Article pubs.acs.org/JPCC
Understanding Energy Transfer in Gas−Surface Collisions from GasPhase Models Juan J. Nogueira,†,§ William L. Hase,‡ and Emilio Martínez-Núñez*,† †
Departamento de Química Física and Centro Singular de Investigación en Química Biológica y Materiales Moleculares, Campus Vida, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain ‡ Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061, United States S Supporting Information *
ABSTRACT: Large-scale trajectory simulations of different projectiles colliding with an organic surface, as well as a gas−surface model for energy transfer, are employed to investigate the effects of the mass, size, shape, and vibrational frequency(ies) of the projectile and of the projectile−surface interaction potential on the energy-transfer dynamics. The gas−surface model employed in this work relies on simple gas-phase scattering models. When energy transfer is analyzed in the limit of high incident energies, the following results are found in this study. The percent of energy transfer to vibration (and rotation) of light diatomic projectiles decreases as the projectile’s mass increases, while this transfer is almost independent of the mass for heavier projectiles. Transfer to final translation of diatomic projectiles is a U-shaped function of the projectile’s mass, as predicted by the hard cube model. For larger projectiles, the partitioning of the energy transferred to the internal degrees of freedom (dof) between vibration and rotation depends on the projectile’s size. In other words, transfer to rotation is more important for the smaller projectiles, while transfer to vibration dominates for the bigger ones, which have more vibrational dof. For small projectiles (less than 10 atoms), transfer to vibration increases as a function of the projectile’s size. However, for larger projectiles, the percent transfer to vibration is nearly constant, a result that can be attributed to a mass effect and also to the fact that only a reduced subset of “effective” vibrational dof is being activated in the collisions. For linear hydrocarbons colliding with the perfluorinated self-assembled monolayer (FSAM), the number of “effective” modes was estimated to be around 18, which corresponds to a percent energy transfer to vibration of 20−22%. The percent transfer to vibration of the more compact cyclic molecules is a bit higher than that for their linear counterparts. mass and energy transfer when atoms strike different surfaces.10 This kinematic model views a gas−surface interaction as a gasphase-like collision between an incident projectile and a region of the surface with an effective mass. An equation with four parameters, inspired from collinear atom + diatom (A + B−C) scattering models,13,14 has been recently employed to predict the average energy-transfer efficiencies in small projectiles (Ar, NO, CO2, and O3) colliding with an organic surface.11,12 This four-parameter model has two contributions, namely, adiabatic14 and impulsive,13 each describing different incident energy (EI) regimes. When applied to the above small projectiles colliding with a perfluorinated self-assembled monolayer (F-SAM), the model provides percent energy transfers to the surface of 90−98% in the limit of high incident energies, and it is able to fit the simulation average energy-transfer values over a 103-fold range of incident energies.12 An interesting property of the four-parameter model is that energy transfer follows a linear dependence on EI in the limit of high incident energies,12 which agrees with previous
I. INTRODUCTION Understanding energy transfer from simple mathematical models is of great interest and may facilitate the interpretation of experimental and simulation data. Several models have been developed in the past to study energy exchange in gas−surface collisions.1−8 The (hard and soft) cube models1−3 have been widely employed to predict trends in experimental scattering results of gas atoms with solid surfaces. In these models, the gas undergoes impulsive collisions with a surface, which is modeled by a cube, with energy transfer taking place only in the normal direction of the cube surface. An extension of the cube models is the washboard model,4,8 which introduces surface corrugation. When the surface is modeled by a sphere, the average energy-transfer efficiencies obtained from classical mechanics differ only slightly from the hard cube model results.5−7 On the other hand, it is very attractive to employ gas-phase scattering models to study a gas−surface interaction, and this has been done by some researchers in the past.9−12 For instance, Hunter studied the interaction of vibrationally excited gases with solid surfaces using a model resulting from the application of time-dependent perturbation theory to gas-phase energy transfer.9 More recently, Alexander et al. employed a kinematic scattering model to interpret the effective surface © 2014 American Chemical Society
Received: November 28, 2013 Revised: January 13, 2014 Published: January 13, 2014 2609
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gas−surface simulation studies.15−18 On the basis of the same A + B−C scattering models,14 equations with fewer parameters are derived to predict average percentages of energy transfer to the surface in the high-EI limit.18 These simplified models predict limiting high-EI percent energy transfers to the surface of ∼90% for two protonated peptides colliding with an FSAM,18 and of 89−98% for Ar and CO2 scattering off several SAM surfaces.19 The same A + B−C models were employed to understand the energy-transfer dynamics in a simulation study of the collision-induced dissociation (CID) of Al6 and Al13 clusters.20 Owing to the success of the above atom + diatom-like models in predicting energy transfer in gas−surface collisions, in this paper we further explore their capabilities. In particular, the effects on the energy-transfer dynamics of the gas−surface interaction potential and mass, size, shape, and vibrational frequency(ies) of the projectile are studied here with the help of the A + B−C models. To this end, chemical dynamics simulations are carried out for two different types of projectiles colliding with an F-SAM. One projectile is a diatom (B2) of variable mass, vibrational frequency, and interaction potential with the surface. The other one is a hydrocarbon of variable size and shape. In particular, the number of carbon atoms ranges from one to eight, and, in some cases, linear and cyclic structures were considered to investigate the effect of the shape. The influence of some of the factors enumerated in the previous paragraph on the collisional activation of polyglycine and polyalanine peptide ions have been previously studied by Meroueh and Hase.21 They found that, in the low collision energy regime, energy transfer for collisions with Ar increases as the peptide size increases. However, at high energies, the peptide size has a negligible effect on the energy-transfer efficiency. Additionally, the more compact α-helix peptides tend to be activated more efficiently than β-sheet structures.21 de Sainte Claire and Hase studied the influence of the repulsiveness of the intermolecular potential on the efficiency of energy transfer in octahedral Al6 and M6 clusters (M being a first-row transition metal atom) colliding with rare gases.22 The influence of the gas−surface interaction potential on the energy-transfer dynamics has been studied in simulation studies of protonated peptide ions scattering off F-SAM surfaces.18,23 The original A + B−C models that inspire the gas−surface model employed in the present paper provide equations that explain the influence of the factors mentioned above on the energy-transfer dynamics.13 For this reason, it is of interest to investigate whether the gas-phase models can be extrapolated to more complex gas−surface scattering problem.
simulation of the monolayer provides a 300 K structure of the surface in close agreement with experiment, i.e., the monolayer forms a hexagonal close-packed structure with the nearestneighbor direction rotated 30° with respect to the Au{111} lattice and the backbone of the CF3(CF2)7S moiety has a tilt angle with respect to the surface normal of ∼12°.24 The intramolecular potential function VProjectile of the diatomic gas B2 reads VB2 =
1 k b(r − r0)2 2
(2)
where a value of 1.00 Å was taken for the equilibrium distance r0. The vibrational frequency ω of the diatom ranges between 300 and 700 cm−1, from which the force constant kb of eq 2 can be obtained: kb = μω2, with ω now in s−1, and the reduced mass μ is mB/2 (mB is the mass of atom B). The total mass of the diatomic gas is mg (= 2mB), and it varies from 4 to 60 au. The values of the vibrational frequencies selected in this work are, in general, lower than typical values for diatomic molecules. This choice was made because previous work shows that lowfrequency modes are more easily excited than high-frequency modes of the projectile.12,15,17,21,26,27 In a similar fashion, energy transfer occurs, primarily, to the low-frequency modes of the surface.28 Additionally, the selected range of vibrational frequencies is sufficiently large to see some trends in the energy-transfer dynamics. A total of nine different hydrocarbons (CxHy) of different sizes and shapes are considered in this work (the structure is included in parentheses): isotopically substituted carbon hydride 15C1 H, methane, ethane (staggered), n-butane (staggered anti), pent-1-ene (linear), cyclopentane (envelope), hex-1-ene (linear), cyclohexane (twist boat), and n-octane (linear). The rationale for the selection of these molecules is explained as follows. With an increase in the size of the projectile, there is a concomitant increase in its mass, which can blur the effect of size on energy transfer. To avoid this problem, the results of 15C1H + F-SAM and CH4 + F-SAM collisions can be compared, because both projectiles have the same mass, and the number of vibrational degrees of freedom increases by a factor of 9 from 15C1H to CH4. The influence of the projectile’s shape on energy transfer can be studied by comparing the results of linear versus cyclic hydrocarbons. To keep the number of degrees of freedom constant in this comparison, the pent-1-ene and hex-1-ene alkenes were selected, instead of the n-pentane and n-hexane alkanes, to compare with cyclopentane and cyclohexane, respectively. The equilibrium geometries of each CxHy molecule are taken from MP2/cc-pVDZ calculations,29 except for CH where r0 is 1.10 Å. A detailed description of the intramolecular parameters of the force field30 is described in the Supporting Information. The interaction between B2 and the F-SAM is modeled by a sum of repulsive two-body potential energy functions:
II. COMPUTATIONAL DETAILS II.A. Potential Energy Surface. The potential energy function of all the projectile + F-SAM systems studied here reads V = Vsurf + VProjectile + Vinter (1)
Vinter,B2 /F‐SAM = 1.5 × 105 ∑ exp( −R ij/L) ij
The first term Vsurf describes the interactions that take place within the F-SAM surface,24,25 VProjectile is intramolecular potential function of the B2 diatom or the hydrocarbon, and Vinter accounts for the interaction between the projectile and the surface. The F-SAM surface consists of 48 chains of CF3(CF2)7S radicals adsorbed on a single layer of 225 Au atoms, which are kept fixed during the dynamics simulations; all atoms are explicitly considered in our model. A molecular dynamics
(3)
The interaction potential of eq 3 is expressed in kcal/mol, i and j represent each of the atoms of projectile and surface, respectively, Rij is the i−j interatomic distance, and L is a parameter that provides the hardness of the two-body interaction, and it varies from 0.33 to 1 Å. To model the interaction between CxHy and the F-SAM, the following sum of two-body repulsive terms, derived by Wang and Hase,23 has been employed: 2610
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a 2 ps molecular dynamics simulation34 in which the atomic velocities are scaled to obtain a surface temperature TS of 300 K. This structure is then used as the initial structure of a 100 fs equilibration run at the beginning of each trajectory. Periodic boundary conditions are also utilized to simulate a larger surface.34 After the integration of each trajectory, the surface energy and the translational, vibrational, and rotational energies of the scattered projectiles are calculated from the atomic Cartesian coordinates and momenta. II.C. Description of the Model. The model employed in this paper to study energy transfer in projectiles colliding with an organic surface is a combination of two gas-phase models for A + B−C collisions.12−14 The gas−surface model assumes that B−C is the gas that strikes A, which now is a region of the organic surface. Each of the two A + B−C models employed here provides a different functional form for the average energy transfer to the projectile ⟨ΔE⟩. One model is valid at low incident energies (adiabatic limit), while the other one is employed at high incident energies (impulsive limit). In the gas−surface model, the adiabatic term is mathematically expressed as11,12,14
∑ Aij exp(−BijR ij) + Cij/R ij5 ij
(4)
where i, j, and Rij are defined as above. The values of Aij, Bij, and Cij as well as the procedure to obtain these parameters is explained in detail in ref 23 and also in the Supporting Information. Although the above interaction potentials do not present attractive terms, they should provide a realistic description of the interactions for most of the incident energies of the present study. The well depths for the interactions between small gases and an F-SAM are typically lower than 0.05 eV,11,12,24,25 which is well below the incident energy range employed in this study (4.3−173.5 eV; see below). Additionally, the same gas−surface model was successfully employed in previous work to fit the simulation data for ozone scattering off an F-SAM surface, with a very accurate gas−surface interaction potential.12 The use of more realistic gas−surface interaction potentials (with the inclusion of attractive terms) could somehow affect energy transfer in the adiabatic (low EI) regime, but the general conclusions of this work would remain unaltered. Taking into account the very high incident energies employed in this work, the projectiles might penetrate through the gold slab, due to the lack of projectile−gold interactions in our potential function. To prevent this unphysical behavior, and in the absence of accurate potential functions for these twobody interactions, the repulsive potential functions of previous work are also employed here (see the Supporting Information for a description of these interactions).12,24,31 II.B. Chemical Dynamics Simulations. Two different projectile types are employed in this work: a diatomic molecule, B2, and a hydrocarbon of general formula CxHy. For the B2 + F-SAM simulations, the following incident energies (in eV) are employed: 4.3, 10.8, 21.7, 32.5, 43.4, 54.2, 65.0, 75.9, 86.7, 108.4, 130.1, 151.8, and 173.5. For each of the above incident energies, a subset of simulations is carried out with mg taking the values 4, 10, 20, 40, and 60 au, while setting ω and L to 300 cm−1, and 0.33 Å, respectively. In another subset of simulations the following values of ω (in cm−1) are considered: 300, 400, 500, 600 and 700, with mg = 20 au and L = 0.33 Å. Finally, five simulations are carried out for L = 0.33, 0.50, 0.67, 0.83, and 1.00 Å, with mg = 20 au and ω = 300 cm−1. The simulation with mg = 20 au, ω = 300 cm−1, and L = 0.33 Å is repeated in all subsets, but it was obviously carried out only once per incident energy, which makes a total of 169 different batches of trajectories. The effects of the size and shape of the projectile are studied in additional simulations of the nine hydrocarbons described above colliding with the F-SAM, for the following values of the incident energies (in eV): 4.3, 8.7, 13.0, 17.3, 21.7, 43.4, 65.0, 86.7, 130.1 and 173.5. The different projectiles collide perpendicularly with the surface, and their initial rotational energy and vibrational quantum numbers are zero. Each batch consists of 2000 trajectories, which are integrated with the Adams−Moulton algorithm, as implemented in the VENUS05 computer program.32,33 Different step sizes are employed for the different projectiles to achieve similar energy conservation errors: 0.7 and 0.2 fs for B2 + F-SAM and CxHy + F-SAM, respectively. The integration terminates when either the scattered projectiles reached a height of 35 Å with respect to the gold slab, or 60 ps elapsed. Before the beginning of each trajectory simulation, the surface is relaxed to a thermodynamic equilibrium structure by
⟨ΔEad⟩ = ⟨ΔE⟩0 + a exp( −b/ E I )
(5)
where a and b are parameters. In the adiabatic A + B−C model,14 b is defined as (mg/2)1/2Lω/2π, where ω, mg, and L have the meanings given above. The term ⟨ΔE⟩0, which is absent in the A + B−C model, provides the amount of energy transfer in the limit of zero incident energy; this term is the only difference between the gas−surface model of eq 5 and the original adiabatic A + B−C model.14 The value of ⟨ΔE⟩0 can be set by assuming that, when EI → 0, the projectile reaches thermal accommodation with the surface.12 Alternatively, it can be treated as a parameter like a and b, which is the procedure followed in the present work. The impulsive term of the gas−surface model is taken from Mahan’s work for A + B−C collisions:13 ⎛ d ⎞ ⎟⎟ ⟨ΔEimp⟩ = cd 2 csch2⎜⎜ ⎝ EI ⎠
(6)
where c and d are parameters. In Mahan’s model they are defined as13 c=
4mA mBmC(mA + mB + mC) (mA + mB)(mB + mC)
(7)
with mA, mB, and mC being the masses of atoms A, B, and C, respectively, and d=
2 π mg Lω 2
(8)
A simpler expression for the impulsive term in the high-EI limit can be obtained if it is approximated by a Taylor series. When the series is truncated after the second term, it shows a linear dependence on EI: ⟨ΔEimp⟩E I →∞ ≅ cE I −
1 2 cd 3
(9)
where the slope c provides the limiting high-EI fractional energy transfer and d2 provides the onset of impulsive behavior. Combining the above adiabatic and impulsive terms, the functional form of the gas−surface model reads12 2611
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⟨ΔEj⟩ = ⟨ΔEjad⟩ + ⟨ΔEjimp⟩ ⎛ dj ⎞ = ⟨ΔEj⟩0 + aj exp(−bj / E I ) + cjdj 2 csch2⎜⎜ ⎟⎟ ⎝ EI ⎠ (10)
where the index j refers to energy transfer to either the projectile’s vibrational (j = V), rotational (j = R), or final translational (j = F) energies. The set of 15 parameters ⟨ΔEj⟩0, aj, bj, cj, and dj (for j = V, R, F) of eq 10 are fit to the simulation data of the present work to obtain the model results for ⟨ΔEV⟩, ⟨ΔER⟩, and ⟨EF⟩. Although eq 10 is employed here to model ⟨EF⟩, in principle the A + B−C models only provide a framework to predict ⟨ΔEV⟩ and/or ⟨ΔER⟩.13,14 The final relative translational energy can still be calculated in a A + B−C scattering event subtracting ⟨ΔEV + ΔER⟩ from the initial relative translational energy. In a more complex gas−surface collision, besides the final translational energy EF of the projectile, there is an extra energy reservoir, the surface vibrational modes ESurf, which prevents the use of the A + B−C models to directly obtain energy transfer to ESurf. Therefore, in this work energy transfer to EF was modeled using eq 10, and for the transfer to the surface the following equation is employed: ⎧ ⎪ ⟨ΔESurf ⟩ = E I − ⎨ a exp( −b VRF / E I ) ⎪ VRF ⎩ ⎛ d ⎞⎫ ⎪ + c VRFdVRF 2 csch2⎜⎜ VRF ⎟⎟⎬ ⎝ E I ⎠⎪ ⎭
(11)
which is obtained from energy conservation E I = ⟨ΔE V ⟩ + ⟨ΔE R ⟩ + ⟨ΔESurf ⟩ + ⟨E F⟩
(12)
and assuming that the average transfer to the projectile, including internal energy and translation, can be grouped together in a single equation with only four parameters (aVRF, bVRF, cVRF, and dVRF); these parameters are fit to the present simulation results for ⟨ΔESurf⟩. From the values of cVRF, the limiting high-EI percent energy transfers to the surface P∞ can be calculated as P∞ = (1 − cVRF) × 100. All the fits are conducted with the help of a genetic algorithm, described in detail elsewhere.35,36
Figure 1. Average energy transfer to vibration (a), rotation (b), and final translation (c) of the projectile, and to the surface (d), for diatomic projectiles of different masses colliding with the F-SAM, obtained in our trajectory calculations (circles) and fits of the gas− surface model (lines). The corresponding percent energy-transfer values are displayed in panels e−h.
III. RESULTS AND DISCUSSION In the following, the dependence of energy transfer on mg, ω, L, and on the size and shape of the projectile will be presented. The simulation and model results are depicted graphically in Figures 1−4, which show the effects of the above factors on energy transfer; Figures 1−3 display energy-transfer results for B2 colliding with the F-SAM, while Figure 4 shows the results for CxHy + F-SAM collisions. The structure of the four figures is the same: panels a−d collect the average amounts of energy transferred to the projectile’s vibration, rotation, and final translation, and to the surface, respectively, while panels e−h display the corresponding percent energy-transfer values. The circles represent the simulation results, and the lines are the fits of eq 10 (for ⟨ΔEV⟩, ⟨ΔER⟩, and ⟨EF⟩) and eq 11 (for ⟨ΔESurf⟩) to the simulation data. The model parameters obtained in the fits are collected in Table 1 and Tables 4−6 of the Supporting Information. Although the average energy values (panels a−d) seem to follow an approximately linear dependence with EI in
Figures 1−3 (in Figure 4 these plots are clearly nonlinear), the percent values (panels e−h) show an interesting structure for low collision energies which is discussed in detail below. III.A. Dependence of Energy Transfer on the Mass of the Projectile. The mass of the projectile has an important effect on energy transfer as can be seen in Figure 1. For instance, at the highest incident energies of this study, a change in the mass of the projectile from 4 to 60 au leads to a 15-fold decrease in ⟨EF⟩ (Figure 1c), and energy transfer to vibration (Figure 1a) decreases by a factor of 8. At all incident energies, energy transfer to the internal energy and final translation of the projectile is much more efficient for the lightest projectiles. As a consequence of energy 2612
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Figure 3. Average energy transfer to vibration (a), rotation (b), and final translation (c) of the projectile, and to the surface (d), for different interaction potentials between the diatomic projectile and the F-SAM, obtained in our trajectory calculations (circles) and fits of the gas−surface model (lines). The corresponding percent energy-transfer values are displayed in panels e−h.
Figure 2. Average energy transfer to vibration (a), rotation (b), and final translation (c) of the projectile, and to the surface (d), for diatomic projectiles of different vibrational frequencies colliding with the F-SAM, obtained in our trajectory calculations (circles) and fits of the gas−surface model (lines). The corresponding percent energytransfer values are displayed in panels e−h.
the percent energy transfers are discussed in the following paragraphs. The analysis of the dependence of the model parameters on mg provides a useful means to understand the energy-transfer dynamics. Of particular interest here is to investigate whether this dependence can be explained by the original meaning of the model parameters (eqs 7 and 8), or by any other simple physical model. While the adiabatic parameters aj and bj decrease with mg (see Figure 5, parts a and b, respectively), the impulsive parameter dj increases (see Figure 5d). Quite interestingly, d in the A + B−C model (see eq 8) is proportional to the square root of mg, which is in qualitative agreement with the dependence shown in Figure 5d. Since d
conservation, the percent energy transfer to the surface increases with the mass of the projectile. As the projectile becomes heavier, the energy-transfer curves of Figure 1 tend to coalesce for most incident energies. However, the high-EI limits of the percent energy transfers, which are numerically provided by the parameters cj, do not necessarily converge for different masses. Figure 5 displays the values of aj, bj, cj, and dj as a function of mg. The values of ⟨ΔEj⟩0 are not shown in this figure because they are not accurate, due to a lack of simulation data for very low incident energies. For instance, Figure 5c shows that, while cV and cR seem to reach a plateau for high values of mg (see the inset), cF does not. The high-EI limits of 2613
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Table 1. Parameters of the Energy-Transfer Model (Equations 10 and 11) for the Different Hydrocarbons (CxHy) Colliding with the F-SAMa ⟨ΔEV⟩0
projectile CH CH4 C2H6 C4H10 C5H10 C6H12 C8H18 c-C5H10 c-C6H12 projectile
0.00 0.00 0.01 0.08 0.04 0.09 0.11 0.03 0.03 ⟨ΔER⟩0
CH CH4 C2H6 C4H10 C5H10 C6H12 C8H18 c-C5H10 c-C6H12 projectile
0.04 0.00 0.12 0.00 0.00 0.00 0.00 0.00 0.00 ⟨EF⟩0
CH CH4 C2H6 C4H10 C5H10 C6H12 C8H18 c-C5H10 c-C6H12 projectile
0.00 1.37 0.00 0.00 0.00 0.00 0.00 0.00 0.00
CH CH4 C2H6 C4H10 C5H10 C6H12 C8H18 c-C5H10 c-C6H12
aV
bV
cVb
dV
56.45 81.01 50.69 13.30 8.93 10.92 15.90 11.47 11.06 aR
28.22 20.01 17.66 10.79 8.27 8.60 9.09 12.65 11.63 bR
0.01 13.95 23.48 22.46 20.98 20.75 22.01 23.46 24.07 cRb
0.00 16.62 17.80 14.81 14.10 14.75 17.33 16.25 16.99 dR
1.23 1.86 1.78 1.40 1.47 1.48 1.59 1.37 1.42 aF
5.71 6.52 5.27 2.66 2.80 2.99 3.39 3.13 3.02 bF
4.45 5.51 3.94 4.91 6.06 0.80 0.80 0.90 2.08 cFb
14.91 19.15 23.07 41.04 46.65 33.17 57.22 21.40 34.11 dF 19.92 23.19 36.33 1.07 1.30
18.27 15.02 15.76 14.55 15.08 14.21 11.92 16.69 17.40
2.64 2.25 2.90 3.55 4.35 4.95 5.72 3.34 3.85
0.97 4.44 0.86 1.22 1.55 1.78 1.96 1.04 1.36
aVRF
bVRF
45.23 46.58 64.76 0.68 0.32 0.00 0.00 0.64 0.37 P∞
3.51 4.49 5.86 9.65 11.45 12.34 13.21 8.02 8.97
1.44 2.02 2.08 2.93 3.17 3.29 3.28 2.67 2.89
46.16 40.57 54.54 73.70 75.98 78.35 81.42 71.42 71.42
0.94 1.17 dVRF
a
Units are such that the energy is in electronvolts (eV). bc is multiplied by 100 so that it is expressed as a percentage (like P∞).
Figure 4. Average energy transfer to vibration (a), rotation (b), and final translation (c) of the projectile, and to the surface (d), for different hydrocarbons colliding with the F-SAM, obtained in our trajectory calculations (circles) and fits of the gas−surface model (lines). The corresponding percent energy-transfer values are displayed in panels e−h.
energy transfer to final translation of the projectile, the dependence of ⟨EF⟩/EI is analyzed here using the hard cube model, which has been widely utilized to understand atoms + surface collisions.1,2 In this model, the tangential component of the incident velocity of the projectile is conserved during collision with the surface, which is modeled by a vibrating cube of effective mass msurf. Additionally, the limiting high-EI fractional energy transfer to EF is2
provides a measure for the onset of impulsive energy transfer, the result of Figure 5d indicates that, for heavier projectiles, energy transfer becomes less efficient, because the impulsive limit is reached at increasingly higher incident energies. This point is discussed in more detail in the following paragraphs. The limiting high-EI percent energy transfers to final translation of the projectile (cF, black symbols in Figure 5c) display a minimum for mg ∼ 20 au. To interpret these results, the definition of the A + B−C model parameters (eqs 7 and 8) cannot be employed, as they are meaningless for energy transfer to EF (see the description of the gas−surface model above). With the A + B−C models unable to provide an explanation for
⎛ ⟨E F⟩ ⎞ (1 − μ)2 = ⎟ ⎜ ⎝ E I ⎠ E →∞ (1 + μ)2
(13)
I
where μ is a mass ratio defined as mg/msurf. The above equation indicates that, in the limit of a projectile with zero mass (mg = μ = 0), energy transfer to translation is 100% efficient. For projectiles with μ < 1, the fractional energy 2
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Figure 5. Model parameters of eqs 10 and 11, obtained from fits to the simulation results, as a function of mg. Parameters aj, bj, cj, and dj are displayed in panels a, b, c, and d, respectively. The subscript j refers to V (vibration), R (rotation), F (final translation), or VRF (overall energy of the projectile). The values of cV and cR are displayed graphically in the inset of panel c.
transfer to translation decreases with μ, and when mg and msurf are equal (μ = 1), transfer to translation vanishes. Finally, for projectiles heavier than the effective surface mass (μ > 1), ⟨EF⟩/ EI increases with μ. Figure 5c shows that the values obtained for cF in our study (black circles) follow this trend. To quantify this dependence, eq 13 is multiplied by a factor cF0 and fit to the parameter values; the result is shown as the solid black line in Figure 5c. The factor cF0, which represents the fractional transfer to translation in the limit of an infinitely light projectile and high EI, and msurf are parameters in the fit. The resulting values of cF0 and msurf are 0.55 and 21 au. Quite interestingly, the effective surface mass is very close to the mass of a fluorine atom. According to the hard cube model, the incident gas undertakes an impulsive collision with a single surface atom isolated from the rest.2 Considering the structure of our F-SAM surface, with the outer surface containing −CF3 groups, the single atom should be a fluorine atom. Similar conclusions were reached in a experimental study of CF+ and CF2+ ions scattering from a perfluoroether liquid polymer.37 In this study, the inelastic scattering of the CF+ ion is determined by a CF+− CF interaction at high incident energies. The limiting high-EI percent energy transfers to vibration and rotation, cV and cR, respectively (see the inset of Figure 5c), display a different dependence on mg: they show an initial decline with mg, and then, they level off. In particular, there is an ∼5% loss of percent energy transfer to vibration, when the projectile increases its mass from 4 au to infinity, and a 3% loss in the percent transfer to rotation. To understand the variation of cV and cR versus mg we can make use of Mahan’s impulsive model,13 upon which our gas−surface model is based. Parameter c in Mahan’s model depends on the masses of atoms A, B, and C, as shown in eq 7. Adapting Mahan’s model to our gas−surface scattering problem, mA is replaced by an effective surface mass msurf, and mB = mC since the diatomic projectile is homonuclear. Using the above definition for the mass ratio μ, eq 7 becomes
c=
4(1 + μ) (2 + μ)2
(14)
This equation shows that the limiting high-EI percent energy transfers decrease with the mass of the projectile: the value of c would be 1 for a projectile of zero mass, and it would be 0 for an infinitely massive projectile. In general the values of cj (j = V, R) obtained in this work follow the trend given by eq 14 (see the inset of Figure 5c). Very good fits to the values of cV and cR are obtained when we employ the following slight modification of eq 14: c′(μ) = c′∞e−α / μ +
4(1 + μ′) (2 + μ′)2
(15)
where c′∞ is a parameter that provides the limiting high-EI fractional energy transfer for an infinitely massive projectile, and the exponential was included to keep the same limiting value of c as in eq 14 for a projectile of zero mass. Finally, μ′ is defined as βμ, with β being a dimensionless parameter. The same surface effective mass of 21 au, obtained in the fit to the cF values, is employed here. The fits of eq 15 to cV and cR provide the following values for c′∞ (expressed as a percentage), α, and β: 1.8%(0.9%), 0.21(0.16), and 410(712) for energy transfer to vibration(rotation). This means that the internal degrees of freedom of an infinitely massive diatomic projectile would receive only 2.7% of the incident energy in the high-EI limit. III.B. Dependence of Energy Transfer on the Frequency of the Projectile. Overall, energy transfer depends very little on the vibrational frequency of the diatom (see Figure 2). The most appreciable influence of ω is on transfer to the internal degrees of freedom of the projectile (see Figure 2, parts a and b): ⟨ΔER⟩, for instance, changes by a factor of 1.6 when the value of ω changes from 300 to 700 cm−1. In general, energy transfer to the rovibrational degrees of freedom of the projectile is enhanced for higher vibrational frequencies as seen in Figure 2, parts a and b. However, a closer look at the percent energy transfers to vibration for EI < 10 eV 2615
dx.doi.org/10.1021/jp4117134 | J. Phys. Chem. C 2014, 118, 2609−2621
The Journal of Physical Chemistry C
Article
The percent transfers to projectile’s vibration (Figure 4e) and rotation (Figure 4f) display an interesting dependency on EI: at incident energies lower than 50 eV, transfer to vibration/ rotation is more efficient for the bigger projectiles, while for EI > 50 eV the behavior is, in general, the opposite. The low-EI behavior of the transfer efficiencies to vibration is explained by the larger number of low-frequency modes in the big projectiles compared to the smaller ones, a well-known effect discussed in detail in previous work.15,17,21,26 At higher incident energies (50−175 eV), the efficiency of transfer to vibration presents a slight tendency to decrease with the projectile’s size, with the exception of the two smallest hydrocarbons (see Figure 4e). This is possibly a mass effect, because the increase in the size of the projectile carries the corresponding increase in its mass. The results discussed above show that the mass effect is not very important for diatomic projectiles, i.e., when the mass of the diatom mg is increased from 4 au to infinity, there is a loss of only 5% in the percent transfer to vibration, while for projectiles with mg > 30−40 au, the percent transfer to vibration is almost independent of mg. Laskin and Futrell, who found the same effect in surface-induced dissociation (SID) studies of small protonated peptides,40−42 gave a successful explanation for this phenomenon. According to these authors, the decrease in the efficiency of transfer to vibration for larger projectiles of mass Mproj can be understood when the incident energy in the lab frame, EI, is converted to an “effective” centerof-mass energy ECM, using an arbitrary neutral partner of mass MN. They propose a model where the percent of ECM transferred to vibration is independent of the projectile’s size. Then, since ECM decreases with the projectile’s mass, ECM = (MN/(MN + Mproj)EI, transfer to vibration will also decrease with the mass of the projectile for a given incident energy. This is a mass effect which provides the same qualitative result as eqs 14 and 15 employed in this work to explain the B2 + F-SAM simulation results. In both cases, an increase in the mass of the projectile leads to a decline in the percent transfer to vibration. However, Laskin and Futrell’s model slightly overestimates the mass effect, i.e., the model predicts a more acute decrease of the percent transfers to vibration than the experimental results. This is probably because they employ a single value for MN for all the projectiles of their study.42 The value of MN, which is the analogous of the effective surface mass msurf employed in this work, should increase with the projectile’s size, as discussed below. If the value of MN increases with the projectile’s size, ECM will be closer to EI, and the mass effect will be less pronounced. Energy transfer to rotation (see Figure 4f) is more efficient for the big projectiles at low incident energies (
