Nucleation in Liquid Ethane with Small Additions of Methane - The

Sep 4, 2012 - The method of measuring the lifetime of a liquid in a metastable (superheated) state has been used to investigate the kinetics of sponta...
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Nucleation in Liquid Ethane with Small Additions of Methane Vladimir G. Baidakov,* Aleksey M. Kaverin, and Artem S. Pankov Institute of Thermophysics, Ural Branch of the Russian Academy of Sciences, 106, Amundsen Street, Ekaterinburg, 620016 Russia ABSTRACT: The method of measuring the lifetime of a liquid in a metastable (superheated) state has been used to investigate the kinetics of spontaneous boiling-up of ethane−methane solutions. The temperature dependence of the solution mean lifetime has been traced at two values of methane concentration (2.1 and 6.0 mol %) and two pressures (1.0 and 1.6 MPa). The results of experiments are compared with classical homogeneous nucleation theory. For pure ethane at small ( ps) was created in the cell, and the measurement cycle was repeated. A total of n = 20−40 values of τ was fixed at given T, p and the concentration of methane in solution x. Since a spontaneous origination of a nucleus is a random event, it is the mean expectation time for the nucleus appearance τ ̅ (τ ̅ ≫ τb) that has a physical meaning. The −1 . nucleation rate in this case is J = (τV) ̅ In experiments use was made of gases with a content of impurities less than 0.25 mol % in ethane and 0.01 mol % in methane. The temperature was measured by a platinum resistance thermometer with an uncertainty of ±0.02 K, and the pressure was measured by spring-pressure and digital gauges with an uncertainty of ±0.005 MPa. The uncertainty of determination of the expectation time for boiling-up was ±0.01 s.

Figure 1. Nucleation rate (a) and mean lifetime (b) in superheated ethane−methane solutions at methane concentration x = 2.1 and 6.0 mol %, respectively: p = 1.0 (1) and 1.6 MPa (2). The height of the rectangles corresponds to the statistical uncertainty of determination of J and τ.̅ Dashed-dotted lines show calculation by classical homogeneous nucleation theory, eqs 1−4 and 6. Dashed lines present heterogeneous nucleation theory, eq 9.

J(T) one can observe a liquids,8,14−17 on all isobars τ(T), ̅ characteristic section of an abrupt decrease of τ ̅ (increase of J) at the approach of a certain temperature (the boundary of attainable superheating).

3. THEORY OF NUCLEATION IN A DILUTED SOLUTION The work of formation of a critical bubble in a solution is determined by the expression1 20459

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Table 1. Attainable Superheating Temperature Tn (Experiment and Theory), Reduced Work of Formation of a Critical Nucleus W∗/kBT, Critical-Nucleus Radius R∗, Total Number of Molecules n∗ and Number of Ethane Molecules n∗1 in a Critical Bubble, Value of the Kinetic Factor B by Different Approximations at Given Methane Concentrations, Pressures, and a Fixed Nucleation Rate J = 107 s−1 m−3 Tn, K x, mol % 0 0 2.1 2.1 6.0 6.0 a

P, MPa

a

1.0 1.6 1.0 1.6 1.0 1.6

calc.

W∗/kBT

R∗, nm

n∗

n∗1

274.42 278.68 271.61 275.87 266.26 270.50

74.3 74.1 74.3 74.1 74.4 74.2

4.7 5.2 4.6 5.1 4.4 4.9

422 644 410 617 384 565

422 644 378 574 306 457

exp. 273.75 278.17a 271.30 275.68 266.88 271.14

B·10−12, s−1 eq 4

B1·10−11, s−1 eq 7

B2·10−9, s−1 eq 8

2.08 2.28 1.98 2.15

2.42 2.04 2.47 2.10 2.56 2.19

2.85 2.08 7.69 5.73

By data of ref 10.

W∗ =

4 16π σ3 πR ∗2σ = 3 3 (p∗″ − p)2

⎛ 3 − b ⎞ 2̃ 1 ⎡⎛ 3 − b 2 ⎞ 1 ⎛ 3 − b1 ⎞⎤ ̃ 1 ⎜ ⎟λ + ⎢⎜ ⎟ ⎥λ − γ ′ = 0 ⎟+ ⎜ ⎝ b ⎠ 2 ⎢⎣⎝ b2 ⎠ 4 γ ′ ⎝ b1 ⎠⎥⎦

(2)

(6)

where p∗″ is the pressure inside a critical bubble, σ is the surface tension. In a diluted solution the vapor pressure in a critical bubble may be determined from the formula obtained for a onecomponent liquid8 p∗″ − p ≈ (ps − p)(1 − ρ″ /ρ′)

where λ̃ = λ(2R∗/3α1υt1), bi = 2σ/R∗p″∗i, γ′ = α1υt1R∗p″∗2/ 4kBTD2ρx. Two limiting cases can be distinguished in the solution of eq 6. If γ′ ≫ 1 − 3/b1, the solution is determined by the volatility of the solvent, and from eqs 4 and 6, it follows that

(3)

in which ps is the pressure of saturated vapors for the case of a flat interface and ρ′ and ρ″ are the orthobaric densities of the solution. At positive pressures the vapor composition in a critical bubble is not much different from the composition of equilibrium vapor over a flat interface, i.e., x∗″ ≈ xs″. The general solution of the problem on the boiling-up of a superheated binary solution is given in ref 8. For the kinetic factor we have B = ρ∗″λ 0R ∗2(kBT /σ )1/2

B1 =

D2ρxb ⎛ kBT ⎞ ⎜ ⎟ B2 = (1 − b/3) ⎝ σ ⎠

1/2

(4)

(8)

In determining the rate of the substance diffusion supply to a growing bubble use was made of the stationary solution of the diffusion equation. If tR = R/|Ṙ | is the characteristic time of change of the bubble radius, and tD = R2/D2 is the characteristic time of establishment of a stationary concentration x(r) around the bubble, then the fulfillment of the inequality tD/tR ≪ 1 guarantees that the mobility of the bubble boundary will be compatible with a stationary approximation for the concentration distribution in its surroundings.18

αiυtiR ∗p∗″ i

4kBTDiρxi

(7)

which with neglect of b/3 coincides with Kagan’s solution for a one-component liquid.5 Otherwise, when the solvent is nonvolatile and the substance dissolved is supplied only by diffusion, we have

where ρ″∗ is the number density of vapor in a critical bubble and λ0 is the parameter determined by the dynamics of the bubble growth near the critical size. Hereinafter we will consider the solution to be diluted, nonviscous, and volatile. The relation between a freely molecular and a diffusion substance supply8 to a growing bubble is determined by the parameter γi =

1/2 α1υt1 ⎛ σ ⎞ 1 ⎜ ⎟ 2 (1 − b/3) ⎝ kBT ⎠

4. COMPARISON OF THEORY AND EXPERIMENT In the calculations employing the formulas of homogeneous nucleation theory (2−8) use was made of data on the thermophysical parameters of pure ethane, methane and ethane−methane solutions.19−26 The nucleation process proceeds in a superheated liquid at a temperature T>Ts(x,p) and a pressure p < ps(x,T). Therefore, strictly speaking, all of the thermophysical parameters of the liquid that appear in calculation formulas refer to the metastable state (T, p, x). However, owing to the weak pressure dependence of the parameters of the liquid, in a first approximation, it is sufficient to know the properties of the liquid along the saturation line. The surface tension of ethane and ethane−methane solutions was measured in our laboratory.19,20 The pressures of saturated vapors and the orthobaric densities of pure ethane and methane have been taken from the monographs,21,22 the coefficient of viscosity from ref 23. The parameters of phase equilibria for

(5)

where αi is the condensation coefficient of the component i (i = 1,2), υti is the average thermal molecular velocity, p″∗i is the partial pressure of the component i in a critical bubble, and Di is the diffusion coefficient. The condition γi ≫ 1 guarantees that the bubble growth will proceed in the regime of diffusion supply to it of the component i. In the case of a diluted solution, the solvent will be characterized by the free molecular regime of the bubble growth, and the substance dissolved by the diffusion one. At positive pressures, when 3/b ≫ 1, where b = 2σ/R∗p″∗ , the solution may also be regarded as a nonviscous one where inertial effects are negligible. Under these conditions a fourth-degree equation (eq (4.44) from ref 8), determining the parameter λ0 transforms into a quadratic one 20460

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the kinetics of spontaneous boiling-up is the free-molecular supply of the solvent molecules to the bubble. The diffusion supply of the dissolved substance may be neglected in a first approximation. This conclusion is indirectly confirmed by the molecular composition of a critical bubble (see Table 1). The case of B2 (eq 8) is not characteristic for an ethane−methane system and leads to essentially different values of the kinetic factor as compared with the common solution of eqs 4 and 6. It should be noted that a change of an order in the kinetic factor changes the attainable superheating temperature by approximately 0.15 K. The relation between experimental and theoretical values of Tn in an ethane−methane system is not characteristic of the gassaturated solutions of cryogenic liquids investigated by us earlier, where the easy-to-volatilize component was helium.14−16 A decrease in the discrepancy between theory and experiment in Tn as the solution concentration approaches the equimolar one was previously observed in an argon−krypton system characterized by a complete solubility of the components in the region of homogeneous nucleation.17 As is shown in ref 8, a systematic “underheating” of a pure liquid to theoretical values can be interpreted by the size dependence of the surface tension of a critical bubble. At a nucleation rate J = 107 s−1 m−3 the surface tension of a critical bubble σ(R *) for condensed inert gases is 5−7% smaller than at a planar interface. Numerical calculations of σ(R∗) for an argon− krypton system in the framework of the van der Waals theory of capillarity have shown that an addition of krypton into argon or of argon into krypton results in the disturbance of the monotonic character of the dependence of surface tension on the size of the bubble σ(R∗). At the approach of the concentration of the solution components to the equimolar one, there appears a maximum on this dependence σ(R∗). The presence of such a maximum of the function σ(R∗) leads to a decrease in the discrepancy between data on Tn of experiment and theory if in theory the value of σ at a flat interface is replaced by the value σ(R∗). It is not improbable that in an ethane−methane system the size effect is expressed more vividly, which may be the reason for the “superheating” of the solution at x = 6.0 mol % beyond the theoretical value.

ethane−methane solutions were calculated from data of refs 13 and 24−26. The diffusion coefficient D2 was evaluated by the Stokes−Einstein formula. The results of calculating the temperature dependence of the homogeneous nucleation rate (mean lifetime) by eqs 1, 2, and 6 at two values of concentration and pressure are shown in Figure 1a,b as dashed-dotted lines. Table 1 and Figure 2 present

Figure 2. Concentration dependence of the attainable superheating temperature Tn for an ethane−methane solution at pressures p = 1.0 (1) and 1.6 MPa (2) and nucleation rate J = 107 s−1 m−3. The values of Tn for pure ethane are taken from ref 10. Dashed-dotted lines show calculation by classical homogeneous nucleation theory, eqs 1−4 and 6. Dashed lines show additive approximation for the attainable superheating temperature.

attainable-superheating temperatures for pure ethane and ethane−methane solutions at the pressure values investigated and a nucleation rate J = 107 s−1 m−3. Table 1 also gives values of the work of formation of a critical nucleus W∗/kBT, the radius of a critical bubble R∗, the number of molecules in a bubble n∗ and the kinetic factor B calculated by eqs 4 and 6−8. As is evident from Figure 1a,b, experimental data agree with predictions of classical homogeneous nucleation theory when certain temperatures Tn are achieved (boundaries of attainable superheating, J ≥ 2.5 × 106 s−1 m−3). Here experimental and theoretical values of the derivatives (∂ lgJ/∂T)px are similar. The discrepancy between experimental and calculated values of Tn does not exceed 0.8 K and has a systematic character. For pure ethane10 the values of the attainable superheating temperature measured in experiment are 0.5−0.7 K smaller than their theoretical values. Dissolution of methane in ethane leads to changes in the relation between theoretical and experimental values of Tn. At a methane concentration in the solution of 6.0 mol %, the superheating temperature Tn attained in experiment exceeds the theoretical value by approximately 0.7 K (Figure 2). Experimental values of Tn at all concentrations are lower than those calculated by the additivity law Tn,add = TnC2H6(1 − x) + TnCH4x, where TnC2H6 and TnCH4 are the temperatures of attainable superheating of pure ethane and methane, respectively. The dashed line in Figure 2 corresponds to this law. From Table 1 it follows that the function B in all approximations has a weak dependence on temperature and pressure as compared with the exponential factor in eq 1. In the case where the nucleation kinetics is determined by the volatility of the solvent (eq 7), the value of B1 is little different from the value of B, calculated by the commonly employed relations, eqs 4 and 6. This result points to the fact that the determining factor in

5. HETEROGENEOUS AND INITIATED NUCLEATION Classical homogeneous nucleation theory predicts a sharp temperature and pressure dependence of the nucleation rate. Experimental data follow this dependence only at nucleation rates that exceed 2.5 × 106 s−1 m−3 (see Figure 1a, b). At lower values of the nucleation rate, J, experimental curves deviate from theoretical lines. This result points to the nonhomogeneous character of nucleation in this region of state parameters. One of the reasons for premature boiling-up of a superheated liquid at J < 2.5 × 106 s−1 m−3 may be the existence of gassaturated solid particles or regions of the walls of the measuring cell with similar properties, poorly wetted inclusions, cracks, etc. No cleaning assures a complete absence of impurities in the sample under investigation. Nevertheless, the contact area of a liquid with the solid walls of the measuring cell is the most probable site of heterogeneous nucleation of the vapor phase. Heterogeneous nucleation theory may be built by the same scheme as homogeneous nucleation theory. Let a liquid contact the inner surface of a measuring cell which is homogeneous and smooth. Then for the stationary rate of heterogeneous nucleation we can write then 20461

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nm and a wetting angle θ ≈ 170° are observed on solid substrates by atomic-force microscopy.29 At T = 264 K, p = 1.0 MPa, and x = 6.0 mol %, the value of B in the case of homogeneous nucleation is equal to 2.6 × 1011 s−1. For heterogeneous nucleation, assuming that B = Bhet, from NsBhet = 1.5 × 107 s−1 m−3, we have Ns ≈ 6 × 10−5 m−3. This signifies the practical absence of heterogeneous centers in the measuring cell. Even if the cell surface possesses incredibly poor wettability (θ ≈ 150°), heterogeneous nucleation of the vapor phase on it at a rate observed in experiment is unlikely. Thus, the bends of experimental curves (see Figure 1a,b) cannot be caused only by heterogeneous nucleation. Another reason for premature (as compared with the prediction of homogeneous nucleation theory) boiling-up of a superheated liquid at relatively large mean lifetimes may be the radiation background and cosmic radiation.3 The phenomenon of initiation of boiling-up of a superheated liquid by ionizing radiation, first discovered by Glaser,30 is widely used in bubble chambers for registration of the tracks of fast particles.31 A phenomenological theory of initiated nucleation close to the boundary of spontaneous boiling-up of a superheated liquid is considered in.3,31 According to this theory, the initiating effect is connected with the formation in the liquid along the particle trajectories of regions of local heating, the so-called “thermal spikes”, on which there form bubbles capable of further growth. An indirect corroboration of such a mechanism of initiation may be experiments on the superheating of liquids in a field of γradiation, in which it has been found that an increase in the intensity of the radiation background leads to a decrease in the mean lifetime of a superheated liquid in the region adjacent to the boundary of spontaneous boiling-up.3,28,32 In this case the character of the dependences τ(T) and J(T) remains the same as ̅ in the absence of additional radiation. In the phenomenological model of initiation of boiling-up of a superheated liquid by ionizing radiation the transformation of experimental dependences τ(T) and J(T) into “the plateau” at ̅ the approach to the boundary of spontaneous boiling-up of the liquid (Figure 1a,b) is explained by the fact that the work of formation of a critical nucleus becomes here much smaller than the average energy of “a thermal spike”, and the probability of boiling-up on every “thermal spike” is close to unity. In this case the rate of nucleation of critical bubbles, which depends on the radiation intensity, is registered in the experiment. If the transfer of the energy of a fast particle expended in the formation of a bubble is hampered, there is no “plateau” on experimental curves, and the deviation of experimental data from theoretical lines occurs at higher values of τ ̅ (smaller values of J). This effect is observed in the series of investigations of condensed inert gases.32,33 For xenon the limiting rate of initiated nucleation is Ji0 = 2.2 × 104 s−1 m−3, for krypton 1.0 × 106 s−1 m−3, and for argon 2.5 × 106 s−1 m−3. In xenon or krypton there is no “plateau”, but in argon it appears at low pressures. The low radiation sensitivity of xenon is caused by its considerable scintillation properties. As a result, the energy of an ionizing particle changes into radiation energy rather than into heat. The scintillation capacities of condensed inert gases increase with increasing atomic number.34

(9)

where Ns is the total number of liquid molecules at the inner surface of the cell related to its volume, Bhet is the kinetic factor at heterogeneous nucleation of boiling centers, which is limited from above by the value of B for homogeneous nucleation B ≥ Bhet, and Ψ is the correction factor which takes into account the reduction in the work of formation of a critical nucleus on the wall as compared with the work W∗ inside the liquid. Since Ψ < 1, the exponential factor in eq 9 is larger than in eq 1. However, the product NsBhet is much smaller than the preexponential factor in eq 1 because Ns ≪ Nv. Owing to this the temperature dependence of the heterogeneous nucleation rate is bound to be weaker than that of the homogeneous one. For nucleation at the wall to compete with that in the body of the liquid the following inequality must be satisfied Ψ ≤ 1 − (W∗/kBT )−1 ln(NvB /NsBhet )

(10)

Let us take the thickness of the liquid wall layer to be equal to the diameter of a critical nucleus. Then for the cell used in this work the volume of this layer is 3.3 × 10−5 of the volume of the liquid being superheated. To the attainable superheatings corresponds a value of the work of critical bubble formation equal to W∗ ≈ 74kBT. Assuming that the kinetic factors B and Bhet are equal, from correlation (10) we have Ψ ≤ 0.86. The value of Ψ depends on the microrelief of the wall and the equilibrium wetting angle θ (inside the liquid). For a solid smooth surface, one obtains Ψ(θ) = 1/4(1 + cos θ)2(2 − cos θ) Then Ψ ≤ 0.86 corresponds to a value of the wetting angle equal to θ ≥ 58°. The solutions investigated in the present paper have an equilibrium wetting angle for glass and most solid surfaces close to zero;27 therefore, the probability of spontaneous nucleation in bulk must be much higher than the probability of boiling-up on the smooth wall of a cell or on the surface of impurity particles. A real solid surface is rough rather than smooth. There are “weak spots” on it. These are microcavities with considerable curvature, poorly wetted areas of a flat surface, on which viable bubbles arise. In processes in which liquids are rapidly transferred from a stable to a metastable state it is difficult to assume the stability of a gas inclusion in a microrecess. For instance, for a conical microrecess with an angle between the axis and the cone generatrix β the surface forces will impede the filling of the cavity with the liquid only on condition that θ > β + π/2.28 On the assumption that the deviation of experimental curves from those calculated by homogeneous nucleation theory is caused only by heterogeneous nucleation experimental data may be used for evaluating the correction factor Ψ and the preexponential factor NsBhet which enter into eq 9. At all investigated pressures and solution compositions the value of Ψ is 0.012− 0.028, which corresponds to a wetting angle θ equal to 145− 150°. The pre-exponential factor in eq 9 in this case is equal to (1−3) × 107 s−1 m−3. The results of calculating the dependences J(T) and τ(T) with these parameters are shown in Figure 1a, b as ̅ dashed lines. In the case of poor wettability of a solid surface a vapor bubble is a sphere segment. At θ = 150° and a critical-bubble radius R∗ = 6 nm the diameter of the base of a sphere segment is a = 2R∗ sin(π − θ) ≈ 6 nm, and its height h ≈ 0.8 nm. As has already been mentioned, such large wetting angles are not characteristic of the liquid under investigation. Nevertheless, long-living nanobubbles of such a form with a base diameter of ∼100 nm, a height of ∼10

6. CONCLUSION A first-order phase transition begins at isolated “points” of a homogeneous system. These “points of growth” or viable nuclei must satisfy the condition R > R∗,and then the growth of a new phase is accompanied by a decrease in the thermodynamic potential. Nuclei may form both as a result of fluctuations in a 20462

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homogeneous liquid (homogeneous nucleation) and at “weak sites” of the system: a gas dissolved in the liquid, poorly wetted sections of the vessel walls, etc. (heterogeneous nucleation). Like homogeneous, heterogeneous nucleation is of fluctuation (activation) character. The present paper investigates fluctuation nucleation in ethane−methane solutions. The data obtained for the attainable-superheating temperature Tn at nucleation rates J ≥ 2.5 × 106 s−1 m−3 are in satisfactory agreement with calculations by classical homogeneous nucleation theory. The derivatives (∂ lg J/ ∂T)p,x that characterize the temperature dependence of the nucleation rate are also similar here. At the same time, a more detailed comparison of theory and experiment reveals a systematic discrepancy in the values of Tn. If for pure ethane the theoretical value of Tn exceeds the experimental one, for a solution (x = 6.0 mol %) theory gives smaller values of Tn than experiment. Such a behavior of Tn may be connected with the adsorption processes at the liquid−vapor bubble interface, which leads to changes in the surface tension. With small nuclear dimensions the dependence of surface tension on the curvature of the separating surface must manifest itself. This dependence may be of qualitatively different character for a pure liquid and a solution. This is confirmed by calculations of the surface tension of vapor bubbles in the framework of the van der Waals capillarity theory.17,35 The striking discrepancy between theory and experiment at J < 2.5 × 106 s−1 m−3 points to the inhomogeneous character of nucleation in this range of state parameters. Here the dependence J(T) cannot be explained only by heterogeneous nucleation of the vapor phase. To describe superheating observed in experiment in the context of heterogeneous nucleation theory, it is necessary to assume incredibly poor wettability of the solid wall, when the wetting angle reaches ∼150°. Besides, at such poor wettability there are practically no centers in the cell at which viable bubbles can be activated. It should be mentioned that in our experiments the beginning of measurements was always preceded by a rather long process of “run-in” of the measuring cell. After the cell was filled with the liquid under investigation several tens of boiling-ups were carried out. The pressure on the liquid decreased to a value p < ps with a subsequent suppression of the boiling liquid by a pressure that exceeded ps. As a rule, in the first boiling-ups the lifetime of a superheated liquid was an order of magnitude smaller than that achieved after the “run-in”. In experiments on krypton there sometimes appeared “a plateau”.8 In the process of depressurization−pressure increase cycles there was an outflow of the gas phase from microrecesses in the cell walls and its subsequent dissolution in the liquid bulk. Another, more probable reason for premature, as compared with homogeneous nucleation theory, boiling-up of liquids at small superheatings is the ionizing action of the radiation background. The effect that initiates nucleation under radiation is connected with the formation of thermal spikes in the liquid, on which there form viable nuclei. The mechanism and details of the initiation process are not sufficiently clear for calculating directly the probability of nucleation from atomic-molecular characteristics of a substance and its thermophysical properties. Therefore, experiment gives only an indirect corroboration of this mechanism.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +7 343 267 88 01. Fax: +7 343 267 88 00. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work has been performed with a financial support of the Ural Branch of RAS (Project No. 12-S-2-1013 of joint investigations of the Ural and Far Eastern Branches of the Russian Academy of Sciences), the Russian Foundation for Basic Research (Project No. 10-08-96043-r_ural_a), and the Government of the Sverdlovsk region.

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