Article pubs.acs.org/JPCA
Smoluchowski Equations for Agglomeration in Conditions of Variable Temperature and Pressure and a New Scaling of Rate Constants: Application to Nozzle-Beam Expansion J. Chaiken,# J. Goodisman,*,# and O. Kornilov§ #
Department of Chemistry, Syracuse University, Syracuse, New York 13244-4100, United States Max-Born-Institute, Max-Born-Straße 2 A, 12489 Berlin, Germany
§
ABSTRACT: The Smoluchowski equations provide a rigorous and efficient means for including multiple kinetic pathways when modeling coalescence growth systems. Originally written for a constant temperature and volume system, the equations must be modified if temperature and pressure vary during the coalescence time. In this paper, the equations are generalized, and adaptations appropriate to the situation presented by supersonic nozzle beam expansions are described. Given rate constants for all the cluster−cluster reactions, solution of the Smoluchowski equations would yield the abundances of clusters of all sizes at all times. This is unlikely, but we show that if these rate constants scale with the sizes of the reacting partners, the asymptotic (large size and large time) form of the cluster size distribution can be predicted. Experimentally determined distributions for He fit the predicted asymptotic distribution very well. Deviations between predicted and observed distributions allow identification of special cluster sizes that is, magic numbers. Furthermore, fitting an observed distribution to the theoretical form yields the base agglomeration cross section, from which all cluster−cluster rate constants may be obtained by scaling. Comparing the base cross section to measures of size and reactivity gives information about the coalescence process.
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volume can be derived from the determined velocity fields. By approximating the general equations by his phenomenological equations (for constant volume and temperature), Smoluchowski opened the way to analytic solutions that could be applied to many different problems and give more physical insight. Assuming all the Kij were equal, Smoluchowski8 was able to obtain an explicit solution to eq 1, nk as a function of t. Explicit solutions also exist9,10 for kernels having a more complicated functional form. It was shown,11 in addition, that an asymptotic solution to eq 1 exists if the kernels scale with monomer number, so that
INTRODUCTION Coalescence growth1,2 is a ubiquitous process in nature, and a better understanding of it would have implications in both applied and basic science. For example, we have previously considered electromigration-induced growth of voids in heated metal films,3 production of metal clusters by laser chemical vapor deposition,4 and production of He clusters by homogeneous nucleation in a supersonic expansion.5 In all cases,6,7 we have found the Smoluchowski equations useful in modeling the evolution of the void or cluster size distribution. Smoluchowski proposed8 that the agglomeration process be modeled as an infinite set of second-order reactions. Smaller particles combine to form larger particles, and dissociation is not considered. Then the rate of change of each number concentration (nk = number of k-mers per unit volume) is given by dnk 1 = dt 2
k−1
Kαi , βj = a μbνK ij
∑ Kkinkni
i=1
i=1
(1)
where the Kij are second-order rate constants, often referred to as kernels. Neglecting dissociation should be a good approximation in the early stages of agglomeration, when the number of larger particles decreases rapidly with particle size. The rigorously correct way to approach the agglomeration problem, especially when local temperature varies with time and position (as in nozzle-beam expansion), is via the reaction−diffusion equation. When solved, this equation produces time- and position-dependent concentrations and velocities for all species. The variations of temperature and © 2015 American Chemical Society
(2)
Then the asymptotic (large k and large t) solution to eqs 1 can be shown to be nk → Akae−bk. Other scalings lead12 to different asymptotic solutions. The general problem of the existence of self-similar solutions to the Smoluchowski equations has been discussed by Niethammer and co-workers.12 An excellent review of the Smoluchowski equations and related equations has been given by Aldous.13 As well as pointing to the wide variety of applications, this 1999 review lists and discusses the different forms for the rate constants that have been investigated and exact, approximate, and asymptotic solutions. Although
∞
∑ K i ,k− inink− i −
( i ≤ j)
Received: January 14, 2015 Revised: May 18, 2015 Published: June 11, 2015 6929
DOI: 10.1021/acs.jpca.5b00408 J. Phys. Chem. A 2015, 119, 6929−6936
Article
The Journal of Physical Chemistry A
We begin by reviewing the form and assumptions of the isothermal and isochoric Smoluchowski model.8 Then we propose a new scaling law for the kernels Kjk and show that it leads to the asymptotic form, nk → Akae−bk, for the number densities. Here, b, but not a, is time-dependent. Next, we show how to account for temperature and volume changes. Although we are particularly interested in the changes that occur during a supersonic expansion, the approach we describe can be adapted to other situations in which temperature and volume vary in a known way. Finally, we discuss issues that can be considered after accounting for the scaling of temperature, monomer mass, and density. For example, fitting an experimental distribution to obtain the parameters in the scaling law yields the “base agglomeration cross section,” to which all rate constants are proportional. Comparing base agglomeration cross sections for different systems can add to our understanding of the collision process.
primarily for mathematicians, it is a valuable resource for everyone involved in this research area. Considering that the Kij must be proportional to collision rates and that collision rate constants are proportional to (iα + jα)μ(iβ + jβ)−ν, we and others anticipated2,5,7,9 that the Kij could be approximated as C(k>)μ(k is the greater of i and j, which is equivalent to eq 2. Applying the model to the results of nozzle-beam expansions of He, we verified5 the asymptotic result, nk → Akae−bk, as well as other predictions of this model. However, the approximation of (iα + jα)μ(iβ + jβ)−ν by (k>)μ(k)2/3(k is the greater of i and j. (Note that Kij must be symmetric in i and j.)
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THEORY: NEW SCALING OF RATE CONSTANTS AND ASYMPTOTIC SOLUTION Figure 1 shows Q (dots) and P for j = 30 and i = 1 ... 60 (C = 1). It is clear that P is not at all a good approximation to Q.
K ij = K11i μj μ
(5)
It must be emphasized that K11, referred to as the base agglomeration rate constant, is not the rate constant for combination of two monomers to form a dimer. Monomer− monomer combination is a unique process. We note that there is only one scaling parameter, μ, in this new scaling; there were two, μ and ν, in the old scaling, (n>)μ(n 3, the error in taking the upper limit as ∞ is 3, the error in replacing the upper limit with ∞ is 0, a/b is negative. To have a peaked distribution, a must be positive; that is, Kjk must decrease with j (or k). Our scaling model, Kjk = Ckμjμ with μ > 0, has Kjk increasing with j or k.. This is in accord with experimental results and also with our intuitions about the kernels. Since each kernel is a reactive rate constant, it must be a product of a collision frequency and a sticking probability. The former is a product of a relative velocity, which decreases with the sizes of the colliding partners, and a geometric cross section, which increases with these sizes. The sticking probability increases with these sizes, as well, but never exceeds unity. One thus expects μ to be small and positive. As seen in Table 1 (a = −2μ) it is either 0.8 or 1.2. The value 1.2 may be an artifact of the data truncation. Coalescence-growth exothermicity could change the mass transfer conditions by influencing the translational energy distribution of newly formed clusters, and this could change the a and b parameters. If peaked distributions occur, they can be fit to give both a and b, as we have previously shown7 Expansions involving greater coalescence-growth exothermicity should be studied to explore this. We stress that the dependence of rate constants on oligomer size, expressed by scaling, arises from mass transfer, size, and the ability to redistribute energy so as to stabilize nascent clusters. The effectiveness of energy redistribution increases with the density of states that participate in the process. The density of states depends differently on monomer mass for solid and liquid clusters, but certainly increases with monomer mass. The deviation of the base agglomeration cross section from that expected from various measures of size gives information about the variation of the internal state density with monomer mass. To extract physical information from the determined parameters, we have employed the ideal gas law, generalized empirical scaling relations known from rarified gas dynamics, and thermodynamic formulas that describe ideal systems. Although these assumptions likely entail only a small error for He, the error may be larger in more complicated systems. There are experiments28−31 in which ejecta from a laserirradiated rod are entrained and semiconfined in a pseudosteady-state “reaction tube” in which coalescence growth produces a distribution of cluster sizes. Adaptation of the equations to different T(t) and V(t) functions would be required; with appropriate ancillary measurements, for example, far-infrared “thermal” imaging of the reaction tube, it might be possible to treat this situation. Irradiation of bulk gases, as in laser chemistry of organometallics4 and flows to produce particles, would also be treatable. From the base cross section K11, one can estimate an infinite number of cluster−cluster rate constants. One might attempt to isolate clusters of a particular size using mass spectrometry and propagate them through a collision or pick-up cell31 to produce larger clusters. Such experiments could yield some rate constants for comparison with the scaling estimation. However, it seems unlikely that many rate constants could be measured. The estimation given by scaling is thus valuable. Another benefit of this analysis is that the deviations of nk from the best-fit function Ajae−bj reveal “magic-number” clusters. These are associated with inhomogeneous reaction kernels that deviate from the scaling formula. From Figure 3,
Table 1. Summary of Experimental Results and Fits pressure (bars)
no. of points
range of k
A=
a=
b=
1.10 1.16 1.22 1.28 1.33
33 35 51 51 51
10−67 11−95 12−121 12−121 12−121
4.465 2.807 0.654 0.482 0.435
−2.323 −2.100 −1.583 −1.558 −1.573
0.0505 0.0298 0.0240 0.0157 0.0120
0.11) and very much the same for the last three (−1.57 ± 0.01). The reason for the discrepancy between the two groups probably relates to the truncation of the data in Figure 3a. Overall, a = −1.83 ± 0.32. It should also be noted that the parameters a and b are not completely independent. To test the linearity of b−(a+1) with pressure, we redetermine A and b for each data set, assuming a = −1.83. The resulting plot (Figure 4) is quite linear, with correlation coefficient, r2, equal to 0.94.
Figure 4. Quantity b−(a+1), calculated from the data of Table 1, is a linear function of the pressure, P (r2 = 0.94), as predicted.
Usually, the source temperatures are known, and S is measured, as well as θ. Then eq 21a or 21b can be used to extract the base agglomeration rate constant, K11. We emphasize again that K11 is not the rate constant for dimer formation from two monomers, but the constant to which the Kjk for large j and k are proportional. It is still expected that K11 is proportional to the monomer cross section and to the relative velocity of the monomers, which is proportional to the square root of the temperature and inversely proportional to the square root of the monomer mass. It also involves a “sticking coefficient” that is related to a density of states. If the cross section and relative velocity factors can be estimated, comparison of K11 values for different monomers gives information about the “sticking probability” involved in agglomeration.
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DISCUSSION We first consider the assumptions of this model and assess how transferable these assumptions may be to monomers other than He. We then consider some other types of coalescence growth conditions and which changes to the above equations might be appropriate. We finally consider the information we might hope to glean from measured size distributions, given various other information. The important point is that measuring the distributions makes it possible to compare systematically different monomers and the clusters they produce. The a and b parameters relate the scaling of mass transfer with temperature and monomer mass to the scaling of the rate constants. 6935
DOI: 10.1021/acs.jpca.5b00408 J. Phys. Chem. A 2015, 119, 6929−6936
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The Journal of Physical Chemistry A one can recognize that “magic-number” clusters exist for k = 13 and 23.
(14) Hutton, K.; Mitchell, N.; Frawley, P. J. Particle Size Distribution Reconstruction: The Moment Surface Method. Powder Technol. 2012, 222, 8−14. (15) Aykroyd, J.; Smith, A. J.; Shirley, R.; McGlashan, L. R.; Kraft, M. A Coupled CFD-Population Balance Approach for Nanoparticle Synthesis in Turbulent Reacting Flows. Chem. Eng. Sci. 2011, 66, 3792−3805. (16) Attarakih, M. Integral Formulation of the Population Balance Equation: Application to Particulate Systems With Particle Growth. Comput. Chem. Eng. 2013, 48, 1−13. (17) Jin, Y.; Mizuseki, H.; Kawazoe, Y. Direct Simulation Monte Carlo for Cluster Growth Process in Rarefied Gas. Mater. Trans. 2001, 42, 2295−2298 and references therein. . (18) Gordiets, B. F.; Bertran, E. Kinetic Model for Generation and Growth of Plasma Dust Nanoparticles. Chem. Phys. Lett. 2005, 414, 423−428 and references therein . (19) Ratner, M. A. Kinetics of Cluster Growth in Expanding Rare Gas Jet. Low Temp. Phys. 1999, 25, 266−273 and references therein . (20) Elminyawi, I. M.; Gangopadhyay, S.; Sorensen, C. M. Numerical Solutions to the Smoluchowski Aggregation-Fractionation Equation. J. Colloid Interface Sci. 1991, 144, 315−323 and references therein. . (21) Gillespie, D. T. An Exac5t Method fior Numerically Simulating the Stochastic Coalescence Process in a Cloud. J. Atmos. Sci. 1975, 32, 1977−1989. (22) Van Dongen, P. G. J.; Ernst, J. H. On the Occurrence of a Gelation Transition in Smoluchowski’s Coagulation Equation. J. Stat. Phys. 1986, 44, 785−792. (23) See, for example: Molecular Beams and Low Density Gas Dynamics; Wegener, P. P., Ed.; Marcel Dekker: New York, 1974, Vol. 4, p 41. (24) Montero, s.; Morilla, J. H.; Teijeda, G.; Fernández, J. M. Experiments on Small (H2)N Clusters. Eur. Phys. J. D 2009, 52, 31− 34. (25) Knuth, E. L.; Fisher, S. S. Low-Temperature Viscosity CrossSections Measured in a Supersonic Argon Beam. J. Chem. Phys. 1968, 48, 1674−1684. (26) Knuth, E. L.; Fisher, S. S. Rarefied Gas Dynamics. AIAA J. 1969, 7, 1174−1177. (27) Michalopoulos, D. L.; Geusic, M. E.; Hansen, S. G.; Powers, D. E.; Smalley, R. E. The Bond Length of Chromium Dimer. J. Phys. Chem. 1982, 86, 3914−3916. (28) Bruhl, R.; Guardiola, R.; Kalinin, A.; Kornilov, O.; Navarro, J.; Savas, T.; Toennies, J. P. Diffraction of Neutral Helium Clusters: Evidence for "Magic Numbers". Phys. Rev. Lett. 2004, 92, 185301− 185302. (29) Powers, D. E.; Hansen, S. G.; Geusic, M. E.; Michalopoulos, D. L.; Smalley, R. E. Supersonic Copper Clusters. J. Chem. Phys. 1983, 78, 2866−2881. (30) Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. C60: Buckminsterfullerene. Nature 1985, 318, 162−163. (31) Telle, H. H.; Gonzalez Urena, A.; Donovan, R. J. Laser Chemistry, Spectroscopy, Dynamics and Applications; John Wiley and Sons: Chichester, 2007; Part 6.
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CONCLUSIONS We have introduced a new scaling of the reaction rate constants Kij into the Smoluchowski equations. This scaling is more justifiable than what has been used previously. We have also shown how to include temperature and volume variations in the Smoluchowski equations. With these changes, the density of jmers for large j and time is proportional to jae−bj. From fits of experimental results for cluster abundances to Ajae−bj, we extract the fundamental base agglomeration rate constant for different monomers.
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AUTHOR INFORMATION
Corresponding Author
*Phone: 315-443-3035. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This paper has benefited from literally years of e-mail discussions with J. Peter Toennies concerning difficulties with applying the Smoluchowski equations to supersonic expansion data, and we gratefully acknowledge his determination to obtain a reasonable description of the effects of temperature and number density changes.
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REFERENCES
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DOI: 10.1021/acs.jpca.5b00408 J. Phys. Chem. A 2015, 119, 6929−6936
