Monte Carlo studies of surface chemistry andnonthermal desorption involving interstellar grainsEric Herbst*†‡§ and Herma M. Cuppen*

Departments of *Physics, †Astronomy, and ‡Chemistry, Ohio State University, Columbus, OH 43210

Edited by William Klemperer, Harvard University, Cambridge, MA, and approved March 28, 2006 (received for review February 28, 2006)

Although still poorly understood, the chemistry that occurs on thesurfaces of interstellar dust particles profoundly affects the growthof molecules in the interstellar medium. The most importantsurface reaction is the conversion of atomic to molecular hydrogen,which is a precursor for all subsequent molecular development andwhich occurs both in diffuse and dense interstellar clouds. Anotherset of surface reactions produces icy mantles of many monolayersin cold and dense regions of the interstellar medium. The mono-layers are dominated by water ice but also contain CO, CO2, andoccasionally methanol. In this work, we first review both ourstochastic approach to the surface chemistry that can occur onsmall dust particles and how it has been applied to the problem ofthe formation of molecular hydrogen. This latter problem isstrongly affected by the pulsed heating of smaller grains byphotons. Photons are not the only source of pulsed heating; cosmicrays also can heat interstellar grains in a pulsed manner. Here, wecalculate the heating by cosmic rays for different grain sizes andcosmic ray components. It is then shown that this mechanism is animportant one for desorption of ice mantles.

interstellar clouds � interstellar medium � dust particles � cosmic rays �H2 formation

The first polyatomic molecules were detected in the gas phaseof interstellar clouds �30 years ago (1). Composed of both

gas and dust particles, these clouds are the cold and neutral-matter component of a complex interstellar medium. At present,�130 different molecular species have been detected by high-resolution spectroscopy in the cold interstellar medium and inrelated circumstellar clouds surrounding old stars (see www.astrochemistry.net). Although most of these species have beendetected only in our own galaxy, the Milky Way, others have beenseen in external galaxies, and a few have been detected in theearly universe. Mainly organic in nature, these molecules rangeup to 13 atoms in size, although the overwhelmingly dominantmolecular species is H2. Interstellar and circumstellar moleculesare detected under a variety of different physical conditions,from diffuse interstellar clouds to heterogeneous dense inter-stellar clouds with significant substructure. Two of the richesttypes of interstellar sources are both located in these denseclouds: (i) cold quiescent cores (T � 10 K), where the inventoryof organic molecules is dominated by unsaturated organic spe-cies, and (ii) hot molecular cores or ‘‘corinos’’ (T � 100 K),where the organic chemistry is dominated by more saturatedorganic species similar to what is common in the terrestriallaboratory. Both of these types of objects are related to theformation of stars, which is far better understood for low-massstars such as the sun. In this case, a cold core collapses in a rathercomplex process that is isothermal at first and changes toadiabatic when an opaque central condensation is formed. Asthis condensation, known as a protostar, heats up to the pointthat nuclear reactions onset, the surrounding envelope warms upand becomes known as a hot molecular corino. In addition,violent winds arise to blow the surrounding interstellar materialaway, and a protoplanetary disk possibly leading to a solar-typesystem forms (2).

Although the formation of interstellar clouds is poorly con-strained, the general picture is that they arise from gas and dustreleased into the interstellar medium by older stars, eitherexplosively, as in the case of supernovae, or more gently, as in thecase of asymptotic giant branch stars and planetary nebulae.Although the dust particles survive in the interstellar medium,molecules produced in stellar atmospheres are photodestroyedin the unshielded medium in relatively short times. The matterthat forms interstellar clouds under the influence of gravity thenmost likely consists of dust particles and atoms, comprisingelements in so-called cosmic abundances measured by the spec-troscopy of stellar atmospheres, in which hydrogen is dominantand the important biogenic elements, carbon, oxygen, andnitrogen, are lower in abundance by 3–4 orders of magnitude.The dust particles, initially ranging in radius from 5 to 250 nm(0.005–0.250 �m) with a population distribution that goes asr�3.5, are thought to consist of two populations, depending onwhether they are produced in the atmospheres of oxygen-rich(elemental O�C � 1) or carbon-rich (elemental C�O � 1) stars:oxygen-rich stars lead to metallic silicates, whereas carbon-richstars lead to some form of carbon. In cold and dense regions ofinterstellar clouds, infrared absorption studies show the exis-tence of mantles of ices, consisting mainly of water, CO, and CO2(3). In protostellar regions, these ices evaporate and change thenature of the chemistry from an unsaturated to a saturated one(4, 5). In regions in the vicinity of newly formed stars, infraredemission pumped by visible and ultraviolet radiation indicatesthe existence of aromatic matter, probably in the form of largepolycyclic aromatic hydrocarbons (6).

The chemical processes that convert the mainly atomic gas-phase matter into molecules consist of gas-phase reactions andreactions that occur on the surfaces of the dust particles (2, 7).Although gas-phase chemistry explains much of what astrono-mers see in the interstellar gas, surface chemistry must beinvoked to explain the efficient formation of H2 and the forma-tion of most species in the icy grain mantles. Surface chemistryhas not been studied to as great an extent as gas-phase processesfor two major reasons: (i) the nature of interstellar grain surfacesis not well determined chemically or physically, and (ii) thechemistry that occurs on cold and nonmetallic surfaces has notreceived a great deal of study in the laboratory. Even theformation of molecular hydrogen from two hydrogen atoms isincompletely understood, despite some promising experimentson the granular surface analogs olivine, amorphous carbon, andamorphous ice (8–11). Part of the problem is that most exper-iments are studied on large surfaces rather than on nanopar-ticles; the small surface area of the latter coupled with low fluxesin interstellar environments means that the average number ofreactive species on grains can be less than unity and that specificdiscrete numbers on an individual grain fluctuate widely, re-

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviation: ML, monolayer.

§To whom correspondence should be addressed. E-mail: herbst@mps.ohio-state.edu.

© 2006 by The National Academy of Sciences of the USA

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quiring a stochastic treatment of the kinetics rather than the useof standard rate laws (12–14). This problem also occurs in cellkinetics and in second-layer nucleation in surface science (15).A variety of simplified stochastic approaches have been used inthe past to model a significant number of surface reactionsoccurring simultaneously under interstellar conditions; of these,the master equation approach and various approximations to ithave been most useful (16–18).

To understand the detailed nature of the diffusive, or Lang-muir–Hinshelwood, chemistry that occurs for specific reactionson small, irregular interstellar particles, we recently developed aMonte Carlo procedure based on the continuous-time random-walk method of Montroll and Weiss (19). In this procedure, thegrain is modeled as a square lattice with sites equivalent to theactual sites, or potential minima, where adsorbing atoms landand possibly stick on the substrate (20). The method allows us tofollow atoms and molecules on the surface and tune propertiesof the surface such as its roughness and temperature to studytheir effect on molecular formation. In this work, we first reviewour previous calculations concerning the formation of molecularhydrogen in diffuse interstellar clouds, and then present previ-ously unreported work on the desorption of icy grain mantles bycosmic ray bombardment.

Monte Carlo MethodLet us start by explaining the Monte Carlo method. Two energyparameters define a site on the grain lattice: Eb, the potentialbarrier between the site and its nearest neighbors, and ED, thelarger energy needed to desorb the adsorbate, often referred toas the binding or evaporation energy. The rates (s�1) fordesorption and hopping from these sites are given by

Rhop,i � � exp��Eb,i

T �, [1]

and

Reva,i � � exp��ED,i

T �, [2]

respectively, where � is a trial frequency, with which adsorbatescollide with the barrier surrounding them, and the index i is usedfor different types of sites, if they arise. Tunneling under barriersis not considered because little or no evidence has been foundfor it (8). Here we use the value of � � 1012 s�1, which is typicalfor physisorption problems (8). The size of the grain is relatedto the size of the lattice by means of the site density of theparticular adsorbate–substrate system (e.g., H atoms on olivine),which, along with the energy parameters, can be determinedthrough temperature-programmed-desorption experiments (8).

In addition to simulating so-called flat surfaces, where eachsite has the same energy parameters, we can simulate roughsurfaces, where the complex topology leads to binding sites thatpossess a discrete number of different energy parameters, oreven amorphous surfaces, where the energy parameters aredescribed by a continuous distribution. A designer surface of aspecific roughness can be ‘‘fabricated’’ on demand, by using aspecial program (21).

In our Monte Carlo method, random numbers are called tomimic the following processes: adsorption of species onto ran-dom sites of the lattice, diffusion of these species throughrandom walk, reaction of two species when they occupy the samelattice site, evaporation of the reactants and products, and bothheating of the grain and photoprocessing of the surface reactantsby the radiation field of the interstellar environment. Becausereaction can occur when an adsorbate lands atop another one,our method also includes the so-called Eley–Rideal mechanism(22). The methods by which the probabilities of the assorted

processes are determined and the sizes of the steps by whichthe clock proceeds forward are discussed in detail in ourearlier papers (20, 21). The key surface features that determinethe efficiency of reactions are the degree of roughness oramorphous nature, the energy parameters, and the so-called trialfrequency �.

Formation of H2 in Diffuse CloudsUnlike dense clouds, diffuse regions are exposed to radiationfrom external stars with very little extinction because of scat-tering and absorption by dust particles. The radiation is generallyreferred to as the average interstellar radiation field, and it isstrong enough to dissociate small molecules within 100 years.Note that astronomers use a parameter known as the visualextinction, AV, which pertains to the visible region of thespectrum and is defined so that 5 units of extinction correspondto a diminution of 100. This parameter is very close to an opticaldepth, and the two can be used interchangeably. Diffuse cloudstypically possess a visual extinction on the order of unity, so thatthe rate of photodissociation is not slowed significantly byextinction. Nevertheless, the chemistry of diffuse clouds is morecomplex than originally thought, with a variety of polyatomicmolecules recently discovered in the gas (23). Still, the dominantchemical reaction in such sources is the formation of molecularhydrogen from hydrogen atoms.

Our first application of the continuous-time random-walkMonte Carlo method was to the formation of molecular hydro-gen in diffuse clouds (20, 21). This molecule has to be formed onthe surfaces of interstellar grains, because the gas-phase radia-tive association reaction is inefficient at low interstellar temper-atures (24). According to an analysis based on recent laboratoryresults on olivine (8), which was treated as possessing a flatsurface, the surface formation of H2 occurs efficiently only overa very small temperature range (6–9 K), which is much lowerthan the estimated surface temperature of �18 K for grains of0.1 �m in radius (25). Note that the efficiency of H2 formationis defined as twice the number of H2 molecules produced per Hatom striking a grain. The main problem is that the residencetime of the H atoms on olivine grains with a standard temper-ature of �18 K is much shorter than the deposition time, becauseof the rapid evaporation of surface H atoms and the low flux ofH atoms striking the grain. Therefore, two hydrogens can rarelymeet to form H2.

In two of our papers (21, 26), we treat olivine as having a roughsurface, including sites with discrete desorption energies higherthan the standard value. The presence of these sites arises fromtopological protrusions on the surface that lead to a higherdesorption energy than the norm (21). Such a rough olivinesurface also can be used to fit the laboratory temperature-programmed-desorption data on H2 formation (21) as well as theoriginal analysis of Katz et al. (8). We also have consideredamorphous surfaces, in which we assumed a distribution ofenergies around the values found experimentally (20). In bothcases, it was found that H2 could be formed efficiently over awider range of temperatures and so better explain the observedabundance of the molecule in diffuse clouds. To reach highefficiency at 18 K requires either a high degree of roughness oramorphism.

However, the concept of a single grain temperature of �18 Kis only a simple approximation (25). In our most recent paper(26), we studied the effect of stochastic heating of olivine grainson the rate of molecular hydrogen formation in diffuse clouds.Grains are heated by the absorption of photons and coolradiatively. Fig. 1 shows the thermal fluctuations of olivinegrains of varying sizes in the presence of a standard interstellarradiation field. Fig. 1 shows clearly that larger grains tend topossess an average temperature, whereas smaller ones possesstemperatures that fluctuate over a large range. Indeed, the

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smallest grains in our sample seem to have a baseline temper-ature of �8 K, although this baseline is not typically reached. Asimilar calculation on carbonaceous grains of the same size (27)shows a lower baseline in a long interval between photon hits.The dependence on size comes about because the smaller grainsabsorb photons at larger time intervals, and these photons havea greater thermal effect.

Because the formation of H2 has a strong dependence on thesurface temperature and because the small grains make a largecontribution to the total surface area, we expect that thesefluctuations will have an important effect on the overall rate ofmolecular hydrogen formation. Our results showed that thishypothesis is indeed the case, although the results depend on theroughness of the material. A variety of different degrees ofroughness were studied, ranging from flat to moderately rough(see ref. 21 for a detailed description of the surfaces). Twocompeting effects on small-to-intermediate grains help to de-termine the efficiency of H2 formation. On the one hand, theefficiency becomes lower because of temperature fluctuationsthat decrease the residence time of the hydrogen atoms on thesurface. On the other hand, the temperature range (inthe vicinity of the modal, or most likely, temperature) at whichthe grain spends most of its time becomes �18 K for smallergrains, which yields an increase in the efficiency because theprocess is more efficient on olivine at temperatures �18 K. Thecomputed results show that for flat and slightly rough surfaces,the efficiency, although �0.1, is largest for the smallest grains,whereas for the roughest surfaces, the efficiency is largest forgrains of �0.02 �m in radius, where it can reach 0.7. Afterintegrating over all grain sizes with the standard size distributionof r�3.5, we determined that the overall production rate for all butflat olivine surfaces is large enough to explain the abundance ofH2 found in the diffuse interstellar medium.

Cosmic Ray HeatingIn both dense and diffuse clouds, grains are also heated byincoming cosmic rays, which are bare nuclei that travel atrelativistic speeds and that have energies mainly between 0.02and 4 GeV per nucleon (1 eV � 1.602 � 10�19 J) (28). Theirelemental composition resembles the cosmic elemental abun-dances to some extent so that the rays are mostly protons but alsocontain other heavier nuclei such as iron, which is abundantbecause the cosmic rays are produced in supernovae. Unlike thecase of ultraviolet and visible radiation, cosmic rays can pene-

trate deeply into molecular clouds. In this section, we will lookat the influence of cosmic rays on both olivine and carbonaceousgrains. Two types of cosmic rays will be considered: protons andiron nuclei. Protons are by far the more abundant, but becausethe energy that is deposited into the grain is proportional to Z2

where Z is the charge of the particle (atomic number of thenucleus), they will not bring about a large temperature gradient.Iron, on the other hand, has a much larger thermal effect (Z �26), but the flux of Fe nuclei is a factor of 1.6 � 10�4 (28) lowerthan the proton flux.

We take the cosmic ray flux in particles cm�2�s�1�ster�1�GeV�1

(where ster is steradian, a unit of solid angle) from Morfill et al. (29)to be

�H(�)��21� 0.02 � � � 0.071.5 0.07 � � � 0.20.3��1 0.2 � � � 10.3��2 1 � �

, [3]

where � is the cosmic ray energy in GeV per nucleon. We onlyconsider particles with energy between 0.02 and 2 GeV pernucleon because the higher-energy ones have much less of aneffect. The cosmic rays deposit an energy Q (MeV) into a grainof radius r (in �m) according to the formula

Q��0.16r(Z�26)2���0.75 0.02 � � � 0.20.25r(Z /26)2�(1 � 0.1��1.5) 0.2 � � � 10 , [4]

where � is the density of the granular material in g�cm�3 (28),which we take to be 3.3 and 2.2 g�cm�3, respectively, for olivineand carbonaceous grains. The heating of the grains is calculatedby using the same procedure as in our previous paper (26), wherewe determined the increase in temperature by using the heatcapacity. We assume the grains to be Debye solids: olivine witha Debye temperature of 500 K (25, 30, 31) and amorphouscarbon with one of 337 K (32). We do not take into account thedamage that is done to the granular material by the impact of thecosmic-ray particle and the exact nature of the transfer of energy,as is done by Bringa et al. (33–35). Our approach corresponds totheir ‘‘late thermal’’ portion (35). We are interested, rather, inusing the temperature fluctuations to simulate the desorption of

Fig. 2. The results of interactions between grains and cosmic rays. (Left) Thecosmic ray collision rate for Fe nuclei as a function of the granular radius.(Right) The distribution of peak temperatures after cosmic-ray bombardmentvs. granular radius with the mode depicted by a cross. Olivine grains are shownin Upper, and carbonaceous grains are shown in Lower. The bars indicate thetemperatures for which 99% of the determinations fall.

Fig. 1. Temperature fluctuations as a function of time for different olivinegrain radii in the presence of the standard interstellar radiation field withoutshielding. [Reproduced with permission from ref. 26 (Copyright 2006, Black-well Publishing).]

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surface species and to demonstrate the dependence of thisdesorption on granular size.

Some results are shown in Figs. 2 and 3. Figs. 2 Left and 3 Leftindicate the rate at which the grains are hit by cosmic rays in therange 0.02–2.0 GeV per nucleon as a function of granular size forFe and H cosmic rays, respectively. The fraction f of grains hitas a function of time is given by the simple expression

f � 1 � exp(�kt), [5]

where k (yr�1) is the collision rate, and t is the time. If weconsider Fe cosmic rays, the rate for grains of radius 0.1 �m is�10�5 yr�1 so that for a young cloud of age 105 yr, f � 0.63.However, because the rate of collision is proportional to thesquare of the granular radius, the time at which f � 0.63 becomesdramatically longer for smaller grains. Because there are manygrains within that size regime, a large number of small grains willstill be heated by Fe nuclei by 105 yr, and the resulting desorptionof many surface species must still be considered. Fig. 3 shows thatH particles hit the grains much more frequently.

Figs. 2 Right and 3 Right give the temperatures to which thegrains are heated as a function of granular radius for Fe and Hcosmic rays, respectively. Figs. 2 Right Upper and 3 Right Upperare for olivine, whereas Figs. 2 Right Lower and 3 Right Lower arefor amorphous carbon. The grains are considered to have aconstant base temperature of 10 K without cosmic rays, which isan appropriate temperature for cold cores in dense interstellarclouds. After a cosmic ray hits a grain, the grain heats up veryquickly. The radiative cooling is slower: after collision with an Fecosmic ray, it takes between 8,000 and 9,000 s for olivine to cooldown again to near 10 K, whereas for amorphous carbon, it takesbetween 1,300 and 1,700 s. The cooling is depicted for a grain ofradius 0.1 �m in Fig. 4 Upper. Because we used a wide spectrumof different cosmic ray energies, different peak temperatureswere reached. The temperature peaks as a function of granularradius are shown in Figs. 2 Right and 3 Right with the maximumof the distribution (the mode) shown as a cross, which is veryclose to the lowest values determined. Fig. 4 Lower shows ahistogram of these temperatures for an olivine grain of r � 0.1�m that is bombarded with Fe nuclei. It clearly indicates thelarge asymmetrical distribution of peak temperatures that aretypically found, which in turn reflects the energy distribution ofthe cosmic rays. The solid line indicates the modal temperatureof the distribution; the two dashed lines give the 99% boundaries,i.e., those temperatures between which 99% of the determina-tions fall.

The peak temperatures of the carbonaceous grains are allslightly higher than for the silicate grains. The graphs furthershow a strong granular-size dependence of the peak tempera-ture. For very small grains, temperatures up to 350 K can bereached, but this drops rapidly to �100 K for r � 0.05 �m grains,and the largest grains are only slightly heated above the startingtemperature of 10 K. For the heating by protons, these temper-atures are all lower and the peak temperatures compare with thephoton heating discussed in ref. 26, mainly because the protons,although much more energetic than the photons, do not disposeof their energy efficiently.

Desorption of Adsorbates in Dense Cold CloudsThere is evidence in cold cores of dense clouds that someinefficient nonthermal desorption mechanisms are operative. Ifonly thermal evaporation is considered, the only species that candesorb at 10 K are hydrogen and helium (7). Yet, heavy speciesare found in the gas, one of which (methanol) can probably onlybe formed by means of surface chemistry (36). Here we focus onthe mechanism of desorption by means of cosmic ray bombard-ment of Fe nuclei. The subisidary, but nonnegligible, role playedby cosmic ray protons in the desorption of relatively weaklybound species such as CO needs further study. Cosmic raydesorption has been discussed before in the astronomical liter-ature (28, 37), and simple estimates have been made for the ratesobtained by means of iron nuclei for various heavy adsorbates(37). In addition, a detailed molecular dynamics simulation hasbeen reported to evaluate the details of the process (35). Herewe use our stochastic method to determine what happens toassorted heavy species such as water ice once the grain is heatedby cosmic-ray bombardment. In particular, we have run fourtypes of simulations with a completely flat substrate. We con-sidered both one and five monolayers (MLs) of water ice on acarbonaceous surface and single MLs of CO and of N2 atop awater ice substrate on a carbonaceous grain. We assumed nolateral interactions between the adsorbates and only verticalinteractions between adsorbate and substrate. At the start of the

Fig. 3. As in Fig. 2, but for protons.

Fig. 4. The temperature profile as a function of time (Upper) and the peaktemperature distribution (Lower) for olivine grains with radius (r) � 0.1 �mupon bombardment with Fe nuclei. The lines in Lower indicate the modaltemperature (solid line) and the two 99% levels (dashed line).

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simulation, the desorption time for each molecule was deter-mined by using a random number to mimic the exponentialdecay. Because the desorption energy is high and the graintemperature is low (10 K), these times are generally longer thanthe simulation time. When the grain is heated by cosmic rays,these times change from tevent

old to teventnew according to

teventnew � (tevent

old �t)Rhop(Told) � Reva(Told)

Rhop(Tnew) � Reva(Tnew) t , [6]

where t is the current time, the rates are defined in Eqs. 1 and2, and Told and Tnew are the temperatures before and afterheating, respectively. During the cooling the times are changed,using the same equation. The grains with mantles do not onlycool radiatively but also by desorption, which extracts energyfrom the grain and can strongly speed up the cooling. We assumethat with every desorption the grain loses ED in energy.

The average surface or mantle abundance and its standarddeviation after a cosmic ray hit was determined, and the resultsare summarized in Table 1. These results were obtained byaveraging �250 runs with cosmic rays of different energies. If wefirst focus on water, we can clearly see a difference between oneand five MLs (Table 1, columns A and B). We chose one and fiveMLs to mimic water desorbing from a bare grain and a grain withan ice mantle, respectively. The reason that the two cases showsuch different behavior is that the first ML of water has aninteraction with the underlying surface of 2,000 K, whereas thewater–water ice interaction is 5,640 K (38–41). In our simula-tions, we assumed that only the top layer could evaporate, so thatin the case of five MLs of ice, the top layer could prevent the restof the layers from evaporating. Based on the results in Table 1,it can be concluded that water desorbs efficiently from baregrains up to a size of 0.1 �m. From ice mantles, on the otherhand, water desorbs efficiently only from grains with a radius of�0.02 �m. The result is that once past the initial stage of growth,water ice mantles can grow significantly in dense, cold sources,as observed, unless there are more efficient nonthermal desorp-tion mechanisms to consider.

For the study of CO and N2 MLs atop a water ice substrate,we used evaporation energies of 1,150 and 1,000 K (42, 43), basedon temperature-programmed-desorption experiments of themolecules on amorphous solid water. We see that in both casesthe single MLs are released completely except for the largestgrains considered, where the release is still partial. The situationwould be more complex if CO and N2 species were to be founddispersed with the dominant water ice, a situation that probablyoccurs in the cold interstellar medium. Nevertheless, our resultsindicate that a ML of either CO or N2 ice is desorbed efficientlyby cosmic-ray bombardment. How this result limits the actualgrowth of ices of these species depends of course on the growthrate as well and must be determined by detailed gas–grain

chemical models that include all relevant chemical and physicalprocesses (36).

With some caution, it is possible to convert the resultsobtained here into first-order rate coefficients for each granularsize that are products of the cosmic-ray bombardment ratemultiplied by the efficiency. Another manner of looking at thedata is to determine a yield, or maximum number of MLs, thatcan be desorbed. Rate coefficients are useful for inclusion intogas–grain kinetic models, but they have their limitations. Sup-pose an Fe cosmic ray strikes a grain with a top ML of COmolecules; the result here is that except for the largest grain theefficiency of desorption is unity, and the first-order rate coef-ficient kcr is simply the cosmic-ray collision rate. But nowsuppose that 50 MLs of CO are present; the efficiency forremoving all of the MLs will probably be much lower, so that therate coefficient will depend on the amount of CO present. (Seealso our water ice results for five MLs in column B of Table 1.)So, perhaps it is best to quantify the desorption by two param-eters: a total yield and the efficiency of removing one ML. Asimple approximation then could be made that the single MLefficiency be used up to the last ML of the total yield.

Let us now consider such a rate coefficient for CO on waterice. Hasegawa and Herbst (37) estimated a first-order cosmic raydesorption rate coefficient kcr caused by Fe cosmic rays bom-barding a 0.1-�m grain for a variety of surface species, assumingthat volatile species comprised most of the mantle. For CO, avalue of 9.80 � 10�15 s�1 was calculated. They estimated the peaktemperature to be 70 K and the time-scale for evaporativecooling to 60 K to be 10�5 s for a grain of 0.1 �m. Our resultsindicate a lower average peak temperature for this case. Toobtain a first-order rate coefficient for the single ML of CO onwater ice from our results in Table 1, we multiply the cosmic raycollision rate per grain by one minus the fraction in the table. Inthis way, the maximum rate coefficient is the cosmic-ray collisionrate per grain. This procedure yields kcr � 5.70 � 10�13 s�1,which is significantly larger than the value estimated by Hase-gawa and Herbst (37). The difference lies mainly in the cosmicray flux assumed: our flux is significantly greater than that usedin the earlier study because we consider a wider range ofcosmic-ray energies. How this larger desorption rate will affectthe partition of CO between gas-phase and mantle in dense coldclouds remains to be studied in gas–grain models. We do note,however, that the molecular dynamics results of Bringa andJohnson (35) also show a small CO desorption rate, composedpartially of a prompt sputtering, early heating, and late (wholegrain) heating. The reason for the discrepancy between the twocalculations is not yet clear because the temperature to which thegrain is heated is comparable with our values, although it shouldbe mentioned that the molecular dynamics simulation is for pureCO ice, which is quite different from our CO � water-ice mantle� carbon-core grains.

The stochastic approach reported here also can be used formore complex surfaces. Interstellar grains are known to befractal-like and porous, and therefore many more individualspecies can be attached to an outer layer and easily evaporate.On the other hand, some of the volatile species can be spread outamong the dominant water-ice MLs, thereby possibly slowingdown and complicating the rate of desorption (42, 44). Beforeincorporating our theoretical results into gas–grain model cal-culations of the chemistry of dense clouds (36), we also shouldconsider these two possibilities.

ConclusionsIn this work, we have discussed our stochastic approach to thesurface chemistry that occurs on interstellar grains and reviewedits use to study the formation of molecular hydrogen. We thenused the approach to study desorption of icy grain mantles aftercosmic ray hits. The formation of molecular hydrogen on

Table 1. Remaining surface/mantle abundance in MLs

r, �m A B C D

0.005 0.00 � 0.00 0.09 � 0.40 0.00 � 0.00 0.00 � 0.000.01 0.00 � 0.00 4.11 � 1.67 0.00 � 0.00 0.00 � 0.000.02 0.00 � 0.00 4.99 � 0.04 0.00 � 0.00 0.00 � 0.000.05 0.02 � 0.04 5.00 � 0.00 0.00 � 0.00 0.00 � 0.000.1 0.91 � 0.25 5.00 � 0.00 0.00 � 0.00 0.00 � 0.000.25 1.00 � 0.00 5.00 � 0.00 0.72 � 0.40 0.20 � 0.27

Uncertainties are assuming a Gaussian distribution, which is incorrect forseveral cases in which the standard deviation exceeds the modal value. Col-umn A, one ML of H2O on a carbonaceous grain heated by Fe nuclei; columnB, five MLs of H2O on a carbonaceous grain heated by Fe nuclei; column C, oneML of CO on a carbonaceous grain with ice mantle heated by Fe nuclei; columnD, one ML of N2 on a carbonaceous grain with ice mantle heated by Fe nuclei.

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irregular surfaces is best handled by a stochastic method in whicheither a continuous or discrete distribution of binding energies,corresponding to amorphous or rough structures, can be treated.Our calculations show that these types of surfaces enlarge thetemperature range over which the formation of molecularhydrogen can be efficient. The stochastic approach also allows usto consider the effect of photons striking the dust particles,leading to heating at irregular intervals for the smaller particles,and so influencing the rate of H2 formation.

Nonthermal desorption mechanisms of low efficiency play arole in maintaining a significant amount of material in the gasphase of cold dense regions (36). Among the many such mech-anisms discussed in the astronomical literature, cosmic-ray bom-bardment by Fe nuclei has been mentioned for some time (28,37). Unlike photons, cosmic rays penetrate large columns ofmatter and so can interact with material deep inside denseclouds. Although the interaction of cosmic rays with dustparticles is a complex one that has profitably been studied bymolecular dynamics simulations for simple systems (35), simu-lations of the more complex granular structures present in theinterstellar medium can profitably be undertaken with stochasticmethods. In this work, we have extended our continuous-timerandom-walk Monte Carlo approach to study the whole-grain

heating of dust particles and the nonthermal desorption thatoccurs as a result of that heating. The particles are struck by aflux of cosmic rays with a wide spread in energy, and theresultant efficiency and rate of desorption of suitable icespresent in cold dense cores are determined. The efficiency is afunction of granular size, because the larger grains are heated tolower peak temperatures. We have specifically considered de-sorption from carbonaceous grains struck by Fe nuclei involving(i) one ML of water ice atop the carbonaceous substrate, (ii) fiveMLs of water ice atop the carbonaceous substrate, (iii) one MLof CO atop a water ice substrate, and (iv) one ML of N2 atop awater ice substrate. Although the single ML of water is efficientlydesorbed for all but the largest grains studied, desorptionproceeds much more slowly for the five MLs because of thestrong bonds between water molecules. Both MLs of CO and N2are desorbed efficiently. Studies of desorption occurring withmore complex structures such as porous material and mixed MLsare also planned. To use these results in gas–grain models ofinterstellar chemistry of cold cores will require converting theresults into rate laws, as initially explained by Hasegawa andHerbst (37).

The research program in astrochemistry of E.H. was supported by theNational Science Foundation.

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