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Apr 9, 2009 - The EVE model is briefly reviewed in this section. The first extension to pure dipole coupling refers to the inclusion of polarization e...
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Predicting the Influence of Shape, Size, and Internal Structure of CO Aerosol Particles on Their Infrared Spectra George Firanescu and Ruth Signorell* Department of Chemistry, UniVersity of British Columbia, 2036 Main Mall, VancouVer, British Columbia V6T 1Z1, Canada ReceiVed: NoVember 19, 2008; ReVised Manuscript ReceiVed: January 21, 2009

The influence of shape, size, and internal structure of CO aerosol particles on mid-infrared spectra is modeled for aggregates in the size range between 1 and 100 nm. Combining the vibrational exciton model with a molecular dynamics approach, we identify spectral features that are characteristic for the shape of the particles and for their internal structure (crystalline, amorphous, and partially amorphous) over the whole particle size range. The characteristic size-dependent patterns in the spectra of small particles (0.1 D), resonant transition dipole coupling turned out to be an excellent sensor for probing the shape, size, phase, and architecture of molecular aggregates. When this type of interaction dominates, it greatly simplifies the computational problem and makes the prediction of infrared spectra from first principles possible even for particles with tens of thousands of degrees of freedom.19,20 Combined with a molecular dynamics approach to determinate the internal structure of the particles, our model allows us to understand infrared spectra of CO aggregates and identify the characteristics specific to their makeup. 2. Computational Approach 2.1. Exciton Model. The vibrational exciton model used in this work to predict the infrared spectra of CO aerosol particles is a quantum mechanical model that mainly describes the resonant transition dipole coupling between all molecules within a molecular aggregate.16,21,22 The current implementationswhich we call the extended vibrational exciton (EVE) modelscontains several extensions to our original implementation,22,23 which have been discussed in detail in refs 19 and 20. The EVE model is briefly reviewed in this section. The first extension to pure dipole coupling refers to the inclusion of polarization effects which were shown to play a significant role in small SF6 clusters24-26 (about 10% of the total contribution to the frequency splitting in the SF6 dimer), followed by similar findings for SF6 particles20 and CHF3 particles.27 The molecular polarizability of CO is roughly half that of CHF3 or SF6 so that we expect similar if somewhat less pronounced effects for CO particles. This is confirmed in Figure 1 by the noticeable re-distribution of intensities in the spectra of small particles (here a crystalline sphere with a radius of 2 nm). A second general effect of induced dipoles is a slight

10.1021/jp8101767 CCC: $40.75  2009 American Chemical Society Published on Web 04/09/2009

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J. Phys. Chem. B, Vol. 113, No. 18, 2009 6367

C(t) ) g(t)

∑

ˆ

〈km|µkme-iHt⁄pµln|ln 〉

(5)

k,m,l,n

where g(t) is the Fourier transform of a suitable window function (line shape). The correlation function C(t) is calculated using a second-order propagation scheme: ˆ

|ψt+∆t 〉 ) e-iH∆t⁄p|ψt 〉 ≈ -i

Figure 1. EVE spectra of a (CO)748 crystalline sphere (r ) 2.0 nm): with (full) and without polarization effects (dashed). The spectra are convoluted with a 0.2 cm-1 full width at half-maximum (fwhm) Gaussian line shape.

ˆ )H ˆ0+ H

∑ µi+Aijµj

σ(ν˜ ) )

∑ σi(ν˜ ) ) ∑ ∫ g(t) ∑ ∑ 〈km|µkme-iHt⁄pµln|ln〉 dt ˆ

λij )

(1 - δij) 4πε0Rij3

∑ λikRkλjk)

(2)

(7) To account for the different sizes of the volume elements, we use the “normalized excitation density”

σ j i(ν˜ ) ) Ni-1σi(ν˜ )

(8)

(3eijeij+ - 1)

(3)

Fi(ν˜ ) )

∑ ∑ |ckm,I|2

with

|EI ⁄ hc - ν˜ | e ∆ν˜

(9)

k∈δVi m,I

k

ˆ 0 is the Hamiltonian for uncoupled molecular oscillators, where H λij are scaled projection matrices, Rk is the polarizability tensor of molecule k, µi is the dipole moment of molecule i, eij is the unit vector pointing from the center of mass of molecule i to that of molecule j, and Rij is the corresponding distance. In the product basis of molecular oscillator eigenfunctions |im〉 the Hamilton matrix elements take the following form: + ˆ |jn 〉 ) δijδmnhcν˜ im + 2µim 〈im|H Aijµjn

k∈δVi m,l,n

where Ni is the number of molecules in δVi. With a local density of states defined as19

with

1 Aij ) - (λij + 2

i

(1)

i,j

(6)

In addition to the calculation of the infrared extinction spectra the analysis of the vibrational wave functions from the EVE model allows us to correlate spectral features with structural properties of an aerosol particle. We can, for example, decide whether a certain spectral feature has its origin in the particle surface or in the core (see section 3.1). To this end, we partition the spectrum σ(ν˜ ) into contributions σi(ν˜ ) associated with individual volume elements δVi in the aerosol particle.19 Technically the σi(t) are obtained from an equivalent partitioning of C(t):19

i

bathochromic shift of the vibrational band as was observed for SF6 aerosol particles. With the inclusion of polarization effects the vibrational Hamiltonian takes the following form:

2∆t ˆ H|ψt 〉 + |ψt-∆t 〉 p

(4)

where ν˜ im and µim are the wavenumber and transition dipole moment, respectively, for transitions from the ground state to the mth level of the ith molecule. |im〉 represents the product function with level m excited on molecule i and all other oscillators in the ground state. Here we have set the energy zero ˆ |0〉. The basis is limited to at the uncoupled ground state 〈0|H near-resonant single-molecule excitations. The second extension in the EVE model accounts for local structural variations within an aerosol particle when it is no longer justified to assume the same values of ν˜ im and µim for all molecules of a given type (as in the original vibrational exciton model). Instead we assign individual transition wavenumbers ν˜ im and transition dipoles µim to each molecule on the basis of the explicit potential and dipole functions described in the next section. For the treatment of particles with more than 10000 degrees of freedom, a direct diagonalization of the Hamiltonian (eq 1) becomes impractical and particle spectra σ(ν˜ ) are computed by Fourier transformation of the dipole autocorrelation function:

σi can be interpreted as the local density of states weighted by the transition probability. Here ckm,I is the contribution of |km〉 to the eigenvector |I〉 with eigenvalue EI of the Hamiltonian (eq 1) and ∆ν˜ is the spectral resolution. Note that the above definition of the local excitation density (eq 7) is equivalent to that of ref 19. 2.2. Potential Model. The EVE model uses as input individual transition dipoles µkm and transition frequencies ν˜ km for each molecule in the particle, which depend on the intramolecular potential and the potential field generated by all other molecules in the particle. Suitable intra- and intermolecular potential models must be computationally efficient to treat particles with tens of thousands of degrees of freedom. This limits the choice for the intermolecular potential to two-body potentials. The most advanced dimer potential has been developed by Vissers et al.28 For our purposes, however, this potential is too complicated. Although the literature provides several simple intermolecular potential models for CO,29-34 a few adjustments were necessary for the application to our CO nanoparticles. The starting point for intermolecular interactions is the model of Nutt and Mewly (NM).29 Dispersion and exchange are modeled using a Lennard-Jones potential:

[( Rσ ) - ( Rσ ) ]

VLJ ) 4ε

12

6

(10)

The C-O interaction parameters are derived as usual:

σCO ) (σCC + σOO) ⁄ 2

and

εCO ) √εCCεOO

(11)

The electrostatics are described by a point charge array composed of three partial charges placed at the atomic sites (qC and qO) and in the center of mass (qCM) with

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Firanescu and Signorell

TABLE 1: Adapted Nutt and Mewly (ANM) Potential Parameters (Harmonic Frequency ν˜ , Lennard-Jones Parameters, and Coefficients for the Variable Partial Charges Placed on the C and O Atoms ν˜ (cm-1) 2143.1

C-C O-O

ε (eV)

σ (Å)

3.4 × 10-3 3.4 × 10-3

3.4 3.4

qC qO

TABLE 2: Dimer Geometry and Dissociation Energy (De) for the Slipped Antiparallel Configurations (S1 and S2; See Figure 2a)a S1 high-level ab initio28 high-level ab initio11 ANM model NM model29 S2 high-level ab initio28 high-level ab initio11 ANM model NM model29

|R| (Å)

θA (deg)

θB (deg)

De (cm-1)

4.31 4.33 4.05 4.26

136.1 134.2 123.3 121.8

43.9 45.8 56.7 58.2

-148.4 -139.0 -131.5 -84.7

3.66 3.67 4.00 3.88

63.6 61.0 52.8 51.5

116.4 119.0 123.6 128.9

-121.8 -123.0 -105.7 -141.5

a R is the vector connecting the molecular centers of mass, θA and θB are the angles between R and the respective molecular axes. θB g 0 corresponds to trans (like atoms on opposite sides) and θB e 0 to cis configurations (like atoms on the same side). The C-O bond length is 1.128323 Å.

qCM ) -(qC + qO)

(12)

30

In extension to previous models the partial charges are allowed to vary with the bond length:

q ) a0 + a1r + a2r2 + a3r3

(13)

Fitted to the high-level ab initio dipole and quadrupole functions of Maroulis,35,36 the variable charges provide a very good description of the electrostatic interactions. There seem to be typographic errors in Table 1 of ref 29 so that we have redetermined the expansion coefficients (a0-a3) given in Table 1. To assess the potential’s quality, we compared its description of the dimer and the crystalline solid with results taken from the literature (Tables 2 and 3). As a reference for the dimer we refer to the high-level ab initio potential of Vissers et al. 28 (see also ref 11). The potential has two minima at planar slipped antiparallel configurations, S1 (carbon atoms closest) and S2 (oxygen atoms closest) (see Table 2 and Figure 2a). The latter represents only a very shallow local minimum at the top of the disrotatory path that connects equivalent global minima S1. The disrotatory S2fS1 conversion is thus almost barrierless, while the conrotatory conversion involves a barrier of about 60 cm-1 with respect to S2 (from Figure 2 of ref 11). The NM model qualitatively reproduces these general features, but with the roles of S1 and S2 interchanged (Figure 2b1, Table 2). As a consequence the dissociation energy for S1 is about 40% too small, which also contributes to the error in the crystal lattice energy, which is calculated with the NM potential 15% below the experimental value (Table 3). We decided to refine the potential model with respect to the S1 and S2 dissociation energies and the lattice energy. Since the description of the electrostatics is already validated independently, we only adjusted the Lennard-Jones parameters to improve the potential. We obtained satisfactory results with a single set of LennardJones parameters for both C and O. The optimal values given in Table 1 are very close to those derived from thermodynamic data by Stoll,37 which provides further validation for our approach. The resulting “adapted NM” (ANM) potential model

a0 (e)

a1 (e Å-1)

a2 (e Å-2)

a3 (e Å-3)

-9.895 -9.701

17.163 18.059

-10.46 -12.21

2.271 2.892

improves the dissociation energy of the S1 configuration and restores the correct energetic ordering (Figure 2b2, Table 2). The disrotatory S2fS1 barrier, which was already tiny in the ab initio potential, has now completely disappeared, so that S2 is now a true saddlepoint. More importantly, the disrotatory motion is correctly described as only weakly hindered (maximum energy difference of ca. 25 cm-1 compared with ca. 20 cm-1 ab initio). To determine the quality of the model in describing the crystalline solid, we compare experimental results for the R-phase of CO with the original NM potential and our ANM potential in Table 3. In both cases the lattice energy was minimized as a function of lattice constant and center of mass shift of the CO molecules from the fcc symmetry positions (fractional coordinates: (0,0,0), (0,1/2,1/2), (0,1/2,1/2), (0,1/2,1/2)). The change in CO bond length is negligible. By design, the ANM model yields very good agreement with the experimental lattice constant and lattice energy. While phonons are not explicitly treated in the EVE model, their frequencies represent another check of how realistic the intermolecular potential is. While there is some uncertainty in the assignment of experimental data38-40 compared with more recent phonon calculations,33,34 the frequency range is clear and well-described within both the NM and ANM models. The calculated results lie approximately 20 cm-1 higher, which is in part consistent with the uncertainties in our potential model. Further the anharmonic red shift is not included since the phonon frequencies were calculated in the harmonic approximation. For the EVE model the intermolecular potential has to be combined with an explicit intramolecular potential function. Instead of the anharmonic intramolecular potential function for CO available in the literature,41 we employ an effective harmonic stretching potential parametrized to reproduce the experimental gas-phase stretching wavenumber. This approach neglects small anharmonic frequency shifts resulting from the change of bond lengths of individual CO molecules in different environments. Our model potentials (see above) yield fully relaxed structures with CO bond lengths shortened by a few 10-4 Å regardless of phase (amorphous, crystalline). This would translate into wavenumber shifts on the order of 10 cm-1. Our choice of an effective harmonic CO stretching potential is justified by the observation that those anharmonic shifts are essentially the same for all molecules within the particulate phase so that there is no net contribution to the spectral band shapes. Note that local transition wavenumbers ν˜ km still depend on the relative position and orientation of individual molecules. They are derived from a normal-mode analysis for each molecule in the field of all others. The same analysis yields local transition dipole moments µkm (within the double harmonic approximation) as the corresponding first derivative of the overall dipole moment function; i.e., the transition dipole moments are directly obtained from the electrostatic contribution to the potential function. In our model overall band shifts remain uncertain to within anharmonic couplings between intra- and intermolecular vibrations. From the above they are expected to be on the order of several cm-1 and largely independent of the molecular environment. We account for the effect a posteriori through shifting all calculated exciton spectra by 5 cm-1 to lower wavenumbers. This constant shift was

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TABLE 3: Crystal Minimum Energy Configuration (Lattice Constant, Molecular Center of Mass Shift from the fcc Symmetry Positions; Fractional Coordinates: (0,0,0), (0,1/2,1/2), (0,1/2,1/2), (0,1/2,1/2)),46 Corresponding Lattice Energy Contributions,33,55, and Phonon Frequencies As Assigned from Experiment 38-40 and Reassigned from Theoretical Work33,34 symmetry geometry lattice constant (Å) shift toward O (Å) xC xO energies (kcal/mol) electrostatic exchange and dispersion total phonons (cm-1)

a

F F F F E E A

experiment

theoretical

NM model

ANM model

5.646 0.1607,a 0.0004a 0.0495, 0.0659 -0.0659, 0.0495

5.7200 0.1128 -0.0537 0.0602

5.627 0.0559 -0.0604 0.0554

-2.480 90.540 85,39 8638 4939/50.538/5240 38b 5840 4440 64.540

-0.509 -1.625 -2.134 123 99 71 52 86 55 60

-0.566 -1.919 -2.485 128 100 88 60 89 66 65

90.533,34 85, 8633,34 58,33 49/50.5/5234 49/50.5/52,33 4434 64.533,34 44,33 3834 5834

The crystal is inverted relative to that in the other publications, which is taken into account. b Not assigned in the experiment.

Figure 2. (a) Dimer structures corresponding to local minima on the ab initio potential surface of Vissers et al.28 (see text). (b) Potential energy surface for planar CO dimer as a function of molecular orientation. θA and θB are the angles between the molecular axes of the monomers and the axis that connects their respective centers of mass. θB g 0 correspond to trans (like atoms on opposite sides) and θB e 0 to cis configurations (like atoms on the same side). At each point the energy is minimized with respect to the intermolecular distance. (b1) Original Nutt and Mewly surface.29 (b2) Adapted Nutt and Mewly surface, this work.

determined by comparing the calculated spectra with the experimental aerosol spectra of our own research group (see section 3.2) as well as experimental aerosol spectra of ref 6.

2.3. Particle Model and Molecular Dynamics Simulations. The particle properties to be studied in the infrared spectra are the size, the shape, and the internal structure (phase, architecture). In the particle ensembles typically formed in experiments

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(collisional cooling, supersonic expansions) these observables are usually distributed over a certain range, which depends on the particle generation method and on the experimental conditions. The purpose of the present contribution is to predict general trends. We discuss the influence of the size and also the size distribution on CO aerosol particle spectra. For the particle shapes, we consider some specific cases in our simulations: equal aspect ratio geometries such as spheres or cubes, which are likely candidates during initial particle formation, and elongated shapes, which are often formed as particles grow over time.20,23,27,42,43 In terms of internal structure, we consider particles ranging from fully crystalline to fully amorphous. The intermediate cases are assumed to be core-shell particles with a crystalline core and an amorphous shell. In the very first stage of particle formation if cooling is fast enough, condensation will initially produce more or less amorphous aggregates. During subsequent growth, as additional molecules condense upon this aggregate, some of the energy released is transferred to the already existing aggregate. Under the right conditions this might lead to annealing of the existing core and eventually to the formation of partially amorphous particles composed of a crystalline core surrounded by an amorphous shell. We have observed the formation of such core-shell particles in the case of nanosized NH3 aerosol particles.19 Bulk CO crystallizes in two phases: The high-temperature hexagonal β-phase (61.5-68.1 K), where molecules rotate freely,44 and the low-temperature cubic P213 R-phase (below 61.5 K), where rotation is hindered.45-47 Since no signs for the occurrence of the β-phase have been found in the available experimental data of CO particles and since it is relevant only in a very narrow temperature range, we will focus entirely on the R-phase. The crystalline particles used in the simulations are cut directly from the bulk structure, which is constructed with the ANM model parameters (Table 1). On the basis of experimental findings,47 head-tail flips of the CO molecules within the crystal are to be expected. In the EVE model CO flips around the molecular center of mass produce only minute changes in the spectra which arise from variations in the local force fields. The spectra of particles with random head-tail flips are virtually indistinguishable from the fully ordered R-phase so that we limit our studies to the latter. The fully amorphous particles as well as the amorphous shells of the core-shell particles are constructed by randomizing the orientations of the molecules and their center of mass positions followed by simulated annealing to relax configurations with unphysically high energy. The range of the center-of-mass shifts is limited to 10% of the nearest neighbor distance (∼0.23 Å) to avoid extremely high energy configurations that would lead to evaporation in the annealing procedure. The simulated annealing does not attempt to reproduce the actual annealing process mentioned previously. It is merely a tool to create physically plausible amorphous structures and is carried out by classical molecular dynamics using a Verlet propagator and the ANM potential. For all annealing runs, the following parameters were being used unless otherwise specified: An initial kinetic energy of 1.5kBTkin. is microcanonically distributed in the system, where Tkin. ) 50 K. The system evolves with a time step of 0.1 fs, and the kinetic energy is reduced by 0.1% of its current value at each time step. Except for slight relaxations, crystalline particles are energetically the most stable configurations so that the annealing ultimately leads to complete crystallization, which must be avoided in order to generate crystalline core-amorphous shell particles. The relaxation of unphysically high energy configurations is so much faster than the eventual recrystalli-

Firanescu and Signorell

Figure 3. Standard deviation of the molecular centers of mass (upper trace) and of the molecular orientation (lower trace) from the crystalline structure (see eq 14) for an ensemble of 10 spheres with a radius of 4 nm and a 40 vol % amorphous shell after 50 fs of simulated annealing.

zation that the two can be separated temporally. We define a structure as sufficiently relaxed if its spectrum has stabilized while still displaying a sharp core-shell boundary. This point was found to be reached universally when the average binding energy per molecule in the amorphous region reached 60% of the value in a fully crystalline shell. As the particles’ size decreases (below 104 molecules) ensemble averages become necessary to account for statistical variations in both their spectral and structural properties. To analyze the internal structure of our particles, we measure their crystallinity, or rather the deviation from it, as the standard deviation of the molecular centers of mass and of the molecular orientations from the perfect crystal (inf over macroscopic rotations):

( ∑ )

σCM ) inf

1 N

( ∑ )

N

(∆xi)2

i

σR ) inf

1 N

N

Ri2

(14)

i

where ∆xi is the shift of the center of mass of molecule i relative to the perfect crystal, Ri is the angle of CO relative to its orientation in the perfect crystal, and N is the number of molecules. On the basis of the aforementioned randomization algorithm, values above 0.23 Å for σCM and 60° for σR indicate essentially amorphous structures. As an example Figure 3 shows the analysis of the internal structure for an ensemble of 10 spheres with a radius of 4 nm and a 40 vol % amorphous shell annealed for 50 fs. The particle was divided into 0.1 nm thick shells. The dips toward the center of the particle (r ) 0 Å) arise from empty shells. The crystalline-to-amorphous boundary is a clearly distinguished step function at 33 Å. Some reorientations and displacements can also be observed in the crystalline core. They arise from the initial kinetic energy distributed in the system and are independent of the core and shell dimensions. 3. Results and Discussion 3.1. Size, Shape, and Surface Effects in Nanosized Particles (10 nm). For large particles, i.e., in the region between 10 and 100 nm, the band shapes in the infrared spectra are converged as a function of size (see refs 18, 22, and 42), leaving only structure and shape as influencing factors. For large CO aerosols, experimental infrared spectra from collisional cooling cells have been measured by us (see Figure 9c) and by Bauerecker et al.6,14,51 This provides us with the opportunity to compare our theoretical results with experimental data and to analyze the experimental data on the basis of the EVE model. There are

three different types of experimental spectra reported: (i) spectra recorded directly after particle formation, which show a single symmetric peak with a fwhm of 1.8-2.5 cm-1, as depicted in Figure 4 of ref 6 (these spectra were thought to arise from spherical crystalline particles, but as our analysis will show, this interpretation is not correct); (ii) spectra with a broad (fwhm ∼ 5 cm-1) band (see Figure 4 in ref 14 and Figure 3 in ref 51) (it was suggested that these spectra arise from amorphous CO particles; our molecular model confirms this interpretation); (iii) a series of spectra recorded as a function of time after CO aerosol particle formation, as depicted in Figure 2 of ref 6. With increasing time, two side bands at higher and lower frequency

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Figure 10. Calculated EVE spectra for mixtures of crystalline particles with different shapes. Spheres, cubes, and cuboids with an axis ratio of 1:1:3 and 1:1:9 are included. The ratio of the different shapes are as follows: full thick line, 21:34:22:23; dashed-dotted line, 14:34:24:26; full thin line, 7:32:28:33; dashed line, 7:21:31:41. The spectra are convoluted with a 0.5 fwhm Gaussian line shape.

evolve. The corresponding simulations are shown here in Figure 10, and the origin of this spectral change is discussed in section 3.2.2. As spectra in this size region (∼10-100 nm) are converged with respect to size, it is sufficient to reach the convergence threshold for the particles in our simulations, i.e., 20000-30000 molecules. Given the experimental resolution of the spectra, we consistently convoluted all calculated spectra in this section with a Gauss function with a full width at half-maximum of 0.5 cm-1. 3.2.1. Phase of the Particles. We start our discussion with spectra of type (i) (see Figure 4 of ref 6). In ref 6 it was argued that these spectra are dominated by spherical crystalline particles because of the symmetric band shape. The bandwidth and the fact that the band lies between the transversal and longitudinal optical modes of the bulk state was taken as a strong hint that these particles are crystalline. The following analysis with our EVE model demonstrates that both interpretations are not correct. In a first step, we assume that the particles are crystalline and have a spherical shape. This leads to the calculated spectrum in Figure 9a. With a fwhm of 0.7 cm-1 it is much narrower than the 1.8-2.5 cm-1 fwhm of the experimental spectra, already a first hint that the particles are not crystalline. The second argument has to do with the crystal structure and the particle shape. Crystalline R-CO has a cubic crystal structure,46 which makes it very unlikely that spherical particles are formed. Much more plausible particle shapes are cubes or cubelike shapes (i.e., cuboctahedra). Both lead to infrared bands with a characteristic shoulder at the high-frequency side of the main peak as depicted in Figure 9b. As demonstrated in previous studies of SF6 particles20 and CHF3 particles,27 cubelike shapes can thereby clearly be distinguished from spherical shapes, experimentally as well as in the calculations. Since the experimental spectra of the CO particles do not show the highfrequency shoulder and since these spectra are by more than a factor of 2 broader than the calculated spectra for crystalline particles, we conclude that the CO particles under discussion are not crystalline. If not completely crystalline, the particles might be amorphous. The comparison with the corresponding calculated spectrum in Figure 9d rules out the possibility of completely

Firanescu and Signorell amorphous particles. The calculated spectrum of a large amorphous sphere (amorphous particles are unlikely to support corners and edges) reveals a much broader band (fwhm ) 5.1 cm-1) than observed in the experiments. In addition, the experimental widths are not constant. By changing the experimental conditions, the width can be varied. We have added as an example an experimental spectrum from our own group (Figure 9c) with a width around 3 cm-1, which lies slightly above the values reported by Bauerecker and co-workers. All this evidence taken together is a strong hint that in all those cases the particles are partially crystalline with varying amorphous contributions, which determine the observed bandwidths. There are several possibilities how these partially crystalline particles could look like. Since in cooling cell experiments particle ensembles are measured, it is possible that the partially amorphous spectra arise from an ensemble with a certain fraction of crystalline and another fraction of amorphous particles which would determine the amorphous contribution to the spectra. Another possibility is the formation of amorphous shellcrystalline core particles. Here the shell volume fraction determines the amorphous contribution. Alternatively, crystalline inclusions could form in an amorphous matrix or vice versa. Lastly, partially crystalline structures could form on a molecular level, e.g., some but not perfect order as in the crystal on a molecular level. We can simulate this last case by controlling the length of the simulated annealing runs. With simulations for the different cases we have tried to distinguish between them. However, for all cases we find essentially a single band and band widths ranging from 1.6 to 4 cm-1, depending on the amorphous contribution, similar to what is observed in the various experiments. Since the general trends are reproduced for all possibilities, it is impossible to say more about the nature of the partially amorphous particles. To give but one quantitative example, the bandwidth of 1.8 cm-1 observed in Figure 4 of ref 6 is reproduced by a large core-shell particle with a 40 vol % amorphous shell. The very broad spectra in Figure 4 in ref 14 for 13C16O (case ii) are indeed very likely to arise from almost completely amorphous particles. This is confirmed by the simulation of a completely amorphous particle in Figure 9d. The calculated as well as the experimental spectrum have a fwhm around 5 cm-1. Note that the band widths are the same for different isotopomers. 3.2.2. Shape Effects. Case iii deals with the evolution of the infrared spectra observed as a function of time (see Figure 2 of ref 6), i.e., the formation of two prominent side bands on the high- and the low-frequency sides of the main peak with increasing time after particle formation. In several previous contributions for various aerosol particles,20,23,27,42,43,48,52 we have shown that analogous spectral changes were due to a change in the particles’ shape with increasing time. The analysis with our quantum mechanical model revealed in all these cases that the particles’ shape changed from an initial shape with equal axis ratios (cubes, spheres) to particles with an elongated shape. This makes it very likely that the same shape effects are also the correct explanation in the case of CO particles. Phase effects, such as a change from the R to the β crystalline phase, can be excluded because the measurements were performed well below the corresponding transition temperature of 61.5 K. We know from our previous studies on shape effects that initially most particles of the ensemble are likely to form shapes of equal axis ratio. We thus assume the same behavior also for CO aerosols. The influence of a spherical and a cubic shape on crystalline CO particle spectra was already presented in section 3, where size convergence was reached for the largest particles

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Figure 11. Refractive index data derived from the calculated EVE spectra of large spherical CO particles. (a) Completely amorphous CO ice. (b) CO ice with 80% amorphous contribution. The spectrum is derived from a particle with a crystalline core-amorphous shell architecture with an 80 vol % amorphous shell. (c) CO ice with 40% amorphous contribution. The spectrum is derived from a particle with a crystalline core-amorphous shell architecture with a 40 vol % amorphous shell. (d) Completely crystalline CO ice.

(see lower traces of Figure 4a, and Figure 5a). The spectra of these shapes with equal axis ratio are dominated by a single band. On the basis of our previous investigations, we also assume that with increasing time elongated CO particles are formed and that their fraction steadily increases with time at the expense of the spherical/cubic particles. Elongated CO particles have spectra with three peaks. The central peak lies close to the band of the spherical/cubic particle and the other two appear at higher and lower frequency of this middle peak (not shown here). In addition, we find that for higher axis ratios the peaks on the lower frequency side are further away from the central peak. Figure 10 shows a series of calculated spectra based on this scenario, i.e., with a systematically increasing fraction of elongated particles (full thick line to dashed line) which produces ever more pronounced side bands. These spectra reproduce the experimentally observed trend very well. We thus conclude that it is indeed the evolution of the particles’ shape which determines the spectral change in the experiment. This statement is based on true predictions from our EVE model, which do not involve any fitting to the experimental data. This is in marked contrast to classical scattering calculations (such as Mie theory or discrete dipole approximation DDA53,54) using experimental refractive index data. Our calculations are not only independent of experimental results but also provide a molecular explanation of the origin of shape effects in infrared spectra, viz., the transition dipole coupling of all molecules in the aerosol particles, which lifts the degeneracy of the uncoupled vibrational states of individual molecules and leads to vibrational eigenfunctions that are delocalized over the whole particle thus probing its shape. Compared with the experimental data (Figure 2 of ref 6) our calculated spectra are slightly more structured. There are several reasons for this: The simulations include only two types of elongated particles (axis ratios 1:1:3 and 1:1:9), whereas in the experiment there will probably be a distribution of different axis ratios. The inclusion of more axis ratios would smooth the

structure in the low-frequency peak around 2138 cm-1. Furthermore the analysis in section 3.2.1 tells us that the particles formed in the experiment are not completely crystalline. Shape effects are less pronounced for partially as opposed to fully crystalline particles. Without information on the exact internal structure of the particles we have performed the simulations for fully crystalline ones. To some extent. we account for deviations from perfect crystallinity by including not only cubes but also spheres in our simulations since partially crystalline particles are unlikely to form perfect cubes. The inclusion of partially crystalline particles would also smooth the band structures as well as slightly broaden the bands. The agreement with experimental spectra could doubtless be perfected by including partially crystalline structures and additional intermediate axis ratios, but we refrain from doing so, as no further physical insight would be gained from such fitting procedures. 3.3. Refractive Index Data. Finally, the EVE model allows us to calculate refractive index data for CO ice with a welldefined phase/internal structure. There are several refractive index data sets available in the literature for CO ice.2-6 The derivation of these data is based on fits to experimental infrared spectra, for which the phase/internal structure is not known. It is therefore not clear whether these data are for crystalline, amorphous, or partially crystalline CO ices. With the EVE model we can now provide optical data for completely amorphous and for completely crystalline CO ices as well, and in principle for any intermediate case. For this purpose, we calculate a spectrum for a large spherical particle with the desired phase/internal structure with the EVE model and use then Mie theory and a Kramers-Kronig inversion53 to extract the refractive index data. The resulting refractive index data for four different phases/ internal structures are presented in Figure 11. Panel a shows the real part n and the imaginary part k of the refractive index of completely amorphous CO ice. Panel d shows the same for completely crystalline CO ice. Panels b and c are derived from amorphous shell-crystalline core particles with 80 and 40 vol

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Figure 12. Refractive index data from the literature derived from experimental spectra of thin CO films at temperatures of 10-16 K: (a) ref 2; (b) ref 3; (c) ref 4; (d) ref 5.

% amorphous shells, respectively. With decreasing amorphous contribution, the maximum values of k and n increase and the minimum value of n decreases. In addition, the “bands” in Figure 11 become sharper. Classical scattering theory53 shows that particle spectra have strong infrared absorption bands and show pronounced shape effects for all particle shapes in regions where n has values close to zero, while k varies strongly. Our EVE model reveals that the classical criterion is equivalent to the case when transition dipole coupling is strong and the molecules in the aerosol particle are well-ordered. It is therefore not astonishing that for CO the refractive index data for crystalline particles in Figure 11d fulfill the “classical” criterion best. Some experimental optical constants from the literature2-5 are depicted in Figure 12. These data were derived from thin film spectra at 10-16 K. The comparison of our calculated optical data with the literature data2-6 reveals that none of these data sets apply to fully crystalline particles. The literature values are for amorphous or partially amorphous CO ices with different degrees of amorphous contribution. 4. Summary We have combined our extended vibrational exciton model with a molecular dynamics approach to predict how the shape, size, and internal structure (surface, phase, and architecture) of CO aerosol particles manifest themselves in infrared spectra. We distinguish two different size regions: (i) particles with diameters between 1 and 10 nm, for which the spectra are sensitive to size, shape, surface, and phase (experimental data are not available for this size range so that our calculations provide important predictions for such experiments); (ii) particles between 10 and 100 nm, whose spectra do no longer depend on the size (in a nontrivial way) or the surface (surface contribution too small), but are dominated by the shape and phase of the particles. In the 1-10 nm region, we find that indiVidual crystalline particles exhibit characteristic size- and shape-dependent spectra, while spectra of particle ensembles with a certain size distribution retain only information on the shape but not on the mean

size. Amorphous particles, in contrast, display a broad asymmetric band regardless of size, only allowing for the determination of the phase. For partially amorphous particles the extent of the amorphous contribution can be estimated from an analysis of the bandwidth of the spectra. In the 10-100 nm size range, we find a similar influence on the spectra for the phase and shape of the particles, while there is no distinct size dependence in this range. The comparison with experimental data shows that the particles generated in collisional cooling cells always have varying amorphous contributions depending on the experimental conditions. The observed change in the band structure as a function of time is identified as a shape effect due to a transformation from globular particles into elongated ones. The success of the EVE model in analyzing the infrared spectra of pure CO particles would warrant an analogous investigation for astrophysically relevant ice mixtures which contain CO. Acknowledgment. We acknowledge Dr. David Luckhaus for his help with the numerical implementation. This project was financially supported by the Natural Sciences and Engineering Research Council of Canada, by the Canada Foundation for Innovation, and by the A. P. Sloan Foundation (R.S.). Allocation of CPU time on WestGrid facilities is gratefully acknowledged. References and Notes (1) Ewing, G. E.; Pimentel, G. C. J. Chem. Phys. 1961, 35, 925. (2) Hudgins, D. M.; Sandford, S. A.; Allamandola, L. J.; Tielens, A. G. G. M. Astrophys. J. Suppl. Ser. 1993, 86, 713. (3) Baratta, G. A.; Palumbo, M. E. J. Opt. Soc. Am. A 1998, 15, 3076. (4) Palumbo, M. E.; Baratta, G. A.; Collings, M. P. Phys. Chem. Chem. Phys. 2006, 8, 279. (5) Ehrenfreund, P.; Boogert, A. C. A.; Gerakines, P. A.; Tielens, A. G. G. M.; van Dishoeck, E. F. Astron. Astrophys. 1997, 328, 649. (6) Dartois, E.; Bauerecker, S. J. Chem. Phys. 2008, 128, 154715. (7) Sandford, S. A.; Allamandola, L. J.; Tielens, A. G. G. M.; Valero, G. J. Astrophys. J. 1988, 329, 498. (8) Bouwman, J.; Ludwig, W.; Awad, Z.; Öberg, K. I.; Fuchs, G. W.; van Dishoeck, E. F.; Linnartz, H. Astron. Astrophys. 2007, 476, 995. (9) Pontoppidan, K. M.; Boogert, A. C. A.; Fraser, H. J.; van Dishoeck, ¨ berg, K. I.; Evans, N. J., II; Salyk, C. E. F.; Blake, G. A.; Lahuis, F.; O Astrophys. J. 2008, 678, 1005. (10) Walker, K. A.; Xia, C.; McKellar, A. R. W. J. Chem. Phys. 2000, 113, 6618.

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