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Measurements and modelling of interfacial tension for water + carbon dioxide systems at elevated pressures
Computational Materials Science 38 (2007) 506–513 www.elsevier.com/locate/commatsci
Measurements and modelling of interfacial tension for water + carbon dioxide systems at elevated pressures Bjørn Kvamme a,*, Tatyana Kuznetsova a, Andreas Hebach b, Alexander Oberhof b, Eivind Lunde a b
a Department of Physics, University of Bergen, All_egaten 55, 5007 Bergen, Norway Institute for Technical Chemistry-CPV, Forschungszentrum Karlsruhe GmbH, Postfach 3640, 76021 Karlsruhe, Germany
Received 1 February 2005; received in revised form 10 September 2005; accepted 17 January 2006
Abstract A novel apparatus, PeDro, was used to measure interfacial tension in a two-component system made of water and compressed carbon dioxide at temperatures ranging 278–335 K and pressures 0.1–20 MPa. Our optimized experimental setup utilized the quasi-static pendant drop method and ensured experimental errors below 2%. The interfacial tension showed a pronounced dependence on pressure and temperature. A regression function was derived that allows to interpolate between the experimental data with high precision. Molecular dynamics simulations were performed for liquid–liquid and liquid–vapor interfaces between water and carbon dioxide at elevated pressures. The interfacial tension was obtained from long constant-volume production runs as the difference between normal and tangential pressure components. The results showed a good agreement with experimental data, with our model system reproducing faithfully the pressure–temperature dependence of the interfacial tension. 2006 Elsevier B.V. All rights reserved. Keywords: Carbon dioxide; Water; Interfacial tension; Pendant drop; Molecular modeling; High pressure
1. Introduction Carbon dioxide (CO2) in its liquid or supercritical state is a promising fluid medium for reactions, separation and surface treatment processes. In multiphase systems, interfacial phenomena play an important role in such applications as mass transfer limitations, mass transfer area, wetting behaviour and formation of micro-emulsions. These properties depend heavily on the density of the phases involved. The density of a highly compressible fluid like CO2 varies rapidly with temperature and pressure, affecting the interfacial properties considerably. One objective of this study was to develop an accurate measuring procedure for this system using the pendant drop method. Another objective was to study the behaviour of the interfacial tension with pressure and temperature in a range of (0.1–20 MPa) and *
Corresponding author. Tel.: +47 55583310; fax: +47 55589440. E-mail address: [email protected] (B. Kvamme).
0927-0256/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.01.020
(278–335 K). A system consisting of water in contact with carbon dioxide, a substance often considered a third type of condensed phase, different from lipophilic and hydrophilic phases, is a worthy target of computer simulation study in its own right. In the last two decades, the demand from industrial separation processes has led to rapid development and frequent application of the theory of inhomogeneous fluids. But certain theoretical progress notwithstanding, determination of the interfacial phenomena at the liquid–liquid water–carbon dioxide interface appears to remain in the realm of experiments and computer simulations. The paper is organized as follows. Section 2 describes the experimental method and apparatus and discusses the obtained results and their accuracy and reproducibility. Section 3 focuses on molecular modeling involving a model interfacial water/CO2 system and compares the simulational results with the experiment. We outline our conclusions in Section 4.
B. Kvamme et al. / Computational Materials Science 38 (2007) 506–513
2. Experimental 2.1. Method and apparatus The interfacial tension data were obtained by using a modified high-pressure pendant drop method, the quasistatic method. The method and apparatus used are described in detail elsewhere [1]. Briefly, the system consists of a high-pressure circuit with a gear pump, a viewing cell, and a sample loop, Fig. 1. The whole system is installed in a climatic chamber, which allows precise and accurate temperature adjustment to within ±0.2 K. The pressure stability is ±0.02 MPa. Isotherms of the interfacial tension are measured using the quasi-static method. This method ensures a time regime where the influence of diffusion and drop aging does not affect the IFT. 2.2. Accuracy and reproducibility of the measurements To determine the accuracy of the equipment, measurements with water and CO2 were compared with recommended values of the ASME at atmospheric pressure. According to this comparison, the experimental accuracy of the PeDro apparatus is greater than 98%. At higher pressures, especially in the presence of a highly compressible phase, the accuracy in density determination of the phases becomes particularly important. Therefore we measured the saturated densities of CO2 and water. Within the limits of the accuracy (better than 0.15%) the carbon dioxide density equals the pure density, but that of the CO2saturated water changes within 1.9% [2]. When compressible fluids are present, dynamic phenomena play an important role, especially in the high-pressure area, and the interfacial tension can change drastically during the life of a pending drop [10], making it necessary to find a reliable experimental procedure. Fig. 2 shows the interfacial tension behavior as a function of time. Three
507
characteristic intervals (A, B and C) can be identified. In the beginning (A), the fast processes—including diffusion, convection due to mixing phenomena and temperature gradients, and eventually chemical reactions—will cause a rapid drop in the measured value of the IFT [11,12]. The extent of the decay in interval (A) can be reduced by presaturation of both phases. After this, the interfacial tension remains nearly constant (B). Static measurements of the interfacial tension are only possible in this interval. The last interval (C) is dominated by long-term processes, generally categorized as ageing effects. Because of these reasons, one only consider the pendant drop method to be a static method in a certain interval of time and we therefore prefer the term quasi-static method. In order to obtain reproducible data, it is essential to perform measurements in regime (B) of the interfacial tension curve. These observations were used as the basis for developing our measuring procedure. Isotherms were measured over the entire pressure range. To prove that there was no interference with drop ageing processes, three different measuring methods were tested:
Fig. 2. Time dependence of the IFT-value with the pendant drop method.
Fig. 1. High pressure pendant drop device PeDro, A-viewing cell, B-inspection window, C-light source, D-CCD camera, E-gear pump, F-filter, G-phase separator, J-thermostat, K-piston pump, L-syringe pump, M-compressor.
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firstly with increasing pressure, secondly with decreasing pressure, and thirdly with randomly selected data points. All values are identical within the limits of experimental accuracy. 2.3. Experimental results Six isotherms reported in this work cover the temperature range of (278–333 K) at pressures ranging from atmospheric to 20 MPa (Fig. 3). A decrease in the interfacial tension was observed as the pressure increases, this phenomenon more pronounced at lower temperatures. At lower pressures, r decreases with rising temperature; the opposite effect is found at higher pressures. At higher pressure and temperature a plateau of r values is reached. Here, the density difference changes to a smaller extent and the r values become comparable to those between water and non-polar organic solvents, such as benzene or hexane under ambient conditions. The isotherms below 304 K, the critical temperature of CO2, are separated into two parts, which are interrupted at vapour–liquid coexistence line of CO2. Both measurements and simulations display a jump in IFT at this pressure point of the isotherm. Below this pressure the isotherm reflects the interfacial tension of water with gaseous CO2, which decreases almost linearly with pressure. The interfacial tension between water and liquid CO2 is measured above this pressure. Isotherms measured above the critical temperature of CO2 continuously decrease with rising pressure, showing a hyperbola-like behaviour.
well as two selected isotherms from this work. The data were taken from Massoudi [6], Chun [7], Jho [8], Da Rocha [9], Heuer [10], Wesch [11] and Jaeger [12]. Except for Chun, Wesch and Jaeger, data had to be extracted from figures. Table 1 summarizes the known publications on interfacial tension values of CO2 + water. They are sorted by measuring method: SP stands for the abbreviation for selected plane method, DP, for the drop profile method of the pendant drop method. The following points out some major characteristics of the various studies: Heuer [10] used the selected plane method and waited 10 s to obtain equilibrium. Chun [7] determined r by means of capillary rise method. Although local equilibrium was achieved within the capillary tube, the entire system was
2.4. Comparison with existing published data Fig. 4 shows the interfacial tension at (308, 311, 313, 314, 318 and 344 K) as measured by several authors, as Fig. 4. Interfacial tension at the CO2 + water interface as a function of pressure at various temperatures: (–n–) da Rocha [9] at 308 K; (–e–) Heuer [10] at 311 K; (- -s- -) Wesch [11] at 313 K; (–v–) Jaeger [12] at 314 K; (–h–) Chun [7] at 318 K; (–,–) Heuer [10] at 344 K. The solid symbols correspond to two isotherms ((j) at 303 K and (d) at 318 K) from this paper.
Table 1 Techniques and temperature–pressure ranges covered by the existing experimental studies on the interfacial tension in the binary water + carbon dioxide system
Fig. 3. Measured isotherms in the CO2 + water system as a function of pressure: 278 K (–h–), 288 K (- -s- -), 298 K (–n–), 308 K (–e–), 318 K (- -,- -), and 333 K (–v–).
Method
p (MPa)
T (K)
Author
CR
0.1–15.7
Chun [4]
CR CR PD-SP PD-SP PD-DP PD-DP
0.4–6.0 0.6–6.1 0.1–20 7.0–24.1 6.6–28 0.4–27.9
278–344 in 5 K steps 285, 298, 318 298 313, 333 311, 344 308, 318 314, 343
Jho [5] Massoudi [3] Wesch [8] Heuer [7] Da Rocha [6] Jaeger [9]
CR, stands for the capillary rise method; PD-SP, pendant drop selected plane method; PD-DP, pendant drop profile method.
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not in equilibrium. Da Rocha [9] applied the profile method, used pre-saturated liquids and measured the data 1 h after drop formation. Massoudi [6] used the capillary rise method, but only with gaseous carbon dioxide. Jaeger [12] assumed the density of water to be constant. He estimated his measuring error to be smaller than ±8%. Wesch [11] determined the interfacial tension using the selected plane method, applying commercial software (Optimas, Stemmer imaging, Puchheim, Germany). The accuracy given by Wesch is ±5%. To the best of our knowledge, all of the above authors measured the temperature at the autoclave walls. In the vicinity of the vapour pressure curve or near the critical point of CO2, where the density varies strongly with small changes in pressure and temperature, large errors may occur. Accordingly, a cusp may be found in this high gradient area, as observed by some authors. These uncertainties can be avoided by placing the thermometer inside the cell and near the pendant drop. In order to verify this assumption, we performed measurements along an isotherm using two different thermocouple positioning: close to the pendant drop (our standard operating procedure), and inside the autoclave insulation. The density difference, needed to calculate the interfacial tension, is determined using this temperature value. The resulting values are shown in Fig. 5. A cusp, similar to the findings of some authors, was observed.
Fig. 5. Interfacial tension values as obtained with the same measurement using two different thermocouple positions: (–h–) in CO2 and (- -s- -) in insulation.
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2.5. Regression function A regression curve was calculated on the basis of the experimental data determined in this work. It allows the data within the experimental range to be interpolated with accuracy greater than 95%. Eq. (1) describes our non-linear regression model: pffiffiffiffiffi r ¼ k 0 1 exp k 1 dd þ k 2 dd þ k 3 dd 2 þ k 4 dd 3 þ k 5 expðk 6 ðdd 0:9958 g2 cm6 ÞÞ
ð1Þ
using dd ¼ ððqH2 O qcorr Þ=1000Þ2 and qcorr ¼
8 b1 > < qCO2 þ b0 ð304 K T Þ p > :
25 kg m3 < qCO2 < 250 kg m3 with in all other cases
qCO2
ð2Þ
where dd is a function of temperature and pressure and represents the squared density difference of the pure components. The units of T, p, and dd are K, MPa and g2 cm6, respectively. The CO2 density has to be corrected for the adsorption of gaseous CO2 (Eq. (2)); qcorr takes this into account. We used the least square fit method without restraints for the fit parameters. The Marquart method was employed for the regression. The fit parameters are compiled in Table 2. The first exponential term is needed to express the increase in r when density differences are small. The second exponential term was introduced to describe the experimental values at high dd values. In this regression function, dd is a function of temperature and pressure and represents the squared density difference of the pure components. The units of T, p, and dd are K, MPa and g2 cm6, respectively. The CO2 density has to be corrected due to adsorption of gaseous CO2 (Eq. (2)); qcorr takes this into account. We used the least square fit method without restraints for the fit parameters. The Marquart method was employed for the regression. The fit parameters are compiled in Table 2. The first exponential term is needed to express the increase in r at when density differences are small. The second exponential term was introduced to describe the experimental values at high dd values. The quality of the regression is shown in a bubble graph of Fig. 6, where the relative deviation of the regression value from each experimental data point is plotted against the squared density difference; only at low squared density differences is the deviation ±5%. This deviation is a result of the experimental error from the regression procedure and from the temperature dependence of the interfacial
Table 2 Coefficients of the regression function (Eqs. (1) and (2)) b0 (g cm3 K1)
b1
k0 (mN m1)
k1 (cm6 g2)
k2 (cm12 g4)
k3 (cm18 g6)
k4 (cm3 g1)
k5 (mN m1)
k6 (g2 cm6)
0.00022
1.9085
27.514
35.25
31.916
91.016
103.233
4.513
351.903
510
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Fig. 6. Deviation of experimental points from the regression function. The bubble size is a measure of the temperature at each point.
tension. The size of the bubble represents the temperature; small diameters correspond to low temperatures. There is a distinctive trend within the deviation. The high and low temperature deviations oppose each other: if one has a positive deviation, the other is negative. One reason for this finding might be the use of pure compound densities instead of actual densities to calculate the interfacial tension. If densities of water saturated with carbon dioxide with deviations below ±2% were available, it would be possible to check this hypothesis. Due to the regression procedure, the deviation of low-density differences is larger. The fit method employed takes the value of the abscissa as absolute. For small differences there is a large deviation in the ordinate, whereas there is a much smaller deviation perpendicular to the regression function. As the exponential function controls the rise of the interfacial tension value in this data range, the deviation is large. In Fig. 7, we have plotted the interfacial tension derived from our regression function as a function of pressure and temperature. The critical isotherm of CO2 is marked sepa-
Fig. 7. Interfacial tension as a function of pressure and temperature, as derived with the regression function.
B. Kvamme et al. / Computational Materials Science 38 (2007) 506–513
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rately. Four distinct areas can be observed: the one following the Eo¨tvo¨s-rule at low pressures, which means that the interfacial tension decreases linearly with temperature [13]. Then, there is the region of the vapour pressure points; here CO2 coexists in the gaseous and the liquid state and the density difference has a discontinuity. As a consequence the interfacial tension has two discrete values; one for liquid and one for gaseous carbon dioxide. The transition occurs at the three-phase line liquid–liquid–gas (llg) in the binary system [14]. Next, a plateau appears at higher temperatures and pressures, and finally there is a region with a significant drop in the interfacial tension, where vanishing density differences occur.
˚ to The resulting systems, ranging from 22 · 22 · 66 A ˚ 22 · 22 · 256 A in size were subsequently equilibrated for several tens of picoseconds before the average collection began. The electrostatic interactions were handled by means of Ewald summation technique [15].
3. Simulations
where Pi is average system pressure in the i direction, hz is box length in the direction normal to the interface.
3.2. Interfacial tension estimation Interfacial tension r was obtained by a common method using pressure tensor in case where there are two interfaces normal to the z-axis: 1 r ¼ 0:5 hz P z ½P x þ P y ; 2
3.1. Details 3.3. Simulational results and comparison with experiment The molecular dynamics used constant-temperature and either constant-pressure or constant-volume algorithm supplied by the MDynaMix package of Lyubartsev and Laaksonen [15] modified by us. The starting interfacial system was constructed from two slabs of bulk water (SPC/E [16], either 256 or 512 molecules, and carbon dioxide [17], 100–200 molecules), thermalized initially at 323 K and set side by side, with the periodic boundary conditions applied.
Figs. 8 and 9 present density profiles characteristic for the systems containing, respectively, liquid and gaseous CO2 as one of the phases. The fine-scale profiles of x y averaged quantities of interest (mass density) (mass density) are generated by partitioning the simulation box into discrete bins in the z direction. We followed the general approach of Davidchak and Laird [18] and applied finite
Fig. 8. Density profiles of the water-gaseous carbon dioxide system at 323 K and densities corresponding to 8.3 MPa. Full line: fine-scale water density; dashed–dot line: fine-scale CO2 density (profiles averaged over 50 ps). Dashed lines: coarse-grain profiles obtained by applying FIR.
Fig. 9. Density profiles of the water–liquid carbon dioxide system at 323 K and densities corresponding to 4.5 MPa. See Fig. 2 for symbols.
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Table 3 Experimental and modeling results for water + CO2 systems at 323 K and various elevated pressures p (MPa)
T (K)
Density sat CO2 (kg/m3)
Density sat water (kg/m3)
Cross-sim IFT (mN/m)
Sim IFT (mN/m)
IFT (mN/m)
1.1 2.1 4.18 6.22 8.26 10.28 12.33 14.4 18.43 20.44 22.45
322.8 322.8 322.9 322.9 322.8 322.8 322.9 322.9 322.8 322.8 322.8
18.8484 37.6033 83.397 143.052 235.565 424.325 608.399 686.864 765.667 791.304 812.725
988.52 990.71 995.07 997.94 1000.51 1002.38 1003.75 1005.11 1007.07 1008.15 1009.13
51.8 48.0 48.6 43.8 41.3 37.0 35.7 33.7 34.3 33.7 31.9
53.2 50.1 46.7 44.5 40.7 36.6 35.0 35.4 35.3 34.2 34.7
63.7 57.1 47.5 44.1 38.4 32.5 31.2 30.6 29.8 29.4 29.1
impulse response (FIR) filters to produce a smooth coarsegrained profile from the oscillating fine-scale profiles. The filtered profile was obtained from the fine-scale profile fn as follows [18]: f n ¼
N X
ð3Þ
wk fnþk
k¼N
The shape of the filter was assumed to be Gaussian from the outset: 2
wk ¼ Aeðk=eÞ ;
k ¼ N ; . . . ; N
ð4Þ
The normalization constant A was determined from the condition Rwk = 1. Two filters, each with its own N and e, were used, the first one aimed to average over the oscillations due to structure but retain the essential features of the interfacial region. The values of N = 50 and e = 36.1 appeared to satisfy both conditions for our system and were used to generate a coarse-grained profile. The second filter was used to smooth out only the momentary fluctuations in the density profiles while maintaining its oscillatory nature. This filter was much narrower, with e = 1.6. The gaseous-CO2 profile of Fig. 8 shows a significant increase in CO2 density at the interface relative to its bulk value, reflecting the expected ‘‘sticking’’ effect. Table 3 shows that except in the lowest-pressure and density region, the simulations (see under ‘‘sim IFT’’) appear to reproduce the experimental data with a fair degree of accuracy, especially given the fact that we used force fields designed mostly for bulk properties at ambient conditions. Still, the simulated data using the standard Lorentz–Berthelot mixing rules appeared to mimic the pressure dependence of the experimental results. To improve the fit of modeling results further, the well depth in the Lennard–Jones part of water–CO2 cross-interactions was modified by factors ranging from 0.8 to 1.2, with the factor of 1.2 yielding the best results. These are listed in Table 3 under the ‘‘Cross-sim IFT’’ column header. As one can see from Table 3, while the non-Lorentx–Berthlot corrections brought the modeling results closer to the experimental ones for the high- and mid-range CO2 densities, they slightly worsened the fit in the low-density region.
4. Conclusions Studies of the interfacial tension in the two-phase CO2 + water system show that a suitable time interval has to be identified to enable reliable values to be determined. It has been found that such an interval exists for a sufficiently long time in relation to the measuring time after initial mixing effects have terminated. Later, ageing effects cause a constant reduction in the interfacial tension. Applying the quasi-static method, as proposed here, precise and reproducible data were obtained as a function of the squared density difference. This difference can be calculated by reference equations using the pressure and temperature values. It was possible to derive a regression function from these data. This function is useful for interpolating the interfacial tension of the CO2 + water system. The accuracy of the regression function can be further improved if the actual densities are used instead of the pure substance values. Comparison with the experimental data shows that atomistic simulations using force fields fitted for ambient bulk properties of water and carbon dioxide were quite successful at evaluating the interfacial systems under a broad range of conditions, with the lowest-pressure and density region being the only exception. A possible solution to this problem may lie in either applying exp-6 potentials derived from latest ab initio studies for both water and CO2, or sacrificing potential transferrability altogether in favor of cross-interaction potential specifically designed for CO2 interaction with water. Acknowledgements Authors AH and AO are grateful to the German Federal Ministry of Education and Research for its financial support. One of the authors (TK) is grateful to the Norwegian Research Council for the grant of a postdoctoral scholarship. References [1] A. Hebach, A. Oberhof, N. Dahmen, A. Ko¨gel, N. Ederer, E. Dinjus, J. Chem. Eng. Data 47 (2002) 1540–1546.
B. Kvamme et al. / Computational Materials Science 38 (2007) 506–513 [2] A. Hebach, A. Oberhof, N. Dahmen, E. Dinjus, J. Chem. Eng. Data 49 (2004) 950–953. [3] A. Hebach, N. Dahmen, H. Ederer, E. Dinjus, A. Koegel, Oberhof, The Carbon Dioxide/Water interface, static and dynamic measurements at elevated pressures. A. In: Proc. of ISASF 8th Meeting on Supercritical Fluids: Bordeaux, (2002) pp. 601–606. [4] J. Zappe, A. Wesch, K.H. Ebert, Measurement of the Mass Transfer into Single Drops in the System of Water/Supercritical Carbon Dioxide, J. Colloid Interface Sci. 231 (2000) 1–7. [5] A. Koegel, Stoffdurchgangsmessungen am ha¨ngenden Wassertropfen in verdichtetem Kohlendioxid; Wissenschaftliche Berichte Forschungszentrum Karlsruhe, (2000) 6476. [6] R. Massoudi, A.D. King Jr., Effect of Pressure on the Surface Tension of Water. Adsorption of Low Molecular Weight Gases on Water at 25 C, J. Phys. Chem. 78 (1974) 2262–2266. [7] B.-.S. Chun, G.T. Wilkinson, Interfacial Tension in High-Pressure Carbon Dioxide Mixtures, Ind. Eng. Chem. Res. 34 (1995) 4371–4377. [8] C. Jho, D. Nealon, S. Shogbola, A.D. King Jr., Effect of Pressure on the Surface Tension of Water: Adsorption of Hydrocarbon Gases and Carbon Dioxide on Water at Temperatures between 0 and 50 C, J. Colloid Interface Sci. 65 (1978) 141–154. [9] S.R.P. da Rocha, K.L. Harrison, K.P. Johnston, Effect of Surfactants on the Interfacial Tension and Emulsion Formation between Water and Carbon Dioxide, Langmuir 15 (1999) 419–428. [10] G. Heuer, Ph.D. Dissertation, The University of Texas at Austin, USA, 1957.
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[11] A. Wesch, N. Dahmen, K. Ebert, J. Scho¨n, Grenzfla¨chenspannungen, Tropfengro¨ßen und Kontaktwinkel im Zweiphasensystem H2O/CO2 bei Temperaturen von 298 bis 333 K und Dru¨cken bis 30 MPa, Chem. Ingen. Tech. 69 (1997) 942–946. [12] P.T. Jaeger, Grenzfla¨chen und Stofftransport in verfahrenstechnischen Prozessen am Beispiel der Hochdruck—Gegenstromfraktionierung mit u¨berkritischem Kohlendioxid, Shaker Verlag, Aachen, 1998. [13] R. Eo¨tvo¨s, Ueber den Zusammenhang der Oberfla¨chenspannung der Flu¨ssigkeiten mit ihrem Molecularvolumen, Ann. Phys. (Leipzig) 263 (1886) 448–459. [14] M.B. King, A. Mubarak, J.D. Kim, T.R. Bott, The Mutual Solubilities of Water with Supercritical and Liquid Carbon Dioxide, J. Supercrit. Fluids 5 (1992) 296–302. [15] A.P. Lyubartsev, A.M. Laaksonen, DynaMix—a scalable portable parallel MD simulation package for arbitrary molecular mixtures, Comput. Phys. Commun. 128 (2000) 565–589. [16] H.J.C. Berendsen, J.R. Grigera, T.P. Straatsma, Potentials The Missing Term in Effective Pair, J. Phys. Chem. 91 (1987) 6269–6271. [17] J.G. Harris, K.H. Yung, Carbon dioxides liquid–vapor coexistence curve and critical properties as predicted by a simple molecularmodel, J. Phys. Chem. 99 (1995) 12021–12024. [18] R.L. Davidchack, B.B. Laird, Simulation of the hard-sphere crystalmelt interface, J. Chem. Phys. 108 (1998) 9452–9462.
Measurements and modelling of interfacial tension for water + carbon dioxide systems at elevated pressures Bjørn Kvamme a,*, Tatyana Kuznetsova a, Andreas Hebach b, Alexander Oberhof b, Eivind Lunde a b
a Department of Physics, University of Bergen, All_egaten 55, 5007 Bergen, Norway Institute for Technical Chemistry-CPV, Forschungszentrum Karlsruhe GmbH, Postfach 3640, 76021 Karlsruhe, Germany
Received 1 February 2005; received in revised form 10 September 2005; accepted 17 January 2006
Abstract A novel apparatus, PeDro, was used to measure interfacial tension in a two-component system made of water and compressed carbon dioxide at temperatures ranging 278–335 K and pressures 0.1–20 MPa. Our optimized experimental setup utilized the quasi-static pendant drop method and ensured experimental errors below 2%. The interfacial tension showed a pronounced dependence on pressure and temperature. A regression function was derived that allows to interpolate between the experimental data with high precision. Molecular dynamics simulations were performed for liquid–liquid and liquid–vapor interfaces between water and carbon dioxide at elevated pressures. The interfacial tension was obtained from long constant-volume production runs as the difference between normal and tangential pressure components. The results showed a good agreement with experimental data, with our model system reproducing faithfully the pressure–temperature dependence of the interfacial tension. 2006 Elsevier B.V. All rights reserved. Keywords: Carbon dioxide; Water; Interfacial tension; Pendant drop; Molecular modeling; High pressure
1. Introduction Carbon dioxide (CO2) in its liquid or supercritical state is a promising fluid medium for reactions, separation and surface treatment processes. In multiphase systems, interfacial phenomena play an important role in such applications as mass transfer limitations, mass transfer area, wetting behaviour and formation of micro-emulsions. These properties depend heavily on the density of the phases involved. The density of a highly compressible fluid like CO2 varies rapidly with temperature and pressure, affecting the interfacial properties considerably. One objective of this study was to develop an accurate measuring procedure for this system using the pendant drop method. Another objective was to study the behaviour of the interfacial tension with pressure and temperature in a range of (0.1–20 MPa) and *
Corresponding author. Tel.: +47 55583310; fax: +47 55589440. E-mail address: [email protected] (B. Kvamme).
0927-0256/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.01.020
(278–335 K). A system consisting of water in contact with carbon dioxide, a substance often considered a third type of condensed phase, different from lipophilic and hydrophilic phases, is a worthy target of computer simulation study in its own right. In the last two decades, the demand from industrial separation processes has led to rapid development and frequent application of the theory of inhomogeneous fluids. But certain theoretical progress notwithstanding, determination of the interfacial phenomena at the liquid–liquid water–carbon dioxide interface appears to remain in the realm of experiments and computer simulations. The paper is organized as follows. Section 2 describes the experimental method and apparatus and discusses the obtained results and their accuracy and reproducibility. Section 3 focuses on molecular modeling involving a model interfacial water/CO2 system and compares the simulational results with the experiment. We outline our conclusions in Section 4.
B. Kvamme et al. / Computational Materials Science 38 (2007) 506–513
2. Experimental 2.1. Method and apparatus The interfacial tension data were obtained by using a modified high-pressure pendant drop method, the quasistatic method. The method and apparatus used are described in detail elsewhere [1]. Briefly, the system consists of a high-pressure circuit with a gear pump, a viewing cell, and a sample loop, Fig. 1. The whole system is installed in a climatic chamber, which allows precise and accurate temperature adjustment to within ±0.2 K. The pressure stability is ±0.02 MPa. Isotherms of the interfacial tension are measured using the quasi-static method. This method ensures a time regime where the influence of diffusion and drop aging does not affect the IFT. 2.2. Accuracy and reproducibility of the measurements To determine the accuracy of the equipment, measurements with water and CO2 were compared with recommended values of the ASME at atmospheric pressure. According to this comparison, the experimental accuracy of the PeDro apparatus is greater than 98%. At higher pressures, especially in the presence of a highly compressible phase, the accuracy in density determination of the phases becomes particularly important. Therefore we measured the saturated densities of CO2 and water. Within the limits of the accuracy (better than 0.15%) the carbon dioxide density equals the pure density, but that of the CO2saturated water changes within 1.9% [2]. When compressible fluids are present, dynamic phenomena play an important role, especially in the high-pressure area, and the interfacial tension can change drastically during the life of a pending drop [10], making it necessary to find a reliable experimental procedure. Fig. 2 shows the interfacial tension behavior as a function of time. Three
507
characteristic intervals (A, B and C) can be identified. In the beginning (A), the fast processes—including diffusion, convection due to mixing phenomena and temperature gradients, and eventually chemical reactions—will cause a rapid drop in the measured value of the IFT [11,12]. The extent of the decay in interval (A) can be reduced by presaturation of both phases. After this, the interfacial tension remains nearly constant (B). Static measurements of the interfacial tension are only possible in this interval. The last interval (C) is dominated by long-term processes, generally categorized as ageing effects. Because of these reasons, one only consider the pendant drop method to be a static method in a certain interval of time and we therefore prefer the term quasi-static method. In order to obtain reproducible data, it is essential to perform measurements in regime (B) of the interfacial tension curve. These observations were used as the basis for developing our measuring procedure. Isotherms were measured over the entire pressure range. To prove that there was no interference with drop ageing processes, three different measuring methods were tested:
Fig. 2. Time dependence of the IFT-value with the pendant drop method.
Fig. 1. High pressure pendant drop device PeDro, A-viewing cell, B-inspection window, C-light source, D-CCD camera, E-gear pump, F-filter, G-phase separator, J-thermostat, K-piston pump, L-syringe pump, M-compressor.
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firstly with increasing pressure, secondly with decreasing pressure, and thirdly with randomly selected data points. All values are identical within the limits of experimental accuracy. 2.3. Experimental results Six isotherms reported in this work cover the temperature range of (278–333 K) at pressures ranging from atmospheric to 20 MPa (Fig. 3). A decrease in the interfacial tension was observed as the pressure increases, this phenomenon more pronounced at lower temperatures. At lower pressures, r decreases with rising temperature; the opposite effect is found at higher pressures. At higher pressure and temperature a plateau of r values is reached. Here, the density difference changes to a smaller extent and the r values become comparable to those between water and non-polar organic solvents, such as benzene or hexane under ambient conditions. The isotherms below 304 K, the critical temperature of CO2, are separated into two parts, which are interrupted at vapour–liquid coexistence line of CO2. Both measurements and simulations display a jump in IFT at this pressure point of the isotherm. Below this pressure the isotherm reflects the interfacial tension of water with gaseous CO2, which decreases almost linearly with pressure. The interfacial tension between water and liquid CO2 is measured above this pressure. Isotherms measured above the critical temperature of CO2 continuously decrease with rising pressure, showing a hyperbola-like behaviour.
well as two selected isotherms from this work. The data were taken from Massoudi [6], Chun [7], Jho [8], Da Rocha [9], Heuer [10], Wesch [11] and Jaeger [12]. Except for Chun, Wesch and Jaeger, data had to be extracted from figures. Table 1 summarizes the known publications on interfacial tension values of CO2 + water. They are sorted by measuring method: SP stands for the abbreviation for selected plane method, DP, for the drop profile method of the pendant drop method. The following points out some major characteristics of the various studies: Heuer [10] used the selected plane method and waited 10 s to obtain equilibrium. Chun [7] determined r by means of capillary rise method. Although local equilibrium was achieved within the capillary tube, the entire system was
2.4. Comparison with existing published data Fig. 4 shows the interfacial tension at (308, 311, 313, 314, 318 and 344 K) as measured by several authors, as Fig. 4. Interfacial tension at the CO2 + water interface as a function of pressure at various temperatures: (–n–) da Rocha [9] at 308 K; (–e–) Heuer [10] at 311 K; (- -s- -) Wesch [11] at 313 K; (–v–) Jaeger [12] at 314 K; (–h–) Chun [7] at 318 K; (–,–) Heuer [10] at 344 K. The solid symbols correspond to two isotherms ((j) at 303 K and (d) at 318 K) from this paper.
Table 1 Techniques and temperature–pressure ranges covered by the existing experimental studies on the interfacial tension in the binary water + carbon dioxide system
Fig. 3. Measured isotherms in the CO2 + water system as a function of pressure: 278 K (–h–), 288 K (- -s- -), 298 K (–n–), 308 K (–e–), 318 K (- -,- -), and 333 K (–v–).
Method
p (MPa)
T (K)
Author
CR
0.1–15.7
Chun [4]
CR CR PD-SP PD-SP PD-DP PD-DP
0.4–6.0 0.6–6.1 0.1–20 7.0–24.1 6.6–28 0.4–27.9
278–344 in 5 K steps 285, 298, 318 298 313, 333 311, 344 308, 318 314, 343
Jho [5] Massoudi [3] Wesch [8] Heuer [7] Da Rocha [6] Jaeger [9]
CR, stands for the capillary rise method; PD-SP, pendant drop selected plane method; PD-DP, pendant drop profile method.
B. Kvamme et al. / Computational Materials Science 38 (2007) 506–513
not in equilibrium. Da Rocha [9] applied the profile method, used pre-saturated liquids and measured the data 1 h after drop formation. Massoudi [6] used the capillary rise method, but only with gaseous carbon dioxide. Jaeger [12] assumed the density of water to be constant. He estimated his measuring error to be smaller than ±8%. Wesch [11] determined the interfacial tension using the selected plane method, applying commercial software (Optimas, Stemmer imaging, Puchheim, Germany). The accuracy given by Wesch is ±5%. To the best of our knowledge, all of the above authors measured the temperature at the autoclave walls. In the vicinity of the vapour pressure curve or near the critical point of CO2, where the density varies strongly with small changes in pressure and temperature, large errors may occur. Accordingly, a cusp may be found in this high gradient area, as observed by some authors. These uncertainties can be avoided by placing the thermometer inside the cell and near the pendant drop. In order to verify this assumption, we performed measurements along an isotherm using two different thermocouple positioning: close to the pendant drop (our standard operating procedure), and inside the autoclave insulation. The density difference, needed to calculate the interfacial tension, is determined using this temperature value. The resulting values are shown in Fig. 5. A cusp, similar to the findings of some authors, was observed.
Fig. 5. Interfacial tension values as obtained with the same measurement using two different thermocouple positions: (–h–) in CO2 and (- -s- -) in insulation.
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2.5. Regression function A regression curve was calculated on the basis of the experimental data determined in this work. It allows the data within the experimental range to be interpolated with accuracy greater than 95%. Eq. (1) describes our non-linear regression model: pffiffiffiffiffi r ¼ k 0 1 exp k 1 dd þ k 2 dd þ k 3 dd 2 þ k 4 dd 3 þ k 5 expðk 6 ðdd 0:9958 g2 cm6 ÞÞ
ð1Þ
using dd ¼ ððqH2 O qcorr Þ=1000Þ2 and qcorr ¼
8 b1 > < qCO2 þ b0 ð304 K T Þ p > :
25 kg m3 < qCO2 < 250 kg m3 with in all other cases
qCO2
ð2Þ
where dd is a function of temperature and pressure and represents the squared density difference of the pure components. The units of T, p, and dd are K, MPa and g2 cm6, respectively. The CO2 density has to be corrected for the adsorption of gaseous CO2 (Eq. (2)); qcorr takes this into account. We used the least square fit method without restraints for the fit parameters. The Marquart method was employed for the regression. The fit parameters are compiled in Table 2. The first exponential term is needed to express the increase in r when density differences are small. The second exponential term was introduced to describe the experimental values at high dd values. In this regression function, dd is a function of temperature and pressure and represents the squared density difference of the pure components. The units of T, p, and dd are K, MPa and g2 cm6, respectively. The CO2 density has to be corrected due to adsorption of gaseous CO2 (Eq. (2)); qcorr takes this into account. We used the least square fit method without restraints for the fit parameters. The Marquart method was employed for the regression. The fit parameters are compiled in Table 2. The first exponential term is needed to express the increase in r at when density differences are small. The second exponential term was introduced to describe the experimental values at high dd values. The quality of the regression is shown in a bubble graph of Fig. 6, where the relative deviation of the regression value from each experimental data point is plotted against the squared density difference; only at low squared density differences is the deviation ±5%. This deviation is a result of the experimental error from the regression procedure and from the temperature dependence of the interfacial
Table 2 Coefficients of the regression function (Eqs. (1) and (2)) b0 (g cm3 K1)
b1
k0 (mN m1)
k1 (cm6 g2)
k2 (cm12 g4)
k3 (cm18 g6)
k4 (cm3 g1)
k5 (mN m1)
k6 (g2 cm6)
0.00022
1.9085
27.514
35.25
31.916
91.016
103.233
4.513
351.903
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Fig. 6. Deviation of experimental points from the regression function. The bubble size is a measure of the temperature at each point.
tension. The size of the bubble represents the temperature; small diameters correspond to low temperatures. There is a distinctive trend within the deviation. The high and low temperature deviations oppose each other: if one has a positive deviation, the other is negative. One reason for this finding might be the use of pure compound densities instead of actual densities to calculate the interfacial tension. If densities of water saturated with carbon dioxide with deviations below ±2% were available, it would be possible to check this hypothesis. Due to the regression procedure, the deviation of low-density differences is larger. The fit method employed takes the value of the abscissa as absolute. For small differences there is a large deviation in the ordinate, whereas there is a much smaller deviation perpendicular to the regression function. As the exponential function controls the rise of the interfacial tension value in this data range, the deviation is large. In Fig. 7, we have plotted the interfacial tension derived from our regression function as a function of pressure and temperature. The critical isotherm of CO2 is marked sepa-
Fig. 7. Interfacial tension as a function of pressure and temperature, as derived with the regression function.
B. Kvamme et al. / Computational Materials Science 38 (2007) 506–513
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rately. Four distinct areas can be observed: the one following the Eo¨tvo¨s-rule at low pressures, which means that the interfacial tension decreases linearly with temperature [13]. Then, there is the region of the vapour pressure points; here CO2 coexists in the gaseous and the liquid state and the density difference has a discontinuity. As a consequence the interfacial tension has two discrete values; one for liquid and one for gaseous carbon dioxide. The transition occurs at the three-phase line liquid–liquid–gas (llg) in the binary system [14]. Next, a plateau appears at higher temperatures and pressures, and finally there is a region with a significant drop in the interfacial tension, where vanishing density differences occur.
˚ to The resulting systems, ranging from 22 · 22 · 66 A ˚ 22 · 22 · 256 A in size were subsequently equilibrated for several tens of picoseconds before the average collection began. The electrostatic interactions were handled by means of Ewald summation technique [15].
3. Simulations
where Pi is average system pressure in the i direction, hz is box length in the direction normal to the interface.
3.2. Interfacial tension estimation Interfacial tension r was obtained by a common method using pressure tensor in case where there are two interfaces normal to the z-axis: 1 r ¼ 0:5 hz P z ½P x þ P y ; 2
3.1. Details 3.3. Simulational results and comparison with experiment The molecular dynamics used constant-temperature and either constant-pressure or constant-volume algorithm supplied by the MDynaMix package of Lyubartsev and Laaksonen [15] modified by us. The starting interfacial system was constructed from two slabs of bulk water (SPC/E [16], either 256 or 512 molecules, and carbon dioxide [17], 100–200 molecules), thermalized initially at 323 K and set side by side, with the periodic boundary conditions applied.
Figs. 8 and 9 present density profiles characteristic for the systems containing, respectively, liquid and gaseous CO2 as one of the phases. The fine-scale profiles of x y averaged quantities of interest (mass density) (mass density) are generated by partitioning the simulation box into discrete bins in the z direction. We followed the general approach of Davidchak and Laird [18] and applied finite
Fig. 8. Density profiles of the water-gaseous carbon dioxide system at 323 K and densities corresponding to 8.3 MPa. Full line: fine-scale water density; dashed–dot line: fine-scale CO2 density (profiles averaged over 50 ps). Dashed lines: coarse-grain profiles obtained by applying FIR.
Fig. 9. Density profiles of the water–liquid carbon dioxide system at 323 K and densities corresponding to 4.5 MPa. See Fig. 2 for symbols.
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Table 3 Experimental and modeling results for water + CO2 systems at 323 K and various elevated pressures p (MPa)
T (K)
Density sat CO2 (kg/m3)
Density sat water (kg/m3)
Cross-sim IFT (mN/m)
Sim IFT (mN/m)
IFT (mN/m)
1.1 2.1 4.18 6.22 8.26 10.28 12.33 14.4 18.43 20.44 22.45
322.8 322.8 322.9 322.9 322.8 322.8 322.9 322.9 322.8 322.8 322.8
18.8484 37.6033 83.397 143.052 235.565 424.325 608.399 686.864 765.667 791.304 812.725
988.52 990.71 995.07 997.94 1000.51 1002.38 1003.75 1005.11 1007.07 1008.15 1009.13
51.8 48.0 48.6 43.8 41.3 37.0 35.7 33.7 34.3 33.7 31.9
53.2 50.1 46.7 44.5 40.7 36.6 35.0 35.4 35.3 34.2 34.7
63.7 57.1 47.5 44.1 38.4 32.5 31.2 30.6 29.8 29.4 29.1
impulse response (FIR) filters to produce a smooth coarsegrained profile from the oscillating fine-scale profiles. The filtered profile was obtained from the fine-scale profile fn as follows [18]: f n ¼
N X
ð3Þ
wk fnþk
k¼N
The shape of the filter was assumed to be Gaussian from the outset: 2
wk ¼ Aeðk=eÞ ;
k ¼ N ; . . . ; N
ð4Þ
The normalization constant A was determined from the condition Rwk = 1. Two filters, each with its own N and e, were used, the first one aimed to average over the oscillations due to structure but retain the essential features of the interfacial region. The values of N = 50 and e = 36.1 appeared to satisfy both conditions for our system and were used to generate a coarse-grained profile. The second filter was used to smooth out only the momentary fluctuations in the density profiles while maintaining its oscillatory nature. This filter was much narrower, with e = 1.6. The gaseous-CO2 profile of Fig. 8 shows a significant increase in CO2 density at the interface relative to its bulk value, reflecting the expected ‘‘sticking’’ effect. Table 3 shows that except in the lowest-pressure and density region, the simulations (see under ‘‘sim IFT’’) appear to reproduce the experimental data with a fair degree of accuracy, especially given the fact that we used force fields designed mostly for bulk properties at ambient conditions. Still, the simulated data using the standard Lorentz–Berthelot mixing rules appeared to mimic the pressure dependence of the experimental results. To improve the fit of modeling results further, the well depth in the Lennard–Jones part of water–CO2 cross-interactions was modified by factors ranging from 0.8 to 1.2, with the factor of 1.2 yielding the best results. These are listed in Table 3 under the ‘‘Cross-sim IFT’’ column header. As one can see from Table 3, while the non-Lorentx–Berthlot corrections brought the modeling results closer to the experimental ones for the high- and mid-range CO2 densities, they slightly worsened the fit in the low-density region.
4. Conclusions Studies of the interfacial tension in the two-phase CO2 + water system show that a suitable time interval has to be identified to enable reliable values to be determined. It has been found that such an interval exists for a sufficiently long time in relation to the measuring time after initial mixing effects have terminated. Later, ageing effects cause a constant reduction in the interfacial tension. Applying the quasi-static method, as proposed here, precise and reproducible data were obtained as a function of the squared density difference. This difference can be calculated by reference equations using the pressure and temperature values. It was possible to derive a regression function from these data. This function is useful for interpolating the interfacial tension of the CO2 + water system. The accuracy of the regression function can be further improved if the actual densities are used instead of the pure substance values. Comparison with the experimental data shows that atomistic simulations using force fields fitted for ambient bulk properties of water and carbon dioxide were quite successful at evaluating the interfacial systems under a broad range of conditions, with the lowest-pressure and density region being the only exception. A possible solution to this problem may lie in either applying exp-6 potentials derived from latest ab initio studies for both water and CO2, or sacrificing potential transferrability altogether in favor of cross-interaction potential specifically designed for CO2 interaction with water. Acknowledgements Authors AH and AO are grateful to the German Federal Ministry of Education and Research for its financial support. One of the authors (TK) is grateful to the Norwegian Research Council for the grant of a postdoctoral scholarship. References [1] A. Hebach, A. Oberhof, N. Dahmen, A. Ko¨gel, N. Ederer, E. Dinjus, J. Chem. Eng. Data 47 (2002) 1540–1546.
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