J. Phys. Chem. B 2008, 112, 12973–12975
12973
Attainable Superheat of Argon-Helium, Argon-Neon Solutions Vladimir G. Baidakov,* Aleksey M. Kaverin, and Valentina N. Andbaeva Institute of Thermal Physics, Urals Branch of Russian Academy of Sciences, Ekaterinburg, 620016, Russia ReceiVed: July 9, 2008; ReVised Manuscript ReceiVed: July 29, 2008
The method of lifetime measurement has been used to investigate the kinetics of spontaneous boiling-up of superheated argon-helium and argon-neon solutions. Experiments were made at a pressure of p ) 1.5 MPa and concentrations up to 0.33 mol% in the range of nucleation rates from 104 to 108 s-1 m-3. The homogeneous nucleation regime has been distinguished. With good agreement between experimental data and homogeneous nucleation theory in temperature and concentration dependences of the nucleation rate, a systematic underestimation by 0.25-0.34 K has been revealed in superheat temperatures over the saturated line attained by experiment as compared with theoretical values calculated in a macroscopic approximation. The revealed disagreement between theory and experiment is connected with the dependence of the properties of newphase nuclei on their size. Introduction The dissolution of a volatile component in a liquid may be accompanied by strong adsorption at the liquid-vapor interface, which leads to considerable changes in the surface tension. The surface tension appreciably determines the work of fluctuation formation of a new-phase nucleus, and consequently the kinetics of the liquid boiling-up. The present paper is devoted to an experimental investigation of nucleation kinetics in argon-helium and argon-neon solutions. The solutions mentioned belong to the class of gassaturated liquids. The surface tension of these solutions was previously measured.1 The results obtained show that helium and neon in argon act as surfactant admixtures, decreasing its surface tension. At a given pressure, the solubility of neon in argon is 3-4× higher than that of helium. The theory of boiling-up of gas-saturated liquids is considered in refs 2 and 3 and in the most general form in refs 4 and 5. As in the case of a one-component liquid, the stationary rate of homogeneous nucleation in a solution is determined by the expression
J ) FBexp(-W* ⁄ kBT)
( ) kBT σ
B)
1⁄2
(2)
where F″* is the density of the number of molecules in a bubble of critical radius R*, σ is the surface tension, and λ0 is the function of thermodynamic and kinetic properties of a solution. At positive pressures and temperatures close to the critical point, the dynamics of growth of a bubble of near-critical size in a superheated gas-saturated liquid is mainly determined by the rate of evaporation and condensation of the mixture * To whom correspondence should be addressed. Tel: +7 343 267 8801. Fax: +7 343 267 8806. E-mail:
[email protected].
2σ ( πm )
1⁄2
(3)
″ -1/2 ″ -2 where m ) [m-1/2 c*] , m 1 and m 2 are the 1 (1 - c*)+m 2 masses of molecules of the first and of the second component, c″* is the concentration of the second component in a critical nucleus, which is close to the concentration of saturated vapor at a flat interface. The value of the kinetic factor depends only slightly on the pressure, temperature, concentration, and the gas dissolved in the liquid. For solutions under investigation, it is (2-6) × 1011 s-1. Taking into account viscous forces, the diffusive supply of the volatile component changes this value by a factor of 1.5 to 2, which leads to changes in the temperature of a solution superheat of no more than 0.03 K. In the Gibbs capillarity theory, the work of formation of a critical bubble is determined by the expression:6
(1)
where F is the particle number density of the solution, B is the kinetic factor, and W/ is the work of formation of a critical nucleus. The kinetic factor takes into account the dynamics of the nucleus growth and, according to ref 5, may be presented as follows:
B ) |λ0|F∗R∗2
components. The viscous forces and the diffusive supply of the volatile component to the bubble may be neglected. Then at small concentrations of helium and neon in liquid argon for the kinetic factor B we can write:
W/ )
16π σ3 3 (p - p)2
(4)
/
If the liquid is incompressible, and the vapor mixture in the bubble is an ideal gas, then at small superheats, when W//kBT . 1, eq 4 for a weak solution may be presented as follows:7
W/ )
16π 3
σ3
( )
(ps - p) 1 + 2
V0
2
(5)
V0
where ps is the saturation pressure of the solution and V′0, V″0 are the specific volumes of a pure solvent in the liquid and the gas phase, respectively. Experimental Section The method of lifetime measurement was used to study by experiment the kinetics of spontaneous boiling-up of argonhelium and argon-neon solutions.8 The experimental setup and
10.1021/jp806048e CCC: $40.75 2008 American Chemical Society Published on Web 09/18/2008
12974 J. Phys. Chem. B, Vol. 112, No. 41, 2008
Figure 1. Temperature dependence of the logarithm of nucleation rate for argon-helium solution: 1 - c ) 0 mol%; 2 - 0.06; 3 - 0.11. Dashed linesscalculation by homogeneous nucleation theory (eqs 1, 3, and 5).
Baidakov et al. dependence of the nucleation rate. On the attainment of a certain temperature (the boundary of attainable superheat) on experimental curves one can observe a section of rapidly increasing J. Here with an increase in the temperature of 1 K, the nucleation rate changes by 10 orders. The considerable deviation of experimental curves from theoretical lines at J < 5 × 106 s-1 m-3 testifies that here, boiling-up is initiated by the action of environmental factors. The determining factors are the natural radiation background8 and the weak spots on the walls of the cell.13 The dashed lines in Figures 1 and 2 show the results of calculating the nucleation rate by homogeneous nucleation theory (eqs 1, 3, and 5) with the use of the surface tension for a flat interface σ ) σ∞1. The discrepancy between experimental and calculated values of attainable-superheat temperatures does not exceed 0.34 K and is of systematic character. Despite good agreement between theory and experiment in the value of the derivative (∂ln J/∂T)p,c experimental values of the superheat temperature are always lower than theoretical. Discussion Nucleation kinetics in superheated pure argon was investigated previously.14,15 Our data on the attainable-superheat temperature for argon within 0.15 K agree with the results of ref 14. From eqs 1 and 5, owing to the weak dependence of the kinetic factor B on temperature, pressure, and concentration, we have:
( ∂ln∂cJ )
p,T
≈-
[(
( )]
W/ 3 ∂σ 2 ∂ps kBT σ ∂c T ps - p ∂c
)
T
(6)
The second term in the right-hand part of eq 6 is 30-40 × smaller than the first, therefore in a first approximation Figure 2. Temperature dependence of the logarithm of nucleation rate for argon-neon solutions: 1 - c ) 0 mol%; 2 - 0.30; 3 0.33. Dashed linesscalculation by homogeneous nucleation theory (eqs 1, 3, and 5).
the experimental procedure are described in detail in refs 5 and 9. A liquid solution is located in a glass capillary of volume V ≈ 83 mm3. The solution is transferred to a metastable state by means of a pressure release (T ) const). The time of expectation of boiling-up of a superheated solution, τ, is measured in experiments at a given temperature, T, pressure, p, and concentration of the second component, c. It is identified with the time of expectation of appearance of the first critical nucleus. The mean lifetime of a superheated solution jτ ) ∑i τi is determined, and the nucleation rate J ) ( jτV)-1 is calculated in a series of N ) 40-80 measurements. The concentration of the solution was determined immediately in the cell by the pressure of saturated vapors before and after the experiments. Data on phase equilibria for argon-helium systems10 and an argon-neon system11,12 were used. Results The experiments were made at one pressure p ) 1.5 MPa and three values of concentration of helium (c ) 0; 0.06; 0.11 mol%) and neon (c ) 0; 0.30; 0.33 mol%) in solution in the range of nucleation rates J ) 104-108 s-1 m-3. The results of the experiments are presented in Figures 1 and 2. As with onecomponent liquids, on the solution isobars one can distinguish two sections that differ in the character of the temperature
( ∂ln∂cJ )
p,T
≈-
3W/ ∂σ σkBT ∂c
( )
T
(7)
In an ideal solution, the derivative (∂σ/∂c)T is proportional to relative adsorption. According to eq 7, relative adsorption is the main parameter determining the concentration dependence of the nucleation rate in a gas-filled solution. For argon-helium and argon-neon solutions (∂σ/∂c)T < 0. With a considerable difference between the solubilities of helium and neon in argon, values of relative adsorption in argon-helium and argon-neon solutions are closely analogous. At the boundary of spontaneous boiling-up (J ) 107 s-1 m-3) values of the derivative (∂ln J/∂c)p,T for the solutions indicated are (3.0 ( 0.1) × 103. In Figure 3, concentration dependences of attainable-superheat temperatures (J ) 107 s-1 m-3) of the solutions investigated and obtained by experiment are compared to those calculated from homogeneous nucleation theory. A decrease in the temperature of a solution superheat with increasing concentration is connected with the decrease of the surface tension at the liquid-vapor interface and the saturation temperature Ts. In this case, the second parameter is the determining one. While losing stability against finite perturbations of the state variables (nucleation process) a metastable solution retains its reducing reaction with respect to infinitesimal variations in composition and density. The boundary of the solution diffusion instabilitysthe diffusion spinodalsis determined by the condition (∂µ/∂c)p,T ) 0, and the boundary of mechanical instability by the condition (∂p/∂V)T,c ) 05. Here, µ is the chemical potential. At a pressure p ) 1.5 MPa in pure argon to the spinodal state corresponds a superheat ∆ Tsp ) Tsp-Ts ) 15.7
Superheat of Ar-He, Ar-Ne Solutions
Figure 3. Concentration dependence of attainable-superheat temperature (J ) 107 s-1 m-3) of solutions: (1) argon-helium; (2) argon-neon. Dashed linesscalculation by homogeneous nucleation theory (eqs 1, 3, and 5).
K, which is approximately 4 K higher than that attained in experiments on nucleation.7 As shown in ref 5 in weak solutions, the position in the phase diagram of the mechanical and the diffusion spinodal differs but little. At a pressure p ) 1.5 MPa and a neon concentration in argon c ) 0.3 mol%, the solution superheat achieved by experiment is ∆Tn ) 14.91 K. The temperature of the diffusion spinodal is 138.9 K. Conclusions Experimental values of the attainable-superheat temperature for solutions, as for pure argon, are systematically lower than those calculated from homogeneous nucleation theory in a macroscopic approximation of σ ) σ∞ (see Figure 3). As shown in ref 5, a systematic discrepancy between theory and experiment for pure cryogenic liquids is connected with the size dependence of the surface tension at the metastable solution-critical bubble interface. For pure liquids taking into account the size dependence, σ(R*), in the context of the van der Waals capillarity theory results in agreement between theory and experiment within their error. It may be suggested that in a solution, too, the systematic underheating to theoretical values has the same reason, i.e., the size dependence, σ(R*). At a helium concentration in argon c ) 0.11 mol% and pressure p ) 1.5 MPa, the theoretical value of the attainable-superheat temperature is Tn ) 135.30 K, which is 0.32 K higher than the experimental value. In this case, the work of formation of a critical nucleus is W// kT ) 75.8, and the radius of a critical bubble R / is equal to
J. Phys. Chem. B, Vol. 112, No. 41, 2008 12975 3.97 nm. A critical bubble contains n″*1 ) 544 argon atoms and n″*2 ) 29 helium atoms. Calculation by formulas 1, 3, and 5 of the critical-bubble surface tension from experimental data on the attainable-superheat temperature Tn and the nucleation rate J ) 107 s-1 m-3 has yielded σ(R*) ) 2.05 mN/m. At given values of Tn and c, the surface tension at a flat argon-helium interface is equal to 2.14 mN/m. The discrepancy between the values of σ(R*) and σ∞ of a solution has the same order of magnitude as in a pure liquid. Thus, an experimental investigation of nucleation in weak solutions of helium and neon in argon at temperatures and pressures close to the critical point of the solvent (argon) with good qualitative agreement between experimental data and homogeneous nucleation theory has revealed their systematic quantitative disagreement. Temperatures of solution superheats attainable in experiments were always lower than their theoretical values. We attribute this discrepancy to the effect of the nucleus size on its surface tension. In this case, the discrepancy between the values of σ(R*) and σ∞ is 4-6%. Acknowledgment. These investigations have been supported by projects of the Programme of Integration Investigations of the Ural and Far-Eastern Branches of the Russian Academy of Sciences and a Programme of the Department of Power Engineering, Machine Building, Mechanics and Control Processes of the Russian Academy of Sciences. References and Notes (1) Kaverin, A. M.; Andbaeva, V. N.; Baidakov, V. G. Zh. Fiz. Khim. 2006, 80, 495. (2) Derjaguin, B. V.; Prokhorov, A. V. Kolloidn. Zh. 1982, 44, 847. (3) Kuni, F. M.; Ogenko, V. M.; Ganyuk, L. M.; Grechko, L. G. Kolloidn. Zh. 1993, 55, 28. (4) Baidakov, V. G. J. Chem. Phys. 1999, 110, 3955. (5) Baidakov, V. G. ExplosiVe Boiling of Superheated Cryogenic Liquids; Wiley-VCH: Weinheim, 2007. (6) Gibbs, J. M. The Collected Works, Vol.2, Thermodynamics; Longmans and Green: New York, London, Toronto, 1928. (7) Baidakov, V. G. Superheat of Cryogenic Liquids; Ural Branch of RAS: Ekaterinburg, Russia, 1995. (8) Skripov, V. P. Metastable Liquids; Wiley: New York, 1974. (9) Baidakov, V. G.; Kaverin, A. M.; Boltachev, G. Sh. J. Chem. Phys. 1997, 106, 5648. (10) Sinor, J. E.; Kurata, F. J. Chem. Eng. Data 1966, 11, 537. (11) Trappeniers, N. J.; Schouten, J. A. Physica 1974, 73, 539. (12) Streett, W. B. J. Chem. Phys. 1965, 42, 500. (13) Kaverin, A. M.; Baidakov, V. G.; Skripov, V. P.; Katianov, A. N. Zh. Tekh. Fiz. 1985, 55, 1220. (14) Baidakov, V. G.; Skripov, V. P.; Kaverin, A. M. Zh. Eksp. Teor. Fiz. 1973, 65, 1126. (15) Skripov, V. P.; Baidakov, V. G.; Kaverin, A. M. Physica 1979, 95A, 169.
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