification of the onset of the instability. Clearly the phys- 00..00 00..11 00..22 00..33 00..44 00..55 00..66 ical origin of this instability stems from the fact that the ρρσσ33 attractive contribution increases unbounded compared to the repulsive one and particles tend to lump up into clusters of multiply overlappingparticles (“blob”). Hence the Fig. 2: Fluid-fluid coexistence line. The solid circle represents energy can no longer scale linearly with the total number thecriticalpoint(ρc,Tc). Inthetoppanel,for∆/σ =0.25and of particles N and the thermodynamic limit is not well ǫa/ǫr =1/6,onehasρcσ3=0.307andkBTc/ǫa =0.762;inthe defined (non-extensivity). We define a cluster in the fol- middlepanel,belowthelimitpenetrability,for∆/σ =0.5and lowing way. Two particles belong to the same cluster if ǫa/ǫr = 1/8, one has ρcσ3 = 0.307 and kBTc/ǫa = 1.241; and there is a path connecting them, where we are allowed to in the bottom panel for ∆/σ =1.0 and ǫa/ǫr =1/11, one has ρcσ3 =0.292 and kBTc/ǫa =2.803. The lines are the result of move on a path going from one particle to another if the centers of the two particles are at a distance less than σ. the fit with the law of rectilinear diameters. The SW results are theones of Vegaet al. [12] The state points belonging to the unstable region are characterizedby asuddendropofthe internalenergyand ofthe acceptanceratiosatsome pointsin the systemevo- elle stable when ǫa/ǫr < 1/f∆ for all values of the ther- lution during the MC simulation. Representative snap- modynamic parameters. For ǫa/ǫr > 1/f∆ the system is shots show that a blob structure has nucleated around a either extensive or non-extensive depending on temper- certain point and occupies only a part of the simulation ature and density. For a given density, one could then box with a few clusters. The number ofclusters decreases expect that there exists a certain temperature Tinst(ρ), as one moves away from the boundary line found in Fig. p-3 Riccardo Fantoni1 Alexandr Malijevsky´2 Andrés Santos3 Achille Giacometti4 3 towards lower temperatures. Upon increasing ǫa/ǫr the 10000 ε /ε=1/7 number of clusters decreases and the number of particles εa/εr=1/8 a r per cluster increases. We assume a state to be metastable 1000 if the energy does not undergo the transition after 107N single particle moves. The fluid-fluid transition above the 100 limit penetrability line of Fig. 1 is not possible because r) 10 thenon-extensivephaseshowsupbeforethe criticalpoint g( is reached. 1 The boundary line of Fig. 3 is robust with respect to the size of the system, provided that a sufficiently large 0.1 size (N 512) is chosen. When the number of particles ≥ inthe simulationgoesbelow thenumberofclusterswhich 0.01 0 1 2 3 4 wouldforminthenon-extensivephasethesystemseemsto r/σ remain extensive. For instance, with N = 1024 particles we obtained under the ǫ /ǫ =1/7,∆/σ =0.5 conditions a r athresholdtemperaturekBT/ǫa 1.15forρσ3 =0.4and Fig. 4: Radial distribution function for the PSW model at 2.25 for ρσ3 =0.8, which are clos≈e to the values obtained ∆/σ = 0.5, kBT/ǫa = 1.20, and ρσ3 = 0.7 for two different with N =512 particles. valuesof thepenetrability ratio ǫa/ǫr. There is an apparent hysteresis in forming and melting the non-extensive phase. For example when ǫ /ǫ = 1/7, a r ∆/σ = 0.5, and ρσ3 = 1.0 the non-extensive phase starts of 0.1. Here we only consider the case of PSW with ≈ ∆/σ =0.5 and ǫ /ǫ =1/8 and 1/15. forming when cooling down to k T/ǫ = 2.75. Upon in- a r B a creasing the temperature again, we observed a melting For the SW system with a width ∆/σ = 0.5 the transitionatsignificantly highertemperatures (kBT/ǫa & critical point is known to be at kBTc/ǫa = 1.23 and 4). We also found the hysteresis to be size dependent; in ρcσ3 = 0.309, its triple point being at kBTt/ǫa = 0.508, thesamestateforρσ3 =0.6themeltingtemperaturesare Ptσ3/ǫa = 0.00003, ρlσ3 = 0.835, and ρsσ3 = 1.28 [15]. k T/ǫ 2.5 for N = 256, k T/ǫ 4.5 for N = 512, No solid stable phase was found in Ref. [15] for temper- B a B a and k T≈/ǫ 6.5 for N = 1024. Th≈is suggests that the atures above the triple point, meaning that the melting B a extensive pha≈se in Fig. 3 is actually metastable with re- curve in the pressure-temperature phase diagram is al- spect to the non-extensive phase in the thermodynamic most vertical. On the other hand, the phase diagram of limit. However the metastable phase can be stabilized by the PSW fluid with the same well width and a value of takingthe sizeofthe systemfinite. Inadditionwecannot ǫa/ǫr = 1/8, just below the limit line of Fig. 1, shows exclude, a priori, the possibility of a true extensive stable that the melting curve has a smooth positive slope in the phase as it is not prevented by the Ruelle criterion. We pressure-temperaturephasediagram. Inordertoestablish notethatthesizedependenceofthehysteresisinthemelt- this, we used NPT simulations to follow the kBT/ǫa = 1 ing could be attributed to the fact that the blob occupies isotherm. FromFig. 5wecanclearlyseethejumpsinden- only part of the simulation box and therefore a surface sitycorrespondingtothegas-liquidandtotheliquid-solid term has a rather high impact on the melting tempera- coexistence regions. The presence of a solid phase can be ture. checkedbycomputingtheQ6 orderparameter[27],calcu- A convenient way to characterize the structure of the lated for the center of mass of individual clusters, that in fluid is to consider the radial distribution function g(r). the present case turns out to be Q6 0.35. The crystal ≈ This is depicted in Fig. 4 for the cases ∆/σ = 0.5, structure is triclinic with a unit cell with a = b = c = σ k T/ǫ = 1.20, and ρσ3 = 0.7 at ǫ /ǫ = 1/8 and and α=β =π/3 and γ =cos−1(1/4). There are possibly B a a r ǫ /ǫ = 1/7. The latter case is in the non-extensive re- other solid-solid coexistence regions at higher pressures. a r gion, according to Fig. 3. We can clearly see that there Moreover the relaxation time of the MC run in the solid is a dramatic change in the structural properties of the regionisanorderofmagnitudehigherthanthe oneinthe PSW liquid. In the non-extensive case, ǫ /ǫ = 1/7, the liquid region. a r radial distribution function grows a huge peak at r = 0 Wealsorunatthesametemperatureasetofsimulations and decays to zero after the first few peaks, which sug- forthePSWfluidwithǫa/ǫr =1/15. Theresults(seeFig. gests clustering and confinement of the system. 5)showednoindicationofastablesolid,inagreementwith In order to study the solid phase of the PSW model the fact that at this very low value of the penetrability below the limit penetrability we employed isothermal- ratio the system is SW-like. isobaric(NPT)MCsimulations. Atypicalrunwouldcon- A peculiarity of the PSW in the region below the limit sist of 108 steps (particle moves or volume moves) with penetrability of Fig. 1, but not in the Ruelle stability re- an equilibration time of 107 steps. We used 108 particles gion,isaviolationoftheClausius-Clapeyronequation[28] andadjustedthe particle movesto have acceptanceratios alongtheliquid-solidcoexistencecurve,whatrepresentsa of 0.5 and volume changes to have acceptance ratios partial lack of thermodynamic consistency ≈ p-4 Phase diagram of the penetrable square well-model plane x−y ble region, and behaves as the corresponding SW model. 10.0 Insummary,byexperimentallytuningtherepulsivebar- rierrelativeto the welldepth onecouldobserve(a)stable 1.0 phases resembling those of a normal fluid, (b) metastable 3σεP/a plane y−z pthheasceosllawpitshe oflfutihde-flsuyisdteamndtofluaidsm-saolllidrecgoioexni.stence, or (c) 0.1 0.0 εaε/aε/rε=r=1/11/58 plane z−x ∗∗∗ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 We thank Tatyana Zykova-Timan and Bianca M. ρσ3 Mladek for enlighting discussions and useful sugges- tions. The support of PRIN-COFIN 2007B58EAB, FEDER FIS2010-16587 (Ministerio de Ciencia e Inno- vacioń), GAAS IAA400720710is acknowledged. 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