Study of the Critical Behavior of Polar Fluids by Renormalization

Honeywell Hi-Spec Solutions, 343 Dundas Street, London, Ontario N6B 1V5, Canada. The critical behavior of polar fluids is studied by a renormalization...
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Ind. Eng. Chem. Res. 2002, 41, 297-302

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GENERAL RESEARCH Study of the Critical Behavior of Polar Fluids by Renormalization Group Theory Li-Ping Duan, Jiu-Fang Lu, Jian Chen, and Yi-Gui Li* Department of Chemical Engineering, Tsinghua University, Beijing 100084, P.R. China

Yi-Ping Tang Honeywell Hi-Spec Solutions, 343 Dundas Street, London, Ontario N6B 1V5, Canada

The critical behavior of polar fluids is studied by a renormalization group (RG) theory in which density fluctuations are taken into account. The Stockmayer potential is adopted to express the interactions between polar molecules. The molecular parameters regressed from the critical temperature are used to predict the behaviors of water and methanol at supercritical and nearcritical temperatures. The calculated pressures are compared with the experimental data. The results show that RG theory combined with the Stockmayer potential is suitable for predicting the thermodynamic properties of water and methanol near and far from the critical region. 1. Introduction The fluctuations of a fluid near its critical region are intense, and one result of these fluctuations is opalescence. The critical behavior of fluids belongs to secondorder phase transition. Near the critical point, some thermodynamic properties, such as the heat capacity, the density difference between vapor and liquid phases, and the correlation length follow power laws1

A CV ) (T - TC)-R R FL - FV ) (B(TC - T)β

(1)

ξ ) ξ0(T - TC)-γ where CV is the heat capacity; T is the absolute temperature; F is the density; ξ is the correlation length, which measures the correlative length of the fluctuations; the subscript C means critical; superscripts L and V indicate liquid and vapor, respectively; R, β, and γ are critical exponents; and A, B, and ξ0 are constants. The exponents above are universal, which means that they have the same values for different systems, and their experimental values are R ≈ 0.1, β ≈ 1/3, and γ ≈ 1.33. Because the critical exponents are all positive, near the critical point, the heat capacity CV diverges strongly, the fluctuations in density are strong, and the correlation length is infinite. These phenomena show that the critical point is a singularity. The occurrence of density fluctuations is the key point around which to build an equation of state for a fluid near the critical point. The density fluctuation is an assembly of waves with different amplitudes and wave* Corresponding author. Tel.: 8610-62784540. Fax: 861062770304. E-mail: [email protected].

lengths. The power law can be deduced from mean field theory (MFT), but the exponents calculated from MFT differ from the experimental values. A correct theoretical equation of state near the critical region should have a singular term that cannot be expressed in analytic form. Because of the long-term aspect of the density fluctuations, the singular term of the theoretical equation of state is a generalized homogeneous function that always satisfies the scaling law. These ideas are the main bases of renormalization group theory. Renormalization group (RG) theory is a theory appropriate for the treatment of the thermodynamic behavior of fluids around their critical points. For Ising and lattice-gas models, RG theory gives nonclassical exponents2,3 that are in good agreement with experimental results. The original RG theory is only applicable very close to the critical point and cannot be implemented globally. Meanwhile, the particles studied in the original RG theory are fixed to particular lattice sites, which is contrary to real molecules of a fluid. White et al. developed a RG theory that it is capable of predicting the thermodynamic properties of real fluids globally, including both at the critical point and away from the critical point.4-8 They applied the new RG theory to predict the thermodynamic properties of n-pentane, square-well fluids, and Lennard-Jones fluids. Tang combined RG theory with the mean spherical approximation (MSA) to describe the thermodynamic behavior of a Lennard-Jones fluid outside and inside the critical region.9 Prausnitz et al. developed an equation of state for fluids close to and far from the critical point by combining liquid-state theory with the RG method and applied this approach successfully to square-well fluids and more realistic fluids including simple-molecule fluids and chain-molecule fluids.10,11 Polar fluids are often used in industry, and among them, water is the most important fluid in the world. It is well-known that water is an associating liquid with

10.1021/ie0103097 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/21/2001

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highly effective hydrogen bonding under ambient conditions, but supercritical water above 647 K and 22.05 MPa shows much less effective hydrogen bonding, as reflected by its lower dielectric constant and dissociation constant.12,13 In this paper, we apply White’s RG method to fluids interacting via a Stockmayer potential function to calculate the thermodynamic properties of water and methanol around the critical point. 2. Renormalization Group Theory According to White’s work, the free energy density is composed of two parts, the repulsive and the attractive contributions -βVf(T,F)

e

)e

-βVfig

∑ exp{-βUrepul[F(r)] -

[F(r)]

βUattra[F(r)]} (2) In the above equation, f is the total Helmholtz free energy density, which is a functional of the density distribution function [F(r)] represented by a simple sinusoidal function. The other parameters in eq 2 are as follows: β ) 1/kBT, where kB is the Boltzmann constant; fig is the ideal-gas free energy density; V is the volume; Urepul[F(r)] is the repulsive interaction contribution to the energy, which is a function of the amplitude of the fluctuation; and Uattra[F(r)] is the attractive interaction contribution to the energy, which is a function of the wavelength as well as the amplitude. In White’s RG work, only the attractive term contributes to the singularity in f at the critical point. At sufficiently high temperatures, different distributions of density fluctuation have almost the same effect on e-βUattra[F(r)], and eq 2 can be rewritten according to Zwanzig’s work14 as

e-βVf(T,F) ) e-βVfig

∑ exp{-βVfrepul(T,[F(r)]) -

[F(r)]

β〈Uattra〉repul} (3) where frepul(T,F) is the free energy density in which only Urepul is taken into account. The summation in eq 3 is extended over all amplitudes of density fluctuations for all wavelengths. 〈Uattra〉repul is the ensemble average of Uattra[F(r)]. According to perturbation theory, if only the repulsive interaction term is taken as the reference, Uattra is the perturbation term in the potential energy, and 〈Uattra[F(r)]〉repul/V ) uattra, which can be expressed as

uattra(T,F) ) -F2a(T,F) )

F2 2

∫Ωdr u(r) grepul(T,F,r)

(4)

where u(r) is the attractive portion of the two-body potential function, grepul(T,F,r) is the radial distribution function (RDF) for the reference system, and the integral domain Ω e V. Perturbation theory can be expressed in increasing powers of 1/T, but it is too complicated and too difficult to evaluate. Here, the renormalization procedure is used to alternate the perturbation expansion. All wavelengths of density fluctuations make contributions to the RG transform results or to the free energy calculations in the RG transform. Different wavelengths have different effects on the repulsive and attractive

parts of the free energy. The repulsive part is more sensitive to the fluctuations of very short wavelengths, whereas the attractive part is more sensitive to longwavelength fluctuations. The wavelength λb is chosen as a criterion, such that the attractive interaction is mainly affected by fluctuations of wavelengths longer than λb and fluctuations of wavelengths shorter than λb are mainly calculated in the repulsive part of the free energy. The RG calculation begins with eq 3, which can be rewritten as

∑ exp{-β∫Ωdr fS(T,[FS(r)]) -

e-βVf(T,F) ) e-βVfig

[FS(r)]

βUS[T,FS(r)]} (5) where [FS(r)] is the portion of [F(r)] that remains when the density fluctuations with wavelengths λ < λs are omitted, and the summations are taken over the amplitudes of the fluctuations with λ > λs. The part of the density fluctuations with wavelengths λ < λs has a trivial effect on the attractive interaction, which is mainly affected by the longer wavelengths, so only the repulsive interaction using the average density or density fluctuation amplitude is taken into account in the repulsive interaction. This means that the integral of the free energy density fS(T,[FS(r)]) over the space Ω is a function of the amplitude of the density fluctuation but not of the wavelength, whereas US(T,[FS(r)]) introduced by the fluctuation of the intermolecular attractions is a function of both the fluctuation wavelength and amplitude. This also means that the summation in eq 5 is performed over the amplitude of all density fluctuations with wavelengths λ > λs. When the density is FS(r), fS(T,[FS(r)]) corresponds to frepul(T,[F(r)]), and US(T,[FS(r)]) corresponds to Uattra(T,[F(r)]), for specified density distribution FS(r). The summation of the density fluctuation in eq 5 is over [FS(r)], whereas in eq 2, the summation region is [F(r)]. If λs is sufficiently large and the attractive portion of the intermolecular potential is sufficiently slowly varying as a function of the density fluctuations, then no summations are needed in eq 5, and it becomes equivalent to eq 2. In general, for a more rapidly varying and shorter-ranged attractive potential, λs must be chosen smaller, such that FS(r) varies appreciably within the space Ω. At the beginning of the RG calculations, λs ) λb. In the case of US * 0, the macroscopic free energy is expressed by the renormalization group procedure, which is to remove successively the attraction force contributions from US and incorporate them into f(T,F), so that the attractive interactions are fully taken into account and the contribution of the density fluctuations with longer and longer wavelengths are absorbed in the repulsive part of free energy. After the RG calculation sequence is completed, US completely disappears, and f has its macroscopic form. Similarly to eq 4, US(T,[FS(r)]) can be expressed as

US(T,[FS(r)]) )

∫Ω′dr′ ∫Ωdr FS(r′) FS(r) ×

1 2

u(|r′ - r|) grepul(r,FS(r),|r′ - r|) (6)

where Ω′ is the subdomain, V′ ) ∫Ω′ dr′. Here, US(T,[FS(r)]) corresponds to the term Uattra in eq 2 in the case of F(r) ) Fs(r) and can be rewritten as an

Ind. Eng. Chem. Res., Vol. 41, No. 2, 2002 299

integral of the local potential energy density over V ) ∫Ω dr

US(T,[FS(r)]) ) where uS(T,FS) ) -FS

2a(T,F

∫

1 2

dr uS(T,FS) Ω

(7)

∫Ωdr u(r) grepul(T,FS,r)

(8)

Equation 8 has the same form as eq 4, but the local density FS(r) is used here instead of the average density F, and FS(r) can be expressed by the simple sinusoidal fluctuation Fs(r) ) F + x cos(k‚r), where x denotes the amplitude of the fluctuation of F and k is the wave vector. The general form of eq 8 is

∫Ωdr cos(k‚r) u(r) grepul(T,FS,r)

1 aλ(T,FS) ) 2

(9)

where λ is the wavelength, k ) |k| is the wavenumber, and λ ) 2π/k. The effects of the density fluctuations on the attractive interaction are taken into account recursively. The recursion begins with the shortest wavelengths, and the phase-space cell approximation is used to calculate the Helmholtz free energy for wavelength fluctuations longer than λs. The RG procedure is described in detail in White’s work.8,10 Here, we sketch it only briefly. At the initial step of renormalization, we have f0(T,F) ) frepul(T,F). After n renormalizations, the free energy density becomes

f(T,F) = fn(T,F) - F2a(T,F)

(10)

fn(T,F) ) fn-1(T,F) + δfn(T,F)

(11)

When n g1

where the increment of the free energy density δfn(T,F) at each renormalization step is

Qn,s(T,F) 1 ln βVn Qn,l(T,F)

(12)

In the above equation, Vn ) V1 × 23(n-1), where V1 ) (λ1/2)3 is the initial averaging volume, and Qn,l(T,F) and Qn,s(T,F) are the density fluctuation integrals for the long-range attraction and short-range attraction, respectively, over the amplitudes of the wave packets at wavelength λ ) λn ) 2n-1λ1, where λ1 ≈ 3σ ) λS and σ is the diameter of a molecule

Qn,i(T,F) )

∫0Fdx e-βV D n

n,i(T,F,x)

, i ) s, l

(13)

where F is the upper limit of the integral. Each Dn,i(T,F,x) is given by

Dn,i(T,F,x) )

ˆfn-1,i(T,F+x) + ˆfn-1,i(T,F-x) - ˆfn-1,i(T,F) 2 (14)

where x denotes the amplitude of the density fluctuation.

ˆfn-1,l(T,F) ) fn-1(T,F)

(15)

ˆfn-1,s(T,F) ) fn-1(T,F) - F2aλn(T,F)

(16)

When i ) s

S)

1 a(T,FS) ) 2

δfn(T,F) ) -

In eq 14, when i ) l

In our work, the dipole-dipole interaction potential is considered, and from perturbation theory, it is known that the average value of udd over the spatial orientation is15

〈udd〉 )

µiµj 4π0r

〈Dij〉 )

3

µiµj 4π0r3

〈3(µ bi‚r bij)(µ bj‚r bij) bj)〉 ) 0 (µ bi‚µ

Thus, the second-order perturbation term 〈(u(r)dd)2〉 is added to eq 9, and eq 9 is rewritten as

∫Ωdr cos(kn‚r) 〈u(r)〉 grepul(T,FS,r) β ∫ dr cos(kn‚r) 〈u2(r)〉 grepul(T,FS,r) (17) 4 Ω 1 2

aλn(T,FS) ) -

where 〈‚‚‚〉 represents the average value over the spatial orientation, |kn| ) kn, kn is the wavenumber, kn ) 2π/ λn, and λn is the wavelength for each recursion. From eqs 10-17, successive recursions to the Helmholtz free energy density fn(T,F) are obtained, which contain the contributions of longer wavelengths λn for the fluctuations of f. Because Vn ) (λ1/2)323(n-1), Vn will increase sharply with the increment of n. From eq 12, we can see that δfn(T,F) in eq 12 will apparently converge. 3. Potential Energy As described above, because the hydrogen bonding in water is weak under critical conditions, the dispersion energy and dipole-dipole interaction energy are taken to be the interaction energies between polar molecules. The Stockmayer potential is used; it is

u(r) ) 4

12

6

µ iµj

[(σr) - (σr) ] - (4π )r [3(µb ‚rb )(µb ‚rb ) 3

i

ij

j

ij

0

(µ bi‚µ bj)] (18) where the first term on the right-hand side is the dispersion term,  is the dispersion energy parameter, r is the intermolecular distance, and σ is the soft-sphere diameter. The second term is the dipole-dipole interaction term

〈(u(r)dd)2〉 )

2µi2µj2 1 (4π0)2 3r6

where 0 is the permittivity of free space, 4π0 ) 1.002 65 × 10-10 C2 N-1 m-2, µ is the dipole moment, and subscripts i and j represent molecules. According to Barker-Henderson perturbation theory,16 the hard-sphere diameter d can be calculated from Cotterman’s expression17

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1.0 + 0.2977TR d ) σ 1.0 + 0.3316T + 0.001 047 7T R

2

(19)

R

where TR ) kT/ is the reduced temperature. The repulsive part of the free energy density is expressed by the Carnahan-Starling equation

βfrepul 4y - 3y2 ) F (1 - y)2

(20)

where y ) (1/6)πFd3. The compressibility, Z ) pV/RT, for a hard-sphere system is expressed as

Zrepul ) F

( )

∂ βfrepul 1 + y + y2 - y3 ) ∂F F (1 - y)3

(21)

In the calculation of aλn, the attractive part of potential in eq 17 is u(r) when the intermolecular distance r g d′(TC), and it is set to be zero when r < d′(TC), where d′(TC) is approximately midway between σ and d(TC), i.e., d′(TC) ) σ + d(TC)/2, and d(TC) is the d evaluated at the critical temperature TC.

Figure 1. Pressure isotherms of water at T gTC (symbols, experimental data19 for 9, 648 K; 0, 698 K; 2, 773 K; 1, 973 K; solid lines, calculated results).

4. Calculations The reduced parameters adopted in the calculation are

FR )

F p , pR ) FC pC

(22)

The free energy density in the region 0 < FR < 4 was calculated. We divided the density domain into many small sections ∆FR and then calculated the corresponding free energy density of each section. The section size was ∆FR ) 0.001 for 0 e FR e 0.1 and ∆FR ) 0.01 for 0.1 < FR e 4. The initial free energy density f0(T,F) is given by

f0(T,F) ) frepul(T,F)

(23)

For iteration numbers n g1, fn(T,F) has no analytical expression and is obtained from the free energy density of the (n - 1)st iteration according to eqs 10-17. At each recursion, after all of the discrete values of fn(T,F) corresponding to each small density section in the region 0 < FR < 4 were obtained, a cubic spline program was used to build a smooth function, which was needed in the next recursion from these discrete free energy densities The integral in eq 13 was performed numerically by using a simple trapezoid method. The term grepul(T,F,r) in eq 8 was obtained from data in the literature.18,19 This numerical method was used to calculate the compressibility factor Z from the equation

Z)F

∂ βf ∂F F T,P

( )

Figure 2. Pressure isotherm of water at T ) 623 K (O, experimental data;19 solid line, calculated results).

(24)

The experimental critical-point data for water20 are TC ) 647.3 K, pC ) 22.1 MPa, and VC ) 56 cm3/mol and those for methanol21 are TC ) 512.2 K, pC ) 8.096 MPa, and VC ) 117.8 cm3/mol. The molecular parameters σ, /k, and µ of the polar fluids were regressed from the experimental pVT data at TC. Then, these parameters were used to predict p,

V, and T at other temperatures. Because of the absence of data on precise critical temperatures, T ) 648.3 K for water and T ) 520 K for methanol were used here. It was found that, near the critical region, the calculated curve of pR versus FR is very sensitive to the values of the parameters. The parameter values obtained are σ ) 2.841 × 10-10 m, /k ) 321.2 K, and µ ) 6.404 × 10-30 C m for water and σ ) 3.73 × 10-10 m, /k ) 401.2 K, and µ ) 6.838 × 10-30 C m for methanol. With these sets of parameters, the predicted curves of pR versus FR in the supercritical region for water are shown in Figure 1, and the curve of pR versus FR at the subcritical temperature T ) 623.3 K for water is shown in Figure 2; the curves of pR versus FR for methanol are shown in Figure 3. The average relative deviation (ARD) is 4.2% for water at 648.3 K and 4.8% for methanol at 520 K. It can be seen from Figure 3 that the pR versus FR curve is a wavy line at T ) 520 K, which means that, in our calculations, the critical temperature is somewhat higher than the true value for methanol. At a certain temperature, the value of δfn(T,F) decreases rapidly with increasing recursion number, and δfn(T,F) ≈ 0 when n g5. At different temperatures, δfn(T,F) has different values. Figures 4 and 5 show the successive renormalization contributions to the pressure

Ind. Eng. Chem. Res., Vol. 41, No. 2, 2002 301

Figure 3. Pressure isotherms of methanol at 500-570K (symbols, experimental data21 for 1, 500 K; 9, 510 K; 0, 520 K; 2, 570 K; solid lines, calculated results).

The pVT properties for near-critical and supercritical water and methanol have also been described by other authors. Smits et al.22 used associated perturbed anisotropic chain theory (APACT) to calculate the pVT properties of water at 473.15-1273.15 K and 2.525 000 MPa and compared the results with those obtained from the Peng-Robinson (PR) EOS. The ARD of the APACT EOS was 2.1748% and that of the PR EOS was 26.4435% for water at 648.15 K and 2.5-100 MPa. The accuracy of the APACT EOS is better, but more parameters are required for it than for RG method. Kiselev et al.23 adopted classical and crossover SAFT EOSs to calculate the pVT properties of methanol in the near-critical and supercritical regions. They found that the ARD of the classical SAFT EOS was very large and that the ARD of the crossover SAFT is 1-2% in the supercritical and liquid regions, but the low-density area was not taken into account at the same time. Thus, it is clear that the RG method used in this work for polar fluids can improve the results in the critical region compared with other equations of state. 5. Conclusions White’s RG method with the Stockmayer potential was applied to calculate the supercritical behavior of two polar fluids in our work. The molecular parameters regressed from the critical temperature were used to predict the behaviors of water and methanol at supercritical and near-critical temperatures. The results are satisfactory for describing the properties of water and methanol under supercritical region and subcritical conditions. Acknowledgment

Figure 4. Pressure isotherms for water at T ) 648 K at orders n ) 0-5 (dashed line, n ) 0; dotted line, n ) 1; dash-dotted line, n ) 2; dash-dot-dotted line, n ) 3; solid line, n ) 5).

The authors appreciate the financial support to this research by the National Nature Science Foundation of China (No. 20006008). Nomenclature

Figure 5. Pressure isotherms for water at T ) 773 K at orders n ) 0-5 (dashed line, n ) 0; dotted line, n ) 1; dash-dotted line, n ) 2; dash-dot-dotted line, n ) 3; solid line, n ) 5).

at different temperatures for water. It is obvious that the density fluctuations become less important at T ) 773 K than at T ) 648 K for water. Equation 1 shows that the correlation length, ξ, decreases when the temperature increases in the region of T > TC, as demontrated by Figures 4 and 5.

A ) coefficient of CV B ) coefficient of density CV ) heat capacity, J/(mol K) d ) hard-sphere diameter of a molecule, m f(T,F) ) Helmhotz free energy density, J mol/m3 g ) radial distribution function grepul ) radial distribution function for hard spheres k ) wave vector k ) wavenumber kB ) Boltzmann constant n ) recursion number p ) pressure, Pa pC ) critical pressure, Pa pR ) reduced pressure Q ) integral in eq 12 r ) intermolecular distance, m R ) universal gas constant T ) temperature, K TC ) critical temperature, K TR ) reduced temperature u ) interaction potential V ) system molar volume, m3/mol VC ) critical molar volume, m3/mol x ) amplitude of density fluctuations y ) packing factor Z ) compressibility factor

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Greek Letters R ) universal critical exponent β ) universal critical exponent β ) 1/kBT γ ) universal critical exponent /k ) dispersion energy parameter of molecule, K λ ) wavelength, m µ ) dipole moment, C m ξ ) correlation length ξ0 ) coefficient for ξ FL ) density of the liquid phase, mol/m3 FV ) density of the vapor phase, mol/m3 FR ) reduced density σ ) soft-sphere diameter of a molecule, m Ω ) density domain Subscripts C ) critical i, j ) molecules l ) long wavelength n ) iteration number R ) reduced value s ) short wavelength Superscripts attra ) attractive term cal ) calculated exp ) experimental repul ) repulsive term

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Received for review April 9, 2001 Revised manuscript received October 29, 2001 Accepted November 1, 2001 IE0103097