J. Phys. Chem. 1995, 99, 8147-8161
8147
Energy Transfer Dynamics in the Collision-Induced Dissociation of Al6 and A113 Clusters Pascal de Sainte Claire, Gilles H. Peslherbe, and William L. Hase* Department of Chemistry, Wayne State University, Detroit, Michigan 48202 Received: November 2, 1994; In Final Form: February 21, 1995@
Using a model analytic potential energy function developed for Al, clusters [J. Chem. Phys. 1987, 87, 22051 and a UMP2(fc)/6-3 lG* potential derived here for the Ar-A1 interaction, classical trajectory simulations are performed to study collision-induced dissociation (CID)of A16 and A113 with argon. For the octahedral Al6 (Oh) cluster the CID threshold is -14 kcal/mol higher than the true threshold. This is because, near the threshold, there are no trajectories which transfer all the reactant relative translational energy to Al6 internal energy. For the planar Al6 (C2h) cluster, the CID threshold is closer to the true threshold. For the spherically shaped Al6 (Oh) and All3 (D3J clusters, T V is the predominant energy transfer pathway. T R energy transfer is important for the planar A16 (C2h), A113 (&), and A113 (&) clusters. T V energy transfer is enhanced as the cluster is softened (i.e,, its vibrational frequencies lowered), the mass of the colliding atom is increased, and/or the relative velocity is increased. These effects are consistent with a previously derived impulsive model [J. Chem. Phys. 1970,52,5221], which says T V energy transfer increases as the collisional adiabaticity parameter is decreased.
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I. Introduction
In the work presented here classical trajectories are used to study the CID of A16 and A l l 3 clusters by high-energy collisions with Ar atoms. The potential energy surfaces for the clusters are represented by the analytic function derived by Pettersson et aL2I from ab initio calculations. The Ar-Al, intermolecular potential is represented by a sum of Ar-Al two-body potentials, determined by ab initio calculations. The trajectories illustrate how the energy transfer to Al, depends on the cluster “stiffness”, cluster structure, and mass of the incident atom. The trajectory results are compared with various models, described previously for translation to internal energy transfer. A model is presented to explain the trajectory results in the translation to vibration (T V) energy transfer regime. The trajectory study is also compared with other studies4 of energy transfer in atom cluster collisions.
Experimental of the collision-induced dissociation (CID) of semiconductor and metal atom clusters have provided information about the bond energies, dissociation pathways, and structures of the clusters. For the most part, ionic clusters have been studied, since specific cluster sizes can be selected by mass spectrometry. Experimental results have been reported for Bn+,l0,ll Al,+,839 C,+,I4-l6 and Si,+ I 2 9 l 3 and for first- and second-row transition metal clusters Ti,+? V,+,2 Cr,+,’ Ni,f,6 Nb,+,’ and Another source of information about the energetics, dissociation pathways, and structures of clusters is ab initio calculations. Calculations of this type have been performed for the Cn+,17,18 Sin, I9x2O B,+,ll and Al,21-29 clusters and for the transition metal clusters Fen+,30C U ~ ,Rh3,32 ~ ’ Pd,,33 Ag3+,34 Pt,,35 and A u ~ . ~ ~ The mechanism of collisional activation of a molecule by an 11. Potential Energy Surface inert atomic gas is strongly dependent on the initial relative A. Analytic Functions. 1. Al6 and All3 Potentials. Two translational energy At high Ere1 (’1 keV) vertical different analytic potential energy functions for Al6 were used electronic transitions are expected. For somewhat lower enerin the calculations reported here. One consists of two-body gies of 10 eV-10 keV, radiationless transitions to a predissoLennard-Jones (L-J) terms ciative state are more likely to occur. For energies in the middle of this range, at 100 eV-1 keV, elastic processes may occur. A B At the lowest energies (several electronvolts), activation is V y = 7 + 7 expected to proceed by direct transfer of translational energy rij rij to cluster internal energy. This energy transfer can occur via a long-lived atom cluster complex or by a direct impulsive and three-body Axilrod-Teller (A-T)4s terms collision with no complexation. It is usually assumed that the internal excitation energy redistributes statistically within the 1 3 cos a,cos a,cos a3 cluster, so that Rice-Ramsperger-Kassel-Marcus (RRKM) LJk = c 3 (2) can be used to calculate the cluster’s unimolecular (‘ij Ijk Iki) dissociation rate c o n ~ t a n t . ~ , ~ . ’ ~ ~ ~ ~ Additional insight into the CID experiments would be gained for which the parameters A = 2 975 343.77 kcal*AI2/mol,B = by understanding the dynamics of the transfer of initial relative -17 765.823 kcal.A6/mol, and C = 81 286.093 kcallmol were translational energy to cluster internal energy and the ensuing determined by a fit to ab initio calculations for Al, (n = 2-6, dynamics of cluster dissociation. Since the CID experiments 13) clusters.21 In eq 2 ry, rjk, rki and a], a2,a3 represent the are carried out at high energies, classical trajectory s i m d a t i o n ~ ~ - ~ ~sides and angles, respectively, of the triangle formed by the are expected to accurately describe the energy transfer and three particles i, j , and k. dissociation dynamics. The second analytic function used for Al6 consists of twobody Morse terms with the exponential @ parameter written as Abstract published in Advance ACS Abstracts, April 15, 1995. a cubic function of rij,46 i.e.
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0022-3654/95/2099-8147$09.00/0
0 1995 American Chemical Society
de Sainte Claire et al.
8148 J. Phys. Chem., Vol. 99, No. 20, 1995 2
Vu = De{ 1 - exp[-P(r, - re)]} - De
/3 = De
+ c2Ar2+ c3Ar3
(3a) (3b)
where Ar = rg - re. Parameters in this Morse function were varied to determine how the “stiffness” of the 4 6 cluster affects energy transfer to Al6 and the resulting CID. The octahedral (Oh) Al6 cluster, described below, was used for this study. Parameters for the Morse potentials, identified as Morsew and Morse/n, were chosen so that they gave the same equilibrium geometry and dissociation energy for Al6 (Oh) as the above L-J/ A-T potential but gave vibrational frequencies for A16 (Oh) which are n times larger and smaller, respectively, than those of the L-J/ A-T potential. The L-J/A-T A16 (Oh) structure (see below) was reproduced by using re = 2.834 A. It was found that a systematic linear scaling procedure could be used to relate the remaining parameters for the Morse*n and Morse/n potentials if the parameters for the “softest” Morse potential considered here, Morse/4, were fit first. Parameters for Morsel4 were chosen by the following steps: (1) De, be,c2, and c3 in eqs 3a and 3b were fit by nonlinear least squares to the L-J/A-T frequencies for Al6 (Oh) divided by four. The frequencies do not simply depend on De and be,since three of the AI-A1 Morse potentials do not have an equilibrium distance equal to re. Adding a linear term clAr to eq 3b had no effect on the fit, since ci was negligibly small. (2) The minimumenergy A15 cluster for the Morse*n and Morseln potentials has a Djh symmetry (see Figure 2), with a nearest-neighbor distance of re = 2.834 A. Given the above parameters, the resulting Al6 (Oh) A l s Al dissociation energy Ed was determined. If EO’matched the dissociation energy EOfor the L-J/A-T Al6 (Oh) cluster, the above parameters De,be,c2, and cj were accepted. If not, De was set to E 40.0 kcallmol. E,,I = 120.8 kcallmol. P(AElnt)is in units of (kcaYmol)-], and the distributions are normalized so that the total probability is unity for transferring energy in excess of the true dissociation threshold EO.The probabilities on the y axis have been multiplied by a factor of 10. TABLE 4: Dependence of A on Mass of An mAb (AEmtYEret 3m~, 0.45 i 0.24 0.43 iz 0.26 2m~, mAr 0.40 i 0.23 0.33 f 0.20 mAd2 mh14 0.25 +c 0.16
+ A& (Oh) Energy Transfer 0CID
24.8 f 0.3 22.7 f 0.3 21.7 f 0.4 18.2 f 0.4 12.6 f 0.3
fraction CIDe 0.64 0.59 0.56 0.41 0.33
a The calculations are for the Lennard-JonesIAxilrod-Teller potential, with E,,I = 120.8 kcallmol. Three thousand trajectories were evaluated for each mass of A. bThe mass of A is in factors of the argon atom mass. The average energy tTansferred, (A&), is for all of the trajectories. See eq 8. b,, = 3.5 A for each of the calculations. e Fraction of the trajectories which result in CID.
choosing b randomly between 0 and b,,,, i.e. N
(AELnJ =
C (mint)/N
]=1
(8)
where (AElnt)]is the energy transfer for thejth trajectory and N is the total number of trajectories calculated, not just those which lead to CID. Using the same b,,, of 3.5 A, the ratio (AEtnJl Erei is 0.38, 0.39, 0.40, 0.41, 0.41, and 0.41 for Erel of 60, 80, 120.8, 170, 200, and 400, respectively. Given the statistical uncertainties, there is no meaningful change in this ratio versus Ere]. Thus, increasing Ereldoes not alter (AElnt)/ER1 but broadens the P(AElnt)distribution so that there are trajectories which transfer all of Ere]to E,,,. a. Effect of Incident Atom's Mass. The effect of varying the incident atom's mass on the energy transfer was investigated at Ere]of 120.8 kcal/mol. The average energies transferred and CID cross sections are given in Table 4. The energy transfer
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Collision-Induced Dissociation of Al6 and A113 Clusters
J. Phys. Chem., Vol. 99, No. 20, 1995 8153
EI 1=170kcaVmol
EIl:ZOOkcaVmol
1
4l d 040
0.09.
EI 14COkcaYmol
80
120
160
200
AEint (kcdmol)
I
130
220
310
400
AEint (kcallmol) Figure 6. Energy transfer distributions for Ar are described in the caption for Figure 5 .
+ A16 (Oh) on the L-J/A-T surface for different Erel’s. The normalization and units for the distributions
dissociations were observed for the first 0.1 ps, and then 3, 86,
44, 13, 9, 1, 1, 0, 0, and 0 dissociations occurred for each of the next consecutive 0.1 ps time intervals. Five percent of the trajectories dissociated, which represents 11%of the trajectories excited above the CID threshold. Animating these dissociating trajectories showed that energy was not redistributed within the cluster, and instead, the A1 atom that was hit by Ar dissociated. dissociations are small in These “apparent” number and appear to be confined to short times. The dissociation probability peaks between 0.2 and 0.3 ps and then falls to zero by 1 ps. One would expect an even smaller nonRRKM effect at lower collision energies, near the dissociation threshold. 2. Planar A16 (C2h) Clusters. To compare with the above CID simulations of the octahedral cluster, simulations of the
planar C2h cluster were performed. The structures of the o h and C2h clusters are compared in Figure 2. As shown in Table 1, the planar cluster has lower vibrational frequencies and a dissociation threshold that is 5.0 kcal/mol larger than that of the oh cluster. For Erel= 120.8 kcdmol, where the different dissociation thresholds for these two clusters is of only minor significance, the CID cross section for the planar cluster is 22.8 f 0.7 A* and statistically the same as for the octahedral cluster (see Figure 4). However, the similar cross sections arise from different attributes. The planar cluster has b,,, of 5.5 A, 2.0 A larger than that for the more compact octahedral cluster, and a larger fraction of the trajectories for the planar cluster do not transfer sufficient energy for CID to occur. These two effects combine to give similar CID cross sections for the two clusters. The
de Sainte Claire et al.
8154 J. Phys. Chem., Vol. 99, No. 20, 1995
2
3. Comparison with Experiment. CID of Alnf clusters has been studied experimentally by Jarrold,8 and coworkers. Jarrold et aL8 determined product branching ratios and cross sections for the CID of Al,+ clusters (n = 3-26) by Ar atoms at a center-of-mass collision energy of 121 kcal/mol. Al+ is observed to be the main product for clusters with fewer than 15 atoms. Our trajectory results (see section IV.A.l.C) agree with this finding. For Al6+, Jarrold et al. determined a cross section of 21 A*, which is similar to the trajectory value of 20-25 A2 in Figure 4. The estimated uncertainty in the experimental cross section is 50% as a result of difficulty in detecting light product fragments, multiple collisions between the translationally excited Al,+ and the Ar atoms, and the finite flight time from the gas cell to the detector, which means that some of the excited Al,+ clusters may not dissociate before reaching the detector. These effects complicate the comparison between the experimental and trajectory CID cross sections. The trajectory CID cross sections are determined by assuming every A16 cluster, vibrationallyl rotationally excited above the A16 ,415 iAl threshold, dissociates. This may not be the case in actual experiments as a result of collisions with other molecules and the cell wall and the finite time for detection. Thus, the trajectory cross sections should probably be considered as “upper limits” to the experimental cross sections. Hanley et aL9 measured fragmentation branching ratios, cross sections, and fragmentation thresholds for CID of A12-7+ with Xe atoms at collision energies less than 230 kcal/mol. They found a dissociation threshold for &+ A15 A1 of 36 kcal/ mol, with an estimated uncertainty of 20%. This value is smaller than EocrDof 52.7 kcal/mol found from the trajectories (see section IV.A.l) but is similar to the actual threshold of 38.8 kcal/mol for the analytic potential energy function used for the trajectory calculation. A total cross section of 6 A* with an uncertainty of 25% was reported by Hanley et aL9 for CID of Al6+ by Xe at a center-of-mass collision energy of 115 kcaV mol. This cross section is smaller than the trajectory upper limit reported here and the experimental value of Jarrold et al., both determined with Ar as the collision partner instead of Xe. Some of the difference in the Hanley et al. and Jarrold et al. cross sections may results from the different cell pressures and flight times for the two sets of experiments. B. Ar A113 COlliSiOnS. CID Of the D3d (fCC), D6h, and D2h A113 clusters (see Figure 3) was studied at Erel = 120.8 k c d m o l and a cluster rotational temperature Trotof 138 K. The dissociation threshold of the spherical type D3d (fcc) cluster is substantially smaller than those of the planar D2h and D6h clusters (see Table 2). The former cluster is compact and has a b,,, for CID equal to 5.0 A, while b,, for the planar less compact clusters is 6.5 A. The CID cross sections calculated from the trajectories are 54.7, 47.7, and 35.4 A2, and the fraction of the trajectories which transfer sufficient energy for CID to occur is 0.70, 0.36, and 0.27 for the D3d (fCC), D6h, and D2h Clusters, respectively. These latter fractions are consistent with the different dissociation thresholds for the clusters. The distributions P(AEinl),of energy transferred to the A113 clusters, are shown in Figure 11. The shape of P(AElnt)for the spherical, compact D3d cluster is similar to that in Figure 5 for the octahedral Al6 cluster. In contrast to the P(hEint)distribution for the D3d cluster, the P(hE,,,J distributions for the planar D2h and D6h clusters decrease more gradually as Eintis increased. As shown in Figure 5, this behavior is also observed for the planar Al6 cluster. For the planar D2h and D6h clusters a large fraction, 39% and 35%, respectively, of the transferred energy is to rotation. For +
..:
!;
0;’ h20
’
40
’
60
’
io
’
100
’
120
AEim,(kcahol)
+
Figure 7. Energy transfer distributions for Ar A16 (Oh) on the L-J/ A-T (solid bar graph) and Morse*! (dashed bar graph) surfaces at impact parameters of 0, 2, and 4 A. E,,! = 120.8 kcavmol. The normalization and units for the distributions are described in the caption
for Figure 5 . energy transfer distribution functions for the planar and octahedral Al6 clusters are compared in Figure 5. For both clusters there are no trajectories which transfer all of Erel (i.e., 120.8 kcal/mol) to cluster internal energy. However, the two distribution functions are quite different. The increase in the energy transfer probability from zero at large Mintis much more pronounced for the compact octahedral cluster. CID cross sections for the A16 (C2h) Cluster versus Ere1 are plotted in Figure 4, where they are fit by eq 6 with A = 44.6 A*, EocrD = 54.9 kcal/mol, and n = 0.98. The statistical uncertainty for these cross sections are nearly the same as those given above for the A16 (Oh) cluster. Thus, as for the A16 (Oh) clusters, the line-of-centers model seems appropriate for the A16 (C2h) clusters. The b,,, value of 3.8 A extracted from the A parameter is considerably smaller than b,,, = 5.5 A for the trajectories, because trajectories with relatively small impact parameters will not collide with the planar C2h cluster if the relative velocity of the collision is parallel to the cluster’s surface plane. The 54.9 kcal/mol threshold, extracted from the fit to the CID cross sections, is significantly higher than the actual dissociation threshold of 43.8 kcal/mol. The accuracy of the fitted threshold was tested by calculating an additional 600 trajectories at each Ere1 of 53.0, 50.0, and 47.0 kcal/mol. The numbers of CID events found for these energies are 10, 3, and 0, respectively, which gives cross sections of 1.3, 0.4, and 0.0 A2. When these three additional low-energy cross sections are included in the fit by eq 6, it is found that A = 37.9 A2, EocrD = 52.7 kcaumol, and n = 1.01. This fitted EocrDvalue is larger than Erel values for which CID occurs. Thus, eq 6 does not extrapolate to the correct CID threshold for the planar A16 (C2h) cluster.
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Collision-Induced Dissociation of A16 and
All3
Clusters
J. Phys. Chem., Vol. 99, No. 20, 1995 8155
0,25
b
0.051
n
mk
0.05, 0.
,
.
8
I
.. L *
0.2,
J 0.2
Q 0.1.
0,l
0 40 Figure 8.
60
80
120
100
40
80
60
100
AEint (kcaYmo1) AE.r t (kcdmol) Energy transfer distributions for A + A16 (Oh)on the L-J/A-T surface for different masses of the A atom.
120 E,I = 120.8 kcaUmo1. The
normalization and units for the distributions are described in the caption for Figure 5 .
the spherical D3d, the fraction is much smaller and only 7%. The distributions of P(AE,ib) and P(AErot),for trajectories leading to CID, are shown in Figure 12. Larger rotational energy transfer is expected for the D2h and D6h clusters in comparison to the D3d cluster, because of the planar, anisotropic structures and large collision impact parameters for the former clusters.
V. Models for T
-
V Energy Transfer
+
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A. Collinear A BC Collisions. To describe T V energy transfer in collinear A BC collisions, Mahan57358 derived an impulsive model, in which Fourier components of the short-range A BC repulsive forces are treated. The repulsive potential is represented by
+
+
V(r)= V, exp(-r/L)
Ere1
where
(i
cosech
AEsuddenlErel = 4 cos2 P sin' ,!?
(13)
For mc much larger than either mA or mB, eq 13 reduces to 'suddenfErel
+ mB)2
= 4mAmB/(mA
(14)
For large E the collision approaches the adiabatic An important parameter for the above model is
(9)
where r is the A-B distance. For this impulsive model, the relationship between energy transferred to vibration A E i n t and the initial relative translational energy Erel is
"int = 4 cos2 ,!?sin2 P
+
where v is the BC vibrational frequency and vrel is the A BC initial relative velocity. For large values of vrelcompared to Y, so that 6 < 1, the collision time is short compared to the vibrational period, and the sudden limit is reached. In this limit eq 10 becomes
-)2t 2
(10)
since it determines whether A and B have a single collision or multiple collisions during the A BC collision event. Secrest6I has shown that, for a harmonic BC oscillator and an exponential repulsive interaction between A and B, multiple collisions between A and B occur if m > 0.697. To approximate the sudden limit for Ar Al, collisions, an interaction between Ar and a single A1 atom is assumed. To identify possible masses for mc, the Ar Al, trajectories were analyzed to determine the number of collisions between Ar and the Al atom hit by Ar. In all cases there was only one collision. Thus, following the above work by Secrest, should be chosen so that the parameter m in eq 15 is less than 0.697, since the features of the Ar Al, potential are qualitatively similar to those used by Secrest. The mass for the C atom of the model
+
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de Sainte Claire et al.
8156 J. Phys. Chem., Vol. 99, No. 20, 1995
0.25
Morsel4
0.2 n
0.15
!A5
;
0.1
'i
0.05
3.051
0 0.2 0.15 n
4I 0.1 z
'I
0.1
44,
0.05 0 0.5
0
: i Morse*(
0.4 n
0.45,
@.I 0.3 U
"
0.3.
0.2
nd
0.1
0
k
0.15,
A 60 80
100
120
AEint (kcdmol)
AEin1 (kcdmol) +
Figure 9. Energy transfer distributions for Ar A16 (Oh) versus the stiffness of the A16 intramolecular potential. E,, = 120.8 kcalfmol. The normalization and units for the distributions are described in the caption for Figure 5.
can then be either that of a single atom or two aluminum atoms. The parameter m for those two mc masses is 0.43 and 0.66, respectively. For a Ar -k Al6 collision, the resulting value of AEsudden/Erel in eq 13 ranges from 0.84 for Ar A1-Al to 0.96 for Ar Al-Al2. This range of AEint/Erel is substantiallylarger than the limiting (AEint)/Erelvalue of -0.5 found from the trajectory calculations (see Table 5). Thus, (mint)from the trajectories is approximately '/2 of A&,, for the model. This result is not particularly surprising, since the model assumes collinear A BC collisions with an impact parameter of zero, while the Ar Al6 collisions are not collinear and do not have a zero impact parameter. What is significant about the above model is that it does reproduce the trends observed in the trajectories for the efficiency of energy transfer versus the incident atom's mass and the cluster stiffness (see section IV.A). As the adiabaticity parameter 5 in eq 12 increases with increase in vibrational frequency Y and/or decrease in initial relative velocity vRl, Mint/ E,I in eq 10 decreases. These are the same effects shown in
+
+
+ +
+
Tables 4 and 5 for Ar Al6 (Oh) collisions. In Figure 13 (AEint)/Erel divided by 4 cos2/3 sin2p is plotted versus 5 for the Ar Al6 (Oh) trajectories with different masses for Ar (results in Table 4) and different (Oh)vibrational frequencies (results in Table 5). It is seen that is a very effective parameter for combining the effects of incident atom mass and cluster stiffness. The frequency used for Y in calculating 5 is the average frequency of the Al6 (Oh) cluster, which is 240 cm-' for the L-J/A-T potential and ranges from 78 to 1244 cm-l for the Morse/4 and Morse*4 potentials, respectively. A value of 0.3 %, is determined for the range parameter L by fitting eq 9 to the ArA1 intermolecular potential in eq 4. To calculate energy transfer for the sudden limit, eq 13, both the mass of a single aluminum atom and the mass of two aluminum atoms are used for mc. The mass of a single aluminum atom is used for mB. Also plotted in Figure 13 is thefle) = [(5/2) cosech(@2)]* term from eq 10, scaled by a factor so that the 5 = 0 intercept fits the trajectory results. The scale factor for mc = mA12 is 0.52 and 0.60 for mc = mAl. As discussed in the next section,
+
Collision-Induced Dissociation of Al6 and A l l 3 Clusters Morse* 1 < AEvib=35.!
..I
1.2.
J. Phys. Chem., Vol. 99, No. 20, 1995 8157
;
I
-*-
q
1
=6.3
1.2.
T
< AEro,r13.f
0
h 60
45
75
90
105
120
A? 3 D3' 0.2
.... < A E p .5 < &r0,>=14.7
i3 d
z
1
%+I
1.
'
'1'
60
'
80 . 100 k120
AEint (kcaVmol)
D6hr and
.............. 4
'..%
20
5.
..,......
40
'
60
'
80
'
100
'
120
AE (kcdmol)
Figure 10. Vibrational (solid bar graph) and rotational (dashed bar graph) energy transfer distributions for Ar Al6 (Oh)versus the stiffness of the A16 intramolecular potential. Ere] = 120.8 kcaymol. The normalization and units for the distributions are described in the caption
+
for Figure 5 .
these scale factors are consistent with the orientation factor which results for collisions with random orientations. The trajectory results agree with the decrease in theAt) term with increase in However, for large 5; the trajectories predict more energy transfer. This is because the term is based on a model which only treats T -.V energy transfer. At large Y , T R is the dominant energy transfer mechanism in the trajectories (see Figure 10). The agreement between the trajectory results and At) of eq 10 is better for mc = A1 than mc = A12, which suggests the preferred model for the mass of mc may be one A1 atom. In summarizing Mahan's impulsive model, it correctly
c.
'
Figure 11. Energy transfer distributions for Ar collisions with the D z ~ , D3d A113 clusters. Ere]= 120.8 kcaymol. The normalization and units for the distributions are described in the caption for Figure
"
0.5. ... I..,
40
-
predicts how energy transfer is affected by cluster stiffness and the mass of the incident atom. However, it overestimates and underestimates energy transfer in the small 5 sudden limit and large 5; limit, respectively. For the former limit it does not include the effect of random orientations, and for the latter it does not treat T R energy transfer. It is noteworthy that Mahan's model, for the sudden limit, gives results in qualitative agreement with experiment for rare gas VO+ CID.62 B. Atom Macromolecule Collisions. A collision theory (CT) has been to describe T -.V energy transfer in collisions between a gas atom with mass mg and a macromolecule with mass mM, which consists of atoms with mass ma. Since there are similarities between properties of macromolecules and clusters, it is of interest to consider this model here. The CT model assumes hard-sphere collisions and accounts for neither the vibrational motions of the macromolecule nor the intermolecular potential between the macromolecule and the gas atom. Collisions are assumed to occur with individual atoms of the macromolecule. AEmax,the maximum energy transferred, is for a collision with an impact parameter
-
+
+
de Sainte Claire et al.
8158 J. Phys. Chem., Vol. 99, No. 20, 1995 0.6
mc=2Al
Al13 Dl.h
0.3 n
2
0.2
0.2 . 0.1 0.1
'
0 0.5 A113 D6h
0.4 0.4.
5
0.3
h
a
z
0.3
0.2 0.1
0 0.4 D3d
5 Figure 13. Plots of the trajectory average energy transferred (AEiOt) divided by the sudden energy transfer for the impulsive model, eq 13 (O), and of fit) = [(@2) cosech( 2k~T,, energy transfer is a maximum when m = M. Applying this model to Ar A16 collisions for 8 = 0" and Ts = 0 K gives AEintIEreI of 0.96,0.98,0.89,0.79,0.71, and 0.63 as the effective mass is varied from Al to six A1 atoms. If these energy transfer values are scaled by an orientation factor of '12 (see previous model), the calculations with M as the mass of one or two Al atoms are in the best agreement with the Ar Al6 trajectories in the sudden limit (see Tables 4 and 5 and Figure 13). Increasing M has the same general effect as increasing the adiabaticity parameter eq 12, in the A BC impulsite model. However, the effects are far from analogous. The gas-surface model does not explicitly consider the influences of surface vibrational frequencies, gas velocity, and range parameter of the gas-surface interaction. For T, = 0 K, m
