Vapor Phase Homogeneous Nucleation of Higher Alkanes: Dodecane

Vapor Phase Homogeneous Nucleation of Higher Alkanes: Dodecane, Hexadecane, and Octadecane. 1. Critical Supersaturation and Nucleation Rate ...
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J. Phys. Chem. B 2001, 105, 11866-11872

Vapor Phase Homogeneous Nucleation of Higher Alkanes: Dodecane, Hexadecane, and Octadecane. 1. Critical Supersaturation and Nucleation Rate Measurements† Mark Rusyniak, Victor Abdelsayed, Jason Campbell, and M. Samy El-Shall* Department of Chemistry, Virginia Commonwealth UniVersity, Richmond, Virginia 23284-2006 ReceiVed: June 5, 2001

The critical supersaturations and isothermal homogeneous nucleation rates of dodecane, hexadecane, and octadecane vapors have been measured over wide temperature ranges (e.g., 285-340 K) using an upward thermal diffusion cloud chamber. Dodecane shows the best agreement between the experiment and the classical nucleation theory. Hexadecane and octadecane exhibit critical supersaturations lower than those predicted by the theory. All three compounds show increasing deviation from theory with decreasing temperature. This trend is also observed in the measured nucleation rates where the predictions of the theory at lower temperatures are shifted to much lower values compared to the experimental data. Temperature dependence correction factors to the calculated rates have been evaluated based on the experimental rates of the studied higher alkanes. The classical theory predicts the correct supersaturation dependence of the nucleation rates of these compounds. Analysis of the supersaturation dependence of the nucleation rate of octadecane indicates that the critical cluster contains 30-35 molecules at the nucleation temperatures of 315-340 K. The classical theory predicts smaller nucleus size as a result of the overestimation of the supersaturation corresponding to the measured nucleation rate.

I. Introduction Homogeneous nucleation from a supersaturated vapor is a process by which the metastable supersaturated vapor state decays by the spontaneous occurrence of thermal fluctuations.1-4 This process results in the formation of droplets of the liquid phase; clusters that are larger than critical sizes grow, and thus the stable phase results. The nucleus is the cluster that has equal probabilities of decay and growth. The rate of nucleation at which the embryos of the liquid phase forms increases exponentially with [-(ln S)-2], where S is the supersaturation ratio. The supersaturation at which the metastable state collapses and the nucleation rate increases explosively is known as the critical supersaturation (Sc). Although vapor phase homogeneous nucleation is the simplest form of a nucleation process, a complete understanding of all of the factors that influence the nucleation rate is still lacking. Most of the studies attempt to compare the experimental measurements with the predictions of the classical nucleation theory (CNT).1-4 This theory is based on the capillarity approximation, which treats the nucleus for condensation as a small fragment of a bulk liquid having the same macroscopic properties such as surface tension and density. Within the framework of the CNT, the rate of nucleation from the vapor phase depends on the surface free energy of the critical clusters, the temperature, and supersaturation of the vapor. Although the CNT has provided significant progress in qualitatively understanding the factors that control the formation of the new phase, it fails to provide a consistent molecular picture of the nucleation process as well as quantitative predictions of the rate of nucleation as a function of temperature and supersaturation. For example, it has been noted that the agreement between experimental data and the CNT is satisfactory only for simple substances such as nonpolar molecules and †

Part of the special issue “Howard Reiss Festschrift”. * To whom correspondence should be addressed.

small alkanes.5-10 However, significant disagreements with the predictions of the CNT have been reported for the homogeneous nucleation of supersaturated vapors of associated,11,12 highly polar,13,14 and hydrogen-bonding molecules.15 This suggests that the CNT has several inherent inconsistencies that become more significant in describing the nucleation behaviors of complex systems. Several models have been suggested to improve the theory, but in most cases, they have retained the capillarity approximation.16-21 Furthermore, none of these models has shown an overall improvement over the CNT for different classes of substances. Other significant developments based on density functional theory, kinetic molecular theory, rigorous statistical mechanics, and molecular simulation methods have been developed with the general goals of providing a molecular level description of the nucleating clusters and inferring the molecularity of the nucleation process from the measured rate data.1,2,22 Several useful reviews on the recent developments of nucleation theories are available in the literature.1,2,22 Systematic studies of homogeneous nucleation of different classes of compounds with related molecular properties are desirable in order to test nucleation theories. Such studies can also provide insight into the role of different molecular properties in the nucleation process. This information is necessary in order to develop scaling laws for homogeneous nucleation, which are not dependent on any particular form of the theory. In this way, general patterns in nucleation behavior can be revealed that cannot be seen on the basis of a substanceby-substance comparison between experiment and theory. For example, homogeneous nucleation studies of n-alkanes,6,7,23-26 aliphatic alcohols,27-29 nonpolar tetrachloride compounds,9,10 and highly polar13,14 and hydrogen bonding liquids15 have provided the necessary data to develop and test scaling models for nucleation.30-34 The homogeneous nucleation of n-alkanes CH3(CH2)n-2CH3 constitutes a special case of both scientific and practical interest.

10.1021/jp012117v CCC: $20.00 © 2001 American Chemical Society Published on Web 11/26/2001

Nucleation of Higher Alkanes The phase transition properties of n-alkanes have received a great deal of interest in recent years mostly due to their importance in the petrochemical industry.35-37 The study of the nucleation behaviors of higher alkanes provides the opportunity for examining the effects of chain length on the barrier height of nucleation and the influence of the dynamics of molecular folding on the rate of nucleation.38 Other issues relating to the packing of long chain alkanes in the critical cluster and the possible orientation of the molecules at the surface of the cluster may also be examined through experimental and theoretical studies of the nucleation of higher alkanes.38 However, vapor phase nucleation was not studied for higher alkanes with more than 10 carbon atoms. In the present study, we investigate the vapor phase homogeneous nucleation of a series of higher alkanes, namely, dodecane CH3(CH2)10CH3, hexadecane CH3(CH2)14CH3, and octadecane CH3(CH2)16CH3. These linear chain alkanes are characterized by their low vapor pressures and relatively high melting points. The long chain alkanes exhibit well-known anomalous behavior in the crystal nucleation from the melt particularly in the range of C15 to C32.39,40 The anomalously low undercooling ratio for crystallization has been rationalized in terms of the transition from short-chain “bundle crystallization” behavior to the folded-chain nucleation of long chain polymers.38,41 The study of the homogeneous nucleation of supersaturated higher alkane vapors can provide, in principle, information on the energetics and kinetics associated with the formation of the critical cluster. The data we present here complements and extends the present body of knowledge concerning vapor-to-liquid phase change for the alkane series. We report the critical supersaturations and the isothermal nucleation rates in the range of 4 × 10-4 to 5 cm-3 s-1 over the temperature range of 290-340 K. Nucleation rates of higher alkanes as functions of both temperature and supersaturation allow a sensitive test of nucleation theories and scaled nucleation models. Theoretical models of the shape, size, and surface properties of the critical nucleus may shed more light on the nucleation behavior of higher alkanes. II. Experimental Section The present data were obtained using an upward thermal diffusion cloud chamber. The general description of this device and the details of its operation are available elsewhere.14,15 The onset of nucleation is determined by observing the forward scattering of light from drops falling through a horizontal HeNe laser beam positioned near the middle of the chamber. These drops originate near the elevation at which the maximum (peak) supersaturation occurs (∼0.7 reduced height). The critical supersaturation measurement consists of setting the temperatures of the top and bottom plates so that the chamber is barely nucleating at a rate of 1-2 drops cm-3 s-1. A photomultiplier positioned to detect the forward scattered light is used with discrimination and counting electronics to measure the number of nucleating droplets within a well-defined volume. Under these conditions, measurements are made of the total pressure Pt, the temperature of the evaporating pool T0, and the temperature of the upper plate T1. The temperatures of both plates are then decreased (or increased) by about 5 °C, and the measurement is repeated. To ensure reproducibility, the experiments were repeated at least three times. To eliminate the interference from ion nucleation resulting from cosmic ray or natural radioactive sources, the measurements were performed with a constant electric field of 70 V/cm applied between the chamber plates. The isothermal rate measurements are made using the same device described above, only now we hold the temperature at

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11867 the nucleation zone constant while varying the supersaturation. Standard deviation in temperature for all of the isotherms reported here is less than 0.2 °C. The viewing area, the intersection of the He-Ne laser and the region in space viewed by the photomultiplier, is kept constant for all of the rate measurements on one isotherm. To minimize boundary effects caused by heating the chamber walls, we use a small viewing area located in the center of the chamber, typically between 2 and 4 cm2. The use of a small viewing area requires very long acquisition times in order to measure low nucleation rates. A single data point may require as long as 2 h in order to count a statistically meaningful number of nucleation events (typically ranges from ≈10 at the lowest rates to ≈1000 at the highest rates). The measured flux of droplets (in units of droplets cm-2 s-1) is converted into rates (in units of cm-3 s-1) using a procedure suggested by Hung et al.25 All chemicals were obtained from Aldrich with stated purity of at least 99% and were degassed repeatedly by the freezepump-thaw method and transferred to the cloud chamber through a vacuum line. The carrier gas was research grade helium (99.999% pure). III. Results and Discussion The chamber parameters T0, T1, and Pt at which a steady nucleation rate of 1-2 drops cm-3 s-1 occurs for dodecane, hexadecane, and octadecane are listed in Tables 1-3 of the Supporting Information, respectively. In all of the experiments reported, no visible convection was observed in the chamber and the total vapor density was a monotonically decreasing function of the chamber height. The thermophysical properties needed to solve the boundary value problem associated with heat and mass flux in the chamber to obtain the supersaturation and temperature profiles are given in Table 1.35,42-46 On the basis of the estimated errors in the thermophysical properties and the uncertainty in measuring T0, T1, and Pt, we estimate the overall error in the experimentally determined supersaturation as 3-5%. The supersaturations at the maximum rate plane in the chamber as a function of temperature are plotted for each experiment and displayed in Figures 1-3 for dodecane, hexadecane, and octadecane, respectively. The prediction of the CNT for the rate of homogeneous nucleation J is computed from

J ) K exp(-W*/kBT)

(1)

where kB is the Boltzmann constant, T is the temperature, and W* is the barrier height for nucleation. The preexponential factor K is calculated from

K ) (2σM/πNA)1/2(P/kT)2/F

(2)

where F is the liquid density, NA is Avogadro’s number, σ is the flat surface tension of the liquid, M is the molecular weight, and P is the pressure of the vapor. The central quantity in the rate equation is the barrier height W* (reversible work expended in the formation of a nucleus) that is given by

1 W* ) n*∆µ 2

(3)

where ∆µ is the difference in chemical potential between the supersaturated vapor and bulk liquid and n* is the number of molecules in the critical cluster (condensation nucleus). Under the assumption of ideal gas behavior, ∆µ ) kT ln S where S is the supersaturation ratio (P/Pe) and Pe is the equilibrium or

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TABLE 1: Thermophysical Properties of Dodecane, Hexadecane, and Octadecanea dodecane M ) 170.34, Tc ) 658, Pc ) 1.82, Tb )489.438 D12b ) 0.156445, sb ) 0.75, ac ) 0.3, ∆Hvapd ) 44.09 de ) 7.624318 × 10-1 - 6.807188 × 10-4t - 3.776708 × 10-7t2 Pef ) 101.325 exp[(1 - Tb/T) exp(3.05854 - 2.018454 × 10-3T + 1.606849 × 10-6T2)] σg ) 27.12 - 0.08843t Cph ) 2.537561 × 102 - 1.132748T + 5.821040 × 10-3T2 - 5.867009 × 10-6T3 ηi ) 5.36 × 10-6T1.5/(T + 330.87) λj ) 4.51467 × 10-3 - 4.157221 × 10-5T + 2.354434E × 107T2 - 1.665414 × 10-10T3 hexadecane M ) 226.46, Tc ) 723, Pc ) 1.40, Tb )559.978 D12b ) 0.1334702, s ) 0.75, a ) 0.3, ∆Hvap ) 51.84 de ) 7.776562 × 10-1 - 6.201299 × 10-4t - 2.773269 × 10-7t2 Pef ) 101.325 exp[(1 - Tb/T) exp(3.18271 - 2.002545 × 10-3T + 1.384476 × 10-6T2)] σg ) 29.18 - 0.08540t Cph ) 3.316097 × 102 - 1.472467T + 7.658429 × 10-3T2 - 7.731156 × 10-6T3 ηi ) 4.40 × 10-6T1.5/(T + 314.10) λj ) 3.249620 × 10-3 - 2.956431 × 10-5T + 1.709124 × 10-7T2 - 1.225110 × 10-10T3 octadecane M ) 254.50, Tc ) 747, Pc ) 1.29, Tb ) 590.023 D12b ) 0.125498, s ) 0.75, a ) 0.3, ∆Hvap ) 55.23 de ) 7.948428 × 10-1 - 6.233769 × 10-4t - 2.545106 × 10-7t2 Pef ) 101.325 exp[(1 - Tb/T) exp(3.24741 - 2.048039 × 10-3T + 1.362445 × 10-6T2)] σf ) 29.98 - 0.08428t Cph ) 3.703554 × 102 - 1.640369T + 8.570411 × 10-3T2 - 8.655875 × 10-6T3 ηi ) 4.40 × 10-6T1.5/(T + 314.10) λj ) 2.765490 × 10-3 - 2.503319 × 10-5T + 1.458922 × 10-7T2 - 1.050322 × 10-10T3 heliumk M ) 4.006 η ) 1.455 × 10-5T1.5/(T + 74.1) λ ) 7.37697 × 10-5 + 1.139222 × 10-6T + 6.343536 × 10-10T2 a Values of M, molecular weight in g/mol; T , critical temperature in K; P , critical pressure in MPa; T , normal boiling point in K; D , binary c c b 12 diffusion coefficient at 273 K and 101.3 kPa; s, exponent of the temperature dependence; a, thermal diffusion factor; ∆Hvap, the enthalpy of vaporization at the normal boiling point in kJ mol-1; d, the density of the liquid in g/cm3; Pe, the equilibrium vapor pressure in kPa; σ, the surface tension in dyn/cm; Cp, the isobaric heat capacity of the vapor in J/mol K; η, the viscosity of the vapor in poise; λ, the thermal conductivity of the vapor in J/(m s K). T ) temperature in K, t ) temperature in °C. b Reference 42. c The thermal diffusion factor is approximated as a ) 0.3 by analogy to the values for other mixtures found. Grew, E.; Ibbs, T. L. Thermal Diffusion of Gases; Cambridge University Press: New York, 1952; pp 128-130. d Reference 35. e Reference 43. f Reference 35. g Reference 45. h Polynomial obtained from fit to data in ref 35, eq 29. i Reference 44, eq 9-4.15, p 400. j Reference 44, eq 10-3.17), p 505. k Reference 46.

Figure 1. Critical supersaturation (Sc) vs temperature for dodecane. Solid line represents classical theory.

“saturation” vapor pressure at the temperature of the vapor T. The Gibbs -Thompson relation is used to calculate n* according to

n* )

32πσ3M2 3∆µ3F2

(4)

By setting J ) 1 in eq 1 and using the literature values of the equilibrium vapor pressure, liquid density, and surface

Figure 2. Critical supersaturation (Sc) vs temperature for hexadecane. Solid line represents classical theory.

tension (as listed in Table 1), the CNT dependence of Sc on T is obtained for each compound. The results are shown as the solid curves in Figures 1-3. It is clear from the results in Figures 1-3 that the predictions of the CNT deviate strongly from the experimental results particularly at lower temperatures. The magnitude of the discrepancy in Sc is shown in Table 2 where we compare the experimental and theoretical Sc for the investigated compounds

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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11869 TABLE 4: Measured and Calculated Data for Rate Experiments with Dodecanea

Figure 3. Critical supersaturation (Sc) vs temperature for octadecane. Solid line represents classical theory.

TABLE 2: Comparison between Experimental, Sexp, and Calculated, SCNT, Supersaturations at Different Temperatures temp, K

Sexp

SCNT

306.00 317.00 328.00 340.00 318.00 326.00 334.00 342.00

14.0 4.4 1.5 7.1 38.2 29.2 21.1 11.6 35.4 28.6 23.5 20.4

TABLE 3: Molecular Properties and Critical Supersaturation Parameters of the Alkane Series at Tra ) 0.45 compound hexanec heptanec octanec nonanec decanec dodecane hexadecane octadecane

To

T1

T

S

Jexp

Jexp(max)

377.05 376.03 375.24 374.43 373.70 365.01 365.05 366.40 367.33 380.20 380.76 381.75 382.69 382.71 383.48 359.66 359.83 361.03 360.33 359.21 358.05 358.14

289.64 289.87 290.16 290.31 290.46 281.32 281.35 281.22 280.98 296.77 296.58 296.40 296.19 296.23 295.90 274.63 274.75 274.55 274.31 274.53 274.67 275.00

308.15 308.18 308.28 309.06 309.06 299.18 299.21 299.29 299.23 315.77 315.73 314.99 315.01 315.04 314.94 291.63 291.75 291.74 291.46 291.50 291.46 291.72

16.84 16.00 15.28 14.38 13.88 18.25 18.24 19.42 20.47 11.93 12.28 13.08 13.65 13.62 14.23 24.16 24.12 25.73 25.41 23.86 22.50 22.08

1.98 4.32 × 10-1 3.56 × 10-2 3.64 × 10-3 2.60 × 10-4 1.01 × 10-3 1.71 × 10-3 6.73 × 10-2 4.72 × 10-1 5.07 × 10-5 1.17 × 10-3 1.79 × 10-2 2.32 × 10-1 2.86 × 10-1 1.30 5.40 × 10-2 8.29 × 10-2 8.38 × 10-1 4.93 × 10-1 4.03 × 10-2 8.88 × 10-4 9.13 × 10-4

3.79 8.43 × 10-1 7.05 × 10-2 7.33 × 10-3 5.32 × 10-4 2.11 × 10-3 3.56 × 10-3 1.36 × 10-1 9.36 × 10-1 1.03 × 10-4 2.35 × 10-3 3.54 × 10-2 4.52 × 10-1 5.58 × 10-1 2.51 1.11 × 10-1 1.70 × 10-1 1.68 1.00 8.37 × 10-2 1.89 × 10-3 1.94 × 10-3

a T , the temperature of the pool surface in Kelvin; T , the temperature 0 1 of the top plate in Kelvin; Pt, the total pressure in kPa; T, the calculated temperature at the maximum rate plane; S, the calculated supersaturation at the maximum rate plane; Jexp the measured droplet flux in drops cm-2 s-1; Jexp(max), derived nucleation rate in drops cm-3 s-1.

% diff.

Dodecane 31.2 36.2 20.1 21.1 13.5 13.3 9.7 9.1 Hexadecane 53.7 86.9 38.9 55.0 28.9 36.7 21.8 24.6 Octadecane 72.9 112.8 56.4 79.0 43.7 57.1 33.8 42.5

285.00 301.00 317.00 333.00

Pt 32.61 32.59 32.49 32.47 32.35 31.08 31.10 31.18 31.24 33.20 33.27 33.37 33.48 33.56 33.62 30.28 30.29 30.34 30.33 30.30 30.18 30.20

T

ωb

Sc,exp

SCNT

% diff.

n*

228.4 243.1 255.9 267.6 278.0 296.1 325.4 336.1

0.299 0.350 0.397 0.443 0.490 0.573 0.737 0.812

11.9 12.9 14.5 16.3 17.1 23.0 31.0 40.8

10.1 12.2 14.1 16.4 18.7 24.1 39.7 50.7

15.1 5.4 2.8 0.6 9.4 4.8 28.1 24.3

51 47 44 41 39 35 29 26

a T ) reduced temperature (T/T ). b Pitzer acentric parameter, ref r c 35. c References 6 and 26.

at four different temperatures within the range of experimental measurements. The general patterns observed from such a comparison reveal that the CNT predictions of Sc are much higher than the measured values particularly at lower temperatures and that the discrepancy appears to decrease as the temperature increases. Table 3 compares the measured Sc with the predictions of the CNT for the normal alkane series (CnH2n+2) with n ) 6-10, 12, 16, and 18. The comparison is made at a constant reduced temperature of Tr ) 0.45 (Tr ) T/Tc, where T and Tc are the nucleation and critical temperatures, respectively) using the Sc data reported in the literature for C6 to C10. The reduced temperature of 0.45 is chosen because the experimental nucle-

ation data for the alkane series is available at nucleation temperatures corresponding to Tr ) 0.45. From the CNT relation between the supersaturation and the size of the critical cluster (n*), given by eq 4, we calculate the values of n* corresponding to the experimental Sc measured for the alkanes listed in Table 3. The correlation between the measured Sc or n* at a given reduced temperature (Tr ) 0.45), and the number of carbon atoms in the alkane molecule are shown in Figure 4. It is clear that ln Sc increases (or n* decreases) with increasing the chain length of the normal alkane. This correlation could be used to predict the critical supersaturations of higher alkanes at Tr ) 0.45. For example, according to this trend, the Sc’s of C20, C24, and C48 are predicted to be 48, 72, and 806 at T ) 346, 358, and 416 K, respectively. Assuming that this correlation holds for smaller alkanes, it would imply that for methane Sc ) 7.2 at T ) 137 K. The observed trend of increasing Sc at a given reduced temperature among the alkane series reflects a general pattern of behavior among any series of structurally related molecules. This is can be seen by comparing the Pitzer acentric parameters ω [ω ) -log Pr (Tr ) 0.7) - 1.00, where Pr is the reduced vapor pressure at a reduced temperature of 0.7]47 for the selected alkanes listed in Table 3. This parameter provides a reasonable measure of the behavior of simple fluids; it is almost zero for a simple fluid and generally positive otherwise.47 The larger the value of ω, the further the departure from the simple fluid behavior since ω is actually a measure of the increase in the entropy of vaporization over that of a simple fluid. As shown in Table 3, the increase in the critical supersaturations of the alkanes at a given reduced temperature is consistent with increasing ω among the alkane series. Tables 4-6 list the results for the isothermal homogeneous nucleation rates as a function of supersaturation at selected nucleation temperatures for dodecane, hexadecane, and octadecane, respectively. Figures 5-7 display the natural logarithm of the measured nucleation rate ln Jexp, versus the natural logarithm of the supersaturation at different nucleation temper-

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Figure 4. (a) Natural log of critical supersaturation (ln Sc), at a reduced temperature Tr ) 0.45, versus the number of carbon atoms in the alkane molecule. (b) Critical cluster size (n*) as a function of the number of carbon atoms in the alkane molecule.

TABLE 5: Measured and Calculated Data for Rate Experiments with Hexadecanea Pt

To

T1

T

S

Jexp

Jexp(max)

15.28 15.31 15.32 15.34 15.33 15.35 15.94 16.03 16.09 16.12 16.14 16.00 16.03 16.11 16.11 16.11 16.95 17.01 17.06 17.11 17.13 17.15 16.95 16.98 17.01 17.00 17.31 17.05 17.06 17.07 17.06 17.08 17.08

377.80 378.80 379.50 380.30 381.10 382.00 385.10 386.10 386.90 387.60 388.70 393.00 394.00 394.90 395.70 396.50 403.10 403.90 404.70 405.50 406.40 407.20 410.00 410.80 411.50 412.50 410.00 410.60 411.60 412.40 413.20 413.90 414.70

293.64 293.61 293.37 293.17 293.05 292.88 300.74 300.58 300.43 300.26 300.22 308.53 308.32 308.04 307.88 307.67 317.86 317.77 317.46 317.23 317.02 316.75 324.67 324.49 324.28 324.04 324.45 324.41 324.26 324.15 323.85 323.75 323.60

308.44 308.55 308.46 308.41 308.43 308.41 316.90 316.93 316.92 316.89 317.00 325.81 325.79 325.69 325.66 325.60 336.11 336.14 336.01 335.94 335.89 335.77 343.06 343.02 342.95 342.90 342.92 342.94 342.95 342.98 342.84 342.87 342.85

44.54 46.61 49.26 51.95 54.43 57.48 33.27 35.20 36.83 38.61 40.55 25.40 26.92 28.46 29.82 31.36 19.81 20.55 21.71 22.78 23.97 25.13 16.88 17.56 18.29 19.36 17.01 17.52 18.39 19.12 20.08 20.79 21.60

7.55 × 10-4 3.26 × 10-3 3.91 × 10-2 2.10 × 10-1 8.70 × 10-1 4.99 2.22 × 10-4 1.51 × 10-3 9.10 × 10-3 4.77 × 10-2 4.33 × 10-1 2.77 × 10-4 8.88 × 10-4 1.21 × 10-2 9.45 × 10-2 5.80 × 10-1 3.55 × 10-4 4.44 × 10-4 5.18 × 10-3 4.66 × 10-2 3.38 × 10-1 1.31 1.78 × 10-4 1.11 × 10-4 3.44 × 10-3 3.96 × 10-2 3.55 × 10-4 5.33 × 10-4 2.66 × 10-3 2.26 × 10-2 1.49 × 10-1 7.71 × 10-1 2.25

1.68 × 10-3 7.13 × 10-3 8.47 × 10-2 4.50 × 10-1 1.84 1.04 × 101 5.09 × 10-4 3.39 × 10-3 2.01 × 10-2 1.04 × 10-1 9.27 × 10-1 6.17 × 10-4 1.93 × 10-3 2.58 × 10-2 1.98 × 10-1 1.19 7.53 × 10-4 9.27 × 10-4 1.06 × 10-2 9.35 × 10-2 6.65 × 10-1 2.53 3.69 × 10-4 2.27 × 10-4 6.92 × 10-3 7.79 × 10-2 7.37 × 10-4 1.09 × 10-3 5.35 × 10-3 4.47 × 10-2 2.89 × 10-1 2.85 × 10-1 1.46

a T , the temperature of the pool surface in Kelvin; T , the temperature 0 1 of the top plate in Kelvin; Pt, the total pressure in kPa; T, the calculated temperature at the maximum rate plane; S, the calculated supersaturation at the maximum rate plane; Jexp the measured droplet flux in drops cm-2 s-1; Jexp(max), derived nucleation rate in drops cm-3 s-1.

atures. The lines in Figures 5-7 are the predictions of the CNT according to eq 1; solid lines correspond to filled symbols and dashed lines to empty symbols. It is clear that the CNT predicts the correct dependence of the nucleation rate on supersaturation. However, the temperature dependence of the nucleation rate is not well predicted. At lower temperatures, the CNT predicts a nucleation rate significantly lower than the measured rate. The disagreement is less pronounced at higher temperatures. Also, the discrepancy between experiment and CNT increases in the order: dodecane < hexadecane < octadecane. The results indicate that temperature dependent correction factors to the CNT are needed in order to quantitatively agree with the

Figure 5. Isothermal rate curves, J (cm-3 s-1), for dodecane. Solid lines represent classical theory predictions.

experimental data. These correction factors can be obtained by fitting the (Jexp/JCNT) ratio as a function of Tr for each of the studied compounds. The temperature correction factors are given in Figure 8, along with the fitted lines for dodecane, hexadecane, and octadecane. The correction factors vary by more than 106 for octadecane at Tr) 0.42 to about 0.1 for dodecane at Tr) 0.48. The observation that the ratio (Jexp/JCNT) is a function of temperature alone is consistent with previous observations48-51 and with the Kelvin scaling theory.52 The nucleation theorem provides direct, model free, and quantitative correlations between the measured nucleation rate and molecular properties such as the number of molecules in the critical cluster and the energy of the critical cluster. The nucleation theorem was derived by Kashchiev, who suggested its general validity beyond the CNT.53-55 For one component nucleus, the theorem may be expressed as

dW*/d ∆µ ) -n*

(5)

The most current version of the theorem uses the “excess” number of molecules (n* - nj) and is generally expressed as

[∂Wn/∂µ]V,T ) -(n* - nj - 1)

(6)

where nj is the average number of molecules that would occupy a volume V centered on a nucleus if that volume contained vapor at the uniform average density, whereas n* is the actual number of molecules in V where the nucleus is present. Therefore, (n* - nj) represents the excess number of molecules present in

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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11871

TABLE 6: Measured and Calculated Data for Rate Experiments with Octadecanea Pt

To

T1

T

S

Jexp

Jexp(max)

13.38 13.40 13.42 13.43 13.45 13.39 13.38 13.39 13.41 13.43 12.68 12.68 12.68 12.71 12.73 12.94 12.94 12.98 13.03 13.04 12.43 12.45 12.50 12.42 12.43 12.45 12.47 12.47 12.47 12.47 12.48 12.49 12.51 12.46 12.47 12.47 12.50 12.52 12.55 12.44 12.45 12.47 12.51 12.55

409.80 410.60 411.50 412.40 413.20 409.70 410.60 411.60 412.30 413.20 398.80 399.70 400.60 401.50 402.10 398.90 399.80 400.60 401.40 402.20 389.30 390.10 390.80 389.20 390.20 391.10 392.10 393.00 387.40 389.30 390.20 391.00 392.00 392.70 393.40 394.30 395.00 396.20 396.90 393.40 394.40 395.20 396.20 396.90

323.73 323.55 323.29 323.11 322.93 323.83 323.68 323.52 323.34 323.17 314.05 313.95 313.76 313.61 313.47 314.18 313.97 313.73 313.62 313.52 304.96 304.84 304.66 305.02 304.88 304.72 304.46 304.32 304.62 304.61 304.40 304.25 304.13 309.48 309.36 309.16 309.03 308.87 308.70 309.33 309.14 309.01 308.87 308.68

340.67 340.64 340.55 340.52 340.48 340.74 340.74 340.74 340.70 340.67 330.51 330.56 330.56 330.57 330.55 330.64 330.62 330.56 330.60 330.63 320.01 320.04 320.01 320.04 320.09 320.10 320.04 320.07 319.42 319.72 319.69 319.71 319.76 325.04 325.05 325.05 325.06 325.11 325.08 325.04 325.04 325.07 325.11 325.07

28.06 29.53 31.23 32.84 34.47 27.80 29.25 30.85 32.26 33.93 36.38 38.08 40.37 42.37 44.04 36.07 38.19 40.38 42.18 43.91 52.10 54.64 57.32 51.69 54.82 57.84 61.89 65.30 48.98 53.66 56.85 59.99 63.25 40.74 42.40 45.02 46.99 50.11 52.51 42.62 45.20 47.46 50.11 52.55

1.26 × 10-3 9.37 × 10-3 5.73 × 10-2 3.60 × 10-1 1.45 3.70 × 10-4 4.62 × 10-3 4.01 × 10-2 1.94 × 10-1 9.92 × 10-1 1.04 × 10-3 6.90 × 10-3 4.55 × 10-2 2.63 × 10-1 1.03 9.62 × 10-4 8.14 × 10-3 4.14 × 10-2 2.22 × 10-1 7.97 × 10-1 3.03 × 10-3 1.55 × 10-2 7.32 × 10-2 1.70 × 10-3 1.26 × 10-2 8.45 × 10-2 4.88 × 10-1 1.53 4.44 × 10-4 6.17 × 10-3 3.22 × 10-2 2.26 × 10-1 9.48 × 10-1 7.40 × 10-5 5.92 × 10-4 8.01 × 10-3 2.70 × 10-2 2.32 × 10-1 8.42 × 10-1 5.92 × 10-4 6.66 × 10-3 4.00 × 10-2 2.36 × 10-1 8.19 × 10-1

2.75 × 10-3 2.01 × 10-2 1.21 × 10-1 7.44 × 10-1 2.95 8.10 × 10-4 9.93 × 10-3 8.46 × 10-2 4.02 × 10-1 2.02 2.40 × 10-3 1.57 × 10-2 1.02 × 10-1 5.76 × 10-1 2.23 2.23 × 10-3 1.85 × 10-2 9.24 × 10-2 4.88 × 10-1 1.73 7.24 × 10-3 3.66 × 10-2 1.70 × 10-1 4.07 × 10-3 2.95 × 10-2 1.96 × 10-1 1.11 3.41 1.09 × 10-3 1.47 × 10-2 7.53 × 10-2 5.21 × 10-1 2.15 1.79 × 10-4 1.41 × 10-3 1.87 × 10-2 6.23 × 10-2 5.24 × 10-1 1.87 1.41 × 10-3 1.56 × 10-2 9.18 × 10-2 5.33 × 10-1 1.82

Figure 6. Isothermal rate curves, J (cm-3 s-1), for hexadecane. Solid lines represent classical theory predictions.

Figure 7. Isothermal rate curves, J (cm-3 s-1), for octadecane. Solid lines represent classical theory predictions.

a T , the temperature of the pool surface in Kelvin; T , the temperature 0 1 of the top plate in Kelvin; Pt, the total pressure in kPa; T, the calculated temperature at the maximum rate plane; S, the calculated supersaturation at the maximum rate plane; Jexp the measured droplet flux in drops cm-2 s-1; Jexp(max), derived nuclation rate in drops cm-3 s-1.

the volume V when the nucleus is present, and if V is large enough, (n* - nj) will be independent of V and it can be used as a measure of the number of molecules in the nucleus.54 This is normally obtained from the measured dependence of nucleation rate on supersaturation, which can be written as

n* ) d(ln J)/d(ln S) - 1

(7)

Using the measured isothermal nucleation rates of octadecane, we calculate the number of molecules in the nucleus according to eq 7. The results are compared to n* values calculated from the CNT as shown in Figure 9. It is clear that the CNT predicts smaller nucleus size for the homogeneous nucleation of octadecane. This behavior is consistent with the CNT predictions of higher Sc as shown in Figure 3. IV. Conclusions The critical supersaturations and isothermal homogeneous nucleation rates of dodecane, hexadecane, and octadecane vapors

Figure 8. Temperature dependent correction factors for the ratios of the experimental to the calculated rates [J(exp)/J(CNT)] for dodecane (O), hexadecane (4), and octadecane (0).

have been measured over wide temperature ranges (e.g., 285340 K) using an upward thermal diffusion cloud chamber. Dodecane shows the best agreement between experiment and the classical theory. Hexadecane and octadecane exhibit critical

11872 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Figure 9. Classical theory prediction of n* (solid line) compared with n* derived from applying the nucleation theorem to the measured nucleation rate of octadecane (squares).

supersaturations lower than those predicted by theory. All three compounds show increasing deviation from theory with decreasing temperature. This trend is also observed in the measured nucleation rates where the predictions of the theory at lower temperatures are shifted to much lower values compared to the experimental rates. Temperature dependence correction factors to the calculated rates have been evaluated based on the experimental rates of the studied higher alkanes. On the other hand, the classical theory predicts the correct supersaturation dependence of the nucleation rates of these higher alkanes. Analysis of the supersaturation dependence of the nucleation rate of octadecane indicates that the critical cluster contains 3035 molecules at the nucleation temperatures of 315-340 K. The classical theory predicts smaller nucleus size consistent with the prediction of higher critical supersaturations. Acknowledgment. Acknowledgment is made to NASA Microgravity Materials Science Program (NAG8-1484) for the support of this research. Supporting Information Available: Three tables of measured and calculated data for each experiment with dodecane, hexadecane, and octadecane. Each table contains the following: T0, the temperature of the pool surface in Kelvin; T1, the temperature of the top plate in Kelvin; Pt, the total pressure in k Pa; Sc, the calculated supersaturation at the maximum rate plane, and T, the calculated temperature at Sc. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Kashchiev, D. Nucleation: Basic Theory with Applications; Butterworth-Heinemann: Oxford, U.K., 2000. (2) Laaksonen, A.; Talanquer, V.; Oxtoby, D. W. Annu. ReV. Phys. Chem. 1995, 46, 489. (3) Nucleation Phenomena; Zettlemoyer, A. C., Ed.; Elsevier: Amsterdam, 1977. (4) Abraham, F. F. Homogeneous Nucleation Theory; Academic Press: New York, 1974. (5) Heist, R. H.; He, H. J. Phys. Chem. Ref. Data 1995, 23, 781. (6) Katz, J. L. J. Chem. Phys. 1970, 52, 4733. (7) Katz, J. L.; Scoppa, C. J., II.; Kumar, N. J.; Virkler, T. L. J. Chem. Phys. 1975, 62, 448. (8) Katz, J. L.; Mirabel, P.; Scoppa, C. J., II.; Virkler, T. L. J. Chem. Phys. 1976, 65, 382. (9) El-Shall, M. S. Chem. Phys. Lett. 1988, 143, 381. (10) El-Shall, M. S. J. Chem. Phys. 1989, 90, 6533.

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