multiple critical points for square-well potential with repulsive shoulder

ARTICLE IN PRESS

Physica A 346 (2005) 347–371

0378-4371/$ -

doi:10.1016/j

�Tel.: +1-E-mail ad

www.elsevier.com/locate/physa

Multiple critical points for square-well potentialwith repulsive shoulder

John A. White�

Department of Physics, American University, Washington, DC 20016-8058, USA

Received 26 April 2004

Available online 19 August 2004

Abstract

A global renormalization procedure used recently to calculate thermal volumetric properties

near and to far from the critical point for several simple square-well and Lennard–Jones fluids

for which comparison could be made with Monte Carlo and molecular dynamic simulations is

here applied to a square-well with repulsive shoulder of width and height that gives ratios of

first and second critical point temperatures and pressures similar to ones proposed recently as

possibilities for water, if indeed water has a second critical point below which, briefly, before

ice forms, it has separate low density and high density liquid phases. Thermal behavior at and

below the second critical point is found to be quite sensitive to choices of height and width of

the shoulder relative to the depth and width of the well, with, for some choices, a subsequent

closing of the liquid–liquid coexistence curve as temperature drops, and emergence at a yet

lower temperature of what appears to be a third critical point.

r 2004 Elsevier B.V. All rights reserved.

PACS: 64.10.+h; 64.60.kw; 64.70.Ja; 05.10.Cc

Keywords: Critical points; Gas–liquid and liquid–liquid coexistence curves; Global renormalization;

Square-well potential with repulsive shoulder; Third critical point

see front matter r 2004 Elsevier B.V. All rights reserved.

.physa.2004.07.039

202-885-2747.

dress: [email protected] (J.A. White).

www.elsevier.com/locate/physa

ARTICLE IN PRESS

J.A. White / Physica A 346 (2005) 347–371348

1. Introduction

In recent papers [1], Stanley and coworkers have discussed the possibility of asecond critical point in water [2,3] which may occur,1 according to simulations byYamada et al. [5], at a temperature Tc2 ’ Tc1=3; pressure Pc2 ’ 15Pc1; and numberdensity (r ¼ N=V Þ of rc2 ’ 3:5rc1: Here, and in the following, the subscript ‘‘c1’’ isused for the low density critical point and ‘‘c2’’ for the high density critical point. Ithad been shown in a somewhat earlier paper [6] that ‘‘liquid–liquid phase transitionphenomena can arise solely from an isotropic pair interaction potential with twocharacteristic lengths’’. Treating the interacting particles as simple spheres, theauthors and some collaborators, Refs. [6–11], considered in particular an isotropicpair potential (Fig. 1) with a hard-core radius r ¼ a; a repulsive shoulder of radiusb4a; and an attractive well from r ¼ b out to radius r ¼ c4b; where r is the distancebetween the centers of the two interacting spheres. For aorob the pair of particlesrepel each other with energy UR40; for boroc they attract each other with energy�UAo0: (Fig. 1 is not drawn to the same scale as in references cited above.)In Refs. [12–15] a ‘‘global’’ renormalization group theory was developed and

applied to some simple isotropic Lennard–Jones and square-well pair potentials forwhich comparison could be made with results of molecular dynamics and MonteCarlo simulations. Agreement was found with simulations to accuracies of severalpercent. The question now arises: can the same (approximate) renormalizationmethods be applied successfully to shoulder-next-to-well potentials? In particular, dothe renormalization calculations give results in good or bad agreement with what hasbeen suggested by Stanley and coworkers [1,5] as a possibility for water?The renormalization approach used here [12], and employed in Refs. [13–15],

begins with a simple mean field approximation, then includes corrections, notconsidered in the mean field approximation, for fluctuations of density ofincreasingly long wavelengths, out to the longest wavelengths that make anyappreciable contribution to the thermal properties of interest. The mean-fieldapproximation begins with an expression for the Helmholtz free energy per unitvolume, f 0ðT ;rÞ; which yields the Carnahan–Starling [16] expression for thecompression ratio, Z ¼ PV=RT ; for a gas of hard spheres. In a first, simplestapproximation, the hard spheres could be taken to have diameter d ¼ constant ¼ a:To the f 0ðT ; rÞ is added the contribution of the repulsive and attractive interactions,characterized by the shoulder height and width and well depth and width of the pairpotential; in calculating this contribution the density distribution is assumed to begiven by the pair correlation function grepulsiveðd;r; rÞ for a gas of hard spheres inPercus–Yevick approximation [17] for spheres of diameter d. The renormalizationcorrections then are made for local deviations (fluctuations) of density atincreasingly long wavelengths, beginning with the shortest wavelengths that makeappreciable contributions to thermal properties not already included in the f 0ðT ;rÞ

1Other investigators, using different approaches, have arrived at appreciably different estimates for the

temperature and pressure at the (conjectured) second critical point. See, for instance, the review article by

Debenedetti, Ref. [21] below, and Ref. [4], and results summarized in Ref. [4] for other approaches.

ARTICLE IN PRESS

Fig. 1. Pair potential UðrÞ for attractive well with repulsive shoulder. Notation is as in Ref. [4]: distance a

is the hard-core radius, b the soft-core radius, and c the cut-off radius. Energy UA is the attractive energy,

and UR the repulsive energy. This figure is drawn to a different scale than in Fig. 1 of Ref. [4].

J.A. White / Physica A 346 (2005) 347–371 349

and grepulðd;r; rÞ; and continuing, doubling the fluctuation wavelength at eachiteration of the renormalization procedure, until longer wavelengths make negligiblecontributions, even at the critical point.The approximation that the spheres have diameter d ¼ a was used successfully in

Ref. [14], which treated simple square well potentials (such as when UR ¼ �UA orb ¼ a in Fig. 1). That approximation was not a good one, however, for treating theLennard–Jones potential, which has a sloping repulsive potential rather than onethat rises abruptly from �UA to infinity at r ¼ a: For the Lennard–Jones potential, atemperature-dependent, ‘‘effective’’ hard-sphere diameter was used [13], employingan expression for d ¼ dðTÞ given by Barker and Henderson [18]. Here, in the spirit ofthat work, the sphere diameter for use in the pair correlation function andCarnahan–Starling approximation will be taken to have the same, Barker–Hender-son, effective diameter dðTÞ; intermediate between a, where the potential becomesinfinitely repulsive, and b, where the potential changes from repulsive to attractive,given at temperature T by

dðTÞ ¼

Z b

0

ð1� e�bUðrÞÞdr ; (1)

where b ¼ 1=kBT ; the kB being Boltzmann’s constant. For work reported in Sections2 and 3 below this dðTÞ is used as the starting point also for the calculation ofcontributions of (the remaining portion of) the repulsive shoulder and attractivesquare well. Alternatively, the starting point for calculation of the contribution fromthe ‘‘attractive’’ interactions can be taken instead to be d 0

ðTÞ; intermediate betweenthe dðTÞ; given by Eq. (1), and b; where the potential changes from repulsive toattractive. A choice for d 0

ðTÞ about half way between dðTÞ and b was found to givebest quantitative agreement with simulation results for the Lennard–Jones potential[13]. Modifications to results in Sections 2 and 3 resulting from use of a d 0

ðTÞ4dðTÞ

are discussed briefly below in Section 4.

ARTICLE IN PRESS

Table 1

Ratios Pr ¼ Pc2=Pc1 and Tr ¼ Tc2=Tc1 of the two critical point pressures and temperatures for shoulder

height h ¼ UR=UA ¼ 0:3 and choices of shoulder width w ¼ 0:3645 4% or of shoulder-plus-well width

(R � 1Þ ¼ ðc � aÞ=a ¼ 1:2141 1%

w R Pr Tr

0.3645 2.2141 15.3 0.333

0.3499 2.2141 28.7 0.205

0.3791 2.2141 13.7 0.468

0.3645 2.2020 23.4 0.341

0.3645 2.2262 3.9 0.421


The method of calculation, outlined above, is described in greater detail inAppendix A.

2. Some results

2.1. A second critical point

Using Eq. (1) and the renormalization procedure in Appendix A, it was found thattwo critical points, with P2r � Pc2=Pc1 ’ 15; T2r � Tc2=Tc1 ’ 0:33; and with r2r �

rc2=rc1 somewhat larger than 4, are obtained for a fairly wide range of choices forthe height of the repulsive shoulder, when the shoulder width and well width areadjusted suitably. In particular, the choices of reduced shoulder height h ¼ UR=UA;shoulder width w ¼ ðb � aÞ=a; and overall range of the potential R ¼ c=a (not to beconfused with the gas constant R ¼ NAvagadrokB that appears in the expression for theratio PV=RTÞ; having values hjwjR ¼ 0:1j0:221j1:990; 0:2j0:288j2:094; and0:3j0:3645j2:2141 all give P2r ’ 15:3; T2r ’ 0:33; and r2r somewhat larger than 4.As the height h increases, it was found that the location of the second critical pointbecomes especially sensitive to the precise choices made for w and R. For example,for h ¼ 0:3; choices wjR ¼ 0:36j2:21; 0:36j2:22; 0:37j2:21; and 0:37j2:22; gave,respectively P2rjT2r ¼ 21:87j0:287; 18:53j0:263; 15:82j0:391; and 8:36j0:429; respec-tively, while wjR ¼ 0:3645j2:2141 gives 15:34j0:333: Table 1 shows the quite widerange of values forP2r and T2r at h ¼ 0:3 that result for somewhat larger changes inw and R, in particular, for values of w that are 4% smaller or larger than the choicew ¼ 0:3645 and for values of (R � 1Þ that are 1% smaller or larger than 1.2141.Figs. 2a and b show the shape of the coexistence curve below Tc2 for hjwjR ¼

0:3j0:3645j2:2141 and also how the pressure P rises on the coexistence curve as thetemperature T drops to a rather low value.2 This latter behavior, dP=dTo0 below

2These and a few other results for the h ¼ 0:3; wjR ¼ 0:3645j2:2141 and wjR ¼ 0:36j2:23 potentialsdiscussed in Sections 2.1 and 2.2 were mentioned in a talk presented by the author at the 15th Symposium

on Thermophysical Properties, held in Boulder, Colorado in June, 2003. See Ref. [19]. With regard to a

third, or even more than three critical points, Ref. [20] notes that a square well with two shoulders of

different heights ‘‘may have three liquid phases of three increasing densities characterized by the

penetration of particles into a soft core of smaller diameter’’.

ARTICLE IN PRESS

Fig. 2. (a) and (b). Temperature dependence of the coexisting densities and pressure on the liquid–liquid

coexistence curve, at temperatures ToTc2 ’ 0:33Tc1; for well-with-shoulder parameters (Fig. 1) h ¼

UR=UA ¼ 0:3; w ¼ ðb � aÞ=a ¼ 0:3645; R ¼ c=a ¼ 2:2141:


Tc2; is as expected for water [21] and similar to that shown by Debenedetti andStanley in their Fig. 5 in the June, 2003 issue of Physics Today [22]. It should benoted that, in the calculations reported here, the qualitative behavior of theliquid–liquid coexistence curve and temperature dependence of the pressure P alongthis coexistence curve below Tc2 is quite sensitive to the height of the repulsiveshoulder. This is seen in Figs. 3a and b, which are plotted for shoulder heighth ¼ 0:25 and wjR ¼ 0:3247j2:1518; which give P2rjT2r ¼ 15:41j0:333; almost thesame as for the hjwjR used for Figs. 2a and b (where h ¼ 0:3), but, in particular,produce a dP=dT curve which is much less strongly negative than was the case forthe shoulder of height h ¼ 0:3 (Fig. 2b). For both the h ¼ 0:2 and 0.1 choicesmentioned above that gave similar values for P2rjT2r the slope dP=dT along thecoexistence curve is positive below Tc2; rather than initially negative as was the casefor hjwjR ¼ 0:3j0:3645j2:2141:A larger value of h, for example, h ¼ 0:35; results in extreme sensitivity to w and R.

Thus hjwjR ¼ 0:35j0:405j2:270 results in a second critical point appearing atP2rjT2r ¼ 32:93j0:273; while hjwjR ¼ 0:35j0:405j2:272 (i.e., for the same h and w, andR just 0.1% larger) produces P2rjT2r ¼ 9:94j0:443: (Something new is going on here,as will be discussed later.)Returning to hjwjR ¼ 0:3j0:3645j2:2141 used for Figs. 2a and b, the Figs. 4a and b

show several isochores where the pressure passes through a minimum (at constantpressure the density is maximum) as the temperature drops, calculated (Fig. 4a) andobtained from water results near 0 C reported by NIST [23] (Fig. 4b). Fig. 5 shows

ARTICLE IN PRESS

Fig. 4. (a) and (b). Some calculated (a) and measured [23] (b) isochores that show a pressure minimum at a

temperature above the normal freezing point of water (0 C; T=Tc1 ¼ 0:422). For (a), calculated forshoulder height h ¼ 0:3 and wjR ¼ 0:3645j2:2141; the isochores shown are, from bottom to top, for

densities r=rc1 ¼ 2:97 and 0.4% and 0.8% higher; for (b) the isochores are for densities of 1 g=cm3

(r=rc1 ¼ 3:1056) and 0.2% and 0.4% higher.

Fig. 3. (a) and (b). Like for Figs. 2a and b, but for height h ¼ 0:25; and wjR ¼ 0:3247j2:1518:


ARTICLE IN PRESS

Fig. 5. The minimum isochoric pressure as in Fig. 4a, plotted as a function of the temperature at which it

occurs, including a much wider range of densities (2:4pr=rc1p4:6). The curve shows, equivalently, ateach pressure the temperature at which the density is a maximum.


the relationship between calculated pressure and temperature at the pressureminimum for densities ranging from rr � r=rc1 ¼ 2:4 to 4.6. Fig. 6 shows, forcomparison with Fig. 4a, some isochores near the pressure minimum calculated forshoulder height h ¼ 0:25 and the wjR used for Figs. 3a and b. For the h ¼ 0:2 and 0.1choices discussed above, for which the slope dP=dT below Tc2 is positive, thepressure at all temperatures above Tc2 simply drops as temperature decreases, on allof the isochores for 2:4pr=rc1p4:6; without reaching a minimum and thereafterrising with decreasing temperature.For h ¼ 0:30 and wjR ¼ 0:3645j2:2141; the first (lower density, lower pressure,

higher temperature) critical point, a3Pc1=UA ¼ 0:0841; kBTc1=UA ¼ 2:001; a3rc1 ¼

0:182; agrees with the Pc1; Tc1 found by experiment in water [23], and to withinabout 0.6% of the rc1 found by experiment for water, when a, b, c, and UA areassigned the values a ¼ 2:572 (A; b ¼ 3:510 (A; c ¼ 5:695 (A; and UA ¼ 0:4998kBTc1 ¼

4:465� 10�21J: Although this choice of values for a, b, c, and UA for h ¼ 0:3 resultsin agreement with the location of the first critical point for water (and a ¼ 2:572 (A isclose to the distance of minimum approach expected for water molecules [24]), and inagreement with the pressure and temperature for the conjectured second criticalpoint for water, it gives less good agreement, as noted previously, for the density rc2

at the second critical point. Also, this square-well with shoulder potential is notoverall a highly accurate choice for the interaction potential for water, as is seen in

ARTICLE IN PRESS

Fig. 6. Isochores for the hjwjR used for Figs. 3a and b, plotted for densities r=rc1 ¼ 3:152; and 0.4% and

0.8% higher than 3.152.


Fig. 7, which shows experimental data for water [23] around the first critical pointfor several temperatures in the range Tr ¼ T=Tc1 ¼ 0:8� 1:2; compared withcalculated isotherms for this hjwjR for temperatures between Tr ¼ 0:72� 1:27: Thisreveals considerable quantitative disagreement, although much of it is compensatedin Fig. 7 by making adjustments (from Tr ¼ 0:8� 1:2 to Tr ¼ 0:72� 1:27) in thetemperatures chosen for the calculated isotherms. The remaining disagreement inFig. 7 shows up especially at densities rr ¼ r=rc142; where the well-with-shoulderisotherms rise less rapidly with increasing density than found by experiment forwater. Similar disagreement at densities rr42 is apparent also at substantially lowertemperatures, 0:42oTro0:52; when comparing Figs. 4a and b, for densities within afew percent of rr ¼ 3:

2.2. A third critical point?

Turning now to more exotic behavior, like that mentioned above for h ¼ 0:35; onefinds at h ¼ 0:30 that for the only slightly different wjR ¼ 0:36j2:23 instead of the0:3645j2:2141 used for Figs. 2a and b, the behavior below the second critical pointchanges dramatically, to that shown in Figs. 8a and b. There the coexistence curvefor two liquid phases below the second critical point closes up again at yet lowertemperatures; and then it reopens at a third critical point, at nearly the same density,after the temperature has dropped some more (see footnote 2). The qualitativebehavior of the pressure of coexisting low-density and high-density liquid as function

ARTICLE IN PRESS

Fig. 7. Pressure isotherms for water from experiment, solid lines, and calculated for h ¼ 0:3 and wjR ¼

0:3645j2:2141; dotted lines. Temperatures, from bottom to top, are T=Tc1 ¼ 0:8; 0.9, 1.0, 1.1, and 1.2 forthe solid lines, and 0.72, 0.86, 1.0, 1.14, and 1.27 for the dotted lines.


of temperature below the second critical point, shown in Fig. 8b, is not greatlydifferent than that shown in Fig. 2b, for the somewhat smaller w and larger R; exceptfor a gap between Tr ¼ T=Tc1 ’ 0:31 and Tr ’ 0:23 where there are no longer twoliquid phases before two liquid-phase behavior resumes at temperatures below Tr ’

0:23: The relatively small change from wjR ¼ 0:3645j2:2141 to 0:360j2:230; has alsocaused the second critical point temperature to rise, to T2r ¼ 0:416; and the pressure,P2r; at that temperature to decrease by a large amount, to P2r ¼ 1:22; instead ofapproximately 15. (Where the coexistence curve closes, near Tr ¼ 0:31; the pressurePr � P=Pc1 ¼ 5:7; and when it reopens, at the third critical point, near T3r ¼ 0:23;the pressure has risen to P3r ¼ 17:5:)Fig. 9 shows pressure isotherms, for the h ¼ 0:3 and wjR ¼ 0:36j2:23; for

temperatures ranging from Tr ¼ 0:47 down to 0.14. The isotherms in the gap(0:314Tr40:23), between where the upper two-liquid phase region closes and whereit reopens again at the third critical point, slope upwards, as in normal single-phaseliquid behavior. For comparison, Fig. 10 shows the pressure isotherms over a similarrange of densities for the somewhat smaller R ¼ 2:2141 and larger w ¼ 0:3645 thatgave just two critical points and results shown in Figs. 2a and b. Fig. 11 shows threeisochores around the pressure minimum (density maximum) for the hjwjR ¼

0:3j0:36j2:23; which can be compared with the isochores shown in Figs. 4a and 6.A close look at the results for h ¼ 0:35 and wjR ¼ 0:405j2:270 or 2:272; mentioned

in Section 2.1, shows that as R is increased from 2.270 toward 2.272, there appears an

ARTICLE IN PRESS

Fig. 8. (a) and (b). Coexistence curve densities and pressure for shoulder height h ¼ 0:3 at temperaturesbelow the second critical point when the width of the shoulder is decreased and the width of the attractive

well increased until wjR ¼ 0:36j2:23 instead of the 0:3645j2:2141 used for Figs. 2a and b. The coexistencecurve has now closed up at Tr ’ 0:31 and then reopened below Tr ’ 0:23 which thus appears as a third

critical point temperature, T3r ’ 0:23: The gap in the pressure curve, (b), is at those temperatures wherethere is only a single liquid phase—instead of the two liquid phases that coexist below T3r; and fromTr ’ 0:31 up to T2r ’ 0:42:


initially quite narrow region of two liquid phases for a small range of temperaturesbelow Tc1 but well above the Tc2 for R ¼ 2:270: That Tc2 for the smaller R thenbecomes the Tc3 for the larger R. The phase diagram for the larger R then lookssomewhat like in Figs. 8a and b, but with initially, as R was increasing toward 2.272, aconsiderably narrower and shorter upper ‘‘loop’’ than that shown in Fig. 8a.

3. Why 2, 3 critical points?

One might ask whether the triple critical point behavior is due to renormalizationcorrections. A quick answer could seem to be ‘‘yes’’, if one uses the hjwjR ¼

0:3j0:36j2:23 but omits renormalization corrections in calculations performed attemperatures ToTc1: That results in Figs. 12a and b, where now, unlike in Figs. 8aand b, there is not a closing up of the coexistence curve below the second criticalpoint and reopening at yet lower temperatures. There is, however, in Fig. 12anarrowing, a kind of neck, in the curve below Tc2: A look at the size ofrenormalization corrections at Tc1 and Tc2; as seen for h ¼ 0:3 and the not fardifferent w and R in Figs. 13a and b, suggests, however, that renormalization,though quite important around Tc1; rc1; maybe does not play nearly as big a role atand below Tc2: This is borne out if the calculation is performed at h ¼ 0:3 and

ARTICLE IN PRESS

Fig. 9. Pressure isotherms for temperatures Tr ¼ 0:14� 0:47 calculated for the h ¼ 0:3 and wjR ¼

0:36j2:23 used for Figs. 8a and b. Short dashed lines are for Tr ¼ 0:47 (upper line) and 0.44 (lower line),then lower solid line for (second critical point temperature) T2r ¼ 0:41; followed upwards by dotted linesfor Tr ¼ 0:38; 0.35, 0.32, where there are two coexisting liquid phases, and then, continuing upwards,medium dashed lines for temperatures Tr ¼ 0:29 and 0.26 in the ‘‘gap’’, 0:314Tr40:23; where there isagain only a single liquid phase, then the upper solid line at (third critical point temperature) T3r ¼ 0:23;followed on upwards, where two coexisting liquid phases have reappeared, by two dotted lines, for

Tr ¼ 0:20 and 0.17, and down again to the dot-dot-dash line for Tr ¼ 0:14.


wjR ¼ 0:358j2:23 (i.e., at a value of w smaller by only 0.002 than the 0.36 used forFigs. 12a and b), with again renormalization corrections omitted for temperaturesToTc1: The resulting coexistence curve without renormalization corrections fortemperatures ToTc1 is shown in Figs. 14a and b, which now look much like theFigs. 8a and b for slightly larger w, which included renormalization corrections forall temperatures.So, why two, three critical points? Sections 3.1–3.5 below address this question in

detail and Section 3.6 illustrates by examples how very sensitive the answer is to theprecise choices made for the height h of the repulsive shoulder and its width w andthe width of the attractive well that extends out to radius R. Section 4 then illustratesthe extent to which conclusions regarding 2, 3 critical points change whencalculations are done in an approximation with d 0

ðTÞadðTÞ; which gave improvedagreement with simulations for a Lennard–Jones potential.

3.1. When mean field approximation is used for temperatures below Tc1

From a fundamental point of view, one can ask: if renormalization corrections arenot mainly responsible for multiple critical point behavior, i.e., for the appearance of

ARTICLE IN PRESS

Fig. 11. Isochores for h ¼ 0:3 and wjR ¼ 0:36j2:23 instead of wjR ¼ 0:3645j2:2141 used for Fig. 4a.Curves from bottom to top are for densities rr ¼ 3:341; and for 3.341 plus 1% and plus 2%.

Fig. 10. Pressure isotherms for temperatures 0:18� 0:58 calculated for h ¼ 0:3 and wjR ¼ 0:3645j2:2141:Short dashed lines, from top to bottom, for Tr ¼ 0:58; 0.53, 0:484T2r; followed by long dashed lines forTr ¼ 0:43 (lower line) and 0.38 (upper line), then solid line at T2r ¼ 0:33 followed, continuing upwards, bydotted lines for Tr ¼ 0:28; 0.23, and 0.18, where two liquid phases coexist below the second critical pointtemperature.


ARTICLE IN PRESS

Fig. 13. (a) and (b). Pressure isotherms for h ¼ 0:3 and wjR ¼ 0:3645j2:2141 at the critical pointtemperatures Tc1 (a) and Tc2 (b). In mean field approximation, dotted line, and after n ¼ 1; 2, 3, and 6renormalization corrections [Appendix A, Eq. (4)] dashed, dash-dot, dash-dot-dot (not visible in (b)), and

solid lines, respectively.

Fig. 12. (a) and (b). Coexistence curve densities and pressure at temperatures below the second critical

point for hjwjR ¼ 0:3j0:36j2:23; as for Figs. 8a and b, but omitting renormalization corrections for thetemperatures below Tc1:


ARTICLE IN PRESS

Fig. 14. (a) and (b). What Figs. 12a and b change to look like when the shoulder width is decreased 0.56%

from w ¼ 0:360 to w ¼ 0:358 (renormalization corrections for all ToTc1 omitted). [Compare Figs. 8a and

b for w ¼ 0:360 when renormalization corrections are included.]


a Tc2 and maybe Tc3 below the Tc1; then what is? In mean field approximation,which is what is being used below Tc1 when renormalization corrections at thoselower temperatures are ignored, the method of calculation employed here uses onlytwo terms: the free energy density for the repulsive interactions, approximated by theCarnahan–Starling plus Barker–Henderson expression; and the contribution of therepulsive shoulder plus attractive interactions, evaluated in mean field approxima-tion.As discussed in Appendix A, the first of these two terms, the free energy density

for the repulsive interactions, f repulðT ;rÞ; is given in Carnahan–Starling plusBarker–Henderson approximation by

bf repul

r¼4y � 3y2

ð1� yÞ2þ ln y ; (2)

where y ¼ 16prd3; the d ¼ dðTÞ being that given by Eq. (1). This gives the

Carnahan–Starling expression for the compression ratio, bP=r; for hard spheres ofdiameter d ¼ dðTÞ (Appendix A, Eq. (13)).The second term, the contribution to the free energy density that comes from the

repulsive shoulder plus attractive interactions, in mean field approximation, is givenby f a � r2aðT ;rÞ; where (Appendix A, Eq. (11), in the limit k ! 0)

aðT ;rÞ ¼Z r¼R

r¼d 0ðTÞ

drU2ðrÞgrepulðd;r; rÞ ; (3)

ARTICLE IN PRESS


in which grepulðd; r; rÞ is taken to be the radial distribution function for hard corerepulsive interactions [17] for spheres of diameter d ¼ dðTÞ; and U2ðrÞ is one half theportion of the two-body potential (repulsive shoulder and attractive well) thatextends from r ¼ d 0

ðTÞ to r ¼ R: As noted after Eq. (1) in Section 1, the simplechoice d 0

ðTÞ ¼ dðTÞ is used here in Sections 2 and 3.

3.2. Combining the f repul and f a

To see how these two contributions, f repul and f a ¼ r2aðT ;rÞ; can combine to givetwo or more critical points, it is useful to look, at each temperature, at the slope ofthe pressure isotherm, or, equivalently, of the chemical potential: m0 � qm=qr ¼

q2f =qr2: (At constant temperature P0 � qP=qr ¼ m0 because m ¼ qf =qr and P ¼

rm� f :Þ Two phases coexist if any portion of the isotherm has a negative slope. Alldensities at which m0ðrÞ is negative lie inside the spinodal; outside the spinodal, thecoexistence curve at equilibrium extends between densities r1 and r2; to the left andright of the spinodal, at which, at the same temperature, both Pðr1Þ ¼ Pðr2Þ andmðr1Þ ¼ mðr2Þ:When the mean field approximation is used for temperatures below Tc1 the slope

m0 of the chemical potential is given simply by m0 ¼ m0r þ m0a; where m0r and m0a are the

second density derivatives of f repul and f a; respectively. Figs. 15a–c show the m0r; �m0a;

and resultant m0 ¼ m0r þ m0a as functions of density calculated at three temperatures,Tr ¼ T=Tc1 ¼ 0:36; 0.27, and 0.19, using Eqs. (1)–(3), i.e., without renormalizationcorrections, for hjwjR ¼ 0:3j0:358j2:23: Here, as always, the upper critical pointtemperature, Tc1; has been calculated including renormalization corrections. (As wasseen in Fig. 13a, renormalization corrections at Tc1 make a relatively largecontribution to thermal properties: not including the renormalization corrections incalculations to determine the location of this—gas–liquid—critical point wouldresult, for the present hjwjR; in Tc1 being about 20% higher than is found whenrenormalization corrections are included.)Sections 3.3–3.6 below present a detailed analysis of what to expect for m0r; m

0a and

their sum m0 ¼ m0r þ m0a for, respectively, dðTÞ ¼ const; for dðTÞaconst; and forrelatively small changes in the shoulder height and shoulder and well widths h, w,and R � b: Readers not interested in this detailed analysis should skip ahead toSections 3.7 and 4.

3.3. Behavior for dðTÞ ¼ d ¼ const

As seen in Figs. 15a–c for hjwjR ¼ 0:3j0:358j2:23 with renormalization correctionsomitted for temperatures below Tc1; the �m0a (solid line), calculated from f a [Eq. (3)],has some oscillation in it as a function of r; owing to the oscillatory nature ofgrepulðd;r; rÞ as a function of r; which oscillation increases in magnitude in anonuniform way as the density increases [17]. If the effective sphere diameter d hadbeen constant, rather than dependent on temperature, as given by Eq. (1), then grepul

and consequently m0a would have been independent of temperature, and Figs. 15a–cwould all have shown the same (somewhat oscillatory) solid line. Meantime, for d ¼

ARTICLE IN PRESS

Fig. 15. (a)–(c). Dotted, solid, and dashed curves show, respectively, m0r; �m0a; and m0 ¼ m0r þ m0a forshoulder height h ¼ 0:3; in mean field approximation for temperatures below Tc1; when wjR ¼ 0:358j2:23:The three plots a, b, c are for temperatures Tr ¼ T=Tc1 ¼ 0:36; 0.27, 0.19, respectively. Where m0 (dashedcurve) drops below zero there are two (gas–liquid or liquid–liquid) phases; when m0 remains above zero forall densities around rr ¼ r=rc1 ’ 4; then there is only a single, liquid, phase. (The vertical scale is inarbitrary units, the same for each plot.)


constant, the m0r would have been simply proportional to temperature, owing to theproportionality of f repul [Eq. (2)] to b�1 ¼ kBT ; and so the dotted line (m0r) in theplots would have had a smaller and smaller amplitude as the temperature drops. As a

ARTICLE IN PRESS


consequence, the m0 ¼ m0r þ m0a (dashed line)—negative for densities in the rangeapproximately 0orr � r=rc1p2; where gas and liquid phases coexist at such lowtemperatures—at somewhat larger densities could still rise above zero, but thenaround density rr ’ 4 would simply have dropped lower and lower as temperaturedecreases, rather than rising a bit around rr ’ 4 as temperature falls from Tr ¼ 0:36(Fig. 15a) to 0.27 (Fig. 15b), before dropping again at still lower temperatures,Tr=0.19 (Fig. 15c). Consequently, for d ¼ constant, at temperatures where, owingto the oscillatory nature of grepul(d,r; rÞ; the m0 at some densities rr42 rises abovezero before dropping to below zero for densities near rr ’ 4; there are as a result two

phase transitions (gas–liquid below Tc1 and liquid–liquid below the Tc2 where m0 justtouches zero at rr ’ 4 before increasing at yet larger densities) but there is not theclosing of the coexistence curve and reopening at a yet lower temperature, Tc3; aswas seen in Fig. 14a. That transition occurred because of the temperaturedependence of dðTÞ; given by Eq. (1).(For the particular choice d ¼ const ¼ a; treated in Section 2 of Ref. [19], the

second critical point occurs at P2rjT2r ¼ 15:6j0:322 for hjwjR ¼ 0:3j0:182j1:946 whenrenormalization contributions for ToTc1 are omitted, changing to P2rjT2r ¼

13:9j0:330 when those contributions are included.)

3.4. When dðTÞa const

The temperature dependent dðTÞ introduces additional temperature dependence inm0r (dotted curve), especially at larger densities, owing to the dependence of y in Eq.(2) on the cube of dðTÞ: The �m0a (solid curve) then also depends on temperature, dueto its dependence on dðTÞ [Eq. (3)], but in a different way than m0r (conspicuous in theplots 15a–c, especially near rr ¼ 2:5 and at large values of rr) resulting from the dðTÞ

temperature dependence of both grepulðd; r; rÞ and the lower limit of integration inEq. (3). The combination of m0r and m0a; for hjwjR ¼ 0:3j0:358j2:23; now results in avalue for m0 ¼ m0r þ m0a (dashed curve) which is negative near densities rr ¼ 4 attemperatures Tr ¼ 0:36 and 0.19, as seen in Figs. 15a and c. But the minimum nearrr ¼ 4 of m0 at the intermediate temperature Tr ¼ 0:27; Fig. 15b, though close tozero, is not negative. This latter behavior is consistent with that shown in Figs. 14aand b, which show that for this hjwjR there are two liquid phases at Tr ¼ 0:36 and0.19, but only a single liquid phase at Tr ¼ 0:27: The temperature Tr below 0.27 butsomewhat above 0.19 where two-liquid phase behavior first re-emerges is T3r; thetemperature of the third critical point. The existence of a third critical point is seen tobe a consequence, at least for this choice of h, w, and R, of the effective spherediameter, d ¼ dðTÞ; not being independent of temperature.

3.5. Result for hjwjR ¼ 0:3j0:358j2:23 of including renormalization corrections for

temperatures below Tc1

Including renormalization corrections at all temperatures, rather than just at Tc1;makes m0 (dashed curve) near rr ¼ 4 a little more positive (less negative) and, e.g., atTr ¼ 0:27 and rr ’ 3:88; where m0ðrÞ40 passes through a minimum, makes Pr a little

ARTICLE IN PRESS


smaller (by about 3%). At Tr ¼ 0:36 and 0.19 the renormalization corrections resultthere, where m0ðrÞo0; in a little narrowing of the coexistence curve(s) (Fig. 14a) andsome decrease in Pr (by about 16% and 1%, respectively, on the coexistence curvesat these two temperatures).

3.6. The rather large effect of even small changes in h; w; or R when near

0:3j0:358j2:23

Returning to mean field approximation for temperatures below Tc1; using theresimply Eqs. (1)–(3), any change of hjwjR away from 0:3j0:358j2:23 generally causeschanges in both m0r and m

0a; and in their sum m0 ¼ m0r þ m0a: The qualitative effect on the

shape of the coexistence curve and the magnitude of m0 and Pr near rr ¼ 0:39 can berather large for even � 1% changes in h, w, or (R � b). The following paragraphshighlight what happens at the single temperature Tr ¼ 0:27; Fig. 15b, for smallchanges separately in h, w, or R away from 0:3j0:358j2:23:First, decreasing h ¼ 0:3 by 1% to h ¼ 0:297 causes m0r near rr ’ 3:9; where m0 is a

minimum, to decrease and �m0a there to increase, with the result that m0 ¼ m0r þ m0a

decreases and changes from positive to negative for a small range of densities nearrr ’ 3:9; so that, unlike in Figs. 14a and b, the coexistence curve has not closed upthere. At this temperature and density the pressure, Pr; has dropped substantially,from about Pr ¼ 10:6 to 8.9. Adding renormalization corrections at temperaturesbelow Tc1 suffices at Tr ¼ 0:27 to make m0 again a little bit positive there (with Pr

somewhat smaller), so that one has again single phase behavior at that temperature,more like in Figs. 14a and 8a. Increasing h has the opposite effect in the mean fieldapproximation, resulting at Tr ¼ 0:27 in m0 becoming more positive around rr ’ 3:9;and m0 becomes yet somewhat more positive when renormalization corrections areadded.Instead increasing w by 1% at Tr ¼ 0:27; from w ¼ 0:358 to w ¼ 0:3616; causes

both m0r and �m0a near rr ’ 3:9 to increase, but �m0a by a larger amount, with theresult again that now m0 becomes negative over a small range of densities nearrr ’ 3:9; resulting in the coexistence curve not closing up around Tr ¼ 0:27: (Suchhad been seen to be the case in mean field approximation for ToTc1 for w ¼ 0:36 inFig. 12a.) The pressure on this coexistence curve at densities near rr ’ 3:9 has nowincreased substantially, to Pr ¼ 11:8: Adding renormalization corrections attemperatures ToTc1 again makes Pr a little smaller and m0 slightly positive at itsminimum near rr ’ 3:9; thus causing the coexistence curve to close up there (muchlike for w ¼ 0:36 in Fig. 8a).Unlike h and w, which, at Tr ¼ 0:27; for 1% changes in their size have only a small

effect on the location of the density, rr;min; at which the minimum of m0ðrÞ occurs,small, �1%, changes in the well width, R � b ¼ c � b for the attractive potentialmove rr;min by several percent. For example, in mean field approximation fortemperatures ToTc1; a 1.1% increase in c � b moves rr;min by somewhat more than3% from 3.9 (more precisely, 3.89) to 3.76, with m0 at rr;min ¼ 3:76 now slightlynegative, rather than positive as it was originally, at rr;min=3.89. (For a smallerincrease of c � b; of 1% instead of 1.1%, the m0 approaches close to zero, but does

ARTICLE IN PRESS


not become negative.) In the case of the 1.1% increase in c � b; the m0r at rr ¼ 4:0increased by almost twice as much as the �m0a at that density, resulting in a sizableincrease in m0 at rr ¼ 4:0: But at rr ¼ 3:5 the m0r increased by only about a third asmuch as �m0a; resulting there in a substantial decrease in m0: And, with the shift to alower rr;min; the m0ðrr;minÞ for 1.1% increase in c � b became smaller (to a bitnegative), as noted above, so that the coexistence curve does not quite close up at thistemperature. The pressure Pr at densities rr ¼ 4 and 3.5 is, respectively, Pr ¼ 8:5 and7.8, and on the coexistence curve, e.g. at rr ¼ 3:76; a little higher, Pr ¼ 8:6:At the gas–liquid critical point, Tc1 (with there, as always, renormalization

corrections included), the small changes in h, w, or R � b considered above hadsubstantially less effect on the pressure, the Pc1 rising 0.6% for the choice h � 1%;dropping by 1.2% for the w þ 1%; and rising 4.3% for the 1.1% increase in R � b:

3.7. Conclusion

As seen from the above examples, even when mean field approximation is used fortemperatures below Tc1 so that basically only the two quantities m0rðT ;rÞ and m0aðT ;rÞare relevant, the behavior of their sum, equal to m0; and of the pressure, Pr; can bequite complex even for rather small changes in h, w, and R.And results become even more complicated when significant nonlinearities show

up, as soon happens for larger changes in h, w, and R:

4. What if d 0ðTÞadðTÞ ?

As noted in the introduction (Section 1) better quantitative agreement withsimulations performed for a Lennard–Jones potential were obtained by making thestarting point, d 0

ðTÞ; for the calculation of the contribution from the ‘‘attractive’’interactions, somewhat larger than the sphere diameter dðTÞ used for the repulsiveinteractions. (Best agreement with simulations for the Lennard–Jones potential wasobtained for d 0

ðTÞ approximately half-way between r ¼ dðTÞob and r ¼ b; wherethe UðrÞ changes sign from positive to negative [13].) For the present, square-wellwith repulsive shoulder potential, a choice of d 0

ðTÞ4dðTÞ requires, for the sameshoulder height h, somewhat smaller choices for w and R to obtain the ratios P2r ’

15:3; T2r ’ 0:33: For the height h ¼ 0:3; the resulting coexistence curve below thesecond critical point has a less rapid increase in pressure with decreasingtemperature, and, by the time d 0

ðTÞ is approximately midway between dðTÞ and b,it has started initially to decrease with decreasing temperature (hence dP=dT40). Aliquid–liquid coexistence pressure vs. temperature curve more like that in Fig. 2b(with initially dP=dTo0Þ is obtained, when d 0

ðTÞ is exactly half-way between dðTÞ

and b; if h is increased to h ¼ 0:4 and wjR set equal to 0:379j2:252: This choice ofh;w; and R makes P2r ¼ 15:5; T2r ¼ 0:329; r2r ¼ 3:9 when renormalizationcorrections are included, and P2r ¼ 10:5; T2r ¼ 0:389; r2r ¼ 0:710=0:1816 ¼ 3:9when renormalization corrections for ToTc1 are omitted. Including the renorma-lization corrections at all temperatures, the isotherms near Tc1 are then nearly the

ARTICLE IN PRESS

Fig. 16. (a) and (b). Coexistence curve densities and pressure at temperatures below the second critical

point, for the d 0ðTÞadðTÞ and hjwjR ¼ 0:4j0:379j2:252 used in Section 4, instead of the d 0

ðTÞ ¼ dðTÞ and

hjwjR ¼ 0:3j0:3635j2:1241 used in Section 3 for, e.g., Figs. 2a and b.


same as shown in Fig. 8, while the densities and pressure on the coexistence curve attemperatures below the second critical point, and the temperature of maximumdensity at each pressure, now are as shown in Figs. 16a, b, and 17 instead of as inFigs. 2a, b and 5.And a third critical point? For h ¼ 0:4; the coexistence curve closes at a

temperature below that of the second critical point and reopens at a yet lowertemperature when R is increased from 2.252 to 2.27 and w is decreased from 0.379 to0.377 or 0.378, i.e., again for values of R and w not greatly different from those (nowfor h ¼ 0:4) that give two critical-point behavior similar to what has been predictedas a possibility for water [5].

5. Discussion

The method of calculation used in this investigation is an approximate one, withunknown uncertainty. Although reasonably good agreement with moleculardynamics and Monte Carlo simulations was found when essentially the samemethod of calculation was used for simple square well and Lennard–Jones potentials[13–15], it would be quite valuable to have molecular dynamics or Monte Carlocalculations to compare with for one or more of the well-with-shoulder potentialsconsidered here. The well-with-shoulder potentials investigated here differ con-siderably from those investigated in Refs. [6–11], having much smaller shoulder

ARTICLE IN PRESS

Fig. 17. Minimum isochoric pressure as function of temperature (equivalently, at each pressure, the

temperature at maximum density) for the d 0ðTÞadðTÞ and hjwjR ¼ 0:4j0:379j2:252 of Section 4.


heights UR=UA than those treated in Ref. [11], and much smaller ratios ðb � aÞ=ðc �bÞ of shoulder-width to well-width than those considered in Refs. [6–10], which gavelarger second critical point temperature and smaller pressure, relative to the Tc1; Pc1

at the lower density, first critical point, than found in the present investigation.3 Itwould also be of interest to see if simulations confirm the possibility of aliquid–liquid coexistence curve at low enough temperatures closing up and thenreemerging at yet a third critical point at still lower temperatures, as predicted herefor some choices of hjwjR: Although attention in this investigation has focussed onhjwjR values not greatly different from ones that, using the present method ofcalculation, yield two critical points with temperature and pressure ratios similar toones estimated [5] for water, the general method of calculation used here may havewider application—for different choices of hjwjR—to different physical systems.Refs. [3,25–27] note a number of substances besides water (e.g., phosphorus, silicon,carbon, Pb–Sn melts, SiO2; :::) which recent investigations suggest might have morethan a single critical point. A general theory capable of approximating the behaviorbeing observed or potentially observable in those systems could be useful. Thepresent approach, though with at present unknown uncertainty for well-with-shoulder potentials, has the advantage that investigations for different choices of

3Different choices for b=a; c=a; and UR=UA were treated in Ref. [11] from which it is possible to see how

P2r; T2r; r2r vary as these parameters are changed. The choices for shoulder width and height in the present

investigation lie off-scale, to the left of the curves shown in Figs. 9f and i and 9e and h in Ref. [11].

ARTICLE IN PRESS


hjwjR can be completed quickly, without the large computer resources and lengthycalculations required for accurate molecular dynamics simulations. Although well-with-shoulder potentials are crude approximations to actual intermolecularinteractions, they may nonetheless yield qualitative insights into what can beexpected in real physical situations. It is possible also that, with some modificationsof the assumed interaction potential, quantitative agreement with experiments onreal systems can be substantially improved.

Appendix A. Details of the calculation

The renormalization method used was basically the same as that described in Ref.[12], and, in present notation, in Ref. [15], with the substitution in a few places ofdðTÞ; Eq. (1), for s ¼ a; and sometimes use of d 0

ðTÞadðTÞ for the starting point ofattractive interactions: see Eq. (11) and remarks after Eq. (13) below. The followingis a brief summary of the procedure.The free energy density, f ðT ;rÞ; of the fluid (Helmholtz free energy per unit

volume) at temperature T and number density r is separated into hard core repulsiveand repulsive shoulder plus attractive portions. Beginning with f 0ðT ;rÞ=f repulðT ;rÞ;renormalization contributions are computed for increasingly long fluctuationwavelengths, beginning with wavelength l1; the shortest wavelength for which anysubstantial contributions are made to thermal properties not already included inf repulðT ;rÞ and the radial distribution function grepulðd;r; rÞ for the hard corerepulsive interactions [13]. After n renormalizations (n � 1 doublings of the initialfluctuation wavelength l1) the free energy density f ðT ;rÞ can be written as

f ðT ;rÞ ’ f nðT ; rÞ þ r2aðT ; rÞ ; (4)

where, for each n (40),

f nðT ;rÞ ¼ f n�1ðT ;rÞ þ df nðT ;rÞ : (5)

The r2aðT ;rÞ is the contribution of the repulsive shoulder plus attractive well to thefree energy density in mean field approximation (i.e., omitting any contributionsresulting from fluctuations of density). The increment df nðT ;rÞ resulting fromfluctuations of wavelength ’ ln is

4

df nðT ;rÞ ¼1

bVn

lnIn;lðT ;rÞIn;sðT ; rÞ

: (6)

Here b ¼ 1=kBT ; where kB is Boltzmann’s constant, Vn is the averaging volume,V n ¼ ðzln=2Þ

3; where z ’ 1; and the In;sðT ; rÞ and In;lðT ;rÞ are integrals over theamplitudes of the wave packets of density fluctuations of wavelengths

4Subscripts l and s in Eq. (6) were (incorrectly) interchanged in Eq. (15) of Ref. [13] and Eq. (3) of Ref.

[14], as well as in Eq. (10) in Ref. [28]. (The error occurred upon attempting to convert Eq. (4c), etc., in

Ref. [12] into some more elegant notation, adapted from Eqs. (20)–(25) in Ref. [29].)

ARTICLE IN PRESS


l ’ ln ¼ 2n�1l1:

In;iðT ; rÞ ¼Z r0

0

dx e�bV nDn;iðT ;r;xÞ; i ¼ s; l : (7)

In Eq. (7) the upper density limit, r0; is the smaller of r or rmax–r; where rmax doesnot exceed the density of closest packing of the molecules. And each Dn;iðT ; r;xÞ isgiven by

2Dn;iðT ;r; xÞ ¼ bf n�1;iðT ;rþ xÞ þ bf n�1;iðT ; r� xÞ � 2bf n�1;iðT ;rÞ ; (8)

where, for i ¼ l;

bf n�1;lðT ;rÞ ¼ f n�1ðT ;rÞ (9)

and for i ¼ s;

bf n�1;sðT ; rÞ ¼ f n�1ðT ;rÞ þ r2alnðT ;rÞ ; (10)

where

alðT ; rÞ ¼Z r¼R

r¼d 0ðTÞ

dr cosðk � rÞU2ðrÞgrepulðd; r; rÞ : (11)

In Eq. (11) grepulðd;r; rÞ is the radial distribution function for the hard core repulsiveinteractions for spheres of diameter dðTÞ; U2ðrÞ is one half the remaining portion ofthe two-body potential (repulsive shoulder plus attractive well), and k is the wavevector of the fluctuation of wavelength l ¼ 2p=k: In the limit n ! 1; for whichln ! 1; the alðT ;rÞ becomes simply the aðT ;rÞ in Eqs. (3) and (4) above.The procedure summarized above is capable of determining, approximately, the

free energy density completely, by taking fully into account details of theintermolecular potential and contributions made by fluctuations at all wavelengths.Specifically, the free energy density f 0ðT ;rÞ ¼ f repulðT ; rÞ of a gas comprised of

hard spheres of diameter d is, apart from a contribution to f repulðT ;rÞ=r dependenton temperature but independent of density, approximately

bf repul

r¼4y � 3y2

ð1� yÞ2þ ln y ; (12)

where y ¼ 16prd3: The pressure P ¼ rqf =qr� f calculated using f ¼ f repul given by

Eq. (12) yields, when multiplied by b=r; the Carnahan–Starling [16] expression forZ ¼ bP=r ¼ PV=RT for hard spheres, namely

Zrepul ¼ rqqr

bf repul

r

� �¼1þ y þ y2 � y3

ð1� yÞ3: (13)

In evaluating Eq. (11), the grepulðd;r; rÞ was approximated as that for a gas of hardspheres, of diameter d; in Percus–Yevick approximation [17], using the same d ¼

dðTÞ; Eq. (1), as for f repul ; Eq. (12). And the U2ðrÞ was taken to be12

UðrÞ in Fig. 1, forthe attractive well with repulsive shoulder, with the integration in Eq. (11) beginningat r ¼ d 0

ðTÞ; where d 0ðTÞ is set equal to dðTÞ in Sections 2 and 4, and midway

ARTICLE IN PRESS


between dðTÞ and radius b in Section 4. Once parameters a; b, c, UR; UA for the wellwith shoulder, and the ratio d 0

ðTÞ=dðTÞ and parameters l1 and z internal to the RGtheory are specified, then the f ðT ;rÞ given by Eq. (4) is completely determined—apart from a contribution [noted above Eq. (12)] that depends only on temperatureand does not contribute to the pressure—upon completion of n renormalizations.For the present calculations, the internal parameters l1 and z were assigned thevalues l1 ¼ 6a and z ¼ 0:85; respectively.In the numerical calculations, the integrations were performed by the trapezoid

rule, using equal size steps. Generally, 5000 or 25 000 steps, depending on theaccuracy desired, were used for the calculation of each alðT ;rÞ; in Eq. (11). For thegrepulðd;r; rÞ appearing in Eq. (11), the table in Ref. [17] was used, with interpolationwhen required. Eq. (11) was evaluated for the twelve (dimensionless) densities rd3 ¼

0:0; 0:1; 0:2; . . . ; 1:1 for which tabulated values of grepul were available [17], and apolynomial of fifth order in rd3 was fitted to each alðT ;rÞ for use in Eq. (10), whichneeds to be evaluated at many intermediate densities in the range 0ord3o1:1:For use in the present investigation, the free energy density f was evaluated, at

(dimensionless, ra3) density intervals of 0:0025; for 0ora3p1:5; for the lower limit,a small value, ra3 ¼ 10�12; was used in place of ra3 ¼ 0 to avoid the logarithmicsingularity in Eq. (12). The integrand in Eq. (7) was evaluated, by trapezoid rule, atthe same dimensionless density intervals, 0:0025; using for the maximum integrationlimit r0a3 ¼ rmaxa3=2 ¼ 1:5=2: Smaller choices for the upper limits of ra3 and r0a3

had little noticeable effect on the results obtained here until rmaxa3 and/or 2r0maxa3

became less than 1.1. Four point interpolation was used to estimate f whencalculating thermal properties at densities intermediate between those at which f hadbeen evaluated. Calculations of f nðT ;rÞ were carried through to order n ¼ 6:

References

[1] H.E. Stanley, M.C. Barbosa, S. Mossa, P.A. Netz, F. Sciortino, F.W. Starr, M. Yamada, Physica A

315 (2002) 281;

H.E. Stanely, S.V. Buldyrev, N. Giovambattista, E. La Nave, S. Mossa, A. Scala, F. Sciortino, F.W.

Starr, M. Yamada, J Stat. Phys. 110 (2003) 1039.

[2] P.H. Poole, F. Sciortino, U. Essmann, H.E. Stanley, Nature 360 (1992) 324.

[3] G. Franzese, M.I. Marques, H.E. Stanley, Phys. Rev. E 67 (2003) 011103.

[4] S.B. Kiselev, J.F. Ely, J. Chem. Phys. 116 (2002) 5657.

[5] M. Yamada, S. Mossa, H.E. Stanley, F. Sciortino, Phys. Rev. Lett. 88 (2002) 195701.

[6] G. Franzese, G. Malescio, A. Skibinsky, S.V. Buldyrev, H.E. Stanley, Nature 409 (2001) 692.

[7] G. Malescio, G. Franzese, G. Pellicane, A. Skibinsky, S.V. Buldyrev, H.E. Stanley, J. Phys.: Condens.

Matter 14 (2002) 2193.

[8] S.V. Buldyrev, G. Franzese, N. Giovambattista, G. Malescio, M.R. Sadr-Lahijany, A. Scala,

A. Skibinsky, H.E. Stanley, Physica A 304 (2002) 23.

[9] G. Franzese, G. Malescio, A. Skibinsky, S.V. Buldyrev, H.E. Stanley, Phys. Rev. E 66 (2002) 051206.

[10] G. Malescio, G. Pellicane, Phys. Rev. E 63 (2001) 020501R.

[11] A. Skibinsky, S.V. Buldyrev, G. Franzese, G. Malescio, H.E. Stanley, http://arxiv.org/abs/cond-mat/

0309632 v2 4 Feb 2004.

[12] J.A. White, S. Zhang, Int. J. Thermophys. 19 (1998) 1019.

[13] J.A. White, J. Chem. Phys. 112 (2000) 3236.

http://arxiv.org/abs/cond-mat/0309632

http://arxiv.org/abs/cond-mat/0309632

ARTICLE IN PRESS



[15] J.A. White, Int. J. Thermophys. 22 (2001) 1147.

[16] N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51 (1969) 635.

[17] G.J. Throop, R.J. Bearman, J. Chem. Phys. 42 (1965) 2408.

[18] J.A. Barker, D. Henderson, J. Chem. Phys. 47 (1967) 4714.

[19] J.A. White, K.P. Tewari, Int. J. Thermophys. 25 (4) (2004) 1005–1024.

[20] S.V. Buldyrev, H.E. Stanley, Physica A 330 (2003) 124.

[21] P.G. Debenedetti, J. Phys.: Condens. Matter 15 (2003) R1669.

[22] P.G. Debenedetti, H.E. Stanley, Phys. Today 56 (2003) 40.

[23] E.W. Lemmon, M.O. McLinden, D.G. Friend, Thermophysical properties of fluid systems, in:

P.J. Linstrom, W.G. Mallard (Eds.), NIST WebBook, NIST Standard Reference Database Number

69, National Institute of Standards and Technology, Gaithersburg Maryland, March 2003, http://

webbook.nist.gov.

[24] D. Eisenberg, W. Kauzmann, The Structure and Properties of Water, Oxford University Press,

Oxford, 1969, p. 159.

[25] S. Sastry, C.A. Angell, Nature Materials 2 (2003) 739;

N. Jakse, L. Hennet, D.L. Price, S. Krishnan, T. Key, E. Artacho, B. Glorieux, A. Pasturel,

M.-L. Saboungi, Phys. Lett. 83 (2003) 4734.

[26] F.Q. Zu, Z.G. Zhu, L.J. Guo, B. Zhang, J.P. Shui, C.S. Liu, Phys. Rev. B 64 (2001) 180203(R).

[27] F.W. Tavares, J.M. Prausnitz, Colloid Polym. Sci. 282 (2004) 620.


[29] L. Lue, J.M. Prausnitz, J. Chem. Phys. 108 (1998) 5529.

http://webbook.nist.gov

http://webbook.nist.gov

multiple critical points for square-well potential with repulsive shoulder

Documents