J. Phys. Chem. 1994,98, 8721-8725
8721
Collisional Stabilization Efficiencies That Control Condensation Flux Rates in Supersaturated Vapors of n-Alcohols and Water S. H. Bauer’ and C. F. Wilcox, Jr. Baker Chemical Laboratory, Cornel1 University, Zthaca, New York 14853 Received: April 13, 1994; In Final Form: July 7, 1994”
In contrast to classical nucleation theory (CNT), wherein J(S;T) magnitudes are expressed in terms of physical parameters of molecular clusters treated as liquid drops, the kinetic molecular model (KMM) focuses on the dynamics of interaction between clusters and colliding monomers, Le. the rates of accretion and efficiencies of stabilization of the transient hot adducts. Instead of providing a predictive model, we accept the reported J(S; T ) values and derive magnitudes for selected kinetic parameters, in particular, (i) a temperature-dependent stabilization factor (SP),which is strongly correlated with (ii) the size-dependent activation energy for evaporation from stabilized clusters and (iii) size-dependent heats of evaporation. Recently published data on water and six n-alcohols, obtained with the double-piston expansion technique, provided the means to evaluate a selfconsistent set of SP values and insights into the proposed mechanism for cluster growth. The results for supersaturated water are of special interest.
I. Introduction In our previous reports14 we described a self-consistentkinetic molecular model (KMM) for rationalizing experimentally determined condensation flux rates from supersaturated vapors. As in any kinetics analysis, the model incorporates molecular parameters that cannot be measured directly. In that respect they are adjustable within ranges of magnitudes that have been derived from a wide variety of physical measurements. The extensively developed and frequently referred-to-classicalnucleation theory (CNT), as generally presented, has no adjustable parameters for predicting condensation flux (J(S;T); cm-3 s-l) values, but unfortunately deviationsof many orders of magnitude are frequently observed. The favored stratagem is to introduce a “replacement factor” to bring predictions and observations into coincidence. Alternatively, an “effective surface free-energy” for the growing clusters is introduced, since it is obvious that such entities are not circumscribed by bulk-like surfaces, as implied in CNT. The effective surface free-energy parameter shows no trends that can be related to the molecular structures of the condensing species. In contrast, with KMM one accepts the published J(S; T ) values and extracts magnitudes of kinetic/ molecular properties by fitting J(ca1c) to J(exp), over a range of supersaturations and temperatures. The validity of KMM hinges on derived values that have physical significance at the molecular level, with values that fall within generally accepted ranges. They should show trends consistent with known temperature dependencies and molecular structure. With the continued evolution of KMM we look forward to uncovering trends with structure types so that limited flux predictions can be made. The underlying distinction between CNT and KMM is that the former focuses on the properties of stabilized clusters, their sizes, and characteristic equilibrium vapor pressures (the offspring of the Kelvin equation), while the latter focuses on the dynamics of cluster growth and the nature of cluster-collider interactions. To date, our analyses of many J(S; 7‘)values reported for diverse systems have been encouraging. The experimental values were obtained via four different techniques that are not as mutually consistent as would be desired. A variety of factors must be considered in comparing the data, but it is clear that the different methods used do not yield consistent sets of data over wide ranges of supersaturations, temperatures, and flux rates (reportedvalues range from 10-4 to 1010 cm-3 s-1). The published data for ethyl Abstract published in Aduance ACS Abstracts, August 1, 1994.
0022-3654/94/2098-8721%04.50/0
alcohol illustrate typical inconsistencies. (Note Figure 1, in ref 2.) Recently, we were greatly pleased to find published data for water and six n-aliphatic alcohols investigated in a single laboratory (uia the two-piston expansion chamber technique), with experimental values presented in a uniform manner, in considerable detail.6.7 These comprise a consistent set of data for exploring trends in KMM parameters for a family of related compounds and permit us to establish ranges of acceptable values for the kinetic parameters. To provide a background for the following discussion, we briefly list the essential equations and justifications for the magnitudes of the parameters we introduced in refs 1-4. Cluster growth occurs via the sequence
K-lu
C,+X+A,+X
(1b)
KUJ
(Xis any collider that stabilizes the hot adduct). Both the forward and reverse of eq lb-deexcitation and excitation-must be considered. The stabilization rate constant K~ = wSP is a product of the probability of an encounter per unit time, per unit concentration, ( w ) and the conditional probability (SP) that sufficient cluster vibrational energy will be drained by X to stabilize C,. Clearly, the derived T-dependent SP value is an average over cluster sizes, the multiple isomericcluster structures (for any u), and the range of AE’s extracted per encounter. For the reverse step we proposed
In our reported studies we setf= 1; g = 3. The probability that an encounter between X and A, will generate C, is given by K - ~ ; C, may either dissociate unimolecularly, per K-, = 0.7937(kbT/ h),or get restabilized by a subsequent collision with X. Equation 2 is an ad hoc but kinetically defensible assumption, being an application of the RiceRamsperger-Kassel model for a bimolecular induced dissociation that involves one degree of freedom (escape normal to the cluster surface). In this equation, e, is the depth of the Lennard-Jones pair potential that accounts for cohesion in the cluster 0 1994 American Chemical Society
8722 The Journal of Physical Chemistry, Vol. 98, No. 35, 1994
Bauer and Wilcox
(3) e, = (1/5){W0,I - lMO,-ll - kbTl where AHo, refers to the hypothetical process uA1- A,. The (1/5) factor in eq 3, which relates to a thermochemical enthalpy (see Figure l), was chosen to bring e2 within close range of experimentally derived magnitudes of e2/k for several species that we investigated. We proposed the empirical eq 4, in which Q D ( r )is the bulk heat of vaporization at T.
I
A?
@ path of,de-exci!ation
(4)
(We shall refer to a and 6 as the thermochemical parameters.) Clearly AiYo,(T) is a smoothed quantity, with a cluster size dependence derived in many theoretical estimates of cluster stabilities.’ Setting a = 0.85 and 6 = 0.333 gave the best correlation of eq 4 with a number of computed AI?,,;,,-,, but this relation does not apply to u = 1, for which the enthalpy is set to 0. Since the u-mer populations calculated under the constrained equilibrium assumption depend sensitively on AH”,( T), accurate values for Q,(T) must be used in eq 4, specifically those that cover the temperature interval over which the condensation experiments were performed.2 In summary, we focus on the two kinetic parameters SP and g and argue that support for KMM rests with the observation that the derived magnitudes of SP generally fall within a physically meaningful range, 0.05 ISP I1.0. Also, one anticipates that SPs show a weak temperature dependence and slowly increase with decreasing temperature, as do typical de-excitationparameters (8) of unimolecular reactions. SPs and gvalues are correlated. The former controls the rate of generating sufficient levels of A, to account for the observed J(S;T), while g controls the rate of vaporization of A,,; hence, increasing g has the same effect on J(S;T) as increasing SP. A6 initio calculations of this energy transfer parameter is a current challenging but unsolved problem. 11. Data Reduction (for mAikyl-OH (c1-C~)and HzO)
Following our standard computer protocol,2we derived SP( r ) values from condensation flux rates for six n-alcohols6and H207, with g set at 3. To present these results with a minimum of overlap, the stabilization parameters were plotted us reduced temperatures (Figure 2). The graphs shown are for the J(S;T) = lO*-lO9 range; for lower condensation flux rates ( J = los) the SPs are somewhat lower, by no more than 0.05 units, with the same Tdependence. This lowering is expected, since smaller Ss are generated at lower supersaturations, where the average cluster sizes are larger. In larger clusters the heat of condensation is more rapidly redistributed, resulting in less efficient vibration/ translation energy transfers. The plot for water has a somewhat lesser curvature than for the alcohols. Were these SPs plotted on a sliding scale, uiz., T - Bi, where Bi is a small shift in temperature so as to bring the curves into coincidence at SP = 0.65, all the curves would fall within a narrow band. The temperature dependence of SP is entirely consistent with molecular dynamics calculations. Schultz et al.8 studied collisional energy transfers of Mos clusters with rare gas atoms. Similarly, quasi-classical trajectory calculations of Ar-Ar recombination rates were made by Howard et al.9 Clearly, sets of mutually consistent condensation flux data, for a family of compounds, show (via KMM) that efficiencies for u/ Tenergy transfers are composition and structure dependent. Even water clusters, which have a distinctive phonon spectrum, show SPs very similar to those of the alcohols. It is reasonable to propose that H-bonding is the common structural controlling factor in this family of cluster types. 111. Reconsideration of the Magnitudes Assigned to Critical Parameters
In the computer code we developed for the KMM there are various options for inserting thermochemical and kinetic pa-
collision during the residence life-time of the transient complex (C,) in the L-J potential --.-.-
I
After yetting trapped in the well intra cluster vibrational redistribution stabilizes the complex, but leaves A, relatively hot.
-
-
Figure 1. Schematicoftheinteractionpotential for theassociation reaction between A,1 and AI, followed by vibration translation de-excitation via an encounter with species X.
rameters. Thus far, the available condensation flux data have been analyzed using the following kinetic parameters:
In eq 2,
g = 3 . 0 ; f = 1.0
We now inquire whether other combinations of g and f would lead to sets of SPs that do not show the large T dependencies indicated in Figure 2. These large ranges are not entirely compatible with our conceptual model. Indeed, for water and the alcohols lower values of SP were derived and their temperature dependence was greatly reduced when we set g = 3.5 and f = 1.O, Figure 3. With these parameters the SPs for water, H&-OH, and C~HS-OH nearly overlap, while those for c3
