Surface Tension of Binary Mixtures Including Polar

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employed to predict the interfacial tension for various quadrupolar (CO2 and .... components, namely nitrogen and carbon dioxide, in various binary mixtures ...
Surface Tension of Binary Mixtures Including Polar Components Modeled by the Density Gradient Theory Combined with the PC-SAFT Equation of State1 Václav Vinš2, Barbora Planková, Jan Hrubý Institute of Thermomechanics AS CR, v. v. i., Dolejškova 1402/5, 182 00 Prague 8, Czech Republic

Abstract In this study, we use the Cahn-Hilliard density Gradient Theory (GT) for predicting the surface tension of various binary mixtures at relatively wide temperature ranges and test the application of the GT for predictions of homogeneous nucleation. The GT was combined with two physically based equations of state (EoS), namely the Perturbed-Chain (PC) Statistical Associating Fluid Theory (SAFT) and its modification for polar substances the PerturbedChain Polar (PCP) SAFT. The GT applied to the planar phase interface was employed to predict the interfacial tension for various quadrupolar (CO2 and benzene) and dipolar (difluoromethane, i.e. R32; pentafluoroethane, i.e. R125; and 1,1,1,2-tetrafluoroethane, i.e. R134a) substances and for five binary mixtures including polar components (n-decane + CO2, benzene + CO2, R32 + R125, R32 + R134a, R134a + R125). The PCP-SAFT EoS combined with the GT provides more accurate results for both the quadrupolar and dipolar substances than the original PC-SAFT EoS. Besides the planar phase interface, the GT was also applied to the spherical phase interface simulating a critical cluster occurring in homogeneous nucleation of droplets. Carbon dioxide was considered, because it has a relatively high quadrupole moment and because of its relevance to natural gas processing. Application of the PCP-SAFT EoS provides a significant improvement compared to the PC-SAFT EoS, and it is clearly superior to the

1

Paper presented at the 19th European Conference on Thermophysical Properties, August 28 –

September 1, 2011, Thessaloniki, Greece 2

Corresponding author, email: [email protected], telephone: +420 266 053 152, fax: + 420

286 584 695

1

classical cubic Peng-Robinson EoS, which is still used for modeling droplet nucleation.

Keywords Density gradient theory, Droplet nucleation, Interfacial tension, PCP-SAFT, Polar substances

2

List of Symbols A

Helmholtz free energy, J

c

influence parameter, J·m5·mol-2

f

general function

J

nucleation rate, m-3·s-1

K

kinetic prefactor, m-3·s-1

kB

Boltzmann constant, J·K-1

kij

binary interaction parameter

N

number of data points

p

pressure, Pa

r

radial coordinate, m

rs

radius of the surface of tension, m

Q

quadrupole moment, DÅ (≈ 3.33564·10-40 C·m2)

s

supersaturation

T

temperature, K

w

liquid mass fraction

x

liquid mole fraction

z

coordinate perpendicular to phase interface, m

Greek letters β

correcting parameter

δ

deviation

ΔΩ

work of formation, J

ε

energy parameter, J

μ

dipole moment, D (≈ 3.33564·10-30 C·m) chemical potential, J·mol-1

ρ

density, mol·m-3

σ

surface tension, N·m-1 segment diameter, Å (10-10 m)

ω

grand potential density, J·m-3

Subscripts cal

calculated data

crit

critical 3

D

dipolar

disp

dispersion

exp

experimental data

G

gas phase

hc

hard chain

id

ideal gas

L

liquid phase

Q

quadrupolar

sat

saturated

Superscripts 0

homogenous average value

1 Introduction Interfacial properties of fluids play an important role in many technical processes such as flow through porous media, distillation, nucleation, and two-phase mass and heat transfer. However, the surface tension of many mixtures is still described only partly, and the experimental data are scarce. Also, the surface tension of a strongly curved surface, needed in nucleation theory, is very difficult to access experimentally. The Cahn-Hilliard [1-3] density Gradient Theory (GT) in combination with an appropriate equation of state (EoS) allows a prediction of interfacial tension both for pure components and mixtures. The GT has mostly been used with a cubic EoS – the van der Waals EoS [4] or the Peng-Robinson EoS [5]. However the cubic EoSs have severe limitations in the accuracy of the predicted thermophysical properties. Consequently, results of the GT were rather questionable, in particular for complex mixtures containing polar and associating substances. During the last twenty years, a new family of the EoSs, based on the Statistical Associating Fluid Theory [6,7] (SAFT), has been successfully used for modeling the vapor-liquid equilibria (VLE) of various systems. Several researchers combined the modifications of the SAFT EoS (SAFT-VR [8], softSAFT [9], PC-SAFT [10,11]) with the GT to predict the temperature and 4

composition dependence of the surface tension. Kahl and Enders [12] demonstrated predicting ability of the SAFT EoS implemented into the GT by comparing its results with the Peng-Robinson EoS. The GT + SAFT successfully predicted the surface tension for methanol, i.e. a substance with strong hydrogen bonding. In their following study, Kahl and Enders [13] verified that the GT + SAFT EoS can also be used for modeling interfacial tension for binary mixtures containing associating components. Gloor et al. [14] achieved relatively good results with the GT combined with the SAFT-VR EoS for pure n-alkanes, alkan1-ols, water, and several refrigerants. Fu et al. predicted surface tension for binary mixtures of methane + n-alkanes [15] and CO2 + hydrocarbons [16] by using the PC-SAFT EoS implemented into the GT. Dias et al. [17] modeled interfacial and second order properties for pure perfluoroalkanes using crossover soft-SAFT. Amézquita et al. [18] predicted the surface tension of two quadrupolar components, namely nitrogen and carbon dioxide, in various binary mixtures with non-polar substances using the GT combined with the Perturbed-Chain Polar (PCP) SAFT EoS. In this study, we focus on the interfacial properties of both the quadrupolar substances, namely carbon dioxide and benzene, and the selected hydrofluorocarbon (HFC) refrigerants (R) representing dipolar components. The GT was combined with the original PC-SAFT EoS [10,11] and with its modification for polar substances, the PCP-SAFT EoS, developed by Gross and Vrabec [19-21]. The surface tension of pure polar components and binary mixtures was calculated from the planar phase interface. In the second part of our study, the GT was applied to the spherical phase interface of pure CO2. The critical cluster, playing a crucial role in the process of homogeneous droplet nucleation, was modeled by the GT combined both with the PC-SAFT EoS, neglecting the polarity effect, and the PCP-SAFT EoS containing the polar term in the Helmholtz free energy expression. In general, consideration of the polarity effect in the EoS resulted in a better description of both the surface tension and droplet nucleation.

2 PCP-SAFT Equation of State The PC-SAFT EoS, in particular its polar modification, the PCP-SAFT EoS, was chosen for our calculations as it provides a relatively accurate prediction of VLE 5

for a large variety of substances [22-24]. The Helmholtz free energy A can be expressed as follows in the PCP-SAFT EoS, A  Aid  Ahc  Adisp  Apolar .

(1)

In Eq. 1, Apolar denotes the free energy contribution due to the polarity. It is given as a sum of the dipolar, quadrupolar, and the combined dipolar-quadrupolar effects [21]: Apolar  AD  AQ  ADQ .

(2)

Further details about the PCP-SAFT EoS can be found in the studies by Gross and Vrabec [19-21] and in the original PC-SAFT papers by Gross and Sadowski [10,11]. The non-polar components are defined with a set of three PC-SAFT parameters, namely the segment number m, the segment diameter σ, and the energy parameter ε. Values of these constants are usually found by correlating the saturated liquid density and the vapor pressure for a pure substance [11]. In the PCP-SAFT EoS, the polar components are described by a fourth parameter: the dipole moment μ for a dipolar component and the quadrupole moment Q for a quadrupolar component. An advantage of the PCP-SAFT EoS, compared to other SAFT EoSs applicable to polar substances, is that the common literature values for both polar moments μ, Q can be used. Table 1 summarizes the pure component parameters m, σ, and ε for the PC-SAFT and PCP-SAFT EoSs used in our study. Besides CO2 and benzene representing quadrupolar components, following HFC refrigerants, i.e. dipolar components, were considered in our study: difluoromethane (R32), pentafluoroethane (R125), and 1,1,1,2-tetrafluoroethane (R134a). Fairly large set of surface tension experimental data was found for all selected substances which allowed us to test predictive ability of the GT + PCP-SAFT EoS. The PCP-SAFT parameters have nonzero polar moments μ or Q. To describe mixtures, the PCP-SAFT EoS uses the Berthelot-Lorentz combining rules, defining the interactions between a pair of unlike segments  ij 

 ii   jj 2

,  ij  1  kij   ii  jj .

(3)

In Eq. 3, kij denotes the binary interaction parameter. Its value has to be correlated to the VLE experimental data available for a given binary mixture. As in a previous study [24], the Levenberg-Marquardt algorithm was employed to 6

evaluate the kij parameter. Values of kij for the selected mixtures together with the reference to the correlated VLE data and their temperature ranges are listed in Table 2. The average values of kij used in our calculations for the binary mixture R125 + R32 are 0.00140 and -0.00847 for the PC-SAFT EoS and the PCP-SAFT EoS, respectively. The binary system benzene + CO2 was modeled with the kij values taken from Gross [19], i.e. 0.088 for the PC-SAFT EoS and 0.042 for the PCP-SAFT EoS.

3 Surface Tension Predicted by the GT In the Cahn-Hilliard density gradient theory [1-4], the interfacial density profile must satisfy the following Euler-Lagrange equation c jk

1    c    2   ij

j

j ,k

j

 j  k 

i

 , i

(4)

where cij  cij  T , 1 ,  2 , denotes the influence parameter, describing the dependence of the density gradients on local deviations of the chemical potentials, and ω marks the grand-potential density defined as follows  d A0   A0      i i0 .  i  di T ,V , 

     A0       i  i

(5)

j

Neglecting the density-dependence of the influence parameter, Eq. 4 can be simplified in the following way for a planar phase interface between two bulk phases [29]

c

ij

j

d2  j dz 2

     0 .

(6)

In Eq. 6, z denotes direction perpendicular to the phase interface. Multiplying Eq. 6 by di dz , summing over i, and integrating, following equation is obtained 1

2c

ij

i, j

d j d i        0   , dz dz

(7)

where 0 = − p0 is the grand-potential density of the homogeneous vapor. 3.1

Pure Substances

After some modifications [29] of Eq. 7, the following relation between the density-independent influence parameter and the surface tension σ of a pure component can be determined 7

1 cii    2 

L



G

2

    d  .  

(8)

In this study, we mostly consider a temperature-independent influence parameter (cii = const.). This simplification is quite common especially when the binary mixtures are solved. Since one of the components may be present at its supercritical conditions at the temperature of the mixture, it would not be possible to derive its cii from the pure surface tension data. Considering constant influence parameter is an adequate simplification used in GT [13,15]. Table 3 summarizes the average values of the influence parameter c both for the GT + PC-SAFT EoS and the GT + PCP-SAFT EoS. The constant value of the influence parameter was obtained by averaging the values of influence parameter determined using Eq. 8 from the experimental surface tension data for various temperatures. Temperature range up to T/Tcrit = 0.986 was considered, as the computed influence parameter decreases steeply as the critical point is approached. Figure 1 compares the experimental data for the surface tension of CO2, i.e. a substance with a high quadrupole moment, with the predictions of the GT combined with both the PC-SAFT EoS and the PCP-SAFT EoS. The results were calculated with the constant values of the influence parameter given in Table 3. As can be seen, the GT + PCP-SAFT EoS provides a significantly better prediction for the surface tension over a wide temperature range. The surface tension modeled by the GT + PCP-SAFT EoS lies in the uncertainties of the experimental data. The results are also quite close to Somyajulu’s [37] correlation for CO2. The PCP-SAFT EoS implemented in the GT achieves better results also for pure dipolar components. An example for R32 is shown in Fig. 2. Marker sizes approximately correspond to the uncertainties of the experimental data given as ± 0.2 mN·m-1. The surface tension predicted by the GT + PCP-SAFT EoS agrees quite well with the experimental data by Heide [34] and Fröba et al. [35]. However, at lower temperatures the predicted surface tension is slightly lower than the experimental values. 3.2

Binary Mixtures

The planar phase interface of a binary mixture has been modeled in this study with the GT in a similar way as that proposed by Kahl and Enders [13], Lin et al. 8

[38], and Mique et al. [39,40]. Profiles of the partial molar densities ρ1 and ρ2, defined in the following way i   xi ,

(9)

have to be calculated before the surface tension can be evaluated. The less volatile component, i.e. the one with the monotonic density profile ρ(z), has to be taken as a reference fluid to assure calculation of the entire density profiles [39]. The density of the second component is then expressed as a function of the reference component, i.e. ρ2 = ρ2(ρ1). Density ρ1 is considered as an independent variable changing from ρGx1G to ρLx1L, where ρG and ρL are densities of the vapor and liquid bulk phases, respectively. For a binary mixture, the density-independent influence parameter c can be determined from the pure-component influence parameters as follows [13] 2

 d  d c  c1  2c12 2  c2  2  , d1  d1 

(10)

where the cross-influence parameter c12 is defined as a geometrical average in the following manner c12  1    c1c2 .

(11)

We set the correcting parameter β equal to zero in our calculations. This common approximation provides a good representation for many mixtures and greatly simplifies the mathematical solution. From Eq. 6, we obtain following set of equations for a binary mixture c1

d 2 1 d 2 2   1  1 ,  2   10 , c 12 2 2 dz dz

(12)

c21

d 2 1 d 2 2   2  1 ,  2   20 . c 2 2 2 dz dz

(13)

Setting β = 0 in Eq. 11 and multiplying Eq. 12 and Eq. 13 by

c2 and

c1 ,

respectively, the following equation can be derived after subtracting Eq. 13 from Eq. 12 [39] c2  1  1 ,  2   10   c1  2  1 ,  2   20   f  1 ,  2   0 .

(14)

The relation between the partial molar densities ρ1 and ρ2 can be obtained by differentiating Eq. 14, i.e. df  1 ,  2   0 , [39] c2  1 1   c1  2 1  d 2  . d1 c1  2  2   c2  1  2 

(15)

9

The dependent density ρ2(ρ1) is calculated from Eq. 15 by an iterative NewtonRaphson method. Once the total density (ρ = ρ1 + ρ2) is known, the grand potential density difference Δω can be calculated as





  1 ,  2    A  110   2 20  p .

(16)

Finally, the surface tension of a binary system at a given temperature and composition can be determined from Eq. 17 [39]. 

 L1



2c  1 ,  2 

(17)

 G1

Figure 3 shows an example of the surface tension of a binary system with one non-polar and one quadrupolar component represented by n-decane and CO2, respectively. The surface tension calculated at two temperatures 344.3 K and 377.6 K continuously decreases with increasing mole fraction of CO2 in the liquid phase. Maximum uncertainty of the experimental data is comparable to the marker sizes in Fig. 3. Results of the GT + PCP-SAFT EoS are in a better agreement with the experimental data than the GT + PC-SAFT EoS which neglects the quadrupolarity of CO2. The surface tension for a mixture of two quadrupolar components, namely benzene + CO2, is plotted in Fig. 4. Neither the GT + PC-SAFT EoS nor the GT + PCP-SAFT EoS provides a satisfactory prediction of the surface tension, especially at higher temperatures. Error of the modeled surface tension is caused by a poor prediction of the VLE of benzene + CO2 in this case. Accuracy of the VLE, especially of the bulk densities providing integration limits in Eq. 17, strongly affects the resulting surface tension. As Gross [19] noted, treatment of mixtures with more quadrupolar components is a difficult task. The crossquadrupolar interactions have to be defined for the specific mixture in this case. Simple mixing rules for the chain lengths mij and mijk proposed by Gross [19] would have to be modified to improve the accuracy of the PCP-SAFT EoS. Besides the usual predictions of the GT + PC-SAFT EoS and the GT + PCPSAFT EoS considering cross-quadrupolar interactions between benzene and CO2, the surface tension was calculated by the GT combined with the PCP-SAFT EoS modified in the manner suggested by Gross [19]. The binary interaction parameter was set equal to zero, and the cross-quadrupolar interactions between benzene and CO2 were neglected in this case. However, even such modification did not result in any improvement of the modeled surface tension. 10

Figures 5 to 7 show results for three different binary mixtures consisting of two dipolar components, namely R134a + R125, R32 + R134a, and R32 + R125. Marker sizes are comparable to maximum uncertainties of the experimental data, i.e. ≤ ± 0.2 mN·m-1 in all three figures. Application of the PCP-SAFT EoS instead of the non-polar PC-SAFT EoS brings a considerable improvement in the predicted surface tension for all three dipolar mixtures. The cross-dipolar interactions, described by the simple mixing rules for the chain lengths mij and mijk [20], seem to be satisfactory for all investigated mixtures. Table 4 summarizes deviations of the surface tension predicted by the GT from the experimental data for all binary mixtures investigated in this study. The deviation δσ is defined as  

1 N   exp,i   cal,i   N i 1  exp

2

  , 

(18)

where  exp denotes an average value of the given set of surface tension experimental data. The difference between the experimental data and the calculated data was normalized by  exp , because some binary mixtures, namely ndecane + CO2 and benzene + CO2, were measured up to high temperatures reaching the critical line. The relative deviation  exp,i   cal,i   exp,i in this region diverges as the experimental surface tension approaches zero. Results in Table 4 show that application of the PCP-SAFT EoS containing the dipolar and quadrupolar contributions considerably improved the accuracy of the surface tension predicted by the GT in all cases studied.

4 Droplet Nucleation An important application of the gradient theory is in modeling homogeneous nucleation. Besides the ability of GT to predict the concentration dependence of the surface tension and the shape of the density profiles, including surface enrichment, it also enables an estimate of the effect of the droplet or bubble radius on the surface tension. Here we focus on the nucleation of droplets. Homogeneous nucleation of droplets in carbon dioxide-rich natural gas appears to be an important phenomenon in some technologies for natural gas processing. As an initial step, we analyze here the predictions of the size-dependence of the surface tension and the resulting nucleation predictions for pure carbon dioxide, 11

represented by the advanced PCP-SAFT and PC-SAFT models, in comparison with the Peng-Robinson equation, which has been used for GT-modeling of nucleation in previous studies [44]. The number of newly formed droplets in a unit of volume (m3) per a unit of time (s) is called the nucleation rate. The nucleation rate depends both on gas kinetics, determining the rate of vapor molecules impinging on the cluster (a microscopic aggregate of several molecules up to several hundreds of molecules) and on thermodynamics, which plays a crucial role in determining the evaporation rate of vapor molecules from the cluster. The so-called classical nucleation theory (CNT), developed by Becker and Döring [45] and others in the 1920’s to 40’s, expresses the nucleation rate J as a product of a kinetic prefactor K and an exponential thermodynamic term, J  K exp   / kBT  .

(19)

Here, kB is the Boltzmann constant and ΔΩ is the so-called work of formation of a critical cluster. A critical cluster is a cluster of specific size which, for a given thermodynamic condition of the surrounding vapor, shows equal probabilities of growth (condensation of a vapor molecule onto the cluster) and decay (evaporation of a molecule from the cluster). This means that there is a zero net flux of mass between the cluster and the surrounding vapor, which – in a thermodynamic sense – means that the chemical potential has a uniform value in the liquid inside the droplet, within the phase interface, and in the surrounding vapor. The thermodynamic meaning of the work of formation ΔΩ is a difference of the grand potential of a system of vapor constrained to contain a critical cluster and the grand potential of a system without the constraint, essentially a homogeneous vapor. In the framework of the gradient theory, the work of formation can be obtained as  





0

2 1  d   2   (  )  c    4 r dr , 2  dr   

(20)

where the influence parameter c and the excess grand-potential density Δω have the same meanings as for the planar phase interface. The magnitude of the work of formation given by Eq. 20 depends on the density profile ρ(r) which is to be determined by considering Eq. 20 as a functional. The density profile of the critical cluster corresponds to a stationary point of the functional. The problem can be reduced to finding a non-trivial solution of the Euler-Lagrange equation 12

d 2 dr

2



2 d 1    (  )  G  r dr c

(21)

with the boundary conditions of a vanishing derivative in the center (r = 0) and at an infinite displacement (r → ∞). Once the density profile is found, the work of formation can be evaluated by a numerical quadrature of Eq. 20. Having computed the work of formation, the nucleation rate, given by Eq. 19, can be evaluated with the kinetic factor K adopted from CNT in the form 1/2

G2  2 sat  K L,sat   M 

.

(22)

This form of the pre-exponential factor corresponds to the original Becker-Döring [45] theory. The exact form of the factor is subject to discussions [46,47] which we do not want to enter here. In order to be able to interpret the results obtained by GT, it is useful to interpret them in terms of Gibbsian surface thermodynamics, which expresses the significant quantities in terms of a reference model comprising a sphere of radius rs (the so-called surface of tension), separating the interior bulk liquid phase from the surrounding homogeneous gas phase. By the above given argument, the chemical potential of the liquid and gas phases should be equal. Assuming also equal temperatures, the liquid pressure pL can be determined from equation

L (T , pL )  G (T , pG ) ,

(23)

where the chemical potentials are expressed with the help of a particular EoS. The thermodynamic state of the metastable vapor is usually characterized with supersaturation s defined as    sat  s  exp  G  .  kBT 

(24)

Assuming ideal-gas behavior of the vapor, the supersaturation simplifies to the familiar form of the ratio of the actual and saturated vapor pressures, s  pG / psat .

(25)

In the case of carbon dioxide nucleation, this approximation is not adequate since the vapor cannot be considered as an ideal gas. For a given supersaturation, the difference of pressures in the liquid and gas phases can be found by solving Eq. 23. In the framework of Gibbsian surface thermodynamics, the pressure difference is also equal to the Laplace pressure Δp 13

pL  pG  p 

2 . rs

(26)

The work of formation can be expressed as

 

1 As , 3

(27)

where As  4 rs2 is the surface area of the surface of tension, and σ is the surface tension, generally dependent on the droplet radius. This dependence is of great interest in nucleation theory. If the work of formation is determined from the GT using Eq. 20, the radius and surface tension of the critical droplet for given p can be obtained by combining Eqs. 26 and 27 1/3

 3  rs     2p 

,

(28)

1/3

 3  p 2      16 

.

(29)

Figure 8 shows the surface tensions of critical clusters of carbon dioxide at the selected temperature of 220 K as a function of the Laplace pressure Δp. The figure shows the typical behavior: initially, the surface tension is increasing. This phenomenon corresponds to a negative initial Tolman length [48]. For greater Laplace pressures (greater supersaturations, smaller critical clusters), the surface tension decays, and ultimately vanishes at the vapor-liquid spinodal. Also shown is the universal asymptotic solution [49] near the spinodal (dashed lines with grey points). However, it is to be noted that the vapor-liquid spinodal is rather a construct of mean field theory. The range relevant to nucleation experiments is typically on the decaying branch of the curve, however still far from the theoretical spinodal. As indicated in Eq. 31, the surface tension appears in the third power in the work of formation, which, in turn, appears in the exponential of the nucleation rate formula Eq. 19. Therefore, the surface tension is the most sensitive quantity in nucleation theory. To represent the surface tension most accurately, we fitted the influence parameter for each temperature to the empirical fit of experimental planar surface tensions by Somayajulu [37]. This is different than the mixture computations, where temperature-independent influence parameters were considered. Consequently, the curves generated with all three equations of state – the Peng-Robinson, PC-SAFT, and PCP-SAFT – start from

14

the same point at Δp = 0. The increasing part is pronounced for the PengRobinson equation, whereas for both the PC-SAFT EoS and PCP-SAFT EoS, it is hardly observable. The ultimate decay is fastest for the PC-SAFT EoS, whereas for the PCP-SAFT EoS and the Peng-Robinson equation, the surface tension decays more slowly with increasing Laplace pressure. The ultimate decay mostly corresponds to the location of the vapor-liquid spinodals of the respective equations of state. For comparison, we also determined the nucleation rates from CNT. CNT approximates the surface tension of the critical cluster with the planar surface tension. The computed nucleation rates for both the GT and the CNT are shown in Fig. 9 for a selected temperature of 220 K. At low supersaturations, the GT predictions almost coincide with CNT. This is clearly due to the fact that the corresponding surface tensions lay in the flat top shown in Fig. 8. With increasing supersaturation, the GT predictions show progressively higher nucleation rates, corresponding to decreasing surface tensions. The difference between the various EoSs is significant, and apparently it remains almost the same for both the GT and the CNT. The differences between predictions by the various equations of state follow predominantly from the differences of saturated liquid densities. The role of the density can be explained as follows. Neglecting the liquid compressibility, the pressure difference can be expressed from Eq. 23 as

p  kBT  L,sat ln s  ( pG  psat ) .

(30)

The term in the parentheses is typically negligible in comparison with Δp.

Equation 30 shows that, for a given supersaturation, the pressure difference is proportional to the liquid density. The work of formation can be expressed with the help of Eqs. 26 and 27 as  

16 3 3p 2

.

(31)

Consequently, a too high liquid density, as predicted by the Peng-Robinson equation, gives, according to Eq. 30, a too high Δp, too low work of formation according to Eq. 31, and too high nucleation rate according to Eq. 19. Figure 10 shows the saturated-liquid densities as computed from the Peng-Robinson, PCSAFT, and PCP-SAFT EoS, compared with the reference EoS by Span and Wagner [50]. Clearly, only the PCP-SAFT EoS provides sufficiently accurate saturated liquid densities. The differences at 220 K directly translate into the 15

differences of nucleation rates shown in Fig. 9. Figure 10 also shows saturated liquid densities computed below the CO2 triple point, corresponding to a supercooled liquid. This range is considered as important for applications, because it is assumed that below the triple point, the supersaturated vapor first condenses into droplets of supercooled liquid, which freeze subsequently. Finally, Fig. 11 shows the temperature dependence of the supersaturation corresponding to a fixed nucleation rate of 1015 m-3·s-1. The temperature dependence of nucleation rates is known to be one of the major weaknesses of CNT. The experimental and predicted dependences were analyzed by Malila et al. [51] for a number of compounds. In the present study, the temperature dependencies of the classical and gradient theories are rather similar. The reason is likely in the fact that the critical clusters in the studied range are relatively large and, consequently, the effect of the droplet size on the surface tension remains moderate. Unfortunately, no experimental data for nucleation rates of CO2 exist which could be used for validation of the predictions.

5 Conclusion The Cahn-Hilliard density gradient theory (GT) has been combined with two physically-based equations of state, the PC-SAFT EoS and its modification for polar substances, the PCP-SAFT EoS. Compared to the original non-polar PCSAFT EoS, the PCP-SAFT EoS considerably improved the accuracy of the surface tension predicted by the GT for pure dipolar and quadrupolar substances. The interfacial tension of a binary mixture with one quadrupolar and one nonpolar component was also modeled quite well. However, mixtures with more quadrupolar components need further attention. The VLE of these systems has not been reproduced satisfactorily with the PCP-SAFT EoS; see Gross [19]. Therefore, the GT could not predict the interfacial properties with a good accuracy in this case. On the other hand, results for three HFC refrigerant-mixtures showed that the GT + PCP-SAFT EoS can successfully model the surface properties of binary mixtures consisting of dipolar components. We demonstrated the ability of the GT combined with a realistic physically-based EoS to predict the temperature and composition dependences of surface tension. These facts support the conjecture that also the dependence on the surface curvature will be predicted correctly. The dependence of the surface 16

tension on droplet radius is an important issue in the theory of homogeneous nucleation. We computed the works of formation of critical clusters for various conditions of the supersaturated CO2 vapor for three EoSs: the PCP-SAFT EoS, PC-SAFT EoS, and the Peng-Robinson EoS, which has already been used in GTmodeling of nucleation [44]. From the works of formation, we predicted the nucleation rates and expressed the surface tensions for critical clusters. The dependence of the surface tension on the Laplace pressure, characterizing the cluster size, is initially very flat for both the PCP-SAFT and PC-SAFT EoSs. This corresponds to a very small negative value of the Tolman length, whereas the Peng-Robinson EoS predicts a considerable initial growth of the surface tension, corresponding to a much more negative Tolman length. The decreasing branch of the surface tension dependence on the Laplace pressure also shows significant differences, corresponding primarily with the various locations of the vapor-liquid spinodal of the three EoSs. We compared the GT-predicted nucleation rates with the predictions by classical nucleation theory (CNT) which neglects the dependence of the surface tension on the interface curvature. The GT predicts higher nucleation rates compared to the CNT. The difference increases with supersaturation, which is explained by the fact that the critical clusters are becoming smaller and their surface tension is reduced. Both GT and CNT showed similar differences when combined with Peng-Robinson, PC-SAFT, and PCPSAFT EoSs. We found that the reason is in the different values of the saturated liquid density given by respective EoSs. Of the tested EoSs, clearly only the PCPSAFT equation provides a satisfactory model for GT modeling of critical clusters.

Acknowledgments

The project has been supported by grant No. IAA200760905 of the GA AS CR, grants GPP101/11/P046 and GA101/09/1633 of the Czech Science Foundation, and by Research Plan of the Institute of Thermomechanics AS CR, v. v. i., No. AV0Z20760514.

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19

TABLES

Table 1

PC-SAFT and PCP-SAFT parameters together with the reference

Substance n-decane CO2 benzene R32 R125 R134a

Table 2

m 4.66325 2.07274 1.51310 2.46520

σ (Å) 3.83840 2.78520 3.18690 3.64780

ε/kB (K) μ (D) 243.8700 169.2100 163.3300 287.3500 -

Q (DÅ) 4.4 -

Reference [11] [11] [19] [11]

2.24630 2.57587 2.47192 3.17812 3.11048 3.24833 3.14704

3.78520 2.76916 2.79714 3.09793 3.11999 3.01572 3.04554

296.2400 180.9438 161.6614 154.9866 153.6956 170.6045 165.3355

5.5907 -

[19] [24] [24] [24] [24] [24] [24]

1.978 1.563 2.058

Values of the binary interaction parameter kij correlated to VLE experimental data taken from the literature

Number PC-SAFT Mixture Reference Tmin (K) Tmax (K) of points kij CO2 + n-decane [25] 344.3 377.6 39 0.11854 R32 + R125 [26] 236.1 323.07 84 -0.00047 [27] 308.15 308.15 34 0.00298 [28] 265.15 303.15 56 0.00169 R32 + R134a [26] 230.91 323.72 74 0.00177 R134a + R125 [26] 232.34 339.95 50 -0.00292

PCP-SAFT kij 0.04332 -0.00977 -0.00812 -0.00753 0.00371 -0.00538

20

Table 3

Values of the influence parameter correlated to the surface tension data taken from the literature for pure substances

Substance CO2 benzene n-decane R32 R125 R134a

Table 4

c (J·m5·mol-2) PC-SAFT PCP-SAFT 1.8636E-20 2.2007E-20 2.0751E-19 2.0721E-19 9.6732E-19 2.2320E-20 2.8128E-20 7.8597E-20 8.1129E-20 7.3843E-20 7.8954E-20

Reference [30,31] [30] [32,33] [34,35] [34,35] [34-36]

Deviation of the surface tension predicted by the GT combined with the PC-SAFT and PCP-SAFT EoSs. δ is given by Eq. 18 PC-SAFT δσ

PCP-SAFT δσ

0.305a

0.250a

[41] 344.3 344.3 16 0.269a [34] 223.15 333.15 21 0.118 [42] 252.93 333.41 118 0.108 R32 + R134a [34] 223.15 333.15 21 0.133 [43] 254.87 333.53 150 0.091 R134a + R125 [34] 223.15 333.15 21 0.081 a Data measured in the high temperature range close to the critical point

0.475a,b, 0.323a,c 0.075 0.074 0.065 0.063 0.060

Mixture CO2 + n-decane

Number Reference Tmin (K) Tmax (K) of points [25]

344.3

377.6

41

CO2 + benzene R32 + R125

b

PCP-SAFT with mixing rule for cross-quadrupolar interactions [19]

c

PCP-SAFT without cross-quadrupolar interactions [19]

21

FIGURE CAPTIONS

Fig. 1

Surface tension of pure CO2 modeled with the GT combined with the PCSAFT and PCP-SAFT EoSs. Comparison with experimental data by Jasper [30], Rathjen and Straub [31], and with the temperature correlation of Somayajulu [37]

Fig. 2

Surface tension of pure R32 (difluoromethane). Comparison of the GT with experimental data by Heide [34] and Fröba et al. [35]

Fig. 3

Interfacial tension of the binary system CO2 + n-decane. Prediction by the GT versus experimental data by Nagarajan and Robinson [25]

Fig. 4

Interfacial tension of binary system CO2 + benzene at 344.3 K. Prediction by the GT combined with the PC-SAFT EoS (kij = 0.088), the PCP-SAFT EoS with cross-quadrupolar interactions (kij = 0.042), and the PCP-SAFT EoS without cross-quadrupolar interactions (kij = 0). Comparison with the experimental data by Nagarajan and Robinson [41]

Fig. 5

Interfacial tension for R134a (1) + R125 (2) at three different mass fractions of R134a in the liquid phase (w1). Comparison between the GT results and the experimental data by Heide [34]

Fig. 6

Interfacial tension of binary system R32 (1) + R134a (2) at three different mass fractions of R134a in the liquid phase (w1). Prediction by the GT versus experimental data by Heide [34]

22

Fig. 7

Interfacial tension for R32 (1) + R125 (2) at three different liquid phase mass fractions of R32. The GT predictions compared to the experimental data by Duan et al. [42]

Fig. 8

Surface tension of a droplet of pure CO2 at 220 K and different supersaturation conditions defined by the pressure difference Δp. Comparison of the Peng-Robinson (PR) EoS with the PC-SAFT EoS and the PCP-SAFT EoS. Dashed line with grey points correspond to an approximate solution near the spinodal

Fig. 9

Fig. 10

CO2 nucleation rate at 220 K calculated by the GT and the CNT

Temperature dependence of the saturated liquid density for pure CO2 calculated by the Peng-Robinson, PC-SAFT, and PCP-SAFT EoS. Comparison with the fundamental EoS for CO2 by Span and Wagner [50]

Fig. 11

Temperature dependence of the supersaturation s of CO2 at a given nucleation rate of 1015 m-3·s-1

23