Nonrandom Hydrogen-Bonding Model of Fluids and Their Mixtures. 1

6592

Ind. Eng. Chem. Res. 2004, 43, 6592-6606

Nonrandom Hydrogen-Bonding Model of Fluids and Their Mixtures. 1. Pure Fluids Costas Panayiotou,* Maria Pantoula, Emmanuel Stefanis, and Ioannis Tsivintzelis Department of Chemical Engineering, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece

Ioannis G. Economou Institute of Physical Chemistry, National Research Center for Physical Sciences “Demokritos”, GR-15310 Aghia Paraskevi, Attikis, Greece

A unified treatment of the phase equilibria and interfacial properties of fluids is presented. This is done through the development of a framework model, which is applicable to nonpolar systems as well as to highly nonideal systems with strong specific interactions, to systems of small molecules as well as to polymers, glasses, and gels, to liquids as well as to vapors and supercritical systems, and to homogeneous as well as to inhomogeneous systems. One key characteristic of this equation-of-state model is its capacity to estimate the nonrandom distribution of the free volume in the system. A quasi-thermodynamic approach of inhomogeneous systems is used for modeling the fluid-fluid interface. The present model is referred to as the nonrandom hydrogen-bonding model. The key differences between this model and the previous quasi-chemical hydrogen-bonding model are the following: (1) The combinatorial term is replaced by the generalized Staverman term. (2) The nonrandomness factor is also generalized. Two alternative expressions are presented in this work. (3) The shape factor, s, is no longer an adjustable parameter. It is set equal to the UNIFAC q/r ratio and obtained from the corresponding UNIFAC compilations in the literature. (4) A most recent quasi-thermodynamic approach is used for the fluid-fluid interface. In the first part of this series of papers, the model is applied for the estimation of basic thermodynamic properties of pure fluids, such as vapor pressures, orthobaric densities, heats of vaporization, surface tensions, and glass transition temperatures. Introduction In many industrial operations, the systems of interest to chemists and chemical engineers are complex in composition, strength of intermolecular forces, and phase behavior. Thus, thermodynamic models/frameworks applicable to multicomponent systems of fluids of varying molecular size and shape, in the liquid, gaseous, or glassy state, to homogeneous systems, and to fluid-fluid interfaces are long-standing goals not only for the coherent insight they provide but also for the industrial practice. In particular, models with a significant predictive capacity of thermodynamic properties of fluids and their mixtures over an extended range of external conditions are most useful today in computeraided product design and process screening. In spite of the availability of high computing speed even at the desktop level, we are far from having, at present, a theoretical model that could properly describe and, preferentially, predict the phase equilibria and related properties of the above systems from first principles or via ab initio calculations. However, during the last few decades, significant progress has been made, especially for the thermodynamics of systems of nonelectrolytes with semiempirical equation-of-state models1-8 or approximate group-contribution models,9-13 which provide with a basic understanding of the physicochemical processes involved and allow for a rational design of processes and products. One especially promising approach, combining in the frame of statistical thermodynamics, an earlier equa* To whom correspondence should be addressed. Tel./fax: +30 2310 996223. E-mail: [email protected].

tion-of-state model,6,14,15 with Guggenheim’s quasichemical model16 for the nonrandom distribution of the free volume and Veytsman’s statistics17-19 for the contribution of hydrogen bonding, is known in the literature as the quasi-chemical hydrogen-bonding (QCHB) model.20-22 The key rationale of this approach resides upon the idea that the major class of systems exhibiting significant nonrandomness contribution to thermodynamic quantities is the class of systems interacting with strong specific forces, or the hydrogenbonding systems. As a consequence, the quasi-chemical approach6,14-16 is used for the nonrandom distribution of the free volume, while the nonrandom distribution of molecular segments is treated with the hydrogenbonding approach.17-19 With this new approach, the calculation scheme is very much simplified because, for example, in a multicomponent mixture of alkanols and inert solvents, such as alkanes, one has to solve one simple analytical equation only in order to account for the nonrandom distribution of molecules in the mixture. Recently,22 it was shown that the equations for the chemical potential of this model, when Staverman’s combinatorial term23 in the partition function (see eq 7 below) is adopted, are identical with the corresponding equations of the Cosmotherm model,13,24-27 an especially promising predictive tool of thermodynamics. The equation-of-state model in its current state is far from complete and fully developed, but, even so, it appears quite versatile because it can be applied to a broad variety of systems over an extended range of external conditions. In addition, it could give insight into a number of physicochemical processes, such as in-

10.1021/ie040114+ CCC: $27.50 © 2004 American Chemical Society Published on Web 08/12/2004

Ind. Eng. Chem. Res., Vol. 43, No. 20, 2004 6593

tramolecular association, hydrogel collapse, and hydrogen-bonding cooperativity, as well as contribute to the understanding of the findings of modern experimental techniques, such as positron annihilation spectroscopy, that the partitioning of the free volume may be nonrandom even in systems of nonpolar substances.28 Another area of ever-increasing interest in the era of nanotechnology is interfacial phenomena. Rational design of numerous processes, notably, emulsion, microemulsion, and suspension processes in food, pharmaceutical, and material technology, in coating, adsorption, and industrial separation processes, and in tertiary oil recovery requires reliable methods for the estimation of interfacial tensions in multicomponent systems. Thus, there is today an ever-increasing interest in understanding interfacial phenomena and in improving estimation methods of interfacial properties. Although there are many approaches for the study of fluid-fluid interfaces,29-42 little progress has been made for the calculations of interfacial tensions and interfacial profiles in highly nonideal mixtures. Our equation-of-state framework could be used for these types of calculations, as was shown recently.43,44 The use of compressed fluids, including supercritical fluids, for processing polymers is recently attracting a lot interest.45-49 These applications range from polymer purification, polymer impregnation and plasticization, and membrane conditioning, to production of microcellular foams, fibers, and microspheres, and to encapsulation of active substances. One of the key issues in these applications is the influence of the compressed fluid on the glass transition of the polymer. Thus, there is increasing interest for models able to describe polymer glass transitions and especially the complex retrograde vitrification of polymers.50 The main objective of this series of papers is to systematically develop and fully test a broader statistical thermodynamic framework along the lines of the lattice-fluid hydrogen-bonding (LFHB)3,18,19 and QCHB approaches20-22 and critically compare it with alternative widely used approaches, especially by chemical engineers and process simulators, for calculations of thermodynamic properties of systems of fluids ranging from supercritical systems up to high polymers and polymeric glasses. This new and broader framework will be referred to as the nonrandom hydrogen-bonding (NRHB) model. Although the development of NRHB follows closely the corresponding development of the QCHB model,20-22 there are key differences that make the new model more advantageous: The combinatorial term in NRHB is replaced by the generalized Staverman23 term. The shape factor, s, is no longer an adjustable parameter. It is set equal to the UNIQUAC ratio of molecular area to molecular volume, q/r, and is obtained from the corresponding DIPPR51 or UNIFAC52 compilations in the literature. The nonrandomness factor is also generalized. Two alternative expressions16,53 are presented in this work. In addition, a most recent quasi-thermodynamic approach is used for the fluid-fluid interface. As a consequence, the key equations of NRHB are different from the corresponding QCHB equations. In the first part of this series of papers, we will lay down the formalism for pure fluids and focus on the calculation of a variety of their thermodynamic properties. The glass transition will be addressed, also, with the new model.

The Model Let us consider a system of N molecules of a pure fluid at a temperature T and an external pressure P, which are assumed to be arranged on a quasi-lattice of coordination number z and of Nr sites, N0 of which are empty. Each molecule is assumed to be divided into r segments of segmental volumes v*. The total number Nr of lattice sites is given by

Nr ) rN + N0

(1)

The average interaction energy per molecular segment is given by

* ) (z/2)

(2)

where is the interaction energy per segment-segment contact. If zq is the number of external contacts per molecule, its surface-to-volume ratio, s, a geometric characteristic of the molecule, is given by

s ) q/r

(3)

There are various ways of calculating s. Modern process simulators have extensive databanks with various molecular properties including UNIQUAC molecular area and molecular volume. Their ratio is just s. In extensive compilations of thermophysical properties of fluids, such as the DIPPR database,51 one may find values for the van der Waals (vdW) volume and surface area of molecules. The parameter s could be obtained by dividing this vdW area with the vdW volume and this ratio by a constant equal to 1.6481 × 107 cm-1. In this series of papers, we will adopt the widely used group-contribution calculation scheme of UNIFAC.52 Thus, s is not an adjustable parameter in NRHB. The characteristic properties or scaling constants, along with s, are reported in Table 1 for a number of common fluids. Having defined s, the total number of contact sites in the system is

zNq ) zqN + zN0

(4)

The same average segmental volume v* is assigned to an empty site. It is assumed that two neighboring empty sites on the quasi-lattice remain discrete and do not coalesce. The total volume of the system is, then, given by

V ) Nrv* + N0v* ) Nrv* ) V* + N0v*

(5)

Let us now incorporate nonrandomness in the distribution of the free volume in the system. For this purpose, we will assume that the partition function of the system can be factorized as follows:

(

Q(N,P,T) ) ΩRΩNR exp -

E + PV RT

)

(6)

where ΩR is the combinatorial term of the partition function for a hypothetical system with a random distribution of the empty sites and ΩNR is a correction term for the actual nonrandom distribution of the empty sites. For the first factor, we will adopt the generalized expression of Staverman:23

Ω R ) ωN

( )

Nr!NrlN Nq! N0!N! Nr!

z/2

(7)

6594

Ind. Eng. Chem. Res., Vol. 43, No. 20, 2004

Table 1. Characteristic Constants of Pure Fluids and of Pure Fluids/Polymersa a. Characteristic Constants of Pure Fluidsa fluid

* ) RT*/J‚mol-1

v* ) *P*-1/cm3‚mol-1

vsp* ) F*-1/cm3‚g-1

s ) q/r

N2 O2 CO CO2 methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane n-octadecane n-eicosane n-docosane n-pentacosane n-octacosane n-triacontane n-dotriacontane n-hexatriacontane 2,4-dimethylhexane 2,2,4-trimethylpentane cyclohexane benzene toluene tetralin acetone n-butyl acetate diethyl ether CCl4 CHCl3 CH2Cl2 methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-octanol 1-decanol phenol dimethylamine NH3 water

1262 1484 1336 3040 1864 2770 3319 3800 4295 4557 4734 4870 4999 5111 5212 5265 5330 5423 5496 5622 5656 5703 5847 5918 5978 6111 6192 6260 6375 4821 4794 5171 5026 5198 6106 4459 4799 3880 5192 4966 4835 4464 4347 4629 4917 5148 5400 5617 5697 6965 4191 3064 4065

6.534 5.048 6.650 5.970 7.342 7.920 9.121 10.748 13.15 13.57 14.00 14.45 14.925 15.317 15.577 15.637 15.839 16.341 16.647 17.36 17.451 17.700 18.566 18.960 19.194 20.132 21.037 21.480 22.666 15.916 17.142 13.040 9.217 10.276 11.463 7.874 11.311 9.200 12.56 9.725 8.700 10.611 10.608 12.358 13.901 15.127 16.540 17.102 17.200 15.543 11.200 7.265 3.750

1.098 0.776 1.119 0.707 2.090 1.574 1.426 1.392 1.373 1.317 1.290 1.283 1.278 1.266 1.259 1.252 1.247 1.244 1.240 1.237 1.233 1.230 1.226 1.221 1.219 1.214 1.218 1.209 1.209 1.301 1.297 1.205 1.075 1.083 0.984 1.113 1.029 1.157 0.579 0.613 0.685 1.141 1.128 1.131 1.142 1.150 1.156 1.157 1.160 0.951 1.324 1.269 0.801

0.931 0.945 0.935 0.909 0.961 0.941 0.903 0.881 0.867 0.857 0.850 0.844 0.839 0.836 0.833 0.830 0.828 0.826 0.825 0.823 0.822 0.821 0.819 0.817 0.815 0.814 0.813 0.812 0.811 0.843 0.857 0.800 0.753 0.757 0.720 0.908 0.869 0.888 0.858 0.840 0.881 0.941 0.903 0.881 0.867 0.857 0.850 0.839 0.833 0.757 0.896 1.039 0.861

b. Characteristic Constants of Pure Fluids/Polymersa fluid (polymer)

* ) RT*/J‚mol-1

v* ) *P*-1/cm3‚mol-1

vsp* ) F*-1/cm3‚g-1

s ) q/r

n-hexatriacontane Wax 2100 polyethylene (linear) polyethylene (branched) polypropylene polyisobutylene polystyrene poly(dimethylsiloxane) poly(vinyl chloride) poly(acrylonitrile) poly(methyl methacrylate) polycarbonate (Bisphenol A) poly(-caprolactone) poly(vinyl acetate) poly(ethylene oxide) poly(propylene oxide) poly(phenylene ether) poly(vinyl methyl ether) Nylon 66

5767 6040 6070 6265 6292 6833 5828 5153 6136 8556 6739 6785 6044 5955 5807 5506 6141 7407 7059

10.021 10.691 10.373 10.090 12.036 9.275 8.418 11.341 6.192 6.377 8.571 7.497 6.777 8.445 5.453 7.645 9.558 11.158 6.605

1.186 - 0.000200P 1.153 - 0.000169P 1.145 - 0.000168P 1.153 - 0.000196P 1.150 - 0.000207P 1.076 - 0.000187P 0.916 - 0.000114P 0.983 - 0.000261P 0.685 - 0.000076P 0.843 - 0.000082P 0.811 - 0.000097P 0.810 - 0.000115P 0.887 - 0.000135P 0.812 - 0.000088P 0.849 - 0.000129P 0.951 - 0.000193P 0.868 - 0.000098P 0.877 - 0.000129P 0.895 - 0.000086P

0.811 0.801 0.800 0.800 0.799 0.839 0.667 0.744 0.780 0.887 0.843 0.728 0.818 0.825 0.829 0.820 0.704 0.819 0.783

a For hydrogen-bonded fluids, the parameters for the hydrogen-bonding interactions (Table 2) are needed also. Pressure, P, in column 4 is in MPa.


where

The total number of intersegmental contacts is

z l ) (r - q) - (r - 1) 2

(8)

In eq 7, ω is a characteristic quantity for each fluid that takes into account the flexibility and symmetry of the molecule. This parameter cancels out in all applications of our interest. However, this is an important parameter dictating the glass transition of polymeric fluids and will be reconsidered later. A nice discussion on parameter l in eq 8 may be found in the original paper by Staverman.23 This parameter is equal to zero for the usual lattice definition of q [zq ) (z - 2)r + 2]. In the latter case and for a lattice coordination number z ) 10, the surface-to-volume ratio, s, becomes equal to 0.8 for high polymers. When l is permitted to deviate from zero, eq 6 can also accommodate molecules with complex geometrical shapes or with largely varying s, as in parts a and b of Table 1. The site fractions f0 and f for the empty sites and the molecular segments, respectively, are related by

f0 )

N0 Nr - rN ) )1-f Nr Nr

(9)

For the second factor, we will use Guggenheim’s quasi-chemical theory16 as proposed previously:6

QNR )

N0rr!N000!

[( ) ] 0 Nr0

! 2 Nr0 Nrr!N00! ! 2

2

[( ) ]

2

(10)

(11)

where

θr ) 1 - θ0 )

q/r q/r + v˜ - 1

(12)

v˜ )

1 V ) V* F˜

(13)

with F˜ being the reduced density. In the random case, the number of contacts between empty sites is given by the equation

N0 z 1 N000 ) N0z ) N0θ0 2 Nq 2

(14)

while the number of contacts between a segment and an empty site is given by

N0 qN 0 ) zqN ) zN0 ) zqNθ0 ) zN0θr Nr0 Nq Nq

z Nrr ) N0rrΓrr ) qNθrΓrr 2 N00 ) N000Γ00 0 Nr0 ) Nr0 Γr0

(15)

(17)

The nonrandom Γ factors in this equation are equal to unity in the random case. These numbers must satisfy the following material balance equations:6,14-16

2N00 + N0r ) zN0 (18)

By combining these equations, we obtain

θ0Γ00 + θrΓr0 ) 1 θrΓrr + θ0Γr0 ) 1

(19)

When one of the nonrandom factors Γ are known, the other two are obtained from eq 19. The reduced density needed in eq 19 is obtained from the equation of state of the system to be derived later in this section. In this work we assume that only first-neighbor segment-segment interaction contacts contribute to the potential energy E of the system and, thus, we may write

-E ) Nrr ) ΓrrqNθr*

and the reduced volume is defined as

(16)

The number of intersegmental contacts Nij in the nonrandom case will be obtained by applying the quasichemical theory.6,14-16 Although most of our calculations will be done with the quasi-chemical approach,6,14-16 it should be stressed that this is not the only approach one could adopt for nonrandomness. An alternative approach for nonrandomness residing upon a recent proposition by Yan et al.53 is presented in the appendix. The following key assumption is now adopted in both approaches: the empty sites are distributed nonrandomly and make no distinction between their neighbor molecular segments. Thus, for the nonrandom distribution the number of intesegmental contacts satisfies the following equations:

2Nrr + N0r ) zqN

In this equation, Nrr is the number of external contacts between the segments belonging to molecules, N00 is the number of contacts between the empty sites, and Nr0 is the number of contacts between a molecular segment and an empty site. The superscript 0 refers to the case of randomly distributed empty sites. In the random case, Nrr takes the form6

1 qN z ) qNθr N0rr ) zqN 2 N0 + qN 2

z qN z N0rr ) qN ) qNθr 2 Nq 2

(20)

The scaling temperature T* and scaling pressure P* of the fluid are related to the intersegmental interaction energy * by

* ) RT* ) P*v*

(21)

while the reduced temperature and pressure are defined as

T ˜ )

T P , P ˜ ) T* P*

(22)

The Gibbs free energy of the system is obtained from the partition function (cf. eq 6) and from statistical thermodynamics as follows:

G ) -RT ln Q

(23)

6596


The reduced density at equilibrium is obtained from the free-energy minimization condition:

(∂G ∂F˜ )

T,P,N,Nr0

)0

(24)

while the nonrandom factor Γr0 or, equivalently, the number of contacts Nr0 is obtained from the free-energy minimization condition:

( ) ∂G ∂Nr0

)0

(25)

T,P,N,F˜

By combining eq 24 with eqs 6 and 23, we obtain the equation of state:

l z q P ˜ +T ˜ ln(1 - F˜ ) - F˜ - ln 1 - F˜ + F˜ + r 2 r z ln Γ00 ) 0 (26) 2

[

[

]

]

On the other hand, eq 25 leads to the equation

4NrrN00 2 Nr0

)

4ΓrrΓ00 2 Γr0

)A (2**/z RT )

) exp

(27)

known as the quasi-chemical condition. Combining eqs 19 and 27, we obtain a quadratic equation for Γr0, with the physically meaningful solution:

Γr0 )

2 1 + [1 - 4θ0θr(1 - A)]1/2

(28)

The above formalism is sufficient for solving phase equilibrium problems and calculating the basic thermodynamic quantities of pure fluids of any molecular size. When the fluid interacts with hydrogen bonds, the above equations change as shown in the next paragraph, where hydrogen bonding is explicitly taken into consideration. Hydrogen Bonding. Following our previous practice,18-22 it will be assumed here that the intermolecular forces may be divided into physical and chemical (or hydrogen bonding). The direct implication of this is that the potential energy, the enthalpy, the volume, and the chemical potential of each component consist of two contributions: one physical, obtained from expressions of the previous paragraph, and one chemical or hydrogen bonding, which will be denoted by subscript H. As previously,18 we consider that there are m types of proton donors and n types of proton acceptors in the system. Let di be the number of donor groups of type i in each molecule and aj be the number of acceptor groups of type j in each molecule. Let Nij,H be the total number of hydrogen bonds between a donor of type i and an acceptor of type j in the system. Using the LFHB procedure,18,19 we obtain for the equation of state of the fluid

(rl - v ) - 2zln(1 - F˜ + qrF˜ ) + z ln Γ ] ) 0 (32) 2

[

P ˜ +T ˜ ln(1 - F˜ ) - F˜

H

00

where vH is the average per segment number of hydrogen bonds in the system and is given by m n

Equations 19, 26, and 28 are coupled equations and must be solved simultaneously for the reduced density and nonrandom factors. The chemical potential is obtained by

µ)

∂G (∂N )

(29)

T,P,Nr0,F˜

vH )

µH RT

] [

]

[

ln Γ00

]

q P ˜ v˜ - +r (30) T ˜ T ˜

(33)

m

) rvH -

∑ i)1

di ln

vd

n

-

vi0

∑ j)1

aj ln

vja v0j

(34)

1 z q µ ) ln - F˜ l + ln F˜ - q ln 1 - F˜ + F˜ + RT ωr 2 r zq q P ˜ [ln Γrr] - + r (30a) 2 T ˜ T ˜

]

The heat of vaporization is given by

vid )

Nid diN di ) ) rN rN r

(35)

vja )

Nja ajN aj ) ) rN rN r

(36)

and

By subtracting eq 26 from eq 30, we obtain the alternative expression for the chemical potential of the pure fluid:

[

∑i ∑j rN

where

q q zq r z ln Γrr + (v˜ - 1) r v˜ - 1 + ln 1 - F˜ + F˜ + 2 r r 2 q

[

Nij,H

The full expression of the chemical potential of our fluid is obtained by adding to eq 30 the following hydrogenbonding contribution:

Combining eqs 6, 23, and 29, we obtain

µ 1 ) ln - l + ln F˜ + r(v˜ - 1) ln(1 - F˜ ) RT ωr

∑i ∑j

m n

vij )

while n

vi0 ) vid -

vij ∑ j)1

v0j ) vja -

vij ∑ i)1

(37)

and

V

H ) (E + PV)vap - (E + PV)liq ) q q ˜ v˜ - θrΓrr - P ˜ v˜ - θrΓrr (31) rN* P r vap r liq

[(

)

(

)]

m

(38)

Ind. Eng. Chem. Res., Vol. 43, No. 20, 2004 6597 Table 2. Parameters for Common Hydrogen Bonds hydrogen bond

EH/J‚mol-1

SH/J‚K-1‚mol-1

VH/cm3‚mol-1

OH-OH NH-N water-water

-25 100 -13 200 -16 200

-26.5 -22.2 -9.2

0.0 0.0 1.13

γ ) [PeV - (µeN - Ψ)]/A ) (Ψ - Ψe)/A

As before,18,19 vij’s satisfy the minimization conditions:

( )

vij -GH ij ) F˜ exp vi0v0j RT

for all (i, j)

(39)

GH ij in eq 39 is the free enthalpy of formation of the hydrogen bond of type i-j and is given, in terms of the energy (E), volume (V), and entropy (S) of hydrogenbond formation, by the equation H H H GH ij ) Eij + PVij - TSij

of our system of volume V and entropy S, we may write29,33

Pe and µe in eq 41 are the equilibrium pressure and chemical potential. Let the z axis be normal to the plane interface, F(z) be the mean segment number density at height z, and ψ(z) be the corresponding mean molecular contribution to Ψ at z. By integrating over the full height H of the system, we have

∫HAdz

(42)

∫HAF(z) dz

(43)

∫HAF(z) ψ(z) dz

(44)

V) N)

(40)

In Table 2 are reported the hydrogen-bonded parameters of three common hydrogen bonds. However, our approach treats the hydrogen-bonding interaction as an interaction “in addition” to the physical interaction of the nonpolar part of the molecule. This nonpolar part of the hydrogen-bonded molecule is essentially the homomorph of the molecule. As an example, the homomorph of ethanol is the hydrocarbon that results by replacing the -OH of the alkanol with one -CH3 group, namely, propane. In view of this, the s ratio of alkanols in Table 1a is equal to s of their homomorph hydrocarbons. The above formalism is sufficient for solving phase equilibrium problems and calculating the basic thermodynamic quantities of hydrogen-bonded pure fluids of any molecular size and of any number of donor and acceptor groups. Extension to Interfaces. Very recently,43,44 the QCHB model has been extended to inhomogeneous systems, that is, to the interfacial region between two phases at equilibrium. The extension has been done by incorporating into the model the density-gradient approximation of inhomogeneous systems.36-42 This latter approximation stems from the pioneering work of vdW36 and his concept that the interface arises from and can be described by the principle of free-energy minimization and takes explicitly into consideration the contribution of density gradients in the various thermodynamic potentials.37-42 In this series of papers, we will adopt the more general extension of ref 44, where densitygradient contributions have been explicitly incorporated in the chemical potential and the equation of state. Because the rationale remains the same, we will confine ourselves here to the essentials and the working equations. Details may be found in ref 44. Let us consider a two-phase system with a plane interface of area A in complete equilibrium, and let us focus on the inhomogeneous interfacial region. Our key assumption is that in an inhomogeneous system it is possible to define, at least consistently, local values of the thermodynamic fields of pressure P, temperature T, chemical potential µ, number density F, and Helmholtz free-energy density ψ. At planar fluid-fluid interfaces, which are the interfaces of our interest here, the above fields and densities are functions only of the height z across the interface. Let us first consider the case where density-gradient terms do not enter explicitly into the expression for the Helmholtz free energy Ψ. For the interfacial tension γ

(41)

Ψ)

Substituting eqs 42-44 into eq 41 we obtain

γ)

∫H[Pe - P′(z)]dz

(45)

where

P′(z) ) F(z) [µe - ψ(z)] ) F(z) µe - ψ0(z)

(46)

ψ0(z) is the local Helmholtz free-energy density of the fluid at T and F(z). For the two bulk phases at equilibrium, we have from classical thermodynamics

h µe ) ψe + PeV

(47)

with V h being the molar volume. We may rewrite eq 45 in a more convenient form by setting the origin z ) 0 within the interfacial region:

γ)

∫-∞+∞[Pe - P′(z)] dz

(48)

∫-∞+∞∆ψ0(z) dz

(49)

or

γ) where

∆ψ0 ) ψ0(z) - Ψe/V ) ψ0(z) - Fµe + Pe

(50)

In a one-component system, the curves F(z), ψ(z), and P′(z) are completely determined by the temperature and Pe, µe, and γ are functions of temperature only. Equation 48 can be used for the calculation of interfacial tension as long as we know the local value of pressure P′ at height z. In the frame of the local thermodynamic approach we, now, assume that the above equation-of-state model can be used to provide the local quantities µ(z), ψ(z), P′(z), and V h (z) connected by the equation

µ(z) ) ψ(z) + P′(z) V h (z)

(51)

Equilibrium, however, requires for the chemical potential to be equal throughout the total volume of the

6598


system. This requirement gives the working form of eq 51 for the local quantities:

∆ψ(z) ) ∆ψ0(z) + F

e

µ ) ψ(z) + P′(z) V h (z)

(52)

Equation 48 can be used now with the understanding that P′(z) is the pressure given by the equation-of-state model for given temperature and number densities (local density and composition). Alternatively, eq 49 can be used, but, now, the equation-of-state model will be used to provide the local Helmholtz free-energy density ψ0(z) for given temperature, local density, and composition. Let us consider now density-gradient contributions. In what follows, it is essential to remember that the properties without density-gradient contributions are indicated by a subscript 0. When density gradients are present, the Helmholtz free-energy density may be written, in general, as

ψ ) ψ0(F) + F

δnm(F) (3nF)m ∑ n,m

(53)

and, thus, the total Helmholtz free energy becomes

Ψ ) V*[ψ0(F) + F

∑δnm(F) (3nF)m] n,m

(54)

This equation implies that there may be densitygradient contributions to both the chemical potential and the equation of state, or

µ(z) ) µ0(z) +

∑δnm(F) (3 F) n

m

[

]

l z q P ˜ +T ˜ ln(1 - F˜ ) - F˜ - ln 1 - F˜ + F˜ + r 2 r z ln Γ00 + ωnm(F) (3nF)m ) 0 (56) 2 n,m

]

∫-∞+∞[∆ψ(z)] dz ) ∫-∞+∞[∆ψ0(z) + Fδnm(F) (3nF)m] dz ∑ n,m

∫-∞+∞[∆ψ(z)] dz ) (1 + 2β)∫-∞+∞[∆ψ0(z) + +∞ dF 2 c ( ) ] dz ) ∫-∞ [Pe - Pl(z)] dz dz

β

2

∆ψ0(z) + c

( )] dF dz

2

(59)

The parameter β is not an external adjustable parameter. It is an internal parameter of the model, a shift factor, which is determined from the consistency requirement that eq 58 holds true. In other words, we require that γ be the same, no matter how it is estimated, either via ∆ψ or via Pe - Pt(z) using the equation of state or the chemical potential for Pt(z). Of course, when β ) 0, the above formalism reduces to the well-known formalism of the density-gradient approximation.37,40,42 The interaction or “influence” parameter c can be estimated if the intermolecular interaction potential of the fluid is known.40 Most often, however, it is treated as an adjustable parameter characteristic of each fluid. From eq 59, we obtain

F

β

δnm(F) (3nF)m ) ∆ψ0(z) + ∑ 2 n,m

( )( ) 1+

β

2

c

dF dz

2

(60)

Applying the usual minimization procedure37,40,42 in eq 58, we obtain

∂∆ψ0 d2F -c 2)0 ∂F dz

(61)

which is the condition subject to which the integral in eq 58 must be evaluated. Multiplying eq 61 by dF/dz and integrating, we obtain

∆ψ0(z) ) c

(dFdz)

2

(62)

Combining eqs 54, 60, and 62, we obtain

[

]

β β Ψ ) Ψ0 + V* ∆ψ0(z) + 1 + ∆ψ0(z) ) 2 2 Ψ0 + V*(1 + β)∆ψ0(z) (63)

(

)

Combining this equation with the equation for the chemical potential from the equation-of-state model (see eq 30), we obtain for the Gibbs free energy:

G(z) ) G0 + V*(1 + β)∆ψ0(z) ) (57)

This is a general expression, and practical calculations cannot be made unless we specify all δ factors. These factors are, essentially, infinite in number and, thus, it is a hopeless task to attempt their specification. We must necessarily resort to an approximation. One convenient approximation is to express the densitygradient contributions as multiples of the quadratic terms,15,16 or

γ)

( )[ 1+

∑

µ0 in eq 55 is the chemical potential without the densitygradient contribution and is assumed to be obtained from the equation-of-state model (eq 30). The detailed expressions for the factors ωnm,i may be obtained only when the corresponding expressions for δnm,i are known. In the general case, eq 49 should be augmented as follows:

γ)

δnm(F) (3nF)m ) ∑ n,m

(55)

n,m

[

In other words, we set

G0 + V*(1 + β)F[µ0(F,Pe) - µe] ) rN[µ0(F,Pe) + (P - Pe)v˜ v*] + (1 + β)rN[µ0(F,Pe) - µe] ) rN[(2 + β)µ0(F,Pe) + (P - Pe)v˜ v* - (1 + β)µe] (64) which implies that the chemical potential per segment is given by

µ(z,P) ) (2 + β)µ0(F,Pe) + (P - Pe)v˜ v* - (1 + β)µe (65) (58)

From the equilibrium requirement (equal chemical


potentials throughout the system for each component), we have

µe ) µ(z,P) ) (2 + β)µ0(F,Pe) + (P - Pe)v˜ v* (1 + β)µe (66) from which we obtain for the pressure at height z

F˜ P(z) ) Pe + (2 + β) [µe - µ0(F,Pe)] v*

(67)

However, because we have the expression for G (eq 64), we may get the equation of state by minimizing it with respect to the reduced density:

∂ {G0(F,P) + (1 + β)[G0(F,Pe) - Ge]} ) 0 (68) ∂F˜ or

{

l z q ˜ +T ˜ ln(1 - F˜ ) - F˜ - ln 1 - F˜ + F˜ + (2 + β) P r 2 r z e ˜ -P ˜ ) 0 (69) ln Γ00 + P 2

[

]}

[

]

Table 3. Influence Parameter and the Shift Factor of Common Fluids fluid

κ

β

O2 CO CO2 NH3 methane propane n-pentane n-hexane n-heptane n-decane benzene CCl4 acetone n-butyl acetate water methanol ethanol 1-propanol 1-butanol polyethylene (linear) polyisobutylene poly(dimethylsiloxane) polystyrene

0.242 0.250 0.210 0.420 0.255 0.185 0.190 0.185 0.174 0.176 0.155 0.190 0.155 0.179 0.409 0.260 0.250 0.241 0.214 0.123 0.055 0.065 0.121

2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.96 (λ ) 4) 1.96 (λ ) 2) 1.97 (λ ) 2) 1.98 (λ ) 2) 1.98 (λ ) 2) 2.0 2.0 2.0 2.0

From classical thermodynamics and eq 23, we have

(∂G ∂T )

S)-

or

l z (1 + β)P ˜ + (2 + β)T ˜ ln(1 - F˜ ) - F˜ - ln 1 - F˜ + r 2 q z ˜ e ) 0 (70) F˜ + ln Γ00 + P r 2

[

[

]

]

Equation 62 can be integrated to give the interfacial profile z(F):

z - z0 )

∫F(z ) F(z) 0

x

c dF ∆ψ0

(71)

Combining eqs 58 and 62, we obtain

∫FF [c∆ψ0]1/2 dF

γ ) (2 + β)

β

R

(72)

where R and β refer to the two phases at equilibrium. The above equations are sufficient for calculating the interfacial tension of the fluid. For the influence parameter c, we may adopt the rationale of Poser and Sanchez,42 leading to the following equation:

c ) *(v*)5/3κ

(73)

This equation satisfies the required dimensionality for c and lets us treat κ as a dimensionless adjustable parameter. Values of this parameter are given in Table 3 for some representative fluids. Generalization of eq 73 leads to eq 80 below, which contains the parameter λ of Table 3. Glass Transition. Following our previous practice,54-56 in order to account for the glass transition of polymeric fluids, we will adopt here the proposition of Gibbs and DiMarzio57 that the entropy of the system is approaching a zero value as the temperature is lowered to the glass transition region. Thus, to proceed, we must derive an expression for the entropy of the system.

P,N

) R ln Q + RT

(∂ ln∂TQ)

P,N

(74)

By replacing Q from eq 6, we obtain

l + ln(rv˜ ) S 1 ) ln ω + (1 - v˜ ) ln(1 - F˜ ) + + rNR r r 2 z q q zq v˜ - 1 + ln 1 - F˜ + F˜ + (1 - θr)Γr0 2 r r 2r zT ˜ SH ln Γrr - (v˜ - 1) ln Γ00 + (75) rNR

[

] [

]

[

]

where SH is the hydrogen-bonding contribution to the entropy. All but the first terms on the right-hand side of eq 75 sum to the external configurational entropy of the system, while the first term accounts for the internal configurational entropy. Following Flory58 and Gibbs and DiMarzio,57 we set

ω ) ηδ

(76)

η is constant, characteristic of the fluid and taking into account the symmetry and the size of the molecule. For convenience, it will be neglected here without any loss in the capacity of the model. The number of internal configurations, δ, available to the semiflexible chain of the fluid is given by57,58

u ln δ ) ln(Z/2) - (r - 2) ln(1 - f) + f(r - 2) RT

(77)

where Z is the number of discrete conformations available to each bond. As before,54,56 this coordination number was set equal to 4. f in eq 77 is the fraction of the r - 2 bonds of the chain molecule that are in “flexed” or high-energy states, while 1 - f is the fraction of these bonds that are in a low-energy state (for example, gauche and trans states). The fraction f is given by

f)

(Z - 2) exp(-u/RT) 1 + (Z - 2) exp(-u/RT)

(78)

6600


Figure 1. Experimental51 (symbols) and calculated (line) vapor pressures of ethane.

Figure 2. Experimental51,59 (symbols) and calculated (line) orthobaric densities of ethane.

Table 4. Glass Transition Temperatures and Flex Energies of Pure Polymers polymer

Tg/K

u/J‚mol-1

poly(dimethylsiloxane) polyisobutylene poly(methyl methacrylate) polystyrene (MW ) 110 000) polystyrene (MW ) 9000) poly(vinyl chloride)

150 243 395 371 364 355

2025 2772 5821 7539 7413 5654

where u is the increase in the intramolecular energy that accompanies the “flexing” of a bond in the chain molecule. This “flex” energy is a characteristic parameter of each polymer and is calculated usually by zeroing S (cf. eq 75) at the glass transition temperature of the polymer at ambient pressure. Values of the flex energies of representative polymers are reported in Table 4. If the polymer is hydrogen-bonded, the contribution SH can be obtained from the general expression56 for a system of m donors and n acceptors:

SH RrN

m

) -vH -

∑i

vid ln

vi0 vid

n

-

∑j

vja ln

v0j vja

(79)

Applications We will apply, now, the model developed in the previous section to the calculation of a number of basic thermodynamic properties of pure fluids. We will calculate, first, the volumetric behavior, the vapor pressures, and the heat of vaporization of representative fluids. Subsequently, we will calculate surface tensions over an extended range of temperature and pressure. Finally, we will calculate the glass transition temperatures as well as the specific volumes at the glass transition of representative polymers over an extended range of external pressures. In Figure 1 are compared the experimental51 vapor pressures of ethane with the calculated ones by the present model. In Figure 2 are compared experimental51,59 and calculated orthobaric densities of ethane up to its critical point. One single set of scaling constants (*, v*, and vsp*) was used for these calculations as well as for the calculation of the heats of vaporization. These scaling constants are reported in Table 1a. One fluid, widely used today in its supercritical state, is carbon dioxide (CO2). The experimental60 and calculated density maps of this important fluid are compared

Figure 3. Experimental60 (symbols) and calculated (line) orthobaric and supercritical densities of CO2. Circles and squares correspond to orthobaric and supercritical densities, respectively.

in Figure 3 over an extended range of conditions from the subcritical up to the supercritical state. Again, one single set of scaling constants was used for these calculations. Another fluid that finds important applications today in its supercritical state is water. This is an example of fluids interacting with strong specific forces. In Figure 4 are compared the experimental51 and calculated vapor pressures of water, while in Figure 5, its experimental51,59 orthobaric densities are compared with the calculated ones. Both figures correspond to the subcritical state of water. The experimental61 and calculated densities of supercritical water are compared in Figure 6 over an extended range of external pressure and temperature. All of these calculations were done by using the scaling constants reported in Table 1 and the hydrogen-bonding parameters (two proton donors and two proton acceptors per water molecule) reported in Table 2. An extensive study of water with the QCHB version of the present framework was reported in ref 20, where the calculated degree of hydrogen bonding is compared with corresponding molecular dynamics results over an extended range of temperature and pressure. This hydrogen-bonding picture remains, essentially, the same in the case of the present generalized NRHB framework.


Figure 4. Experimental51 (symbols) and calculated (line) vapor pressures of water.

Figure 5. Experimental51,59 (symbols) and calculated (line) orthobaric densities of water. Squares and triangles correspond to the liquid and vapor phases, respectively, at equilibrium.

Figure 6. Experimental61 and calculated densities of supercritical water, as calculated by the present equation-of-state model.

As already mentioned, the scaling constants reported in Table 1 were obtained by simultaneously correlating vapor pressures, orthobaric or liquid densities, and heats of vaporization. A least-squares minimization procedure through a Powel or Levenberg-Marquardt nonlinear optimization algorithm was applied. To examine the consistency of the procedure, we treated the series of normal alkanes up to hexatriacontane. In

Figure 7. vdW (hard-core) molar volumes of normal alkanes as a function of the number of chain carbons.

Figure 8. Molar interaction energies of normal alkanes as a function of the number of chain carbons.

Figure 7 are presented the hard-core molar volumes of normal alkanes (V* ) rv* ) Mvsp*) as a function of their number of carbon atoms. As observed, these data fall on an almost perfect straight line (correlation coefficient R2 ) 0.999 98). An analogous picture is obtained when the logarithm of molar interaction energies (E* ) r*) of normal alkanes is plotted as a function of the logarithm of the number of chain carbons, as shown in Figure 8. Again, the data fall on a straight line (R2 ) 0.999 49). However, when plotting the interaction energies per segment, *, of normal alkanes, one may distinguish three domains, as shown in Figure 9. The first five alkanes exhibit a rapid increase in their segmental interaction energy as the number of chain carbons increases. This increase is much smaller for all alkanes above n-pentane. This may be related to the “stiffness” or the “liquid-crystalline” behavior that imposes the linear character of higher normal alkanes. At about NC ) 15, a slight change in the slope is observed, probably indicative of the chain folding that comes into play as the length of the linear chain increases. Let us now turn to high polymers. It is common practice to correlate volumetric properties of polymers for obtaining their scaling constants because their vapor pressures or heats of vaporizations are impractical to measure. Fortunately, there are extensive compilations of high-accuracy PVT data in the literature over an

6602


Figure 11. Experimental (symbols)62 and calculated (lines) specific volumes of poly(acrylonitrile). Figure 9. Segmental interaction energies of normal alkanes as a function of the number of chain carbons.

Figure 12. Nonrandomness factors for PMMA at 20 MPa, as calculated by the present model. Figure 10. Experimental (symbols)62 and calculated (lines) specific volumes of PMMA.

extended range of temperature and pressure. Often, the pressure range extends from ambient atmospheric pressure up to 200 MPa or even higher pressures. To correlate the experimental data over the full pressure range, we have used a small “compressibility” correction of the scaling constant vsp*, which is reported in Table 1b along with the three scaling constants of the polymeric fluids. Of course, this compressibility correction is not needed in applications at low to moderate pressures (say, below 10 MPa). In Figure 10 are compared the experimental62 PVT data with the calculated ones for poly(methyl methacrylate) (PMMA) over a rather extended range of temperature and pressure. An analogous plot is shown in Figure 11 for poly(acrylonitrile). In both cases, the model can reproduce, rather well, these extensive data. One of the merits of the present model is its ability to estimate the nonrandom distribution of free volume in the fluid. This is important information when the polymer is used as a membrane material and in separation processes. In Figure 12 is shown a typical plot of the variation of the nonrandomness factors Γrr, Γr0, and Γ00 with temperature for PMMA. As observed, there is a drastically different behavior of Γ00 compared to Γrr or Γr0. The latter are close to 1 and change slightly with temperature. On the contrary, Γ00 departs significantly from unity and falls gradually as the temperature

Figure 13. Nonrandomness factor Γ00 for poly(acrylonitrile). The lines passing through the circles were calculated by the present model when Guggenheim’s quasi-chemical approach is adopted. The lines passing through the squares were calculated when the approach of the appendix is adopted.

increases. Γ00 reflects the tendency of “segregation” of empty sites or their preference to be surrounded by empty sites rather than molecular segments. In Figure 13 is shown the variation of Γ00 with temperature and pressure for poly(acrylonitrile). Two approaches of nonrandomness were used in these calculations. In the first, the quasi-chemical model of


Figure 14. Experimental51 (symbols) and calculated (solid lines) surface tensions of representative non-hydrogen-bonded fluids.

Guggenheim16 (cf. eq 10) was adopted, while in the second, the approach that is exposed in the appendix was adopted instead. In both approaches, Γrr is near 1.00, while Γr0 varies from ca. 0.970 to ca. 0.985. On the contrary, Γ00 exceeds 1.70 at lower temperatures, as shown in Figure 13. Lower values are obtained for this factor when the quasi-chemical approximation is adopted. This behavior was observed in all studied polymers. In addition, an increase in the pressure was observed to lead to an increase in Γ00. As already mentioned, large departures from unity for the factor Γ00 indicate a strong tendency of freevolume segregation. These findings could be combined with modern positron annihilation spectroscopic measurements for an understanding of the free-volume distribution in polymers.28 In addition, such systematic data could form the basis for comparison of the alternative approaches for the description of a nonrandom distribution of free volume in the fluids. The phases of our systems were treated so far as homogeneous. Let us now turn to nonhomogeneous systems and apply our model to the estimation of surface tensions (liquid-vapor interface) of pure fluids. In Figure 14 are compared the experimental surface tensions of representative nonpolar or weakly polar fluids with calculated ones by the present equation-ofstate model. One experimental datum (usually at the lowest temperature) is used for determining the influence parameter κ for each fluid. This parameter is then used for the prediction of the surface tension at higher temperatures. Values of the parameter κ are reported in Table 3 for representative fluids. In the same table are also reported the values of the internal parameter β. The striking result is that β turns out to be equal to 2 for all non-hydrogen-bonded fluids. Once again, it is stressed that this parameter is not an adjustable parameter, but it is fixed internally by the consistency requirements of eq 58. In Figure 15 are compared the experimental51 and calculated surface tensions of three representative hydrogen-bonded fluids. In the case of hydrogen-bonded systems, however, eq 73 must be modified in order to account for the hydrogen-bonding contributions to the volume and enthalpy of the system. The following simple generalization of eq 73 is adopted here:44

[

](

vH H vH (E - TSH) v* + λ VH 2 2

c ) * + λ

)

5/3

κ (80)

Figure 15. Experimental51 (symbols) and calculated (solid lines) surface tensions of representative hydrogen-bonded fluids.

Figure 16. Experimental63 (symbols) and calculated (solid lines) surface tensions of representative pure polymers.

The values of λ are reported in Table 3 along with the κ and β parameters for a number of hydrogen-bonded fluids. It is worth pointing out that, at least for alkanols and water, λ is equal to the maximum number of hydrogen bonds in which each molecule of the fluid may participate. The expressions for the surface tension may be used unaltered for high polymers as well. In Figure 16 are compared the experimental63 and calculated surface tensions of four representative polymers. Once again, the agreement is rather satisfactory. As a last application, let us consider the glass transition of polymeric fluids. An interesting application is the prediction of the effect of pressure on the glass transition and the densities or specific volumes at the transition. The flex energy, u, which is needed for these types of calculations, can be estimated from the glass transition temperature, Tg, at atmospheric pressure. These temperatures are easily available in the open literature for most polymers. In Figure 17 are compared the experimental62 and calculated glass transition temperatures of two polystyrene samples of different molecular weights over an extended range of pressures. The flex energies of the two samples were calculated from the Tg’s at atmospheric pressure. As observed, the effect of pressure can be accounted for properly by the model. In Figure 18 are compared the experimental62 and predicted specific

6604


Figure 17. Experimental62 (symbols) and calculated (line) glass transition temperatures of two polystyrene samples (M ) 110 000 and 9000).

Figure 18. Experimental62 (symbols) and predicted (line) specific volumes at glass transition temperatures of two polystyrene samples.

volumes at the glass transition for the same polystyrene samples. The agreement is again satisfactory. Among others, these types of calculations are most useful in designing a process known as glass densification. Discussion and Conclusions In the previous paragraphs, we presented the working equations of NRHB, an equation-of-state framework, and applied them to a number of properties in a variety of systems. This is indeed a framework rather than a specific model. Going back to eq 6, it becomes clear that one has a multitude of choices for the basic factor ΩR (Flory’s,64 Guggenheim’s,16 Staverman’s,23 etc.). One may select for nonrandomness either the quasi-chemical approach (eq 10) or the approach presented in the appendix. In a similar manner, one could select the hydrogen-bonding approach or the approach for determining the surface-to-volume ratio s. In view of this multitude of choices, the QCHB model that was reviewed recently21 is just one special case of this equation-of-state framework. The applications of the previous section prove that this is a rather versatile framework, applicable to a broad spectrum of thermodynamic properties from the supercritical state down to the glass transition. This framework can be used for correlation purposes and, to some extent, for prediction purposes

as well. The significant advantage of the models associated with this framework over the original purely quasichemical version of it6 is the simplified calculations involved and its good performance even with systems of molecules interacting with strong specific forces. In the appendix, we have presented an alternative approach to nonrandomness. Preliminary calculations for pure fluids as well as mixtures have shown that the quasi-chemical approach6,14-16 is slightly superior. The approach in the appendix is, however, even simpler in its application, and this warrants its further exploration. One key characteristic of this equation-of-state framework/model is its capacity to estimate the nonrandom distribution of free volume in the system. Strictly speaking, the model can calculate the nonrandomness factors Γ, on the basis of which one can calculate the probabilities that a site next to an empty site is a molecular segment or an empty site. On the basis of these probabilities, one could calculate the probabilities of occurrence of clusters of empty sites. The framework presented here resides upon the mean-field approximation and, as a consequence, it has all drawbacks associated with it. As observed in Figures 2, 3, and 5, there is still a discrepancy between the experimental and model calculations near the critical point, although there is a good agreement in the subcritical and supercritical regions, as shown in Figures 3 and 6. This indicates that the discrepancy in the near-critical region is not a matter of inappropriate scaling constants but rather an inappropriateness of the very approach for the critical phenomena. Although the model takes into consideration the segregation of free volume through the nonrandomness factors, this seems to be not sufficient enough to cope with the cooperativity character associated with critical phenomena. Another drawback of our approach is the dependence of the values of scaling constants on the data used for correlation. Different sets of data result in different scaling constants. Thus, if we use PVT data only for n-hexatriacontane, we obtain the scaling constants reported in Table 1b. As observed, these constants are significantly different from the corresponding constants reported in Table 1a, which were obtained from liquid density data along with vapor pressure and heat of vaporization data. This is an important point because PVT data are used almost exclusively by the various equation-of-state models in order to obtain the scaling constants for high polymers. We will address this problem in more detail later in this series of papers. In spite of these drawbacks, the present framework can be used for calculations of a variety of thermodynamic properties, including interfacial properties of fluids of varying polarity, including molecules interacting with strong specific forces and of any size. Both of the above drawbacks point to the approximate character of our approach. Yet, alternative less approximate approaches, such as modern computer simulations or semi ab initio calculations, are still limited in scope from the perspective of a chemical engineer seeking realistic calculations for his/her complex systems. Acknowledgment Financial support for this work has been provided by the Greek GSRT/PENED 2001.

Ind. Eng. Chem. Res., Vol. 43, No. 20, 2004 6605 Table 5. Scaling Constants of Pure Fluids When Equation A-6 Is Adopted fluid

* ) RT*/J‚mol-1

v* ) *P*-1/cm3‚mol-1

vsp* ) F*-1/cm3‚g-1

s ) q/r

n-hexane toluene CCl4 polyethylene (linear) polyethylene (branched) polyisobutylene polystyrene poly(dimethylsiloxane) poly(vinyl chloride) poly(acrylonitrile) poly(methyl methacrylate)

4587 5155 5049 6472 6703 7275 6199 5396 6519 8991 7206

12.240 9.900 11.49 9.879 9.859 8.674 7.998 10.709 5.807 5.828 8.085

1.170 1.053 0.556 1.150 - 0.000181P 1.160 - 0.000214P 1.081 - 0.000202P 0.919 - 0.000123P 0.980 - 0.000245P 0.687 - 0.000080P 0.844 - 0.000087P 0.815 - 0.000108P

0.857 0.757 0.858 0.800 0.800 0.839 0.667 0.744 0.780 0.887 0.843

By combining eqs A-5 and A-6, we obtain

Appendix: An Alternative Expression for Nonrandomness In this appendix, we will present the working equations of the model by adopting an alternative expression for the nonrandom factors. Recently, Yan et al.,53 on the basis of computer (Monte Carlo) simulation results, have proposed the following linear expression for the nonrandom factors in a binary mixture of simple monosegmental hard-sphere fluids:

2N11 x1 ) f12 N12 x2

and

2N22 x2 ) f21 N12 x1

(A-1)

Γ12 )

1 1 - θ1θ2(1 - g)

(A-7)

Γ11 and Γ22 are then obtained by substituting eq A-7 in either eq A-5 or eq A-6. The working equations are now obtained from eq 6 of the main text but with ΩNR ) 1. When θ1 ) θr and θ2 ) θ0, these equations are as follows:

Equation of state: l ˜ ln(1 - F˜ ) - F˜ P ˜ + θrΓr0(Γrr + Γ00 - 1) + T r z q ln 1 - F˜ + F˜ ) 0 (A-8) 2 r

[

where

f12 ) x1 + x2g

and

f21 + x2 + x1g

(A-2)

)]

(

Chemical potential:

and

(

)

11 + 22 - 212 g ) exp kT

(A-3)

We could adapt the above assumption to our case of a binary mixture of multisegmental molecules 1 and 2 as follows: We may write for the numbers of intersegmental interactions (by combining equations analogous to eq 17 of the main text)

z N22 ) q2N2θ2Γ22 2 (A-4)

which lead to the following conservation equations (analogous to eq 18 of the main text):

θ1Γ11 + θ2Γ12 ) 1 θ2Γ22 + θ1Γ12 ) 1

(

) [

]

In Table 5 are reported the scaling constants of some representative fluids when the formalism of this appendix is adopted. Literature Cited

z N11 ) q1N1θ1Γ11 2

N12 ) zq1N1θ2Γ12 ) zq2N2θ1Γ12

µ ) -ln δr - l + ln F + r (v˜ - 1) ln(1 - F˜ ) RT qθrΓrr q q z P ˜ v˜ r v˜ - 1 + ln 1 - F˜ + F˜ +r (A-9) 2 r r T ˜ T ˜

(A-5)

With these definitions, we could generalize eqs A-1 and A-2 as follows:

θ1 θ1 2N11 θ1Γ11 ) ) f12 ) (θ1 + θ2g) or N12 θ2Γ12 θ2 q2 Γ22 Γ11 ) θ1 + θ2g and ) θ2 + θ1g (A-6) Γ12 Γ12

(1) Patterson, D.; Delmas, G.; Somcynsky, T. A Comparison of Lower Critical Solution Temperatures of Some Polymer Solutions. Polymer 1967, 8, 503. (2) Flory, P. J. Fifteenth Spiers Memorial Lecture. Thermodynamics of Polymer Solutions. Discuss. Faraday Soc. 1970, 49, 7. (3) Sanchez, I. C.; Lacombe, R. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145. (4) Donohue, M. D.; Prausnitz, J. M. Perturbed Hard Chain Theory for Fluid Mixtures: Thermodynamic Properties for Mixtures in Natural Gas and Petroleum Technology. AIChE J. 1978, 24, 849. (5) Kleintjens, L. A.; Koningsveld, R. Liquid-Liquid-Phase Separation in Multicomponent Polymer Solutions. XIX. MeanField Lattice-Gas Treatment of the System n-Alkane/Linear Polyethylene. Colloid Polym. Sci. 1980, 258, 711. (6) Panayiotou, C.; Vera, J. H. Statistical Thermodynamics of r-mer Fluids and their Mixtures. Polym. J. 1982, 14, 681. (7) Jackson, G.; Chapman, W. G.; Gubbins, K. E. Phase Equilibria of Associating Fluids. Spherical Molecules with Multiple Bonding Sites. Mol. Phys. 1988, 65, 1. (8) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (9) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. GroupContribution Estimation of Activity Coefficients in Non-Ideal Liquid Mixtures. AIChE J. 1975, 21, 1086.

6606


(10) Kojima, K.; Tochigi, K. Prediction of Vapor-Liquid Equilibria by the ASOG Method; Elsevier: Amsterdam, The Netherlands, 1979. (11) Gmehling, J.; Li, J. D.; Schiller, M. A Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178. (12) Lee, B.-C.; Danner, R. P. Prediction of Polymer-Solvent Phase Equilibria by a Modified Group-Contribution EoS. AIChE J. 1996, 42, 837. (13) Klamt, A.; Eckert, F. COSMO-RS: a Novel and Efficient Method for the a Priori Prediction of Thermophysical Data of Liquids. Fluid Phase Equilib. 2000, 172, 43. (14) Panayiotou, C.; Vera, J. H. The Quasi-Chemical Approach for Non-Randomness in Liquid Mixtures. Expressions for Local Surfaces and Local Compositions with an Application to Polymer Solutions. Fluid Phase Equilib. 1980, 5, 55. (15) Panayiotou, C.; Vera, J. H. Local Composition and Local Surface Area Fractions: a Theoretical Discussion. Can. J. Chem. Eng. 1981, 59, 501. (16) Guggenheim, E. A. Mixtures; Oxford University Press: Oxford, U.K., 1952. (17) Veytsman, B. A. Are Lattice Models Valid for Fluids with Hydrogen Bonds? J. Phys. Chem. 1990, 94, 8499. (18) Panayiotou, C.; Sanchez, I. C. Hydrogen Bonding in Fluids: an Equation-of-State Approach. J. Phys. Chem. 1991, 95, 10090. (19) Sanchez, I. C.; Panayiotou, C. Equations of State Thermodynamics of Polymer and Related Solutions. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S., Ed.; Marcel Dekker Inc.: New York, 1994. (20) Vlachou, Th.; Prinos, J.; Vera, J. H.; Panayiotou, C. Nonrandom Distribution of Free Volume in Fluids and Their Mixtures: Hydrogen-Bonded Systems. Ind. Eng. Chem. Res. 2002, 41, 1057. (21) Panayiotou, C. The QCHB Model of Fluids and their Mixtures. J. Chem. Thermodyn. 2003, 35, 349. (22) Panayiotou, C. Equation-of-State Models and Quantum Mechanics Calculations. Ind. Eng. Chem. Res. 2003, 42, 1495. (23) Staverman, A. J. The Entropy of High Polymer Solutions. Generalization of Formulae. Recl. Trav. Chim. Pays-Bas 1950, 69, 163. (24) Klamt, A.; Jonas, V.; Burger, Th.; Lohrenz, J. C. W. Refinement and Parameterization of COSMO-RS. J. Phys. Chem. 1998, 102, 5074. (25) Klamt, A. Comments on “A Priori Phase Equilibrium Prediction from a Segment Contribution Solvation Model”. Ind. Eng. Chem. Res. 2002, 41, 2330. (26) Klamt, A.; Krooshof, G. J. P.; Taylor, R. COSMOSPACE: Alternative to Conventional Activity-Coefficient Models. AIChE J. 2002, 48, 2332. (27) Lin, S.-T.; Sandler, S. I. Reply to Comments on “A Priori Phase Equilibrium Prediction from a Segment Contribution Solvation Model”. Ind. Eng. Chem. Res. 2002, 41, 899. (28) Rane, S. S.; Gujrati, P. D.Characterization of the EndGroup/Middle-Group Free Volume Distribution in a Polymer Melt. Polym. Mater. Sci. Eng. 2002, 87, 157. (29) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, U.K., 1982. (30) Gibbs, J. W. Collected Works; Longmans Green: New York, 1906; Vol. I. (31) Sprow, F. B.; Prausnitz, J. M. Surface Thermodynamics of Liquid Mixtures. Can. J. Chem. Eng. 1967, 45, 25 (32) Rusanov, A. I. Problem of Surface Thermodynamics. Pure Appl. Chem. 1992, 64, 111. (33) Hill, T. L. Statistical Thermodynamics of the Transition Region between Two Phases. I. Thermodynamics and QuasiThermodynamics. J. Chem. Phys. 1952, 56, 526. (34) Toxvaerd, S. Perturbation Theory for Nonuniform Fluids: Surface Tension. J. Chem. Phys. 1971, 55, 3116. (35) Haile, J. M.; Gray, C. G.; Gubbins, K. E. Theory of Surface Tension for Molecular Liquids. II. Perturbation Theory Calculations. J. Chem. Phys. 1976, 64, 2569. (36) van der Waals, J. D. Thermodynamische Theorie der Kapillarita¨t unter Voraussetzung Staetiger Dichteanderung. Z. Phys. Chem. 1894, 657. (37) Cahn, J. W.; Hilliard, J. E. Free Energy of a Nonuniform System I, Interfacial Free Energy. J. Chem. Phys. 1958, 30, 1121.

(38) Widom, B. Surface Tension and Molecular Correlations Near the Critical Point. J. Chem. Phys. 1965, 43, 3892. (39) Yang, A. J. M.; Fleming, P. D., III; Gibbs, J. H. Molecular Theory of Surface Tension. J. Chem. Phys. 1976, 64, 3732 (40) Bongiorno, V.; Scriven, L. E.; Davis, H. T. Molecular Theory of Fluid Interfaces. J. Colloid Interface Sci. 1976, 57, 462. (41) Evans, R. Nature of the Liquid-Vapor Interface and other topics in the Statistical-Mechanics of Nonuniform, Classical Fluids. Adv. Phys. 1979, 28, 143. (42) Poser, C. I.; Sanchez, I. C. Interfacial Tension Theory of Low and High Molecular Weight Liquid Mixtures. Macromolecules 1981, 14, 361. (43) Panayiotou, C. Interfacial Tension and Interfacial Profiles of Fluids and Their Mixtures. Langmuir 2002, 18, 8841. (44) Panayiotou, C. Interfacial Tension and Interfacial Profiles: an Equation-of-State Approach. J. Colloid Interface Sci. 2003, 267, 418. (45) McHugh, M. A.; Krukonis, V. J. In Encyclopedia of Polymer Science and Technology; Mark, H. F., Bikales, N. M., Overberger, C. G., Menges, G., Eds.; Wiley-Interscience: New York, 1989. (46) Kazarian G. S. Polymer Processing with Supercritical Fluids. Polym. Sci., Ser. C 2000, 42, 78. (47) Kiran, E.; Saraf, V. P.; Sen, Y. L. Solubility of Polymers in Supercritical Fluids. Int. J. Thermophys. 1989, 10, 437. (48) Zhang, Z.; Handa, Y. P. An in Situ Study of Plasticization of Polymers by High-Pressure Gases. J. Polym. Sci., Polym. Phys. 1998, 36, 977. (49) Stafford, C. M.; Russell, T. P.; McCarthy, T. J. Expansion of Polystyrene Using Supercritical Carbon Dioxide: Effects of Molecular Weight, Polydispersity, and Low Molecular Weight Components. Macromolecules 1999, 32, 7610. (50) Kikic, I.; Vecchione, F.; Alessi, P.; Cortesi, A.; Eva, F.; Elvassore, N. Polymer Plasticization Using Supercritical Carbon Dioxide: Experiment and Modeling. Ind. Eng. Chem. Res. 2003, 42, 3022. (51) Daubert, T. E.; Danner, R. P. Data Compilation Tables of Properties of Pure Compounds; AIChE Symposium Series 203; American Institute of Chemical Engineers: New York, 1985. (52) Fredeslund, A.; Sorensen M. J. Group Contribution Estimation Methods. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S., Ed.; Marcel Dekker Inc.: New York, 1994. (53) Yan, Q.; Liu, H.; Hu, Y. Analytical Expressions of Helmholtz Function of Mixing for Ising Model. Fluid Phase Equilib. 2004, 218, 157. (54) Panayiotou, C. Glass-Transition Temperatures in Polymer Mixtures. Polym. J. 1986, 18, 895. (55) Condo, P. D.; Sanchez, I. C.; Panayiotou, C.; Johnston, K. P. Glass Transition Behavior Including Retrograde Vitrification of Polymers with Compressed Fluid Diluents Macromolecules 1992, 25, 6119. (56) Prinos, J.; Panayiotou, C. Glass-Transition Temperatures in Hydrogen-Bonded Polymer Mixtures. Polymer 1995, 36, 1223. (57) Gibbs, J. H.; DiMarzio, E. A. Nature of the Glass Transition and the Glassy State. J. Chem. Phys. 1958, 28, 373. (58) Flory, P. J. Statistical Thermodynamics of Semi-Flexible Chain Molecules. Proc. R. Soc. London, Ser. A 1956, 234, 60. (59) Perry, R.; Green, D. In Chemical Engineers’ Handbook; CD Ed.; McGraw-Hill: New York, 1999. (60) Lemmon, E. W.; McLinden, M. O. NIST Standard Reference Database 23: NIST Reference Fluid Thermodynamic and Transport Properties, version 7.0; NIST: Gaithersburg, MD, 2001. (61) Saul, A.; Wagner, W. A Fundamental Equation for Water Covering the Range from the Melting Line to 1273 K at Pressures up to 25000 MPa. J. Phys. Chem. Ref. Data 1989, 18, 1537. (62) Zoller, P.; Walsh, D. PVT Data for Polymers; Technomic Publishing Co.: Lancaster, PA, 1995. (63) Wu, S. Interfacial and Surface Tension of Polymers. J. Macromol. Sci. 1974, C10, 1. (64) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.

Received for review April 9, 2004 Revised manuscript received June 28, 2004 Accepted July 2, 2004 IE040114+

Nonrandom Hydrogen-Bonding Model of Fluids and Their Mixtures. 1

Recommend Documents