Configurational Properties and Corresponding States in Simple Fluids

5106

J. Phys. Chem. B 1999, 103, 5106-5116

Configurational Properties and Corresponding States in Simple Fluids and Water Isaac C. Sanchez,* Thomas M. Truskett,† and Pieter J. in ’t Veld Chemical Engineering Department, UniVersity of Texas, Austin, Texas 78746 ReceiVed: February 8, 1999; In Final Form: April 16, 1999

Using accurate equations of state, contributions to the configurational entropy and energy of water are calculated and compared with six simple fluids and a model Lennard-Jones (LJ) fluid. The reorganizational entropy is calculated for the first time for water and other fluids. Comparisons are made at equal reduced densities from the triple point to the gas-liquid critical temperature. A corresponding states principle (CSP), suggested by the van der Waals model, is satisfied for certain configurational properties among the simple and LJ fluids. Saturated simple liquids in states of equal free volume fractions have comparable configurational properties. Water deviates significantly from the CSP in some but not all cases. The normal boiling point of eight simple fluids is seen to be an isofree volume state that occurs at a reduced density (F/Fc) of 2.63 ( 0.02, or equivalently, an occupied volume fraction of 0.42. A nearly invariant entropy of vaporization at the normal boiling point among eight simple fluids ((9.1 ( 0.3)k) is one manifestation of the CSP (Trouton’s rule). The LJ model fluid accurately describes the configurational properties of argon and methane. This result lends credibility to the wide spread use of the LJ potential to describe atom-atom (or site-site) interactions in large and complex molecules.

I. Introduction In a Monte Carlo or molecular dynamics simulation of a model molecular fluid, the configurational contribution to the chemical potential can be determined.1,2 Other configurational properties, such as the configurational energy and entropy, can also be calculated. To effect a comparison of model properties with those of a real fluid, it is necessary to calculate the corresponding configurational properties of the real fluid. Herein, we present recipes for calculating configurational properties from the equation of state. Using statistical mechanics, these configurational properties are also given a microscopic interpretation. In the last 10 years, accurate equations of state have become available for water,3 argon,4 methane,5 nitrogen,6 oxygen,6 ethylene,6 and ethane.7 One of the most well-studied models by computer simulation is the Lennard-Jones (LJ) fluid (6-12 potential). Using simulation results, an accurate equation of state for the LJ fluid has also been developed.8 These equations can be used to accurately calculate configurational properties. There are a number of points that will be made in this paper: (1) The configurational entropy of a fluid has four fundamental contributions. The “reorganization entropy” and the other entropy contributions are calculated (some for the first time) for the above mentioned eight fluids at saturation from the triple point to the gas-liquid critical temperature. (2) In dense liquids, the chemical potential is dominated by the configurational energy. (3) The LJ model fluid adequately describes the configurational properties of the simple fluids (especially argon). This lends credence to the wide spread use of the LJ potential to describe atom-atom (or site-site) interactions in large and complex molecules. (4) A corresponding states principle is satisfied among the simple and LJ fluids for most properties, but water deviates significantly in some cases. The most striking †

Current address: Chemical Engineering Dept., Princeton University, Princeton, NJ 08544-5263.

departures occur in the “relaxation” and reorganizational properties. (5) The existence of a corresponding states principle argues persuasively that the fraction of space occupied by the molecules (or the related free volume fraction) largely governs configurational properties of saturated liquids. The normal boiling point of simple fluids appears to be an iso-free volume state where the reduced density (F/Fc) equals 2.63 ( 0.02 and the occupied volume fraction equals 0.42. II. Configurational Properties A configurational property is as an excess property (positive or negative) relative to an ideal gas at the same temperature (T) and pressure (P):

x)

|

∂X ) X(N,T,P) - X(N - 1,T,P) ∂N T,P

) xideal(T,P) + ∆xconf

(1a) (1b)

where x is an intensive property that can be the chemical potential µ or the molar values of the entropy s, internal energy u, enthalpy h, or volume V. If the molecular interactions in a real fluid at a given T and P are “turned off”, the fluid becomes an ideal gas; thus, ∆xconf measures the effect on x caused by molecular interactions relative to an ideal gas at the same T and P. The extensive property X associated with the intensive property x is given by X ) Nx where N is the number of molecules. Using standard thermodynamic formulas, all configurational properties are derivable from the configurational chemical potential ∆µconf. The configurational chemical potential is given by9,10

β∆µconf ) -ln zB

(2)

where β ) 1/kT has its usual significance, z is the compressibility factor (z ≡ PV/NkT ≡ βP/F), and B is the “insertion factor”. Conceptually, the insertion factor can be determined

10.1021/jp9904668 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/02/1999

Simple Fluids and Water

J. Phys. Chem. B, Vol. 103, No. 24, 1999 5107

by randomly inserting a molecule into a fluid and measuring the interaction energy ψ of this “test molecule” with the other molecules. The average value obtained for the Boltzmann factor e-βψ is the insertion factor B; it can be calculated in two ways:10

B ) 〈e-βψ〉0 ) 1/〈eβψ〉

(3)

The subscript zero indicates that the average is taken under the condition that the other molecules in the system do not sense the presence of the “test molecule”. The average without the subscript zero indicates a normal ensemble average where the presence of the molecule of interest interacts and influences the other system molecules. As has been recognized for some time, it is the insertion factor that is intimately related to solvation thermodynamics.11 Using the standard isothermal thermodynamic formula,

d∆µconf ) ∆VconfdP ) (V - kT/P)dP

(4)

If the liquid is in equilibrium with its vapor (saturated), the pressure that appears in eq 12 is the equilibrium vapor pressure. Note that ∆µcond ) 0 (the configurational energy of an ideal gas is zero), ∆hcond ) ∆(PV) ) k∆T ) 0 and ∆Vcond ) ∆Vconf ) V - kT/P. Also note that ∆Vp ) 0. If the molecular interactions in a liquid were “turned off”, it would expand and become an ideal gas. Conversely, if we start with an ideal gas and “turn on” the interactions, the gas would “condense” to a smaller volume; ∆scond contributes to the entropy of vaporization (see later). Thus, the self-solvation entropy at constant pressure is given by

[

∆sp ) ∆sconf - ∆scond ) k ln B +

|

(5)

1 F

(

∆srelax ) V

∫0Fln B dF

(6)

where as in eq 5 the integration is along an isotherm. The latter equation is obtained from eq 2 and the standard isothermal relation dP ) F(∂µ/∂F)T,NdF. The nonideal behavior of a fluid is contained completely in the microscopic quantity B. For an ideal gas, B ) 1. The configurational entropy is given by

∆sconf ) -

| [

|]

∂∆µconf ∂ ln zB ) k ln zB + ∂T P ∂ ln T P

(7)

|

∂(β∆µconf) ∂β

|

∂ ln zB ) kT P ∂ ln T

P

(8)

(9)

∆xcond ) xideal(T,Fcond) - xideal(T,Fideal)

(11)

The entropy change associated with this compression of an ideal gas is given by

∆scond ) k ln(Fideal/Fcond) ) k ln(P/FcondkT) ) k ln z (12)

- k ≡ Vγ - k

(15)

(13b)

|

∂ ln B ∂ ln T V

(16)

Using eqs 5 and 16, it can be shown that ∆sp is also given by

∫0F∆srelaxdFF

∆sp ) -

(13c)

Equation 13c was obtained by a different argument over 50 years ago.12 Similarly, for the configurational energy

∆uconf ) ∆uv + ∆urelax ) ∆up

(17a)

where

(10)

where ∆xcond is the change in x for an ideal gas when it is compressed isothermally from its initial density Fideal ) P/kT to the density Fcond of the condensed phase (fluid or solid) of interest at T and P:

V

where

∆sv/k ) ln B +

where xideal(T,F) is the value of x for an ideal gas at the same temperature T and number density F as the condensed phase (fluid or solid). Equation 9 should be compared with eq 1b. Thus,

∆xconf ) ∆xcond + ∆xp

|

∂P )V ∂T

T

∆sp ) ∆sv + ∆srelax

A physically interesting contribution to ∆xconf is ∆xp, the “self-solvation” at fixed pressure P. It is defined as

x ) xideal(T,F) + ∆xp

)

∂∆sconf ∂V

and γ is the thermal pressure coefficient. ∆sp has two contributions. First, the molecule is added at fixed total volume V which yields the constant volume self-solvation entropy ∆sv (and raises the pressure by δP) followed by a relaxation of the system back to the initial pressure P (volume increases by δV ) V/N ) V on relaxation). Thus,

and the configurational enthalpy by

∆hconf )

(14)

where ∆srelax is the “relaxation entropy”:

The inverse relation yields the equation of state

z ) 1 - ln B +

|

∂ ln zB ∂ ln B ) + ∆srelax/k ∂ ln T P ∂ ln T V

∫0F(1 - z)dFF

(13a)

The required derivative at constant pressure can be related to one at constant volume:

and eq 2, we obtain

ln B ) 1 - z +

|]

∂ ln zB ∂ ln T P

∆urelax )

|

∂∆uconf V ) (Tγ - P)V ≡ PiV ∂V T

(18)

and Pi is the internal pressure and ∆uv is the self-solvation energy at constant volume:

|

(19a)

∫0FPiVdFF

(19b)

∂ ln B ∆uv ) kT ∂ ln T V ) -PiV -

The last equation (eq 19) is obtained from eq 5. Thus,

5108 J. Phys. Chem. B, Vol. 103, No. 24, 1999

∫0FPiVdFF ) -∫0F∆urelaxdFF

∆uconf ) ∆up ) -

Sanchez et al.

(17b)

∆ureo ) -[∆uint/2 + ∆urelax] ) -[∆up + PiV] )

which should be compared with eq 13c. From the definition of B, we can define an interaction energy, ∆uint, and interaction entropy ∆sint:

(20)

where 〈ψ〉 ≡ ∆uint is the average interaction energy of a molecule with all other molecules in the system and

∆sint ) -k ln[〈eβ(ψ-〈ψ〉)〉] e 0

(21)

Using the inequality 〈eβψ〉 g 〈eβ〈ψ〉〉, we see that ∆sint is always negative or zero. As will become clear later, ∆sint is the entropy penalty (intrinsically negative and unfavorable) that is required to form a cavity large enough to accommodate an additional molecule; ∆uint is the average interaction energy (intrinsically negative and favorable) of a molecule with all other molecules in the system when it occupies a cavity. From eq 19a, we also have

∆uv ) 〈ψ〉 + ∆ureo ≡ ∆uint + ∆ureo

(26)

) -(∆up + ∆urelax) ) T∆sreo Finally, note the following:

-kT ln B ) kT ln 〈eβψ〉 ≡ 〈ψ〉 + kT ln 〈eβ(ψ-〈ψ〉)〉 ≡ ∆uint - T∆sint

∫0FPiVdFF - PiV

(22a)

and ∆ureo is the “reorganization energy”.11,13,14 In words, ∆ureo is the change in configurational energy of N molecules when an extra molecule (the “interloper”) is added to the system at constant volume. The N molecules must “reorganize” to accommodate the interloper. Similarly, the constant volume solvation entropy can also be expressed as

∆sv ) ∆sint + ∆sreo

(22b)

-kT ln B ) ∆uint - T∆sint ) ∆uv - T∆sv

(23)

-kT ln zB ) ∆hconf - T∆sconf -kT ln B ) ∆hp - T∆sp ) ∆uv - T∆sv ) ∆uint - T∆sint

∆sconf ) ∆scond + ∆sp ) ∆scond + ∆sv + ∆srelax ) ∆scond + ∆sint + ∆sreo + ∆srelax

(24)

A similar hierarchy exists for the configurational energy and enthalpy (recall ∆ucond ) ∆hcond ) 0):

III. Results and Discussion A. van der Waals Fluid and Corresponding States. It is instructive to calculate the configurational properties of the van der Waals fluid (VDW). The VDW equation of state in reduced variables is

z)

3 9F˜ 3 - F˜ 8T˜

(25a)

|

∂ ln zB ∆hconf ) ∆hp ) kT ∂ ln T P ) ∆up + ∆(PV) ) ∆up + kT(z - 1)

ln B )

9F˜ F˜ + ln[1 - F˜ /3] 4T˜ 3 - F˜

(29a)

6 - F˜ - 2z 3 - F˜

(29b)

) ln[1 - F˜ /3] +

The remaining configurational properties can be easily obtained by taking appropriate temperature derivatives at fixed density on eqs 28 and 29a and using the preceding relations:

β∆ureo ) ∆sreo/k ) 0 ∆sv/k ) ∆sint/k ) ln[1 - F˜ /3] ∆srelax/k )

) ∆uv + ∆urelax + kT(z - 1) ) ∆uv + T∆srelax In the Appendix, we show that ∆uconf ) 〈ψ〉/2 ) ∆uint/2. Thus,

(30) F˜ 3 - F˜

F˜ 3 - F˜

∆sp/k ) ln[1 - F˜ /3] β∆uv ) β∆uint ) β∆urelax )

9F˜ 6 )+ 2z 4T˜ 3 - F˜

9F˜ 3 ) -z 8T˜ 3 - F˜

β∆uconf ) β∆up ) (25b)

(28)

where F˜ ) F/Fc and T˜ ) T/Tc are the reduced density and temperature, respectively. From eq 5 we obtain the following:

∆uconf ) ∆up ) ∆uV + ∆urelax ) ∆uint + ∆ureo + ∆urelax

(27b)

Equation 27 indicates that both the reorganizational and relaxation quantities cancel in forming the chemical potential. Thus, the canonical energy-entropy pairs are (∆hconf, ∆sconf), (∆hp,∆sp), (∆uv,∆sv), and (∆uint,∆sint). From the microscopic viewpoint, (∆uint,∆sint) is the fundamental pair, but from the experimental viewpoint, (∆hp,∆sp) is the fundamental pair.

But since

we see that ∆srec ) ∆ureo/T and we have perfect energy-entropy compensation in the reorganizational terms (∆ureo - T∆sreo ) 0). Note the hierarchal relationships:

(27a)

β∆hconf ) β∆hp ) -

3 9F˜ )+z 8T˜ 3 - F˜

3 6 - F˜ 9F˜ + )+ 2z 4T˜ 3 - F˜ 3 - F˜

(31) (32) (33) (34) (35) (36) (37)

Notice that the VDW equation of state is recovered when eq 29a is substituted into eq 6 and that eqs 13c and 17b are satisfied. Also notice that ∆hconf ) ∆uconf + P∆Vconf as it should.



As can be seen, the configurational properties of a VDW fluid depend on the reduced density only or on the reduced density and the compressibility factor z. Thus, the configurational properties of all VDW fluids with the same reduced density and compressibility factor are in corresponding states. Saturated liquids in equilibrium with their vapors will have very small values of the compressibility factor z. Thus, all saturated VDW liquids (z ≈ 0) will be in corresponding states when their reduced densities are equal. Below we will see that this corresponding states statement holds to a large degree for simple fluids. B. Corresponding States in Real Fluids. All results presented below are for saturated liquids in equilibrium with their vapors and at their respective saturation pressures. The equation of state and the chemical potential were used to determine the liquid-vapor coexistence curve and the configurational properties were then calculated for the saturated liquid. The conditions of equilibrium on the coexistence curve are (pressure and temperature are fixed):

(zB)liq ) (zB)vapor

or

(F/B)liq ) (F/B)vapor (38)

and

Pliq ) Pvapor

or

(Fz)liq ) (Fz)vapor

(39)

which are two equations in two unknowns (Fliq and Fvapor). As the density of the vapor approaches zero, Bvapor f 1 and zvapor f 1 so that Bliq f 1/zliq . 1. From eq 5, and the aforementioned equations of state, the insertion factor B was calculated as a function of reduced density from the triple point to the gas-liquid critical point for water, the LJ fluid, and 6 simple fluids. The results are illustrated in Figure 1. The LJ fluid solidifies at about T˜ ) 0.53 at a reduced density of about 2.7.15 The highest density points illustrated in Figure 1 are at the fluid’s triple point. Up to densities of 2.7, the LJ fluid describes the solvation contribution to the chemical potential (-ln B) of the six simple fluids very well. The corresponding states principle suggested by the VDW fluid model seems to hold quite well over the entire saturated liquid range, but quantitatively it is deficient [for the sake of clarity, the VDW results (eqs 30-37) are not illustrated in this or other figures]. It is important to note in Figure 1 that B . 1 (favorable solvation thermodynamics) for a saturated liquid far from the critical point. For a system of hard molecules (molecules that infinitely repel one another if they overlap), it is easy to show that B is equal to the “insertion probability”. This is the probability that a randomly inserted molecule into a system of N hard molecules will not overlap with another molecule. This probability is intimately related to the availability of molecular size cavities in the fluid (see later). Thus, for a dense, hard molecular fluid, B , 1 (unfavorable solvation thermodynamics). For hard molecules, B is completely determined by repulsive forces and is entirely entropic in origin. In dense real liquids, the probability of finding a molecular size cavity is also very small, so it may come as a surprise to some that B . 1. This illustrates that favorable attractive forces dominate the value of B (and thus, the chemical potential) and unfavorable entropic contributions play a secondary role. In Figures 2-8, the configurational entropy contributions are illustrated (∆sconf, ∆scond, ∆sp, ∆sv, ∆srelax, ∆sint, and ∆sreo). Note that the spread in the ∆sp values appears greater because the scale of the vertical axis has been amplified by a factor of 2. To check the calculation of ∆sconf, the absolute entropy of water

Figure 1. Insertion factor (B) for the indicated saturated liquids. The solvation configurational chemical potential is given by -kT ln B. A large value of the insertion factor is thermodynamically favorable. The reduced density (F/Fc) is a direct measure of the molecular occupied volume (see text). The highest density shown for each liquid is the triple point. The reduced triple point densities are LJ fluid (2.7), argon (2.67), methane (2.77), nitrogen (2.78), oxygen (3.00), ethylene (3.06), ethane (3.15), and water (3.11). The triple point is not a corresponding state.

was calculated by adding the ideal gas contribution (translational + rotational) to ∆sconf and it was compared to published absolute entropy values. Excellent agreement from the triple to the critical point was obtained. If the calculations are replotted as a function of the reduced temperature T/Tc, the observed corresponding states principle (CSP) weakens considerably (not shown). That is, a better correspondence states law is observed when saturated liquids are compared at equal reduced densities rather than at equal reduced temperatures. The observation of a CSP for the LJ and simple fluids has a physical basis. A LJ fluid has a critical density of σ3Fc ) 0.31 where Fc is the critical number density and σ is the LJ parameter that scales the repulsive interaction in the LJ potential.8 As is customary, if σ is taken as the effective diameter of the LJ particle, then ηc ) (π/6)σ3Fc ) 0.16 is the fraction of space η occupied by all LJ type fluids at the critical point (water is not an LJ fluid). Comparing configurational properties at equal Values of F/Fc ) η/ηc is equiValent to comparing properties at equal free Volume fractions. As can be seen from the figures, the configurational properties of monatomic argon are accurately captured by the LJ model. To a very good approximation, the other simple fluids are also well-described by the LJ model up to density ratios of 2.7. At 2.7, the LJ fluid undergoes a liquidto-solid transition (T/Tc ) 0.53) at η ) 2.7 × 0.16 ) 0.43. Hard spheres crystallize at η ) 0.497.16 For reference, the other liquid-to-solid (triple point) density ratios are argon (2.67), methane (2.77), nitrogen (2.78), oxygen (3.00), ethylene (3.06), ethane (3.15), and water (3.11). The existence of a CSP argues


Figure 2. Configurational entropy (∆sconf) of the indicated saturated liquids. The configurational entropy has four fundamental contributions: ∆sconf ) ∆scond + ∆sint + ∆sreo + ∆srelax.

Figure 3. Condensation entropy (∆scond) of the indicated saturated liquids. When an ideal gas is isothermally compressed from its initial volume Videal ) kT/P to the volume Vcond of the condensed phase of interest, the corresponding entropy change is ∆scond ) k ln(PVcond/kT) ) k ln z. The associated changes in the internal energy ∆ucond and enthalpy ∆hcond are zero.

persuasively that occupied volume fraction (η), or equivalently, the unoccupied (free) volume fraction (1-η) largely governs configurational properties. Thus, the CSP can be expressed this

Sanchez et al.

Figure 4. Self-solvation entropy at constant pressure ∆sp of the indicated saturated liquids. This is the entropy change associated with adding a molecule to a system at fixed T and P and number density F; ∆sp ) ∆sint + ∆sreo + ∆srelax. Note that the vertical scale here is amplified by a factor of 2 compared to Figures 2 and 3.

Figure 5. Self-solvation entropy at constant volume (∆sv) of the indicated saturated liquids. This is the entropy change associated with adding a molecule to a system at fixed T and V; ∆sv ) ∆sint + ∆sreo. The observed minima are caused by the large reorganizational entropy contributions (see Figure 8).

way: saturated simple liquids in states of equal free Volume fractions haVe comparable configurational properties.



Figure 6. Relaxation entropy (∆srelax) of the indicated saturated liquids. This entropy change occurs after a molecule is added at fixed system volume V (raises the pressure). The relaxation of the system back to its initial pressure P is accompanied by an entropy change of ∆srelax. Note that the critical point is a corresponding state for all fluids with ∆srelax ) (0.8 ( 0.1)k.

Figure 7. Interaction entropy (∆sint) of the indicated saturated liquids. When a molecule is added to a fluid, there are two effects that contribute to ∆sint. First, a cavity must exist that can accommodate the added molecule. This is determined primarily by the fluid density and the fraction of space occupied by the molecules. After a molecule is accommodated, there is a second contribution to ∆sint related to the fluctuations in the molecular interaction energy ψ.

Although the triple point is not a corresponding state, the normal boiling point seems to be. The reduced density ratios at the normal boiling point are argon (2.63), methane (2.60), nitrogen (2.58), oxygen (2.62), ethylene (2.65), and ethane (2.63). In addition, we have krypton (2.63) and xenon (2.65).17 The average reduced density for these eight simple fluids at their normal boiling points is 2.63 ( 0.02 which corresponds to an occupied volume fraction of 0.42. Water is exceptional with a reduced density of 2.97 at its normal boiling point of 373 K. This ratio for a broad range of organic liquids is 2.7 ( 0.1. At a reduced density of 2.63, the saturated LJ liquid has a configurational entropy of ∆sconf ) -8.9 k. Since the equilibrium vapor of a liquid at the normal boiling point is nearly ideal, the entropy of vaporization [∆sconf(vapor) - ∆sconf(liquid)] at the boiling point equals ∆sconf(liquid). The values of the entropy of vaporization in units of k are: argon (8.9), methane (8.8), nitrogen (8.7), oxygen (9.1), ethylene (9.6), ethane (9.6), krypton (9.2), and xenon (9.2).17 This nearly invariant value of the entropy of vaporization is well-known Trouton’s rule. Although it is not the normal boiling point, ∆sconf ) -9.0 k at F/Fc ) 2.63 for water. C. Configurational Energy and Pairwise Interactions. In Figures 9 and 10, the configurational energies ∆up (∆uconf) and ∆uv are illustrated. As can be seen in these and other figures, the LJ fluid mimics the configurational properties of argon exceptionally well. This is somewhat surprising. The LJ fluid is a model with pairwise additive interactions. In a dense fluid, it is believed that three-body interactions are important. The three-body interaction raises the configurational energy (Axilrod-Teller contribution). In crystalline argon, the lattice energy is raised about 10%, which might be an upper bound for this

effect in liquid argon.18 One possible explanation for the good agreement is that it is fortuitous. The repulsive part of the LJ potential (the r-12 law) does not have a sound theoretical basis.18 If the correct (but unknown) repulsive potential was used, then the effect of three-body interactions would become manifest. A second explanation is that the effect of three-body interactions in dense disordered fluids is not as significant as in ordered (crystalline) systems. The aforementioned correction of 10% for the configurational energy of argon is based on lattice calculations of the energy where the number of atomic triads and their specific contribution is known exactly. It is possible that in a disordered system the effect of three-body interactions is mitigated to a large degree (see the Appendix). Whatever the reason, it is encouraging that a simple pairwise additive model, such as the LJ model, yields an accurate description of a real monatomic (argon) and a spherically symmetric polyatomic (methane) fluid. This result lends credibility to the widespread use of the LJ potential to model atom-atom or sitesite interactions in large and complex molecules. D. Relaxation Entropy. In Figure 6 we see that ∆srelax for water is significantly smaller than the other fluids (or equivalently, the corresponding relaxation energy, ∆urelax is smaller). Above we have already commented on the significance of the relaxation entropy [see eq 15]. So why does water have such a small ∆srelax value relative to other fluids? One possible explanation is this: icelike clusters are more expanded (less dense) than disordered water clusters and have a lower entropy. Allowing the liquid to “relax” by expansion would favor the formation of low entropy icelike structures. Ice at the triple point has a reduced density of about 2.8. Notice that the maximum in ∆srelax for water occurs near 2.8 and then precipitously drops


Figure 8. Reorganization entropy (∆sreo) of the indicated saturated liquids. The reorganization entropy is the entropy change suffered by the molecules of a fluid when an extra molecule is added to the system at constant volume. The molecules of the fluid must “reorganize” to accommodate the added molecule. For a van der Waals fluid, ∆sreo ) 0 at all conditions. There are two opposing contributions to ∆sreo that are discussed in the text. The large values of ∆sreo near a fluid’s triple point may signal the impending liquid-solid phase transition. See Figure 11 for an expanded look at ∆sreo.

to negative values as the liquid density increases to its value of 3.1 at the triple point. Note that maxima in ∆srelax are also exhibited by oxygen and ethylene. Since both of these fluids (unlike water) become more dense upon solidification, these maxima are more inexplicable. There is a corresponding relaxation energy and enthalpy [see eqs 18 and 25]. The relaxation terms have no effect on the value of B. There is perfect energy-entropy compensation in the relaxational terms in forming the chemical potential (-kT ln B). The relaxation energy and entropy also play a role in the reorganizational properties discussed below. E. Reorganization Energy and Entropy. There are two effects that contribute to ∆ureo, one global and one local. The first is energetically favorable. The presence of the interloper globally reduces the volume (δV ) -V) available to the N molecules and the associated favorable energy change is -(∂∆up/∂V)TV ) -∆urelax < 0. But locally, the presence of the interloper interferes with the N molecules sampling some favorable compact configurations that it was able to sample in the absence of the interloper (an excluded volume effect). We have shown that the associated energy penalty is -∆up ) -∆uint/2 > 0. In a VDW fluid, these two opposing effects exactly cancel and ∆ureo ) 0 and thus ∆sreo ) 0. From the entropy perspective, the interloper forces the N molecules closer together with an associated negative global entropy change, -(∂∆sp/∂V)TV ) -∆srelax. But the presence of the interloper eliminates the possibility of some compact configurations among the N molecules. Losing some compact configurations means losing some local structure with a corresponding entropy gain. Globally, there is an entropy lost because

Sanchez et al.

Figure 9. Configurational energy ∆uconf (or ∆up) of the indicated saturated liquids. Notice that the LJ fluid satisfactorily mimics the configurational energy of argon and methane even though it is a pairwise additive potential (see text). The average molecular interaction energy is given by 〈ψ〉 ≡ ∆uint ) 2∆up.

Figure 10. Constant volume solvation energy (∆uv) of the indicated saturated liquids. Notice that the vertical scale has been reduced by a factor of 2 compared to Figure 9.

the molecules are forced closer together (on average) by the interloper, but locally, there is a entropy gain because the interloper disrupts structure by interfering with the formation of compact configurations.



We see in Figure 8 that ∆sreo is small over a large part of the temperature range for the simple and LJ fluids (consistent with the VDW result that ∆sreo ) 0). Note however, that ∆sreo increases (especially oxygen, ethylene, and ethane) near the triple point. The sharp rise in ∆sreo appears to be a signature of the impending liquid-to-solid phase transition. The local effect (structure disruption) appears to become dominant near the triple point. For water, the local structure disruption effect is anomalously large over the entire temperature range. One important caveat must be mentioned. Below 4 °C, water has a negative thermal expansion coefficient and a negative thermal pressure coefficient [∂S/∂V|T ) ∂P/∂T|V < 0]. This behavior continues down into the supercooled liquid range. In this condition, both the global and local entropy reorganizational changes are positive; both lead to a disruption of structure. Above 4 °C, these two effects oppose one another and the preceding arguments apply. In Figure 5 we see that at high densities, ∆sv is much smaller for water than the other fluids. This result is a consequence of the large reorganizational entropy contribution for water. (see Figure 8). The minima in ∆sv observed for water, oxygen, and ethylene can also be traced back to increasing large values of ∆sreo as the triple point is approached. The constant volume energy ∆uv is similarly affected (see Figure 9) so that the difference (∆uv - T∆sv ) -kT ln B) is unaffected by the reorganizational terms, i.e., the configurational chemical potential does not depend on ∆sreo because it cancels with ∆ureo/T (exact energy-entropy compensation). The same cancelation occurs when considering the interaction of a solute with the solvent.13 Since the solubility of a solute depends on the chemical potential, the question of “solvent reorganization” around a solute is a nonissue for determining solute solubility. This fact does not seem to be fully appreciated in the literature. What directly affects the chemical potential is the “ interaction entropy” (see next section). Some of the finer details of the behavior of the reorganization entropy are illustrated in Figure 11 for the LJ fluid and four selected simple fluids. Note that ∆sreo for the simple fluids passes through a minimum at a reduced density of about 2.2 and a maximum at about 1.3. As can seen, the LJ fluid qualitatively captures this interesting nuance in the behavior of ∆sreo. As Figure 8 illustrates, water does not exhibit this qualitative behavior at all. F. Interaction Entropy. Although the interaction entropy is rigorously given by eq 21, the physics inherent in this quantity is not transparent. We can obtain a better appreciation for the interaction entropy by using a new interpretation of B. From ref 10, we have

P(ψ) )

P0(ψ)e-βψ B

B)

∫-∞0 βψP(ψ)dψ

Pins ≡

∫-∞0 P0(ψ)dψ

where Pins is the probability that a randomly inserted molecule into a system of N molecules will experience an attractive or zero interaction (ψ e 0) with the N molecules (aka the “insertion probability”). This requires that a cavity exist that is large enough to accommodate the randomly inserted molecule. At high densities, this probability is very low. We now defined a new average calculated only over those states where the interaction energy is attractive (ψ < 0):

∫-∞0 f(ψ)P(ψ)dψ ∫-∞0 f(ψ)P(ψ)dψ 〈f〉a ) ≡ Pa ∫-∞0 P(ψ)dψ

(43)

Thus,

B)

Pins βψ

Pa 〈e 〉a

)

Pins β〈ψ〉

Pae

〈eβ(ψ-〈ψ〉)〉a

(44)

or

(40)

where P0(ψ)dψ is the probability that a randomly inserted molecule into a fluid will experience an interaction energy of ψ, and P(ψ)dψ is the probability that any molecule among the N molecules in the fluid (it already occupies a cavity) has an interaction energy ψ. The above equation allows us to obtain an alternative and useful expression for B:

Pins

Figure 11. Detailed look at the reorganization entropy (∆sreo) of the indicated saturated liquids. Figure illustrates that ∆sreo passes through a maximum and minimum. Methane and ethylene also exhibit this qualitative behavior (not shown) but water does not.

ln B ) -β〈ψ〉 + ln Pins - ln 〈eβ(ψ-〈ψ〉)〉a - ln Pa (45) Under normal circumstances, Pa (the probability that a molecule in the fluid will have an attractive interaction energy) will be very close to unity and the ln Pa term above can be ignored. Note that, for hard molecules with attractiVe interactions, 〈ψ〉 ) 〈ψ〉a and Pa ) 1. So without any approximation

ln B ) -β〈ψ〉 + ln Pins - ln 〈eβ(ψ-〈ψ〉)〉a

(46)

(41) for hard molecules with attractive tails. Substituting this result into eq 21a yields

(42)

∆sint/k ) ln Pins - ln 〈eβ(ψ-〈ψ〉)〉a


) ln Pins - [〈ψ2〉 - 〈ψ〉2]aβ2/2 + ...

Sanchez et al.

(47)

So we see that the interaction entropy is governed by the probability that a molecular cavity exists in the fluid plus a term that is a measure of the fluctuations in the attractive energy. Notice that both terms contribute negatively to ∆sint. In a mean field approximation, fluctuations in energy are ignored and the second term above would be set equal to zero. Equation 47 is rigorously correct for molecules with hard repulsive potentials and attractive tails and should be an adequate approximation for more realistic and “softer” repulsive potentials. The effect of attractive interactions is to further reduce configurational space available to a randomly inserted molecule. If a cavity is larger than required to accommodate the molecule, the added molecule will not sample the cavity volume with equal probability. Each fixed system configuration with the added molecule is weighed by a Boltzmann factor e-βψ. This weighting factor varies (fluctuates) within the cavity. In the jargon of statistical mechanics, the available configuration space is biased by attractive interactions. This loss of configuration space corresponds to a loss of entropy which is compensated by the lowering of the configurational energy (ψ). Thus, k ln Pint is an upper bound on ∆sint; attractive interactions will lower ∆sint even more as indicated above through the fluctuation term. Since ∆sint always contributes a negative contribution to ln B, we conclude that energetics (not entropy) control the chemical potential in pure dense fluids where ln B is positive (B ) 1 corresponds to exact energy-entropy compensation). This dominance of energy over entropy can also be seen in Figures 5 and 10 where at high densities the magnitude of β∆uv is about a factor of two larger than ∆sv/k. This energy dominance for self-solvation was noted earlier and has been noted by others.11 Notice in Figure 7 that ∆sint becomes more negative as the density increases. This is consistent with the availability of molecular size cavities becoming less probable. We can assess the importance of energy fluctuations on ∆sint by comparing the LJ value of ∆sint with that of a hard sphere (HS) fluid. Using the Carnahan-Starling equation of state for hard spheres19 and eq 5, we find for a hard sphere fluid:

ln B ) ln Pins ) ∆sint/k ) -

η(3η2 - 9η + 8) (1 - η)3

(48)

where η is the fraction of space occupied by the hard spheres (η ) πσ3F/6). To effect a comparison with an LJ fluid at equal reduced densities, we must assign a value to η at the critical point (F/Fc ) η/ηc). The fraction of space occupied at the critical point by the LJ fluid is 0.16. The comparisons are shown in Figure 12 for ηc ) 0.16 and 0.15. As can be seen, the comparisons are relatively sensitive to the value of ηc that is chosen. (Since the repulsive part of the LJ potential is not infinitely repulsive, the corresponding ηc for a hard sphere potential is not necessarily equal to 0.16.) This comparison argues strongly that the value of ∆sint in LJ and simple fluids is dominated by the insertion probability up to reduced densities of 2.7. A similar conclusion is also reached if scaled particle theory is used to compute ∆sint. G. Supercooled Water and Divergences. The sharp increase in ∆sreo near the triple point of water suggests that ∆sreo might diverge at some unique temperature T0 in the supercooled liquid region. This possibility is known as the “stability limit conjecture”.20 Other apparent divergent properties (e.g., heat capacity) have been suggested for water.20,21 Reported divergent properties for water fall in the range 220-230 K with 228 K commonly

Figure 12. Interaction entropy (∆sint) of the LJ fluid compared with that of a hard sphere fluid compared at two different values of the critical density (see text). As can be seen, most of the contribution to ∆sint comes from ln Pins where Pins is the insertion probability or the probability that a cavity of sufficient size is available to accommodate an added molecule (cf. eq 47).

cited.22 For water, T0 can be predicted by extrapolating the equation of state into the supercooled liquid regime where we find T0 ) 227 K. At T0, the liquid branch of the coexistence curVe crosses the spinodal line into the unstable regime. Importantly, ∆sreo and the other configurational properties do not diVerge but remain finite at T0. This resembles the behavior of the configurational properties at the liquid-vapor critical point where they remain finite; but the heat capacity, compressibility, and thermal expansion coefficient diverge. If the other fluid equations of state are extrapolated into the supercooled liquid regime, we find a density maximum and a T0 for all fluids except nitrogen and methane. For the LJ fluid a density maximum is predicted at reduced temperature of 0.441 and density of 2.82 with T0 ) 0.291. We have performed NPT Monte Carlo simulations on the LJ fluid in the supercooled regime and see no evidence of a density maximum. The density monotonically increases with decreasing temperature. Thus, our conclusion is that the saturated LJ fluid does not have a density maximum or a T0 in the supercooled region. Extrapolation of the LJ equation of state into the supercooled liquid regime yields an aphysical result. IV. Summary Convenient relations have been derived to calculate the configurational properties of a pure fluid from its equation of state. These were applied to the VDW fluid for the purposes of illustration and to demonstrate that the VDW model suggests a corresponding states principle (CSP) for configurational properties. A hierarchy of relations exist for the configurational entropy and energy. These configurational contributions were calculated for six simple fluids, water, and the LJ model fluid from the

Simple Fluids and Water triple to the critical point. As suggested by the VDW model, a CSP was found to exist for simple, classical liquids at saturation when certain configurational properties are compared at equal reduced densities rather than at equal reduced temperatures. The LJ model fluid describes the configurational properties of simple fluids surprisingly well. It is surprising because the LJ fluid is a pairwise additive model, whereas it is believed that three body and higher order interactions are important for dense liquids. The good agreement helps to support the use of the LJ potential in modeling atom-atom (or site-site) interactions in large polyatomic molecules. Using the LJ model, we argue that comparing configurational properties at equal reduced densities is equivalent to comparing properties at equal occupied volumes (or equivalently, equal free volumes). The normal boiling point of eight simple fluids is a corresponding state at a reduced density of 2.63 ( 0.02. This density corresponds to an occupied volume fraction of 0.42. The entropy of vaporization of the LJ model at a reduced density of 2.63 is 8.9k. The average entropy of vaporization of eight simple fluids at their normal boiling points is (9.1 ( 0.3)k. Thus, Trouton’s rule (invariant entropy of vaporization) is one manifestation of the CSP. All configurational properties remain finite (no divergences) at the liquid-gas critical point. The “relaxation entropy” approaches a nearly universal value of (0.8 ( 0.1)k at the critical point for all eight fluids we have examined (including water). The reorganizational entropy and energy of water are anomalously large (compared to simple fluids) over the entire saturated liquid range (see Figure 8). The reorganizational entropies of some simple fluids (oxygen, ethylene, and ethane), although small over most of their liquid range, increase dramatically as the triple point is approached. In contrast, a VDW fluid has a zero reorganizational entropy under all conditions. To our knowledge, the reorganizational entropy has never been calculated before for water or for any other fluid. The interaction entropy (∆sint) is controlled by the probability (Pins) that a molecular size cavity exists in the fluid (see Figure 12). A secondary and smaller contribution to ∆sint comes from fluctuations in the single molecule interaction energy ψ. These fluctuations contribute negatively to ∆sint [see eq 47]. The empirical equation of state developed for water3 captures the density maximum at 4 °C. When this equation of state is extended into the supercooled liquid regime, the density of the supercooled water continues to decrease (negative thermal expansion coefficient) until 227 K (denoted as T0). At T0, the liquid-vapor coexistence line crosses the spinodal line (defined by where the isothermal compressibility diverges). Although divergences occur in the thermal expansion coefficient and compressibility at T0, the calculated configurational properties all remain finite. The reorganizational and relaxation energy and entropy contributions exactly compensate in forming the chemical potential. The fundamental energy-entropy pair is the interaction quantities (∆uint,∆sint). It has been illustrated that certain configurational properties expressed in dimensionless form satisfy a CSP. The choice of how to reduce the properties to dimensionless form is not arbitrary. An energy divided by kTc (where Tc is the gas-liquid critical temperature) is dimensionless but does not satisfy a CSP. When energy (or free energy) is divided by kT, it satisfies a CSP. Similary for the entropy properties, (T/Tc)∆s/k does not satisfy a CSP, but ∆s/k does. There is much virtue in expressing energies and entropies in the correct dimensionless form.

J. Phys. Chem. B, Vol. 103, No. 24, 1999 5115 Acknowledgment. This work was financially supported by a NSF-DMR grant. Appendix Mediated Pairwise Interactions and Interaction Energy. Representing the total configurational energy of N particles by a sum of pair potentials N

UN )

N

∑ ∑ u(Rij) i)1 j)i+1

(A1)

where Rij is the vector distance from the ith to the jth particle, is well-known to be an approximation because three-body and higher order interactions and their effect on the energy are ignored. However, these higher order interactions can be formally taken into account as follows: N

UN )

N

N

u(Rij|RN-2) ) ∑ψN-i(Ri|RN-1) ∑ ∑ i)1 j)i+1 i)1

(A2)

where u(Rij|RN-2) is the interaction potential between the ith and jth particles when the remaining N-2 particles are in configuration RN-2 and we use the shorthand notation RN ≡ R1R2...RN. The interaction between the i and jth particle not only depends on the distance of separation |Rij| but also on the positions RN-2 of the other N-2 particles. The pair interaction is “mediated” by the surrounding particles. Only in the limit of low density will eq A2 become eq A1. The quantity ψN-i(Ri|RN-1) is the interaction of a ith particle at Ri with N-i other particles at Ri+1,Ri+2,...,RN and the remaining particles are in configuration R1,R2,...,Ri-1. Thus, the ensemble average of the configurational energy can be expressed as

〈UN〉 )

∫VUN exp[-βUN)]dRN ∫

N

)

exp[-βUN)]dRN V

〈ψN-i〉 ∑ i)1

(A3)

where N

〈ψN-i〉 )

〈u(Rij|RN-2)〉 ) (N - i)〈ψ1〉 ∑ j)i+1

i ) 1, 2,...,N (A4)

and

〈ψ1〉 ) 〈u(Rij|RN-2〉

for all i, j pairs.

(A5)

The latter follows because all N particles are identical and translational invariance is assumed for the pair interaction. The average interaction energy of a particle with the remaining N-1 particles in the system 〈ψN-1〉 is an intensive property; i.e., in the thermodynamic limit (N f ∞, V f ∞, N/V ) F ) constant), 〈ψN-1〉 approaches a limiting finite value independent of N. Equation A4 can now be expressed as

〈ψN-i〉 )

N-i 〈ψ 〉 N - 1 N-1

(A6)

and the average configurational energy eq A3 becomes N

〈UN〉 )

N

〈ψN-i〉 ) 〈ψN-1〉 ∑ 2 i)1

(A7)


Sanchez et al.

In the main text, UN f N∆uconf, 〈ψN-1〉 f 〈ψ〉, thus

〈UN〉/N ) ∆uconf ) ∆up ) 1/2〈ψ〉 ) 1/2∆uint

(A8)

References and Notes (1) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: New York, 1996. (2) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987. (3) Hill, P. G. J. Phys. Chem. Ref. Data 1990, 19, 1233. (4) Stewart, R. B.; Jacobsen, R. T. J. Phys. Chem. Ref. Data 1989, 18, 639. (5) Setzmann, U.; Wagner,W. J. Phys. Chem. Ref. Data 1991, 20, 1061. (6) Vargaftik, N. B.; Vinogradov, Y. K.; Yargin, V. S. Handbook of Physical Properties of Liquids and Gases; Begell House, Inc.: New York, 1996. (7) Friend, D. G.; Ingham, J.; Ely, J. F. J. Phys. Chem. Ref. Data 1991, 20, 275. (8) Johnson, J. K.; Zollweg, J. A.; Gubbins,K. E. Mol. Phys. 1993, 78, 591.

(9) Widom, B. J. Chem. Phys. 1963, 39, 2808. (10) Widom, B. J. Phys. Chem. 1982, 86, 869. (11) Ben-Naim, A. SolVation Thermodynamics; Plenum Press: New York, 1987. (12) Micheals, A.; Geldermans, M.; De Groot, S. R. Physica 1946, 12, 105. (13) Yu, H.-A; Karplus, M. J. Chem. Phys. 1988, 89, 2366. (14) Guillot, B.; Guissani, Y. J. Chem. Phys. 1993, 99, 8075. (15) Chokappa, D. K.; Clancy, P. Mol. Phys. 1988, 65, 97. (16) Alder, B. J.; Hoover, W. G.; Young, D. A. J. Chem. Phys. 1968, 49, 3688. (17) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 75th ed.; CRC Press; 1994. (18) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces; Clarendon Press: Oxford, 1981; Chapter 8. (19) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (20) Debenedetti, P. G. Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, NJ, 1996. (21) Speedy, R. J.; Angell, C. A. J. Chem. Phys. 1976, 65, 851. (22) Sorensen, C. M. Nature 1992, 360, 303.

Configurational Properties and Corresponding States in Simple Fluids

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