The Kinetic Monte Carlo Method as a way to solve the masterequation for interstellar grain chemistry

H.M. Cuppen,∗,† L.J. Karssemeijer,† and T. Lamberts†,‡

Theoretical Chemistry, Institute for Molecules and Materials, Radboud University Nijmegen, and Sackler Laboratoryfor Astrophysics, Leiden Observatory, Leiden University

E-mail: h.cuppen@science.ru.nl

Contents1 Introduction 1

2 Theoretical background 2

3 Historical and theoretical overview of grain mod-eling techniques 43.1 Rate equations . . . . . . . . . . . . . . . . . 43.2 Master equation method . . . . . . . . . . . 63.3 Macroscopic Kinetic Monte Carlo . . . . . . 73.4 Microscopic Kinetic Monte Carlo . . . . . . 8

4 Technical aspects of KMC 84.1 Representation of the grain . . . . . . . . . . 94.2 Input parameters in grain models: filling the

table of events . . . . . . . . . . . . . . . . . 94.2.1 Sticking Fractions . . . . . . . . . . 104.2.2 Thermal desorption rates . . . . . . . 104.2.3 Diffusion rates . . . . . . . . . . . . 114.2.4 Theoretical determination of rates . . 13

4.3 Kinetic Monte Carlo algorithms . . . . . . . 144.4 Simulations with varying rates . . . . . . . . 16

5 Simulations of experiments 175.1 H on graphite . . . . . . . . . . . . . . . . . 175.2 Ice chemistry . . . . . . . . . . . . . . . . . 19

6 Applications of the KMC technique to astrochem-istry 216.1 H2 formation . . . . . . . . . . . . . . . . . 21

6.1.1 The effect of inhomogeneity . . . . . 216.1.2 Models including chemisorption sites 226.1.3 Temperature fluctuations . . . . . . . 23

6.2 Ice chemistry . . . . . . . . . . . . . . . . . 24

7 New directions 277.1 Unified Gas-Grain KMC models . . . . . . . 277.2 Combining different models . . . . . . . . . 287.3 Off-lattice models . . . . . . . . . . . . . . . 28

7.3.1 Continuum Kinetic Monte Carlo . . . 29∗To whom correspondence should be addressed†Theoretical Chemistry, Institute for Molecules and Materials, Radboud

University Nijmegen‡Sackler Laboratory for Astrophysics, Leiden Observatory, Leiden Uni-

versity

7.3.2 Adaptive Kinetic Monte Carlo . . . . 297.4 Coarse graining . . . . . . . . . . . . . . . . 31

8 Conclusions and outlook 32

1 IntroductionThe interstellar medium (ISM) is far from empty, ratherit contains large molecular clouds consisting of dust andgas. It is in these clouds that new stars form — possiblywith habitable planets — with molecules playing a crucialrole.1,2 At the moment, over 170 different molecules havebeen detected: stable molecules, radicals, cations, and an-ions. Many molecules only consist of a few atoms and arefamiliar to us from a terrestrial point of view, like water,molecular hydrogen, methanol, formaldehyde, formic acid,and dimethyl ether. Other molecules are more exotic andconsist, for instance, of very unsaturated carbon chains. Gasphase chemical models show that these latter molecules canbe quite easily formed through gas phase reactions.3 The sat-urated molecules are however only formed in small quan-tities through gas phase chemistry, since they require ei-ther high temperatures to be formed or three-body reactions,which are both not available in cold molecular clouds. Dustgrains can act as a third body, facilitating addition reactionsthat lead to the formation of saturated molecules. Already in1949 van de Hulst 4 realized the importance of grain surfacechemistry and he suggested surface formation routes for H2from two hydrogen atoms and the formation of H2O from O2with hydrogen peroxide as intermediate.

From various sources of information, we know that ap-proximately 1% of the mass of the material in the ISM, and10−10 % in terms of number, consists of small, nano-sizedsilicate and carbonaceous particles called dust grains. Al-though this seems a small portion of the total amount ofmatter, dust grains play an important role in the ISM, notonly by acting as catalytic sites for molecule formation butalso by shielding molecules from dissociating UV radiation.Depending on the physical conditions, the dust grains canbe coated with a layer of “dirty” ice consisting of a mixtureof different species. Their main components are water, car-bon monoxide and carbon dioxide, but also more complexmolecules like methane, methanol, formaldehyde and am-monia may be present. Some of the ice species accrete from

1

the gas phase, but, as mentioned before, many of the simpleimportant molecules like H2 and H2O and also several com-plex organic molecules are not formed in the gas phase, butrather on the grain surfaces themselves. As a result, manymodelers have begun to include grain-surface processes intheir models to describe chemical evolution during, for in-stance, star formation.1,5,6 Originally, the development ofgrain surface models was mostly driven from the astrochem-ical modeling community7–9 without much input form spe-cialists in ice chemistry. Gas phase models were extendedwith a surface phase and the grain surface chemistry wastreated in a similar way as gas phase chemistry. This wasfound to lead to problems in the regime where the surfaceabundance is low – the so-called “accretion limit” – whereformation rates could be orders of magnitude off. Since then,different modeling methods have been applied to tackle thisand other modeling problems.

From the physico-chemical point of view, refinementshave been made as well. Many of the initial expressionsand input parameters originated from surface science orwere extensions from gas phase experiments. In surfacescience, mostly metal or semiconductor surfaces are stud-ied on which absorbates are deposited within the monolayerregime, around room temperature. This is quite differentfrom the rough grain surface on which inhomogeneous icelayers of tens of monolayers are formed at very low temper-atures. Fortunately, technological developments have nowenabled the experimental verification of grain surface chem-istry. Many of the assumptions within the models are nowput to the test by applying to experimental surface sciencetechniques to real interstellar analogs. With this new experi-mental information comes the need for detailed surface mod-els to understand the physico-chemical processes behind theexperimental results and to translate the experimental find-ings to astrochemical timescales.

Different modeling techniques have been applied to grainsurface astrochemistry, covering a large range of time- andlength scales. More molecular detail comes at the expenseof more CPU time to cover the same evolution in real time.How different techniques relate to one other in this respectis indicated in Figure Figure 1. Molecular dynamics simula-tions trace the exact location and orientation of the moleculesincluding lattice vibrations and in some cases even the in-tramolecular movement, but they typically stay within thepicosecond to nanosecond time frame. Rate equations on theother hand can easily handle 108 years – much longer thanthe lifetime of a molecular cloud – but adsorbed moleculesare only treated in terms of numbers and their exact loca-tions are not known. The present review aims to discussthe application of the kinetic Monte Carlo technique (KMC)to grain surface chemistry. This method was initially intro-duced into astrochemistry as a solution to the “accretion”limit, but is now more and more applied to gain insight intothe physico-chemical surface processes. The method has,in principle, not a real restriction in how molecules are rep-resented: in terms of number densities or with their exactlocation and orientation. The different implementations ofkinetic Monte Carlo can therefore cover a huge time- andlength scale range.

Molecular

Mol

ecul

arD

etai

l

Dynamics

off−lattice kMC

lattice−gas kMC

Macrosc. kMC

Master eq. Rate eqs.

Simulated time / CPU time

Sys

tem

Siz

e

Figure 1: Overview of the different simulation methods men-tioned in the present review.

The kinetic Monte Carlo technique is a way to solve themaster equation and this review will start by deriving thisexpression in Section 2. Several modeling methods havebeen used to simulate grain surface chemistry and Section 3gives a historical overview of the introduction of these tech-niques into astrochemistry and their main advantages anddisadvantages. The reason to include a short discussion onthese other techniques is that this allows us to put the resultson kinetic Monte Carlo into a better perspective. The re-view will then continue by discussing some of the technicalaspects involved in constructing a KMC model and the in-put parameters that are needed for such a model. Sections 5and 6 summarize the application of the kinetic Monte Carlotechnique in astrochemistry. Here the distinction is madebetween simulations that try to reproduce astrochemicallyrelevant experimental results and simulations that attemptto predict astrochemical environments. Here, only modelsthat treat the grain with some microscopic detail will be dis-cussed, starting from the first model in 2005 until beginningof 2013. Finally, some recent advances in the application ofkinetic Monte Carlo are discussed. We will go in detail onthe challenges and opportunities involved.

2 Theoretical backgroundKinetic Monte Carlo is a method to solve the master equa-tion. We will give a brief derivation of this equation in or-der to discuss the underlying assumptions to the derivationof this expression, the different ways in which this equationis solved, the approximations involved, and their advantagesand disadvantages. A more thorough derivation can be foundin textbooks such as Refs. 10–12.

Many processes occurring on the surfaces of grains are inprinciple stochastic processes. We can describe these sys-tems by a state vector x and a time coordinate t. In thissection, x is an abstract quantity which may be some lo-cal minimum on the potential energy surface (PES), whichis schematically drawn in the left panel of Figure Figure 2.Here the black circles represent different states. In manygrain surface models a PES is however not explicitly consid-ered and in those cases a state x can be represented by, e.g.,the species on the grain, their position, the temperature ofthe grain, etc., or x can simply contain the coordinates of allparticles on the grain.

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Let us call p(x, t) the probability density for the system tobe in state x at time t. The probability density can be writtenas

p(x, t) =dP(x, t)

dx, (1)

where P(x, t) is the probability that the system is in an in-finitesimal volume element dx around x at time t. Since theprobability to find the system in any x at a given time t isunity, we have ∫

dxp(x, t) = 1. (2)

The definition of the probability density, Eq. (1), canbe extended to an n’th order probability density func-tion p(xn, tn;xn−1, tn−1; . . . ;x1, t1) which gives the probabil-ity density for the system to be in x1 at t1 and in x2 at t2 etc.To describe the time evolution of the system the conditionalprobability density is introduced

p(xm, tm; . . . ;xn+1, tn+1|xn, tn; . . . ;x1, t1)=p(xm, tm; . . . ;x1, t1)p(xn, tn; . . . ;x1, t1)

,

(3)which gives the probability density to be in states xn+1through xm at time tn+1 through tm given that it is in statesx1 through xn at times t1 through tn. The higher order con-ditional probability densities are complicated functions andtherefore one usually assumes a Markov Chain. To define aMarkovian system we imply time-ordering (t3 > t2 > t1) andimpose the following restriction on the conditional probabil-ity density functions

p(xn, tn|xn−1, tn−1; . . . ;x1; t1) = p(xn, tn|xn−1, tn−1). (4)

This means that the probability of arriving in xn, tn is onlydependent on xn−1, tn−1 and not on the states before tn−1. AMarkovian system is completely defined by an initial state,say x1, t1, and the first order conditional probability densi-ties since any other probability density function can be con-structed from these. For example

p(x3, t3;x2, t2;x1; t1) = p(x3, t3|x2, t2;x1; t1)p(x2, t2;x1, t1)

= p(x3, t3|x2, t2)p(x2, t2|x1, t1)p(x1, t1),(5)

where we used Eqs. (3) and (4).The Markov chain assumption is valid for memoryless

processes. When simulating grain surface chemistry, we aretypically interested in rare events, such as diffusion and des-orption. This type of events occur at much longer timescalesthan the (lattice) vibrations and the event timescale and thevibrational timescale become decoupled. This makes rareevents effectively memoryless, since all history about whichstates where previously visited is lost due to the vibrationsbetween two transitions. This is schematically depicted inFigure Figure 2. The residence time of the system in a localminimum of the potential energy surface is several orders ofmagnitude larger than the vibrational timescale. The systemwill therefore typically proceed according to a trajectory likethe one drawn in the right panel of Figure Figure 2. At thetime that the system leaves the potential well, all informationabout the direction from which it initially entered this poten-

Figure 2: Schematic representation of a potential energy sur-face (PES). The states, local minima on the PES, are indi-cated by black solid spheres. (left) The system moves fromstate to state or (right) the vibrational movement is includedas well and the trajectories spend most of their time aroundthe local minimum.

tial well, is lost. In kinetic Monte Carlo this is approximatedby a trajectory as drawn in the left panel. In some cases,when the residence time in a state is short as compared tothe vibrational timescale, this approximation breaks down aswe will discuss in Sections 5.1 and 5.2. Examples of theseare H atoms diffusing on a graphite surface without feeling apotential (Section 5.1) and the “hot” reaction products whichmove away from each other with a certain momentum uponreaction (Section 5.2). For these cases, Molecular Dynam-ics simulations, which use Newton’s equations of motion todetermine the molecular trajectories, are better suited. Thesetypes of simulations would result in trajectories plotted in thepanel on the right. A drawback of this technique is howeverthat their simulation timescales do not match astrochemicaltimescales – they are shorter by roughly twenty orders ofmagnitude – which makes them less suitable for astrochemi-cal simulations. Individual processes with astrochemical rel-evance can however be treated with Molecular Dynamics.

The transition probabilities of Markov processes obey theChapman-Kolmogorov equation

p(x3, t3|x1, t1) =∫

dx2 p(x3, t3|x2, t2)

× p(x2, t2|x1, t1). (6)

This equation describes the transition from x1, t1 to x3, t3through all possible values of x2 at t2. Let us now assumethat the conditional probability densities are time indepen-dent and so we can write

p(x2, t2|x1, t1) = p∆t(x2|x1), ∆t = t2− t1, (7)

because there is now only a dependence on the time differ-ence ∆t. The expression p∆t(x1,x2) denotes the transitionprobability density from state x1 to state x2 within the timeinterval ∆t. This can be expressed as a rate, k, by

k(x2|x1) =p∆t(x2|x1)

∆t. (8)

Using the Chapman-Kolmogorov equation (6) for p∆t , we

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get

p∆t+∆t ′(x3|x1) =∫

dx2 p∆t ′(x3|x2)p∆t(x2|x1). (9)

After some manipulation of this expression, we arrive at themaster equation for the transition probability densities

dd∆t

p∆t(x3|x1) =∫

dx2k(x3|x2)p∆t(x2|x1)

−∫

dx2k(x2|x3)p∆t(x3|x1). (10)

If we want to obtain a master equation for the state probabil-ity density itself, we can multiply this equation by p(x1, t),integrate over x1. This will ultimately lead to the masterequation

ddt

p(x3, t) =∫

dx2k(x3|x2)p(x2, t)−∫

dx2k(x2|x3)p(x3, t).

(11)

On a discrete state space, which is usually used for grainsimulations, the equation is written in terms of probabilitiesin the following way:

ddt

P(xi, t) = ∑j(k(xi|x j)P(x j, t)− k(x j|xi)P(xi, t)) . (12)

The first part of the sum represents the increase in P(xi, t)because of events that enter state xi from any other state x j;the second part is the decrease in P(xi, t) due to all eventsleaving xi. Most chemists will construct this expression in-tuitively. Here we made the formal derivation of the masterequation to show the assumptions involved at the root of thisexpression which could be a severe restriction, for instancethe Markov chain assumption.

3 Historical and theoretical overviewof grain modeling techniques

3.1 Rate equationsOne of the most applied methods in astrochemistry is solv-ing rate equations numerically. These equations model theabundances of species and can be seen as the deterministiccounterpart of the stochastic master equation. They do notdescribe the probabilities, P, to be in individual states, butrather the surface abundances, ns, of the different specieson the grain. Since a similar quantity is used to describethe gas phase chemistry (the gas phase abundance ng), thesetype of equations can be easily coupled to gas phase chem-istry networks. The first gas-grain simulations were there-fore rate equation based.8,13–15 Rate equations are often ap-plied in chemistry to describe macroscopic (experimental)effects and account for many-body effects with a mean-fieldapproach. A set of rate equations to describe surface reac-tions in astrochemistry was first constructed by Pickles andWilliams 16 to describe the formation of molecular hydrogenfrom H atoms and water from O2 and H through HO2 and

H2O2.Grain species can generally undergo four types of pro-

cesses: accretion onto the surface, desorption, diffusion,and reaction. This leads to the following expression for thechange in surface abundance of species A at time t

ddt

ns(A, t) = ∑B,C

kreact,B,Ckhop,B + khop,C

Nsitesns(B, t)ns(C, t)

−ns(A, t)∑B

kreact,A,Bkhop,B + khop,A

Nsitesns(B, t)

+ kacc,Ang(A)

− [kevap,A + knon−th,A]ns(A, t), (13)

with Nsites the number of sites on the grain. Here the sub-scripts s and g of n represent the surface and gas phase abun-dances of the species, respectively. The first two terms inthis expression denote the gain and loss of species A due toreactions which form and destroy A. These are describedin terms of reaction rates kreact and hopping rates khop. Thethird term accounts for the accretion of new species from thegas phase, in terms of the accretion rate kacc. The final termrepresents the desorption of the species into the gas phase.This can either be due to thermal desorption (kevap) whenthe temperature is high enough to overcome the binding en-ergy or be due to non-thermal effects (knon−th) like photodes-orption,17–20 sputtering by cosmic rays or grain heating bycosmic rays.21,22 Here first order kinetics are assumed forthe desorption, although experimentally zeroth order kineticsare often observed for the desorption of ices in the multilayerregime. We will come back to this in Section 4.2.2.

The reactions included in Eq. (13) are assumed to oc-cur through the Langmuir-Hinshelwood mechanism. Gen-erally, the distinction is made between three types of surfacereactions – schematically depicted in Figure Figure 3: theLangmuir-Hinshelwood (LH) mechanism, which is a diffu-sive mechanism in which the species move over the surfaceand try to react upon meeting, the Eley-Rideal (ER) mecha-nism where one (stationary) reactant is hit by another speciesfrom the gas phase, and the hot-atom mechanism which isa combination of both mechanisms where non-thermalizedspecies travel some distance over the surface finding reac-tants on their way. In all cases, the excess heat which be-comes available through the reaction can be used for des-orption of the products. Since in dense clouds the temper-ature of the gas phase is similar to the grain temperature,the hot-atom mechanism is generally not considered. It canbe important in experimental conditions or for instance inshock regions where the gas phase is often much warmerthan the grain. The diffusive LH mechanism leads to secondorder rate kinetics as can be seen from the first two terms ofEq. (13).

Eley-Rideal is generally considered to be only importantfor high surface coverages or low surface mobility23 and istherefore not explicitly considered in most models. How-ever, catastrophic freeze-out of CO in prestellar cores resultsin a layer of CO ice,24 i.e., a high surface coverage of reac-tive species. Under these circumstances, Eley-Rideal couldbe an important mechanism in the formation of methanol.

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Langmuir−Hinshelwood Eley−Rideal Hot Atom

Figure 3: Schematic representation of the three mechanismsfor surface reaction: the Langmuir-Hinshelwood mecha-nism, which is a diffusive mechanism in which the speciesmove over the surface and try to react upon meeting, theEley-Rideal mechanism where one (stationary) reactant ishit by another species from the gas phase, and the hot-atom mechanism which is a combination of both mecha-nisms where non-thermalized species travel some distanceover the surface finding reactants on their way.

Rate equations that include surface reactions following theEley-Rideal mechanism contain an additional term that de-pends on the gas phase abundance of one reactant, the sur-face abundance of the other reactant and the rate of reactionbetween them.25 The Langmuir-Hinshelwood expression forthe same reaction contains the surface abundance of both re-actions, the reaction rate and a meeting rate, as can be seen inEq. (13). The meeting rate due to the movement of speciesA is given by khop,Ans(A, t)/Nsites. The hopping rate khop,Aused here is the rate of an individual hop of species A. Of-ten, however, the hopping rate quoted in literature dealingwith rate equations, is the hopping rate of an individual hopdivided by the number of sites on the grain. It is worth men-tioning that the reaction rate is different for Eley-Rideal andLangmuir-Hinshelwood: for ER the two reactants have oneattempt to cross the reaction barrier, whereas for LH the tworeactants will remain adsorbed in close vicinity until they re-act or one of them diffuses or desorbs. Reaction is thereforein competition with diffusion and desorption.26 Rate equa-tion models, however, often ignore this.

To relate the deterministic rate equations to the stochas-tic master equation, one can think of the abundance of thespecies A in terms of the expectation values of the number ofparticles of type A

ns(A, t) = 〈NA(t)〉 ≡∞

∑m=1

mP(xA[m], t) (14)

where the brackets denote the expectation value and NA(t)is the number of particles of type A on the grain at time t.On the right of the expression we have the probability of thestate vector xA[m] which describes a state with m particlesof species A. An obvious disadvantage of the rate equationmethod is that it does not treat statistical fluctuations. Oneof the assumptions underpinning the rate equations leads tofollowing assumption:

〈NA〉〈NB〉 ≈ 〈NANB〉 . (15)

When the number of species on the surface is small, this as-sumption breaks down and the systems reaches the accretionlimit as it was called by Charnley et al. 9 . In this case therate limiting step for reaction is not diffusion but the arrivalof species on the surface from the gas phase. Figure Fig-

ure 4 shows the evolution of several surface species as cal-culated by Stantcheva and Herbst 27 using the rate equationmethod and the “master equation method”, which will bediscussed in the next section. The main difference betweenthe two methods is that the rate equations use expectationvalues whereas the master equation method does not. Weshow these models here to illustrate the effect of the accre-tion limit. The results from Stantcheva and Herbst showthat the abundances of the probabilistic species, such as Hatoms and other reactive species, are highly overestimatedby rate equations. The most extreme case is for atomic H,shown in the top panel of Figure Figure 4, where the dif-ference is many orders of magnitude. Since these reactivespecies are minority species which cannot be observed di-rectly, we are usually more interested in stable reaction prod-ucts as plotted in the bottom panel for the CO hydrogenationroute leading to methanol. Also here the effect is striking,with CH3OH being underproduced by orders of magnitudedepending on the age of the cloud. In the master equationmodel, all CO molecules appear to be converted almost im-mediately to methanol, while in the rate equation model theconversion occurs much later. This discrepancy arises fromthe huge difference in surface abundance of atomic H and theinefficiency of the H + CO reaction, which proceeds with abarrier.

To account for the accretion limit problem in the rate equa-tions, alternative methods have been proposed which eitheradjust the rate equations in the accretion limit regime to re-sult in the correct rates or do not use the expectation value as-sumption and are so-called stochastic methods. The masterequation method, applied in Figure Figure 4 is an exampleof the latter. Caselli et al. 28 were the first to propose a wayto adjust the rate equations. They applied a semi-empiricalapproach to scale down the reaction rates for those caseswhere the surface migration of atomic hydrogen is signifi-cantly faster than its accretion rate onto grains. This methodappeared to give a good agreement with stochastic methodsfor a limited number of cases. Unfortunately, due to the ad-hoc way in which the rates were scaled down, the validity ofthe results outside the tested regime cannot be guaranteed.

More recently, a new modified-rate approach was sug-gested by Garrod 29 . This method is more rigorously derivedthan the original modified-rate approach. A modified rate isdetermined by the accretion rate of one of the reactants mul-tiplied by the surface abundance of the other reactant timesan efficiency term that takes into account the competition be-tween diffusion across the surface for the reactants and thedesorption of the reactants. The expressions for the reactionsare adjusted if the modified reaction rate is smaller than theclassical diffusive rate. The main advantages of these mod-ified rate equation models are that they are computationallyinexpensive as compared to the stochastic methods and thatthey automatically switch to the normal rate equations in theregime where those are still valid.

3.2 Master equation methodThe accretion limit problem led to the realization that grainsurface processes can be better described by a master equa-

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Figure 4: (top) Absolute surface abundance of atomic H at10 K, 15 K, and 20 K and (bottom) fractional abundances(relative to H2 ) for CO, H2CO, CH3OH and CO2 at 10 Kas calculated by master equation (meq) and rate equation(req) methods. Reprinted from Ref. 27. Credit: Stantcheva,Herbst, Astron. Astrophys., 423, 241, 2004 reproduced withpermission c© ESO

tion. This resulted in the introduction of several techniquesthat solve this master equation. One of these constructsdifferential equations for the probabilities to have a certainnumber of species on the surface and uses a numerical inte-grator to obtain the average number of species on the grainas a function of time. This method was introduced into as-trochemistry to circumvent the accretion limit problem30–32

and is usually referred to as the “master equation method”.This can be confusing since the kinetic Monte Carlo ap-proach, which is discussed in the next section, also solvesthe master equation. A better description, although ratherlengthy, would perhaps be “numerical integration of masterequations”, but in line with previous authors we will stick tousing the term “master equation method” in the remainder ofthis review.

Since most gas phase models also use numerical integra-tion, the master equation approach can again be easily inte-grated within a gas phase astrochemical code. If only twotypes of species, A and B, are considered, the expressions

that have to be integrated become of the type

ddt

P(xA,B[k, l], t) = kreact,A,Bkhop,A + khop,B

Nsites

× [(k+1)(l +1)P(xA,B[k+1, l +1], t)− klP(xA,B[k, l], t)]

+ kacc,Ang(A)[P(xA,B[k−1, l], t)−P(xA,B[k, l], t)]

+ kacc,Bng(B)[P(xA,B[k, l−1], t)−P(xA,B[k, l], t)]

+ [kevap,A + knon−th,A][(k+1)P(xA,B[k+1, l], t)− kP(xA,B[k, l], t)]

+ [kevap,B + knon−th,B][(l +1)P(xA,B[k, l +1], t)− lP(xA,B[k, l], t)]. (16)

Here P(xA,B[k, l], t) denotes the probability that k speciesA and l species B reside on the grain at time t. Onecan again recognize the reaction, accretion, and desorptionterms. The factors (k + 1) and (l + 1) in the first reactionterm reflect the number of possible combinations to formnew molecule AB of k+1 species A and l +1 species B andleave k species A and l species B behind. In principle, oneshould construct these expressions for all possible combi-nations of species A and B (P(xA,B[0,0], t), P(xA,B[1,0], t),P(xA,B[0,1], t), P(xA,B[1,1], t), etc.). Obviously, these ex-pressions become much more involved if more types ofspecies are included in the model. Theoretically the num-ber of expressions to solve is infinite. However, cutoffs canbe used to exclude the higher order terms, if their probabil-ity of occurrence becomes very small. In other words, onecan set upper limits on the maximum number of particlesper species that can reside on the grain, Nmax(A). Stantchevaet al. 32 used this approach for a small network containing11 species connected through 10 surface reactions. Onlythe five minority species O, H, OH, HCO, and H3CO weretreated probabilistically; the other species were described us-ing rate equations. For the species O and H the maximumnumber on the grain considered never exceeded five. Theresulting expressions, both master equations and rate equa-tions, were integrated numerically using a numerical integra-tor. In this way, they were able to simulate a small networkof species under conditions where the abundance of theseminority species remains low.

If the number of particles expands, either by a change inphysical conditions (Nmaxgrows) or by increasing the num-ber of considered species in the model, the number of equa-tions blow up rapidly. Algorithms have been suggested tomake the effect less severe and to extend the range in whichthe master equation method is applicable.33,34

The lowest possible cutoff would be Nmax = 2 for speciesthat form homonuclear diatomic molecules (H2, O2, etc.)and Nmax = 1 for all other species, but obviously this wouldlead to a large error. To make this less dramatic one canconvert the original master equations into moment equa-tions34–36. Let us consider moments of the distribution P.The kth moment is defined as⟨

Nk(A, t)⟩≡

∞

∑m=1

mkP(xA[m], t) (17)

and the expectation value used in the rate equation (Eq. (14))

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is therefore simply the first moment. Differently from themaster equation approach, which is expressed in differen-tial equations of the probabilities P(xA,B[k, l], t), the momentequations are differential equations of the different moments〈Nk

ANlB〉. These equations can be directly obtained by ex-

panding the time derivatives of the moments using Eq. (17)and inserting the expression for P(xA[m]) from the masterequation. Obviously, still an infinite number of equations canbe composed; with the order of the moment running to infin-ity instead of the number of species. But the expressions forthe first moments, the population sizes we are interested in,are completely determined by first and second moments andwe could use the second order as a upper cutoff. The secondmoment equations, however, are determined by first, secondand then unknown third moments. In order to close the set ofmoment equations, Barzel and Biham 34 imposed the follow-ing constraint on the moment equation: at any given time,at most, two atoms or molecules can be adsorbed simulta-neously on the surface. Furthermore, these two atoms ormolecules must be from species that react with each other.In a later paper, Barzel and Biham 37 introduced a diagram-matic way to visualize a surface reaction network and to aidin the construction of the moment equations to make themethod more applicable for grain surface chemistry in gen-eral. This formulation was applied in Ref. 38 to incorporatestochastic chemistry in the form of moment equations into anastrochemical code to simulate Photon Dominated Regions.The specific focus was on the formation of H2, D2, and HD.

While the constraint that at any given time, at most, twoatoms or molecules can be adsorbed simultaneously on thesurface can work for a small network under specific physicalconditions where the surface density is low, this constraintcannot be obeyed in the general case and application of mo-ment equations to large grain surface networks could resultin undesirable effects. For this reason, Du and Parise 39 in-troduced a hybrid method that combines moment equationsand rate equations. In general, both gas phase and surface re-actions are treated by rate equations, but if the average popu-lation of a certain surface species becomes smaller than oneper grain, the algorithm switches to moment equations.

3.3 Macroscopic Kinetic Monte CarloThe first astrochemical application of simulations based onthe Monte Carlo method was performed by Tielens and Ha-gen 7 . They used the method to simulate the evolution of thegrain mantle by one-by-one accretion of gas phase speciesand calculating their fate (reaction, desorption, trapping) be-fore a new species would land on the surface. In this way,the time evolution is determined by the accretion rate and theMonte Carlo aspect determines the fate of accreted species.Steady-state gas phase abundances were used to determinethe relative accretion rates for each of the gas phase species.These were obtained from a pure gas phase model and gas-grain interactions were not directly included in this calcula-tion other than using a depletion parameter which decreasesthe relative elemental abundance of the heavier species likeO, N and C towards clouds with higher densities. Using theresulting gas phase abundances, the mantle composition was

calculated as a function of visual extinction AV or radiationfield, cloud density and depletion factor. Since the results ofthe simulations were presented as the steady-state fractionalcomposition of the grain mantles, the time evolution of thegrain was not discussed.

Charnley 40,41 was the first to introduce the Kinetic MonteCarlo method to determine the chemical evolution of a cloud,including both gas phase and grain surface. This was donein a more rigorous way than by Tielens and Hagen 7 andbuilt strongly on the work by Gillespie 42 , who used theKMC method to numerically simulate the stochastic timeevolution of coupled chemical reactions. Both Charnley andGillespie started from the master equation (Eq. (12)). Us-ing this equation one can derive the time interval betweentwo events, which is one of the fundamental expressions inkinetic Monte Carlo, since it describes how time progressesduring a KMC simulation. If one starts from state i at time t(so P(x j, t) = 0 ∀ j 6= i), then for small ∆t we have

ddt

P(xi, t +∆t) =−∑j

k(x j|xi)P(xi, t +∆t) (18)

from which we can calculate the probability that the systemhas not left state xi at time t +∆t

Pnot left(xi, t +∆t) = exp(−∫ t+∆t

tdt ′ktot(xi, t ′)

), (19)

where ktot(xi, t) is the total rate leaving state xi. If we assumestationary rates than Eq. (19) becomes

Pnot left(xi, t +∆t) = exp(−ktot(xi)∆t) . (20)

Stationary rates are a reasonable assumption for grains ofconstant temperature. If the surface temperature fluctuates,for instance due to stochastic heating of the grains by incom-ing UV photos or cosmic rays21,22,43,44 or in a TemperatureProgrammed Desorption experiment, the rates change overtime and the time interval ∆t should be determined differ-ently.45 This will be discussed in Section 4.

In practice, this means a KMC cycle starting from state xiconsists of three steps if all possible events are known: firsta final state x f is picked with probability

pi→ f =k(x f |xi)

ktot(xi), (21)

using a random number. Next, the time is advanced wherethe values for ∆t are chosen such that they follow the dis-tribution dictated by Eq. (20). This can be numericallyachieved by generating a uniform random number Z in range(0,1] and equating this to the probability that the reaction hasnot yet occurred:

Z = Pnot left(xi, t +∆t), (22)

which leads to

∆t =− ln(Z)ktot(xi)

(23)

for stationary rates. Finally, the picked transition is evaluated

7

and the transition rates leaving this new state, x f , are deter-mined. The consequence of Eqs. (21) and (23) is that at eachgiven time all possible transition events leaving the currentstate and their corresponding rates should be known. Thiscombination of events and rates is often referred to as “ta-ble of events”. For the systems discussed so far, the table ofevents can be easily constructed, since like in the rate equa-tion approach and numerical integration of the master equa-tion approach all possible events are known at the start of thesimulation. If we want to look at grain surface chemistry inmore microscopic detail, this can become problematic as wewill discuss in Sections 4 and 7.

The KMC realization of the master equation is not as eas-ily implemented in a gas-grain astrochemical code as the rateequation and master equation methods. The reason for this isthat for the gas phase usually rate equations are used and theMonte Carlo algorithm and numerical integrators cannot beeasily coupled. Chang et al. 46 presented an iterative solutionwhere the grain surface chemistry is treated by a microscopicKMC (see next section) method and the gas phase by meansof rate equations. Attempts to couple gas phase models tograin chemistry models are discussed in more detail in Sec-tion 7.

3.4 Microscopic Kinetic Monte CarloThe methods discussed so far only considered the abun-dances or numbers of the different surface species and nottheir exact location and exact movement on the grain. Thediffusion of the different species is included in some aver-age way. In microscopic KMC simulations, the location ofthe particles is. They undergo a random walk and they canrevisit sites multiple times. This revisiting is often referredto as back diffusion. Although it is possible to express thiseffect in rate equations,47 it is included in a more straight-forward way in microscopic KMC simulations. In these kindof simulations, accretion, desorption, diffusion, and reactionare again considered, with similar rates to the macroscopicKMC method, but with the difference that for each individ-ual atom or molecule on the surface its specific position onthe grain is monitored. Diffusion now occurs through indi-vidual hops of the species from one site to the next. Thisincludes hopping back to where it came from. The informa-tion of the position of the atoms/molecules on the surface isan advantage over macroscopic KMC.

Another advantage is that the surface structure can be in-cluded. The rate of diffusion and desorption can be madesite-specific, mimicking steps on the surface, amorphicity ofthe grain, and/or inhomogeneity of the grain material. Thespecies might undergo a hindered random walk that does notresult in an exact second order for reaction. Since in mi-croscopic KMC, only species either in neighboring sites oroccupying the same site are allowed to react and the reactionorder is not included specifically beforehand, this deviationfrom second order will come out naturally. In the same way,crossing a reaction barrier is treated in competition with dif-fusion and desorption automatically in microscopic KMC.48

One can imagine that while the grain mantle grows,species can become trapped in the lower layers and are not

able to participate in the chemistry that occurs on the surfaceof the formed ice, or that these species will not be able todesorb. Again this will come out naturally from the micro-scopic Monte Carlo simulations. We will come back to thispoint at a later stage (Sections 4, 5, and 7).

The first implementation of microscopic KMC in astro-chemistry was focused on H2 formation and was calledcontinuous-time, random-walk (CTRW) Monte Carlo49 fol-lowing the algorithm by Montroll and Weiss 50 . We will referto this application as microscopic KMC throughout this re-view to set it apart from macroscopic KMC. When referringto the algorithm, we will still use the term continuous-time,random-walk KMC. In their study, Chang et al. used the mi-croscopic aspect of the method to study the effect of backdiffusion and of site-specific diffusion and desorption rates.The site-specific diffusion and desorption rates were appliedto mimic either the amorphous character of the grain or theeffect of composite grains of carbonaceous and silicate mate-rial. The simulations start with an empty grain, but depend-ing on the temperature and surface structure of the grain, theH-atom surface abundance slowly builds up until a steady-state value has been reached. Chang et al. determined thesteady-state recombination efficiency

η =2NH2

NH(24)

with NH the number of arriving H atoms and NH2the num-

ber of formed H2 molecules as a measure for the formationrate of H2. Back diffusion was found to be of importance,through a comparison of the η obtained with a rate-equationmodel, a master equation model (both from Biham et al. 30 )and two microscopic KMC models differing in the amount ofnearest neighbors (Figure Figure 5). Since the likelihood ofrevisiting a site decreases with number of nearest neighbors,more H2 is formed for surfaces with a higher coordinationnumber, although still less than predicted by the rate equa-tion and master equation models. Since the latter modelsmimic the limit in which no back diffusion is present, theyeasily overestimate the efficiency. Figure Figure 5 shows anoverestimation of a factor of 2–3 at 10 K. Additional sim-ulations were performed with site-specific rates. The inho-mogeneity of the diffusion and desorption rates was foundto have a strong effect on the temperature window, whereH2 can be efficiently formed; much stronger than back dif-fusion. Molecular hydrogen could be produced up to highertemperatures than in the case of homogeneous rates.

4 Technical aspects of KMCThis section will discuss the technical aspects of the Ki-netic Monte Carlo technique applied to surface astrochem-istry. Sections 2 and 3 discussed the theory behind KMC. Inthis section, we will go more in detail on the implementationof the theory and its technical limitations. As discussed inSection 2, the theory behind KMC demands the formationof a table-of-events. We will discuss the type of informationthat is needed to construct such a table, how this is obtained,and what the common assumptions are. Different applica-

8

Figure 5: Hydrogen recombination efficiency as calculatedby the microscopic KMC approach for homogeneous olivineand amorphous carbon as a function of surface temperature.For olivine, the circles represent a lattice with four nearestneighbors while the squares represent a lattice with six. Forcarbon, the triangles pointed upwards and downwards rep-resent, respectively, lattices with four and six nearest neigh-bors. The master equation/rate equation results of Bihamet al. 30 for olivine are shown as diamonds. Reproduced fromRef. 49. Credit: Chang, Cuppen, Herbst, Astron. Astro-phys., 434, 599, 2005 reproduced with permission c© ESO

tions of the KMC technique require different algorithms, andtherefore a wide selection of different algorithms has beendeveloped, usually with only slight variations. Here, threealgorithms that are most commonly used within astrochem-istry will be discussed, together with their advantages anddisadvantages. Additionally an algorithm which is able tohandle non-stationary rates will be discussed as well. But letus start by discussing the representation of the grain in thesimulations.

4.1 Representation of the grainAs discussed in Section 2, KMC follows the evolution ofa Markov chain from one state, or local minimum on thepotential energy surface, to the next. These states are usuallyput on a regular grid represented by a 2D or 3D array wherefor each grid point the occupancy is monitored. A reactionor a diffusion from one site to another will be modeled as asudden change in the occupancy of the sites. A model thatputs sites on a grid and only registers their occupancies, isoften referred to as a lattice-gas model.

One of the largest disadvantages of using a lattice model isthat a large part of the molecular detail is lost in these mod-els. Because of the amorphous character of the grain and ice,the local environment of the sites probably varies across thesurface and the states are most likely not at exact equidis-tant locations. Nevertheless, the bare grain or its ice mantleis usually considered to have well-defined positions on itssurfaces where other species can adsorb and these sites areprobably distributed with some limitations in terms of num-ber of neighbors. In this sense, there is some regularity, evenfor amorphous grain mantles, which would justify the use of

lattice-gas models. Deviations from the regular character ofthe grid can be accounted for by using site-specific rates aswas done in Ref. 49.

On the other hand, since the molecules are confined to lat-tice positions, using realistic molecular potentials can be-come problematic. Furthermore, making guesses on thepossible events is far from straightforward and we can eas-ily miss important mechanisms. So, ideally, one wants tomove away from lattice models. In computational chemistryand statistical physics so-called off-lattice models are morecommonly used for Monte Carlo simulations. These MonteCarlo simulations are however not kinetic Monte Carlo sim-ulations, but what we will refer to as Metropolis MonteCarlo simulations based on the early work by Metropolisand Ulam 51 , Metropolis et al. 52 . These type of simulationsdo not follow the time evolution of a system but rather ob-tain an ensemble average of certain properties of the systemand their algorithms only involve in one trial configurationat each point in the simulation, which does not necessarilyneed to be a minimum on the PES. Kinetic Monte Carlo onthe other hand allows to simulate the evolution of a system intime, but the drawback is that the simulation runs from stateto state and all accessible processes at a given cycle in thesimulation need to be known. This difference in approach,has a direct effect on the effort it takes to use an off-latticemethod. In Metropolis Monte Carlo this is can be achievedfairly easily whereas for Kinetic Monte Carlo this is not soobvious. Examples of off-lattice Kinetic Monte Carlo tech-niques will be discussed in detail in Section 7.3. For now,we will limit ourselves to lattice-gas simulations, since theseare most commonly used in astrochemistry.

A clear advantage of a lattice-gas model is that one canwork with a predefined event table. Since the molecules areconfined to a lattice, only a limited amount of events is pos-sible of which the rates can be determined at the start of thesimulation. The limited amount of possible events in lattice-gas models allows one to cover very large timescales. Thesimulations can really mimic laboratory experiments of ices,covering hours of simulation time and tracing similar prop-erties.

4.2 Input parameters in grain models: fillingthe table of events

A requirement for KMC is to have a table of events withcorresponding rates. For macroscopic KMC this table con-tains similar processes and input parameters as rate equationand master equation models. These include sticking frac-tions, diffusion barriers, binding energies and activation en-ergies for reaction. The sticking fraction is the fraction ofimpingements that leads to adsorption. Fractions lower thanone effectively decrease the flux of incoming particles. Thebinding energies determine the temperature regime in whichthe species are on the surface of the grain and are avail-able for reaction. Diffusion barriers are crucial in determin-ing the rate of reaction, since most reactions occur throughthe Langmuir-Hinshelwood mechanism and should also begiven for each species. Since these are intrinsically hard todetermine experimentally often a fixed fraction of the bind-

9

ing energy is taken. Originally H atoms were assumed todiffuse through quantum tunneling. Nowadays models favorthermal hopping after experimental studies that showed thatthe diffusion of H atoms is rather slow.53–56 Finally, the ac-tivation barrier for reaction should be given for all reactionsin the network. These are often guessed from analogous re-actions or taken from gas phase data, ignoring the influenceof the grain. This ignores the catalytic effect of the grainhowever, which can lower the barrier and change the branch-ing ratios of reaction products. Often quantum tunneling isassumed using an approximation with tunneling through asquare barrier.8

In summary, macroscopic KMC requires at least two inputparameters per species (diffusion barrier and binding energy)and one per reaction (reaction barrier). If other types of des-orption are considered, like photodesorption,17–20 sputteringby cosmic rays, grain heating by cosmic rays21,22 or des-orption upon reaction using the excess energy,2,57 more in-put parameters are needed. For microscopic KMC the list ismuch longer, since the location of each species is known andthe exact environment can influence binding energies andbarriers.

4.2.1 Sticking Fractions

The sticking fraction SA of a species A to the surface de-pends both on the grain and gas temperature and is one ofthe quantities determining the accretion rate of said specieson the grain

kacc,A = SAvAngrainπr2grain (25)

with rgrain the average radius of an interstellar grain (∼0.1 µm) with number density ngrain, and with vA the aver-age gas phase speed

vA =

√8kTgas

πmA. (26)

which is determined by the gas phase temperature Tgas, themass of the species mA and the Boltzmann constant k. Stick-ing fractions are often assumed to be equal to unity, whichis a good approximation for species other than hydrogen atlow gas and grain temperatures.58–60 They can deviate fromone if the incoming species cannot convert their momentuminto phonons or if a barrier for sticking, typically restrictedto chemisorption, exists. Sticking fractions can be deter-mined by Molecular Dynamics,58–62 perturbation and effec-tive Hamiltonian theories, close coupling wavepacket, andreduced density matrix approaches63 or by the much-more-approximate soft-cube method.64,65 Experimental studies onthe sticking of H, H2, D, and D2 have been performed.66,67 Itwas found that at low coverage, the sticking coefficient of H2increases linearly with the number of deuterium moleculesalready adsorbed on the surface. Similar trends were foundfor the hydrogenation of CO, where the KMC simulationscould only explain the experimental trends at intermediatetemperature if a H2-coverage-dependent sticking fraction ofH atoms was assumed.68

4.2.2 Thermal desorption rates

The residence time of a species A on the surface is pre-dominantly determined by their desorption rates. The ther-mal desorption rate, in turn, depends on the binding energyEbind,A

kevap,A = ν exp(−Ebind,A

kT

), (27)

where ν is an attempt frequency. In an off-lattice simula-tion, one should be able to obtain the binding energy fromthe potential energy surface; in lattice-gas models this is notpossible and the binding energies should be taken from someexternal source. One of these sources are Temperature Pro-grammed Desorption experiments. Since this is such an im-portant technique for the determination of input parametersand since also KMC simulations of TPD experiments havebeen reported, we will here briefly explain the technique.See Ref. 69 for a review. These experiments are usuallyperformed in an ultra-high-vacuum set-up equipped with aquadrupole mass spectrometer. The temperature of the sub-strate of interest can be carefully controlled. The experi-ment consist of two phases: first, the substrate is kept at aconstant low temperature and a known quantity of one ormultiple species is deposited; during the second phase thetemperature is linearly increased and the desorption of sur-face species is monitored using the mass spectrometer. Themeasured desorption rate of the species is then plotted as afunction of temperature.

One can perform a series of these experiments, varying theinitial deposited amount, the deposition temperature, and/orthe heating ramp. Often, the resulting TPD spectra are thenfitted by the Polanyi-Wigner equation

ddT

ng(A) = ns(A)o ν exp(−Ebind,A/kT )β

(28)

where o is the order of the desorption process and β is theheating rate. Generally the order is assumed and the prefac-tor and binding energy are used as fitting parameters. Un-fortunately, these two parameters are correlated and it is notstraightforward to get accurate values for both parameters atsame time. For this reason, sometimes the prefactor is as-sumed as well and only the binding energy is obtained fromthe fit.

Figure Figure 6 shows synthetic TPD spectra using thisexpression with different desorption orders. The left panelshows three TPD spectra of zeroth order desorption with dif-ferent initial deposited amount. The leading edge of thesethree curves coincide. Zeroth order desorption generally oc-curs when multiple layers of the same species are deposited.During the desorption, the number of surface species avail-able for desorption (the top layer) remains the same. Themiddle panel shows first order desorption. Here, the tem-perature for which the desorption rate peaks is independentof the deposited amount. First order desorption generallyoccurs in the monolayer regime, where all surface speciesare also available for desorption. Please notice that most rateequation, master equation and macroscopic KMC models as-sume first order desorption in all cases, even in the multilayer

10

40 50 60 70

Des

orpt

ion

rate

(a.

u.)

40 50 60 70Temperature (K)

40 50 60 70

Figure 6: Synthetic TPD spectra using Eq. (28) for ze-roth (left), first (middle) and second (right) order desorption.Three different initial deposition amounts have been used.

regime. Section 7 discusses some more recent models thatmove away from this assumption.

Finally, the right panel of Figure Figure 6 shows secondorder desorption. Here the trailing edges coincide. Secondorder desorption is mainly due to two reasons: either des-orption of reaction products, that were formed in a secondorder surface reaction, or the surface exhibits a distributionof binding sites. In the latter case, the stronger binding sitesare filled first and the average binding energy decreases withcoverage, resulting in TPD curves that resemble second or-der behavior. Second order behavior is also found for thedesorption of HD that is formed by reaction of H with D.Katz et al. 56 obtained binding energies and diffusion barriersfor H2 and H by fitting a rate equation model to the exper-imental data where the binding energies and diffusion bar-riers are used as fitting parameters. A diffusive model wasapplied to fit the results of Pirronello et al. 53,54,55 , whichcould only give a lower limit for the desorption energy of Hatoms. Higher values for this energy were found to give de-generate results. In general, one could state that the desorp-tion energies of stable species like H2 and D2 are relativelystraightforward to determine whereas desorption energies ofunstable species like H and D are much harder to determine.For most cold dark cloud simulations, this is however nota problem for most species other than H and D, since des-orption of these species hardly occurs because of the lowtemperature. Unfortunately, binding energies of H and D areespecially important in the determination of the temperaturewindow in which efficient hydrogenation can occur. For thediffusion barrier, the situation is the reverse: they are easierto obtain for reactive species than for non-reactive species.Unfortunately, the ratio between the found diffusion barrierand lower limit of the binding energy of H is often gener-alized for other species as well. This is however an upperlimit.

The binding energies of a wide collection of stable specieshave been determined using the TPD technique. Examplesare N2,70 CO,70 O2,71 H2O,72,73 and CH3OH.74 The bind-ing energies have been mostly determined for the desorptionof pure ices from different substrates. The differences be-tween the different substrates are rather small and becomenegligible in the multilayer regime.74 The experiments showthat the molecules indeed desorb with a (near) zeroth or-der rate in the multilayer regime whereas they desorb with a(near) first order rate in the monolayer regime, as explainedabove.73

Since interstellar ices are not homogeneous the desorption

of mixed layers is more relevant for astrochemical modeling.However, the introduction of more species in the ice makesthe desorption process immediately much more complex.First, the binding energy of a particle can change dependingon its surrounding material. Second, the dominant mantlespecies can prevent other species from desorbing. Collingset al. 75 showed, for instance, that molecules like CO andCO2 can become trapped in an ice mantle which consists pre-dominantly of water ice since the desorption of water occursat much higher temperatures than for CO and CO2. How-ever, at the long timescales available in the ISM, some thesetrapped species might be able to escape because of a segre-gation process where the two main fractions of the mantleslowly separate.76 This process depends on a large numberof parameters including surface temperature, ice composi-tion and mixing ratio. The effect of trapping due to layer-ing can be easily simulated using microscopic KMC simula-tions, segregation is harder to model, since the exact mecha-nism is not yet understood.76

4.2.3 Diffusion rates

TPD experiments can also be applied to obtain informationon diffusion barriers. This information is however indirectand the result is rather sensitive to the formation mecha-nism/reaction order that is implicitly assumed in the modelused for fitting. For example, Katz et al. 56 , Cazaux andTielens 77 and Cuppen and Herbst 78 were able to fit thesame experimental results from Refs. 53–55 applying dif-ferent models. Katz et al. had a homogeneous model withone binding site, Cazaux and Tielens applied a binary modelwith both physisorption and chemisorption binding sites, aswell as quantum mechanical tunneling and thermal hoppingbetween sites. Cuppen and Herbst introduced surface rough-ness in their model.

Recently, TPD experiments were applied to obtain the dif-fusion barrier of D atoms on porous amorphous water.79

The barrier of 22 ± 2 meV is in agreement with earlier ex-perimental80 and theoretically results.58 The determinationof diffusion barrier remains however experimentally chal-lenging and is one of the largest reasons of uncertainty inthe modeling of surface chemistry in general. Experimen-tal information about site-specific barriers is practically non-existent. Computational efforts to obtain more insight havebeen reported.81–83 A few of these studies applied KMC toobtain diffusion rates and these will be discussed in Sec-tion 7.

A requirement for every rate is that it should fulfill macro-scopic reversibility. In equilibrium, the master equationshould result in a steady state

∑x f

(k(xi|x f )Peq(x f )− k(x f |xi)Peq(xi)

)= 0, (29)

where k(x f |xi) denotes the equilibrium rates, which are equalthe transition probabilities times some unit of time. The eas-iest way to fulfill this equation is to require detailed balanceor microscopic reversibility, i.e., the net flux between every

11

pair of states is zero

k(xi|x f )

k(x f |xi)=

Peq(xi)

Peq(x f ). (30)

For a canonical ensemble

Peq(xi)

Peq(x f )= exp

(−

E(xi)−E(x f )

kT

). (31)

For KMC models with homogeneous diffusion and desorp-tion rates, this requirement is met naturally. Only when dif-ferent types of binding sites are introduced, one should becareful to check whether especially the diffusion rates obeythis detailed balance criterion. Not for all models with in-homogeneous binding this requirement is met. For instance,the model with the distribution of diffusion barriers in Ref.49 or the work by Dumont et al. 84 do not obey detailed bal-ance. For some models it not clear whether they follow mi-croscopic reversibility since the barriers are not mentionedin the paper explicitly.85,86 Fortunately, most models do andin general two types of expressions for the diffusion barri-ers have been applied. The first expression for the hoppingbarrier from site i to site f is

E i→ fdiff = Ehop +E i

bind−E0bind, (32)

where Ehop is an input parameter determining the diffusionbarrier on a flat surface, i.e., between sites of equal bindingenergy, E i

bind the binding energy at site i, and E0bind is some

reference binding energy, usually the binding energy of theabsorbate without lateral interactions. The second is

E i→ fdiff = Ehop +

12

(E f

bind−E ibind

). (33)

Both expressions hold microscopic reversibility. but the ex-act choice influences the final results. Figure Figure 7 plotsthe recombination efficiency for H2 formation. This is thefraction of H atoms that hits a grain and leaves again as H2.The top panel plots the results as presented in Ref. 78, wherethe four surfaces mentioned in the legend have different de-grees roughness, starting from completely flat (Surface a) toa very rough surface (Surface d). Here Eq. (32) is used todescribe the diffusion barrier. The bottom panel shows theresults for the other choice of diffusion barrier. The choiceof barrier clearly has an influence on the exact value of theefficiency, but more importantly it influences the exact tem-perature range in which H2 can be efficiently formed, withEq. (33) leading to a wider range.

Expression (32) was introduced by Cuppen and Herbst 78

to introduce hampered movement and desorption because ofsurface roughness. Eq. (33) was later introduced based ona study of hydrogen atom diffusion on graphite87 (see Fig-ure Figure 8). Here the barrier for diffusion was found to belinearly dependent on the energy difference between the ini-tial and the final state. For H on graphite, the binding energyof an atom is affected by the specific arrangement of otherhydrogen atoms chemisorbed in its vicinity. The same de-pendence of Ediff and ∆E is found for dimer structures, wheretwo H atoms are involved, and for tetramer structures with

four atoms. This very nice correlation is probably an effectof the isotropic nature of the system and possibly due to thestrong chemisorption interactions that are at play. For dif-fusion of physisorbed CO on a hexagonal water ice surface,no correlation was found as can be seen from the top panelof Figure Figure 9. Again the energy difference betweenthe initial and the final state is plotted versus the diffusionbarrier. These barriers are obtained in off-lattice KMC sim-ulations83 and will be discussed in more detail in Section 7.Even less correlation is found for diffusion on an amorphouswater substrate (bottom panel Figure Figure 8 based on un-published results). Checks for the validity of Eq. (32) fordiffusion on interstellar ice, show a similar lack of structure.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50Temperature (K)

0

0.2

0.4

0.6

0.8

1

Surface aSurface bSurface cSurface d

Rec

ombi

natio

n ef

fici

ency

Figure 7: Molecular hydrogen recombination efficiency asa function of temperature on olivine substrates of differentsurface roughness. The top panel is reproduced from Figure6a in Ref. 78 with permission of Oxford University Press onbehalf of the Royal Astronomical Society. It uses Eq. (32) tocalculate the diffusion barriers. Bottom panel uses Eq. (33).

For both Eqs. (32) and (33), a choice for Ehop needs tobe made. This is often chosen as a fraction of the bindingenergy, because of lack of information on the exact barrier.This fraction is preferably chosen to be rather high, since alow diffusion barrier slows down the simulations substan-tially. In that case, the diffusion timescale becomes verysmall as compared to, for instance, accretion or desorptionrates and the simulation spends most of its CPU time on dif-fusion steps that do not lead to a fast evolution of simulatedtime. On first instance, one would expect the diffusion rateto play a crucial role in the overall evolution of the systems.This depends however strongly on the circumstances. ForH-atom bombardment of a CO ice under laboratory condi-tions the effect was found to be rather small68 since a bar-rier exist for the reaction between CO and H. A faster dif-

12

-1 0 1∆E (eV)

0

0.5

1

1.5

2

Edi

ff (eV

)

dimerstetramers0.5 ∆E + 1.04

Figure 8: The diffusion barrier for chemisorbed H atomson graphite as a function of the energy difference betweenthe initial and final configuration. The diffusion barriers be-tween dimer and the tetramer configurations follow the samelinear dependences (Eq. (33)). Figure reproduced with per-mission from Ref. 87. Copyright 2008 AIP Publishing LLC.

Ice 1h

0

30

60

90

120

150

180

-100 -50 0 50 100

ΔE (meV)

Edi

ff (m

eV)

Amorphous

0

30

60

90

120

150

180

-100 -50 0 50 100

Figure 9: Similar to Figure Figure 8 for CO diffusion oncrystalline (top panel) and amorphous (bottom) water ice.The black solid line indicates the minimum barrier possi-ble to ensure microscopic reversibility (∆E). Top panel con-structed on the basis of data produced for Ref. 83.

fusion rate leads to more frequent meetings between H andCO, but per encounter the time to react is shorter with re-spect of lower diffusion rates. Under interstellar conditionsthe effect of the diffusion/desorption ratio on the CH3OHformation by CO hydrogenation was found to be somewhatstronger. Chang and Herbst 88 developed a macroscopic gas-microscopic grain Monte Carlo model which treats a moder-ately complex grain surface chemical network that leads tothe formation of the stable molecules CH4, CO2, CH3OH,and H2O. They found that the diffusion ratio has a mod-erate influence on the abundance of most surface species;CO and CH3OH were most affected. The main differencebetween their simulations and the laboratory simulations,68

apart from the very different fluxes, is that CO is a minorityspecies in the ice. The laboratory experiments, on the otherhand, start with a pristine CO ice surface and only after sometime when H2CO and CH3OH become CO and H encountersmore rare. This difference in CO distribution could explainthe difference in sensitivity towards the height of the diffu-sion barrier between the laboratory and ISM simulations.

But even in simulations of the hydrogenation of pure ices,diffusion rates can have a significant effect on the final yieldof many of the species, as is the case for O2 hydrogena-tion under laboratory conditions.89 For this particular sys-tem, bulk diffusion is crucial in determining the final yield.90

Simulating bulk diffusion using lattice-gas KMC simulationsis however hard, since the exact mechanism is unclear. Thediscrepancy between simulations and experiments both inthe case of CO68 and O2

89 hydrogenation is probably mainlydue to a missing bulk diffusion mechanism. KMC simula-tions on segregation of ice mixtures, a direct measure of bulkdiffusion, showed that experimental results can only be qual-itatively reproduced if both surface diffusion and bulk diffu-sion by swapping of molecules between sites is included.76

The exact molecular mechanism behind this process is un-clear though.

4.2.4 Theoretical determination of rates

Results from theoretical calculations are also often used as asource of rate constants, especially when experimental datais scarcely available. Binding energies can be determined ina relative straightforward way using different levels of the-ory. The Density Functional Theory calculations mentionedin Section 5.1 are an example. Wave packet propagation andMolecular Dynamics techniques can be applied to determinerates or branching ratios. A review of these applications ishowever beyond the scope of this review. Here, we will focuson reaction path finding methods since these are – in combi-nation with Transition State Theory (TST) – frequently ap-plied to determine rates for KMC simulations. With the useof TST it is possible to calculate process rates if a suitablediving surface (the transition state) between the initial andthe final state of the process is known. The identification ofthis transition state is the crucial step in the procedure andcan be handled using a variety of techniques.91 In the often-used classical harmonic approximation to TST, the transitionstate is simply a first order saddle point on the potential en-ergy surface. In this case, the Nudged Elastic Band (NEB)

13

method92 is one of the most applied methods for locatingthe transition state, if the initial and final state of the reactionare known. The NEB algorithm can determine the minimumenergy path between the two minima on the PES. A NEBcalculation starts from a sequence of images of the system,initially placed along some pathway between the two min-ima, usually just along a straight line connecting the two.Every image is then connected to its two neighboring sys-tems by fictitious springs and the energy of this constrainedsystem is then minimized using any minimization algorithm.Analysis of the coordinates and energies of the highest en-ergy image, after minimization, then gives an estimate of theprocess energy barrier and the transition state itself. With theimproved, Climbing Image NEB93 (CI-NEB), algorithm, itis also possible to find the location of the saddle point witharbitrary precision.

Within an astrochemical context, the NEB method has forexample been used to determine the diffusion rate of atomichydrogen in a CO ice environment.68 These rates were thenused as input in microscopic KMC simulations. One of themajor advantages of using theoretical calculations to obtainrate constants is that the environment can be precisely con-trolled, which can greatly add to the physical accuracy ofthe KMC simulations. Fuchs et al. 68 for example found astrong dependence of the barrier for a CO and H to swap onthe depth from the surface where the process occurs.

The determination of diffusion barriers is a typical casewhere NEB calculations can be of use because, as said be-fore, experimental data for these processes is often hard toobtain. An example where this has been employed is thework by Batista and Jónsson 81 . Here diffusion rates for H2Oon hexagonal ice were determined using NEB and subse-quently included in a lattice-gas KMC model to determinethe diffusion constant which is in agreement with the experi-mental diffusivity at the same temperature. The NEB methodcan in principle be used for any kind of process however, in-cluding chemical reactions. The only prerequisites being thatthe initial and final states of the process are known and thatan adequate theory is available to describe interactions. Inthis spirit, the studies by Goumans et al. 94,95 are certainlyworth mentioning. They used an embedded cluster methodto study the reaction pathways of H2O and H2 formation onolivine surfaces using the CI-NEB method. Also here, theatomistically detailed information about the environment ofthe reaction revealed important aspects of the reaction path-way.

Summarizing, the NEB method is a powerful theoreticaltool which can be of great value when creating a rate tablefor KMC simulations. Also when accurate experimental datais available, NEB calculations can often still reveal importantaspects of the processes under consideration which can thenalso be included in the KMC simulation.

4.3 Kinetic Monte Carlo algorithmsThe theory behind KMC can be implemented into an algo-rithm in several ways. Here, we will discuss a few algorithmsthat are most commonly used for microscopic simulationsin astrochemistry. Figure Figure 10 shows a flow diagram

Start Input rates

Initialize surfaceTime=0, M(i,j)=0

N=0

Calculate rates:DesorptionDiffusion

Deposition

Chooseevent

Desorption or diffusion

Deposition

Choose siteChoose site

Perform event

Calculate Δt = -ln(Z)/ktot

Time = Time+Δt

Stop

Finished?

Sample?

Take averages

Write output:snapshot

surface abunanceaverages

no

yes

no

yes

Easy to implement for homogeneous

rates, sinceRdes = N kdes

Rdif = N khop

and not all individualevents need to be

considered

Figure 10: Flow diagram of the KMC algorithm adaptedfrom Ref. 96 to simulate grain surface chemistry with per-mission Copyright 1992 American Institute of Physics.

of the KMC algorithm as proposed by Nordmeyer and Za-era 96,97 which is based on the original paper by Gillespie 42 .This algorithm is mainly developed to study the kinetics ofgas adsorption on solid surfaces and it is used to simulatehomogeneous surfaces, in other words the desorption anddiffusion rates are site independent and also only one typeof species is considered. The algorithm starts by inputing allrates, i.e., creating the table of events. Then, the surface isinitialized by creating an empty matrix, M, where each en-try represents a site and the occupancy of this site can bechanged by changing the number. In the next step the des-orption, diffusion and deposition rate are calculated. Since ahomogeneous surface is assumed this corresponds to

Rdesorb = Nkdesorb, (34)Rdiff = Nkhop, (35)Rdep = Nsiteskacc, (36)ktot = Rdesorb +Rdiff +Rdep, (37)

where N is the number of species on the grain, Nsites the num-ber of sites (the size of matrix M) and kx the rate for an indi-vidual atom or molecule to undergo process x. Then one ofthe three possible events is chosen by picking a random num-ber between 0 and ktot, a second random number is picked tochoose the site where the event will occur and a third oneto increment the simulated time according to Eq. (23), andif the termination criterion is met, the simulation will finish;otherwise the loop is entered again.

The termination criterion can be in terms of the numberof iterations or of a certain final time. In astrochemistrythe time steps are often very inhomogeneous: there can bea long time span between two depositions but once, for in-stance, a hydrogen atom lands on the surface it can make

14

Start Input rates

Initialize surfaceTime=0, M(i,j)=0

N=0

Calculate rates for all individual processes:

ktot = ΣiNΣj

p kij

Chooseevent +

site

Perform event

Calculate Δt = -ln(Z)/ktot

Time = Time+Δt

Stop

Finished?

Sample?

Take averages

Write output:snapshot

surface abunanceaverages

no

yes

no

yes

Rates can be site specific.

Update by taking changes in table of

events.Changes occur

at changed positionand its neighbours

Figure 11: Flow diagram of the n-fold way KMC algorithmderived from Ref. 98 and adjusted to simulate grain surfacechemistry.

many fast hops before it reacts or desorbs. Setting a ter-mination time is therefore a better criterion than simply thenumber of iterations, although this is traditionally more com-monly used in for instance catalysis. The same applies to thesampling criterion, which is checked of the end of every loopin Figure Figure 10. One would like to take some averages,snapshots or write the current surface abundance at regularintervals. These intervals can again be defined in terms ofsimulation time or number of iterations. When taking aver-ages of, for instance, the average production rate under cer-tain conditions, one should be careful that steady state hasbeen reached and that the time between two samples is longenough for them to be independent.

Upon reentering the simulation loop, the following itera-tions simply start by recalculating the desorption, diffusionand deposition rates. For homogeneous surfaces this can bedone in a straightforward way and this step does not requiremuch effort. Extending the model to systems with morespecies, would already be more involved and if one wouldlike to treat inhomogeneous surfaces, smart updating rou-tines should be applied to keep the computational cost of thisstep limited. In that case, however separating the determina-tion of the overall event and the specific location becomesunnecessarily cumbersome and a different algorithm wouldbe preferred, for instance the commonly-used n-fold way al-gorithm.98

Figure Figure 11 shows a flow diagram based on the n-fold way KMC algorithm.98 The main difference betweenthis algorithm and the algorithm of Figure Figure 10 is thatthe event and the location are chosen in one step. This makesthis algorithm more appropriate for systems where the sur-face is inhomogeneous or where the adsorbed species inter-act which each other, since not all sites are equally likely tohave an event. Only two random numbers are needed per it-

Start Input rates

Initialize surfaceTime=0, M(i,j)=0

N=0, tevent=0

Determine first deposition timetdep = -ln(Z)/kacc

tdep < tevent?

yes

noSelect event species (i)

Choose site

Perform eventTime=Time+tevent

Calculate rates for species i anddetermine tevent

i

Stop

Finished?

Sample?

Take averages

Write output:snapshot

surface abunanceaverages

no

yes

no

yes

The rates of only one species is considered. Works well if species

do not interact.If species interact, tevent

i

of neighbours need to be adjusted as well

Determine tevent=min(tevent

i)

Perform deposition

Determine tdep=time-ln(Z)/kacc

Figure 12: Flow diagram of the continuous-time, random-walk KMC algorithm derived from Ref. 50 and adjusted tosimulate grain surface chemistry.

eration in this routine, but the determination of which eventwill occur is more computationally intense as well as de-termining the total rate, since the table of events is longer.Updating for the next iteration can be optimized by only re-calculating those entries in the table of events that refer tothe changed species or, in the case of interacting species, toits neighbors. The total rate can be determined by subtract-ing the old rates at these entries and adding the new ones.This works well for systems where the rates are rather ho-mogeneous. For systems where certain events can result inlarge changes of the rates, numerical errors can occur. Anexample would be a case where the reactivity changes dra-matically between sites. If an atom or molecule would hopup and down between this reactive site and a less reactivesite, large rates would be constantly added and subtracted,resulting in a numerical error of the same order as the rate ofthe slow process. In the selection process, the wrong eventcould then be selected.

These numerical issues do not occur in the continuous-time, random-walk algorithm.50 The first microscopic KMCsimulation in astrochemistry used this particular algorithmand its flow diagram is depicted in Figure Figure 12. Thisalgorithm is designed to describe non-interacting randomwalkers on a surface. The main difference between the previ-ous two algorithms is that a separate time line for each indi-vidual random walker is determined. The time increment istherefore not determined by the total rate for the whole sys-tem, but the total rate for an individual random walker. Theevent times for each walker are then compared, together withthe next deposition time, and the first event is then executed.Only one random number per iteration is needed; two in thecase of deposition. In this way when updating the systemfor the next iteration only the random walker which movedin the previous iteration has to be considered. This random

15

walker will get a new event time and will be placed in thespecies list which is sorted according to its event time. Thisalgorithm works very well for non-interacting species. Ob-viously when species interact, a move of one random walkercan effect the total rate of a neighboring random walker and,in turn, its event time. A good example would be when astationary species, that has an event time well above the ter-mination time of the simulation, gets a mobile neighbor withwhich it can undergo reaction. The event time of the station-ary species changes drastically. A way to adjust for this is tostore for each species, the random number Zi that was drawnto determine its initial event time, its total rate, and the timeat which its last event occurred t i

event,old. The adjusted eventtime, t i

event,adjust can than be calculated using

− ln(Zi) =kitot,old(tcurrent− t i

event,old)

+ kitot,adjust(t

ievent,adjust− tcurrent). (38)

Again numerical errors can occur in adjusting this eventtime, if large time differences are subtracted. In this case,it is best to determine the time according to usual schemesince the first term will probably be small any way.

Algorithms, that simulate the master equation in an ap-proximate manner, have been suggested as well with the aimof speeding up the simulation. One such suggestion is bysplitting by physical processes where each of the basic pro-cesses – deposition, diffusion, reaction and desorption – isconsidered as an independent Markov chain.99 This algo-rithm was tested for H2 formation on interstellar grains andsimilar results were obtained as in Refs. 30,49. Exact timingdifferences with other algorithms are not given; it is there-fore unclear how large the final speed up will be.

4.4 Simulations with varying ratesTemperature Programmed Desorption is one of the maintechniques within solid state laboratory astrophysics. Assuch it is interesting to simulate these experiments usingKMC. There are also advantages of using KMC over thePolanyi-Wigner equation (Eq. (28)) for simulating TPDspectra. First, the order of desorption might change dur-ing the experiment, for instance if one goes from multilayerto monolayer desorption. KMC will automatically take thisinto account. Second, there might be lateral interactions be-tween the adsorbates, either repulsive or attractive, and thismight result in non-linear effects that the Polanyi-Wignerequation cannot account for, but which can be included ina KMC model. Finally, the Polanyi-Wigner equation pro-vides an empirical description of the data and does not givereal microscopic insight into the mechanism behind the des-orption.

As described in Section 4, the two main equations onwhich KMC algorithms rely are Eqs. (19) and (21). Whilethese are relatively straightforward for stationary rates, theybecome more complex in the case of a TPD experimentswhere the changing surface temperature causes the rates tochange in time. The good news is however that we knowin advance how the temperature will change. For TPD ex-periments this is usually T = T0 + β t where T0 and β are

constants. Jansen 45 has described a way to account for thistime dependence and to obtain values for ∆t that follow thecorrect distribution. Both Eqs. (19) and (21) have the term∫ tk

tk−1

k(t ′)dt ′. (39)

For thermal rates and assuming a linear increase in tempera-ture this becomes∫ tk

tk−1

ν exp(− E

k(T0 +β t ′)

)dt ′ = Ω(tk)−Ω(tk−1) (40)

with

Ω(t ′) =ν

β(T0 +β t ′)E2

[E

k(T0 +β t ′)

]. (41)

Here E2 is an exponential integral.100 Since tk−1 is known,it is the current time, and β and T0 are experimental parame-ters, i.e., the heating ramp and the initial deposition temper-ature, respectively. A value for tk can be obtained by solving

Z = exp(−Ω(tk)+Ω(tk−1)). (42)

This effectively means finding the root of

Ω(tk)−Ω(tk−1)+ ln(Z). (43)

This can be most conveniently done with the Newton-Raphson method.101 Since Ω and dΩ/dt are both monoton-ically increasing, the method always succeeds.

The derivation here used a linear increase of the surfacetemperature. Different time dependences are also possi-ble. The numerical effort depends on the integration of∫ t

0 ν exp(− E

kT (t)

)dt ′. One such application could be the

thermalization of species upon reaction or deposition. Herethe surface temperature does not really change, but one couldconsider that each species has its own internal “temperature”which starts at some high value and then decreases until itreaches the surface temperature. Cuppen and Hornekær 87

applied this approach to account for the thermalization of Hatoms landing on graphite. A discussion of the results ofthese simulations is provided in the next section. In the ex-periments that they aimed to reproduce, the H atoms arrive atthe graphite surface at a temperature around 2000 K, whichis much higher than the surface temperature. Furthermore,the atoms will gain energy due to the high binding energy ifthey chemisorb. The atoms will not be thermalized instan-taneously, but will most likely gradually lose their energy tothe substrate. The dissipation of the excess energy into thesubstrate is expected to be exponential.102 When using anexponential function, the expression for Ω in Eq. (43) be-comes rather complex and computationally expensive, thetime dependence of the energy loss was therefore emulatedby a simpler expression. The following function was used todescribe the temperature

T ′(t) = max

(Ts,

Tstart

(1+B(t− ta))2

)(44)

where Ts and Tstart are the surface and starting temperature

16

respectively, ta is the time at which the atom has adsorbed onthe surface, and B is some thermalization parameter. A valuefor tk can now be obtained using

Ω(t ′) =ν

2B

√πTstart

Eerf((1+B(t ′− ta))

√E

Tstart

)(45)

with erf being the error function. Several functions of differ-ent order in t or with different functional form were tested,but this appeared to have a minor effect on the results. Forhigher orders in t solving Ω becomes more computationallyexpensive.

5 Simulations of experimentsAn advantage of KMC over, for instance, Molecular Dynam-ics, is the relatively long timescales that can be simulated.Laboratory surface science experiments of several hours canbe easily reached with the possibility to include atomisticdetail in the simulations. For this reason, KMC is oftenapplied in surface science and catalysis to simulate experi-mental conditions and determine experimentally measurablequantities. These can then be directly compared to experi-ments. The previous section made very clear that lattice-gasKMC is far from being an ab-initio method, in the sense thatall input parameters need to be given. This can also be anadvantage however, since the effect of these parameters onthe output can be tested and different mechanisms can beswitched on and off by changing these parameters. By test-ing different mechanistic scenarios in this way and compar-ing the results to experiments, insight into the mechanismscan be obtained. This section will discuss the application ofthe KMC technique to gain more insight into experimentalobservations. Since this is a huge area of research with manydifferent applications, we will limit ourselves to the simu-lation of experiments that are astrochemically relevant. Allsimulations discussed in this section will either use the KMCtechnique to verify the experimental mechanism or to param-eterize the KMC model using experimental constraints. Inthe next section, we will discuss KMC simulations under as-trochemical conditions: low fluxes and long timescales. Thesimulations in both sections will generally involve the samesystems: H2 formation on different substrates or ice chem-istry, but in Section 6 the model and mechanism are consid-ered a given and are not tested against experiments; they arerather applied to predict abundances in the ISM.

5.1 H on graphiteThe formation of the most abundant molecule in the Uni-verse, H2, occurs mainly through surface reactions and assuch it received considerable attention. These reactions havebeen the topic of several astrochemical modeling studies asmentioned in previous sections, but also of experiments andquantum chemical calculations. A model system for H2 for-mation on carbonaceous grains is the deposition of H atomson a graphite surface or a graphene layer. Figure Figure 13shows a schematic representation of the honeycomb lattice

p

o

Figure 13: A schematic representation of the honeycomb lat-tice of a graphene layer with a hydrogen atom attached (graycircle). Stable dimer configurations can be formed by attach-ment of a second hydrogen atom at the o(rtho)- or p(ara)-position.

of a graphene layer. Hydrogen atoms can be chemically at-tached to the carbon atoms in this hexagonal lattice. Thegray circle represents such an atom. Molecular hydrogen canbe formed by a process called abstraction where the atomchemisorbed to the surface reacts with another H atom toform H2. Hydrogen abstraction from graphite has been stud-ied extensively both experimentally103,104 and by DensityFunctional Theory (DFT) and Quantum Wave packet calcu-lations.105–110

Experimental111,112 and theoretical111,113–116 studiesshow that hydrogen atoms preferentially adsorb in clus-ters on graphite. Particularly preferred clusters are so-calledortho- and para-dimers, where two hydrogen atoms adsorbto carbon atoms that are within the same hexagonal ring inthe graphite structure at the ortho- and para-position, indi-cated in Figure Figure 13 by an o and a p, respectively. DFTcalculations show that not only the binding energy of thesedimers is higher than for two individual monomers, but alsothat the sticking into dimer configurations is more favorable.More complex structures consisting of three or more hydro-gen atoms are found to be favorable as well.115,117 A recentstudy has applied ab-initio Molecular Dynamics to study thecompetition between the Eley-Rideal abstraction reactionand dimer formation.118

Performing these type of competition calculations at ahigh level of theory is however computationally demanding,and could therefore only recently be investigated. Typically,either none or only a single C atom on the simulated graphitesurface is allowed to relax during the reaction, which can ar-tificially close other low barrier reaction channels. Anotherapproach is to apply Kinetic Monte Carlo models to simulatethe experimental measurements and to connect the atomisticresults from the quantum calculations to the more macro-scopic experimental observations. Cuppen and Hornekærhave used such a hybrid approach by using binding energiesand barriers derived from DFT and wavepacket calculations

17

as input parameters for Monte Carlo simulation and allowingdifferent mechanisms for H2 formation. Independent of theexact values of the input parameters they found that certainmechanisms could better explain the experiments than oth-ers. Fast diffusing physisorbed atoms119 could explain thehigh percentage of H atoms present on the surface as part ofa cluster (dimer, trimer etc.)115 The fast moving physisorbedatoms can scan a large part of the surface, even consider-ing their low binding energy, and find chemisorbed atoms toform a cluster. Some excess energy will probably be releasedin the binding process of this atom. The experimental data ofZecho et al. 103 could be explained by an abstraction mecha-nism in which part of this excess energy is used to overcomethe barrier for abstraction.

The combination of these mechanisms together with theparameter settings that reproduced the experiments best,were subsequently used by Gavardi et al. 120 to study thedesorption behavior of H2 from graphite, testing the KMCsimulations against experimental Temperature ProgrammedDesorption curves.104 Again the various clustered hydrogensurface structures were taken into account, using binding en-ergies based on DFT calculations and accompanying esti-mations. Experimentally two desorption peaks are observedas well as a shoulder on the trailing edge of the first peak.The top panel of Figure Figure 14 plots the simulated TPDspectrum at a coverage of 0.1 ML and shows to differentcontributions to the TPD spectrum. This graph is a differentrepresentation of the data from Ref. 120 to allow for bettercomparison with Ref. 84. The main graph shows the cov-erage loss of the different configurations in time, whereasthe inset shows the desorption of H2 and the configurationsthat are directly responsible for this. From the simulationsit becomes apparent that the first peak arises from H2 des-orption from the para-dimer or larger clusters containing apara-configuration as a substructure. The second peak wasattributed to the two-step diffusion of hydrogen atoms in anortho-dimer into para-dimer. This para-dimer can then des-orb again in the form of H2, resulting in the second peak.More complex structures were found to be responsible forthe third peak, experimentally observed at high coverage.120

Gavardi et al. 120 also recognized the possible existence ofa directional bias for the diffusion of physisorbed H atoms.Due to their fast mobility, the Markov chain assumption ofmemory-loss is no longer valid. It is in fact more likelythat an atom proceeds in the same direction for some time.Therefore, the six possible directions of diffusion on the hon-eycomb structure were assigned different weights with thehighest probability for the atom to continue in its originaldirection. This directional bias was found to have very lit-tle effect on the final results. Only when no deviations froma straight line were allowed, was a lower H2 formation rateobtained.

Dumont et al. 84 performed a similar study in which theysimulated the TPD experiment by Zecho et al. 104 using akinetic Monte Carlo algorithm. Although from their letterit is not clear how they accounted for the temperature de-pendence of the rates. Their explanation for the two TPDpeaks agrees with Gavardi et al., despite some differencesin implementation of the H-graphite system in terms of al-

400 500

400 5000

2×10-4

4×10-4

tetra. 1tetra. 2tetra. 3tetra. 4tri. 9p-dimero-dimertotal

cove

rage

loss

rat

e (M

L/s

)Temperature (K)

H2 d

esor

ptio

n ra

te

temperature (K)

400 450 500 550 600 650

coverageloss(MLs-1 )

0.00030totalorthopairsparapairs

0.00025

0.00020

0.00015

0.00010

0.00005

0.00000

Figure 14: KMC TPD simulations of coverage loss rate fromgraphite as determined by Gavardi et al. 120 (top) and Du-mont et al. 84 (bottom). The inset in the top panel indicatesthe H2 desorption rate and the different contributions. Bot-tom panel is reproduced with permission from Ref. 84 Copy-right 2008 American Physical Society.

18

gorithm. Dumont et al. allowed absorption, desorption anddimerization but they did not include the formation of clus-ters larger than two atoms and diffusion of physisorbed Hatoms on the surface. The latter was found to be a domi-nant mechanism for the formation of clusters in the study byCuppen and Hornekær. Consequently, the surface snapshotspresented by the Dumont group show only dimer configu-rations. Their results are reproduced in the bottom panelof Figure Figure 14. The first peak is again attributed topara-dimer desorption, whereas the second peak is due tothe disappearance of the ortho-dimer. The latter can proceedthrough different routes: through the evaporation of individ-ual H atoms in the dimer and through hydrogen diffusioninto different dimer structures.

5.2 Ice chemistryIn dark regions of the interstellar medium ices have beenabundantly observed. The name “ices” in astrochemistry isnot only reserved for water ice, but for all solids that con-sist of volatiles (molecules that are liquid or gaseous at roomtemperature). Since frozen molecules are not isolated whenthey react with each other, gas-phase reaction dynamics can-not be directly extended to the solid phase; although the ini-tially proposed surface reaction networks were largely con-structed based on analogies with gas phase data.7 In the pastfew decades, coinciding with improvements in (Ultra) HighVacuum techniques, more and more ice processes have beenstudied experimentally, to test and improve the postulatedgrain surface reaction networks. These include both ener-getic processing initiated by Hagen et al. 121 as well as lower-energetic processes such as surface formation of interstellarrelevant molecules by atomic addition pioneered by Hiraokaet al. 122 . To translate the experimental results into reactionrates or barriers, however, is far from trivial. The formationof new species is the result of an interplay between diffu-sion, desorption and reaction. This is where KMC comesinto play as it can explicitly take the microscopic structureinto account and reactions and diffusion can be evaluated intime.

In the field of experimental ice chemistry, a large varietyof processes can be explored. Here we discuss only ther-mal processing and atom addition reactions since they haveresulted in follow-up studies involving KMC. Thermal pro-cessing consists of probing a reaction and/or diffusive behav-ior that is temperature driven without other energetic input.Experimentally, there are two types of atomic addition ex-periments: pre-deposition and co-deposition. For the formertype, the ice under investigation is grown to obtain the de-sired number of monolayers (ML). Subsequently, the ice canbe bombarded with atoms. This is beneficial for studyingstable final products. For a co-deposition experiment, how-ever, all species are allowed to freeze-out simultaneously andcan therefore be isolated in the ice. Hence, it is possible toinspect also radicals and intermediate species.

Let us first consider the thermally induced process of seg-regation in mixed H2O and CO2 ices, where the key parame-ters are bulk and surface diffusion and the difference in bind-ing energy between a water-rich phase and a carbon-dioxide-

rich phase. The latter drives the segregation process; the firstdetermines the rate by which this occurs. Astronomical ob-servations show the presence of pure CO2 ice in protostellarenvelopes whereas CO2 is predominantly mixed with H2Oice in the precursor clouds.123 This has been attributed tothe higher temperatures in protostellar envelopes that wouldallow segregation of H2O and CO2 ice mixtures to occur.In order to be able to use the fraction of pure CO2 ice asa probe for the thermal history towards a protostar, mecha-nistic information about how this process occurs as well asquantitative information on the barriers involved are neces-sary. This motivated a combined experimental and computa-tional study, where three sets of experiments concerning thesegregation in H2O:CO2 and H2O:CO mixtures under differ-ent vacuum conditions were complemented by KMC simu-lations.76 The experimental data could be fitted with two ex-ponential functions, suggesting the existence of two first or-der mechanisms, a fast and a slow one. Subsequently, KMCwas applied to test different segregation mechanisms. Thelattice-gas model included desorption, hopping and swap-ping. The interactions between molecular species of simi-lar kind were set to be higher than the interactions betweenH2O and CO2, partially corresponding to values found inTPD experiments. Hopping could occur from site to site andthe rate followed Eq. (33). Finally, swapping is a process inwhich molecules exchange position, accounting for the bulkdiffusion. Also for this, Eq. (33) was applied but with a bar-rier Eswap instead of Ehop, much higher (factor ∼3-7) thanfor surface diffusion. It was found that hopping is responsi-ble for segregation at the surface, which is described by anexponential function and reaches steady state. Swapping re-sults in some surface segregation which is slow and does notreach steady state (at the simulated timescales). When bothmechanisms are included, the experimental temperature de-pendence of the segregation rate is correctly reproduced aswell as the fact that at higher temperatures, the bulk segre-gation seemed to play a larger role. In a following paper, therole of segregation was further investigate through TPD stud-ies of the desorption of mixed ices, where segregation deter-mines the amount of more volatile species to be trapped. Athree-phase (gas–ice mantle–bulk ice) rate equation modelwas constructed with which good experimental agreementwas obtained.124

Vasyunin and Herbst 125 used these TPD experimentsto calibrate their multilayer macroscopic KMC modelMONACO. In this model only the outer most layers arechemical active, i.e., chemical reactions and desorption canonly occur from these layers. The exact amount of layerswhere these processes can occur, is determined by compar-ing simulations and the TPD experiments in Ref. 124. Theyfound that MONACO treats zeroth- and first-order desorp-tion correctly and further concluded that the experimentalcurves are well reproduced assuming four monolayers tobe chemically active. This number is probably larger thanin reality since the authors did not allow bulk diffusion tooccur.

Examples of surface reaction experiments that are mod-

19

eled using KMC are the hydrogenation of solid CO

CO +H−−→ HCO +H−−→ H2CO +H−−→ H3CO +H−−→ CH3OH (46)

and the hydrogenation of molecular oxygen

O2+H−−→ HO2

+H−−→ H2O2+2H−−−→ H2O. (47)

The CO hydrogenation scheme was the first ice reac-tion route to be studied extensively by a number ofgroups68,126,127 and leads to the formation of methanolwhich is an important precursor molecule for more com-plex organics.128,129 The latter route has also been studiedextensively experimentally,79,90,130–133 but the exact forma-tion route was found to be more involved than the simplethree-step process mentioned here.133

For the KMC modeling of the experimental CO hydro-genation temperature-dependent reaction rates for the firstand third reaction steps were used a fitting parameters tofit the experiments.68 Quantum chemical calculations haveshown that these two reaction steps posses a barrier for re-action134 and that these barriers can probably be crossedthrough quantum tunneling.135 The temperature dependenceof the rates found in the KMC study was indeed in agreementwith this. Experimentally, surface abundances of CO, H2COand CH3OH were followed in time by means of RAIRS ina sequential experiment: CO was first deposited and subse-quently hydrogenated for three hours. Good agreement wasobtained between KMC simulations and experiments for lowsurface temperatures where only the top few monolayers ofthe deposited CO ice was chemically altered as can be seenin the top left panel of Figure Figure 15. At higher temper-atures, the hydrogen atoms were found to able to penetratethe CO ice deeper, but this penetration effect was not fully re-produced by the KMC simulations, which resulted in a lowerfinal yield in the simulations. The initial reaction rates of thepristine CO ice, when penetration does not play a role yet,was however in good agreement with experiment.

Considering simulations of water formation under exper-imental conditions, Lamberts et al. 89 used the reaction net-work proposed in Ref. 133 to reproduce the sequential andco-deposition experiments reported in Refs. 90,133. Sincethis reaction network contains many more reactions than inthe case of CO hydrogenation, constraining the rates of thesereactions proved much harder. They therefore performed asensitivity analysis to understand which reactions had thelargest effect on the experimental results and how well thesereactions could be constrained. A reasonable agreement be-tween KMC simulations and the different types of experi-ments was obtained. Experimentally it was found that hy-drogen atoms can penetrate into tens of monolayers of O2,unlike the case of CO where only a maximum of four mono-layers is affected. Again this penetration mechanism provedhard to reproduce in the KMC simulations. Only a fast Hdiffusion combined with access to interstitial sites was capa-ble of reproducing some kind of penetration. The agreementwith the co-deposition experiments where O2 is constantlydeposited, was much better. For astrochemical purposes,these co-deposition experiments are in closer agreement with

0 50 100 150

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Figure 15: Time evolution of the surface abundance of CO,H2CO and CH3OH during H-atom bombardment of a pre-deposited CO ice with at four different surface temperatures.Experimental data (symbols) and Monte Carlo simulationresults (solid lines) are shown. Reproduced from Ref. 68.Credit: Fuchs, Cuppen, Ioppolo, Romanzin, Bisschop, An-dersson, van Dishoeck, Linnartz, Astron. Astrophys., 505,629, 2009 reproduced with permission c© ESO

20

the conditions in the interstellar medium where all speciesland on the grains simultaneously and penetration effects areof less importance. However, the diffusion processes cover-ing the penetration of reactive species into the ice can be verysimilar to the ones that drive segregation of ices, discussedearlier, and as such they are interesting to study.

One aspect of diffusion mechanism that is missing in bothatomic addition studies, mentioned above, is the directional-ity of the movement of hot species upon formation. If hot re-action products are formed in the ice with some momentum,the Markov chain assumption breaks down which results inan effective directionality of diffusion. Gavardi et al. 120 dis-cussed something similar for the diffusion of physisorbed Hatoms on a graphite surface as mentioned in Section 5.1. Anexample for ice chemistry is the formation of two OH radi-cals from H + HO2. The two fragments are expected to moveaway from each other and therefore, recombination is lesslikely. We refer for a more in-depth discussion to Ref. 89.Currently, this directionality is not included in the models.All ice KMC simulations, reactive and non-reactive, haveshown that bulk diffusion is important in reproducing the ex-perimental results, but that this is still an ill-understood pro-cess. Whether directionality or swapping are good handleson this complicated process is yet to be determined by futuretheoretical and experimental studies.

6 Applications of the KMC techniqueto astrochemistry

The previous section discussed KMC simulations which aimat modeling (experimentally observed) behavior in order toobtain relevant parameters, e.g., activation barriers. In thissection, Kinetic Monte Carlo simulations under astrochemi-cally relevant conditions will be passed in review. Both H2formation in the monolayer regime and ice mantle growthand accompanying chemistry will be discussed. Typical as-trochemical environments, e.g., dark or diffuse clouds, aresimulated to predict or reproduce molecular abundances.

6.1 H2 formationHydrogen is the most abundant element and molecularhydrogen the most abundant molecule in the InterstellarMedium (ISM). H2 is largely responsible for shielding ofthe denser regions from the interstellar radiation field, thusallowing more complex molecules to survive. It further-more plays an important role in the cooling of the interstellarmedium. Molecular hydrogen is clearly a critical moleculein astrochemistry and its formation routes have thereforebeen intensely studied. Its most obvious gas phase formationroute, H+H−−→ H2, is not efficient under interstellar condi-tions, since there is no third body to act as a heat sink and ra-diative association should occur through a forbidden transi-tion.136 Because of the high observed abundances of H2, for-mation must be driven by another process, such as formationon grain surfaces.4 H2 is assumed to be formed through theLangmuir-Hinshelwood mechanism where two physisorbedhydrogen atoms scan the surface through thermal hopping or

tunneling. Molecular hydrogen can be formed barrierlesslywhen the two atoms reach the same site. Commonly it is as-sumed that the new hydrogen molecule immediately desorbsback into the gas phase. Although this mechanism was al-ready postulated in 1949, laboratory experiments were onlyperformed at the end of the 20th century.53–55 Although themechanism works at 10 K and lower temperatures, a modelbased on these experimental results showed that the forma-tion efficiency was too low to account for the observed abun-dances at typical grain temperatures of roughly 20 K in dif-fuse clouds.56 Most of the H2 formation studies are devotedto understand this discrepancy. Inhomogeneity in terms ofbinding sites was introduced to increase the temperaturerange in which H2 can be formed towards higher tempera-tures. This was initially done by introducing chemisorptionsites using a rate equation model,77 but since microscopicKMC is a more obvious way of introducing site-specific in-formation, later mostly this technique was used.

6.1.1 The effect of inhomogeneity

The absence of substructure in the observed 9.7 µm bandleads to believe that interstellar silicates are amorphousrather than crystalline44 and Refs. therein. As such, it is unlikelythat the adsorbate behavior on these grains can be modeledby a single diffusion barrier and binding energy. Inhomo-geneous surfaces can be modeled in different ways. Herewe will discuss four implementations of surface inhomo-geneity. The first is due to amorphicity. Amorphous sur-faces have binding energies and barriers that can be thoughtof as following a (yet unknown) distribution, with slightlychanging barriers depending on the small differences in lo-cal environment due to the lack of long range order. Changet al. 49 studied the Langmuir-Hinshelwood mechanism forphysisorbed hydrogen atoms on (i) flat, homogeneous car-bonaceous and olivine surfaces with a single binding en-ergy and diffusion barrier, (ii) ‘inhomogeneous surfaces’ ofamorphous carbon and olivine in which binding energiesand diffusion barriers were taken as continuous distributions(Gaussian and exponential) and (iii) ‘mixed’ grains, con-sisting of both amorphous carbon and olivine. In agree-ment with earlier studies,30,56 perfectly flat homogeneoussurfaces (both olivine and carbon) result in narrow temper-ature ranges within which H2 formation is efficient. Thesetemperatures were too low to explain the observed H2 for-mation rate by Jura 137 . Molecular hydrogen was found toform at 10 K and below, whereas the dust in diffuse clouds,where H2 formation is observed, is closer to 20 K. For the in-homogeneous systems the sites with higher binding energyact as sinks in which H atoms can reside up to higher tem-peratures; thus increasing the surface abundance of atomichydrogen. The net result is an increase in reaction efficiency.Especially amorphous carbon appears a good candidate forsurface H2 formation where recombination efficiencies of atleast 0.1 were obtained for temperatures as high as 21.8 K,high enough to cover the grain temperatures of interstellardust in diffuse clouds.

Since amorphous carbon is not the main component of in-terstellar dust, it is useful to consider mixed grains consisting

21

of both olivine and amorphous carbon (< 40%). In all mixedcases the temperature window for adequate recombinationis expanded. 40% of clustered, inhomogeneous carbon wasfound to give an upper bound temperature of 23.8 K.

Inhomogeneities can also occur due to surface roughness.Even for crystalline systems it is well known that surfacesare usually not flat but exhibit islands with steps and kinks.Cuppen and Herbst studied this effect by using surfaces ofdifferent roughnesses.78 In this case, one can distinguish be-tween vertical interactions, Ev, of a deposited hydrogen atomto the substrate as well as lateral bonds, El , in case the atomis located near a protrusion. The total binding energy, EB canthen be expressed as

EB = Ev + iEl (48)

with i the number of lateral bonds. The top panel of Fig-ure Figure 7 shows the results for four different surfaces withincreasing roughness. The surfaces with high surface rough-ness have more sites with higher values of i and thus more Hatoms stay adsorbed which can react with newly depositedspecies, leading to an extended temperature range in whichmolecular hydrogen can be efficiently formed. For thesesimulations, the Eley-Rideal mechanism is included as well,this is especially important at low temperature where the sur-face coverage is high and the atom mobility low. The lateralinteractions in these simulations are chosen to be equallystrong as the vertical interaction. In reality, these interac-tions are probably weaker. Since the temperature windowfor efficient hydrogen recombination scales with El , the as-sumption for the exact value of El is of key importance. Forolivine surface d, still an efficiency of 50 % could be ob-tained for El = 0.4ED at grain temperatures representativefor diffuse clouds.

A fourth way to model inhomogeneity is to introduce asecond type of binding site with higher binding energy thanstandard, instead of using a continuous distribution or ex-plicit surface characteristics. Wolff et al. constructed sucha system with two binding sites.138 If the energy differencebetween the two types of binding sites is chosen to be large,two temperature regimes exist in which H2 is efficientlyformed. The authors were especially interested in circum-stances where the presence of the deep sites extends the tem-perature regime belonging to the shallow sides until the tworegimes are bridged. They obtained a rate equation modelwhich describes their KMC results with a high accuracy. Astrong effect of the formation efficiency on the arrangementof the deep sites was found. The border length which sepa-rates the deep and shallow binding sites appeared to be cru-cial in determining the final H2 formation rate. Surfaces withrandomly placed deep sites will have a longer border lengththan surfaces with one island of deep sites, which results inmore efficient H2 formation.

Cuppen and Garrod 139 constructed a similar KMC modelwith two binding sites. The aim was to develop a method thatcan grasp some of the surface inhomogeneity that is mostlikely present on interstellar grains, but that is, at the sametime, simple enough to describe by modified rate equationsthat can then be applied to simulate large grain chemistry

networks over large timescales and a wide range of differ-ent circumstances. A different choice for the diffusion bar-rier was used – Eq. (33) instead of Eq. (32) used in Ref.138 – and the focus lied on a different regime in parameterspace – low surface coverage, where standard rate equationsdo not apply. Wolff et al. 138 considered the regime withhigh efficiencies and surface coverage. As mentioned ear-lier, traditional rate equation models incorporating the sur-face formation of H2 failed to take into account the accretionlimit problem, first ventilated by Tielens and Hagen 7 . Thesize limit for this to occur was found to be between 100 and10000 sites per grain at temperatures from 10 to 22 K. Theexact limit depends on the composition of the grain, and theamount of surface inhomogeneity, likely due to the diversityof surface diffusion and binding energies.49 Note that gen-erally the lower bound of the grain-size distribution is con-sidered to be 0.005 µm,140,141 which corresponds to roughly3000 sites.

Cuppen et al. found that the H2 formation efficiency in-creases with increasing binding energy, fraction of the sec-ond binding site, and with decreasing temperature. Theyfurthermore managed to construct a modified rate equationmethod that was in reasonable agreement with the KMC re-sults, if the second type of binding sites were randomly dis-tributed. For distributions other than random (clusters withdifferent aspect ratios, clusters of different sizes, or dispersedin lines) the agreement was not obtained and the results var-ied by as much as four orders of magnitude. As stated before,surface inhomogeneity is most likely due to the amorphouscharacter of the grain and the irregular surface topology ofthe grain. Since in both cases the strong binding sites willbe randomly distributed across the grain, the authors recom-mend that the modified rate equation model with the intro-duction of the second type of binding site can be used.

6.1.2 Models including chemisorption sites

The existence of chemisorption wells at the surface can alsobe viewed as inhomogeneity. Depending on the surface,there can either be a barrier into this well, or not. Cazauxand Tielens 77 showed that this type of surface inhomogene-ity can play a role in the formation of H2 and that the pres-ence of chemisorption sites can extend the temperature rangein which reaction occurs to quite high temperatures, depend-ing on the height and width of the barrier into the well.The study was initially performed using only rate equations,but in follow-up studies, KMC simulations were applied tocheck the validity of the rate equation model. In Ref. 143 therole of carbon grains in the deuteration of H2 was studiedusing both rate equations and KMC simulations. A modelwith a high barrier for chemisorption was applied to simu-late polycyclic aromatic hydrocarbons, which represent thesmall grains in the grain size distribution. An amorphouscarbon model with no barrier to chemisorption was used todescribe large grains, following experimental results on thehydrogenation of nano-sized carbon grains.144 Small grainswere found to play an important role for dust temperaturesbelow 25 K; large grains for the high temperature range. Anempirical formula was obtained to describe the efficiency as

22

a function of grain size and D/H ratio. Since the interstellarD/H ratio is roughly 10−5, it is very hard to obtain enoughstatistics on the formation of HD and D2, since most of thesimulation time is spend on H deposition and H2 formation.For this reason the D/H ratio is increased with a factor 100or 1000 and the results are then extrapolated.

In a follow-up study,145 the model was used to study H2and HD formation at high redshift where the dust abundanceis low and gas phase routes compete with the surface forma-tion of these molecules. Three different grain types whereconsidered: amorphous carbon, silicates and graphite to rep-resent PAHs. The latter model included preferred stickingat para-sites as discussed in Section 5.1, although a squarelattice was used instead of an hexagonal grid. Rate equationresults were compared with microscopic KMC results. Agrain size dependence of the formation efficiencies was onlyfound for large D/H which is not so astrochemically relevant.For this reason, rate equations were used in the remainder ofthe paper to construct fitted expressions that can be used ingas phase models and give the H2 and HD formation rate as afunction of redshift, gas and dust temperature. Even for highredshift where little dust is present an enhancement of H2was obtained by surface chemistry with respect to modelswithout surface chemistry. HD was predominantly formedthrough gas phase route D+ +H2 and therefore the enhance-ment of H2 lead indirectly to an enhancement of HD as well.

The model by Cazaux and Tielens was further extendedto olivine by Iqbal et al., who used chemisorption sites forthis type of grains as well.146 Again, both rate equationsand KMC were used to describe this system. They per-formed simulations with different heights and widths for thephysisorption to chemisorption barrier for both olivine andcarbonaceous material and obtained efficient H2 formationover a wide temperature range. They further found that therate equations overestimate the KMC results over the entireinvestigated regime, since for all cases the number of atomson the surface is less than one. They concluded that coveragecan be misleading as an indicator for efficient H2 formation,because low coverage can also mean that all H atoms on thesurface immediately convert to H2, resulting in a very highefficiency.

The hydrogen abstraction model described in Section 5.1using a honeycomb graphite structure87,120 was used to studythe onset of H2 reformation in post-shock regions.142 Dur-ing J-type shocks the velocities and temperatures are highenough to dissociate molecular hydrogen and cooling takesplaces through, subsequently, H, O and H2 as soon as it is re-formed. This reformation takes place on the surfaces of dustgrains, but the exact conditions under which this occurs areimportant components in the interpretation of observationaldata. Observations of O and H2 emission are complementaryand they can be translated into physical conditions using ac-curate predictions and models. One of the ingredients in suchmodels is the H2 reformation rate. Using KMC the prod-uct of the sticking efficiency and the H2 formation efficiency(S×η) was determined on grains at a constant temperatureof 15 K. The sticking efficiency was first calculated for baregrains, incorporating the effect of chemisorption, leading tothe enhancement of S×η at higher temperatures (>500 K)

with respect to the original model. Running a shock modelincluding these new results leads to H2 reformation at earliertimes since chemisorption is efficient at high temperatures,i.e. on short timescales after the shock. The higher rate hasa substantial effect on the predicted line fluxes of O and abetter agreement was found between the model and observa-tions.

6.1.3 Temperature fluctuations

Most studies consider a grain at constant temperature, whichis a good approximation for large grains and in dark regions.However, since the grain size distribution follows a powerlaw140

ngrain(r) ∝ r−3.5, (49)

small grains are responsible for a considerable fraction of thetotal grain surface area. These grains can be easily heated byUV photons or by cosmic rays. Like grain surface chemistry,this grain heating is a stochastic process and can thereforebe perfectly studied by means of KMC simulations. Cuppenet al. 43 studied the effect of photon heating on H2 formationin this way whereas Herbst and Cuppen 22 investigated theeffect of cosmic ray heating on the desorption of interstellargrain mantles of different composition. Since the latter con-siders ices, it will be discussed in Section 6.2. UV photonshit the grains in diffuse clouds with a similar frequency as Hatoms. If these hits lead to heating, they have a strong effecton the surface chemistry.

When the temperature of the grain fluctuates, the ratesbecome time dependent like in the case of the TPD exper-iments. Section 4 discussed an elegant algorithm for such acase, using the fact that the time dependence of the rate isknown. Unfortunately, this information is not known in thecase of stochastic fluctuations. Now, species can land on thegrain when it is in its cold phase at which time the event timeis determined using Eq. (23). This time can easily be longerthan the inverse frequency with which the grain is heated.When the grain is heated by cosmic rays or photons, thesetimes change from tevent,old to tevent,new according to

tevent,new = (tevent,old− t)khop(Told)+ kevap(Told)

khop(Tnew)+ kevap(Tnew)+ t (50)

where t is the current time and Told and Tnew are the tempera-tures before and after heating, respectively. During the cool-ing the times are changed, using the same equation. Coolingoccurs through radiative cooling applying a Debye model forthe heat capacity and empirical emission coefficients.

Stochastic heating was found to have the strongest effectfor smaller grains; they have fewer incoming photons pertime interval and the temperature increase per hit is largersince it is distributed over less volume. The temperature fluc-tuations can be best described in terms of modal temperature,the temperature at which the grain is most of the time, andthe temperature range. For photon heating, the modal tem-perature is low for small grains and larger values of AV andpeaks around r = 0.02 µm and AV = 0.43 The variation oftemperature is the largest for the smallest grains. Adding H2formation to the equation shows that the temperature fluc-

23

tuations decrease the residence times of H atoms and thusdecrease the efficiency. The efficiency is substantially lessthan the 100% formation efficiency that is often assumed tobe required to account for the observational H2 formationrate found by Jura 137 . This assumption however based ona very narrow grain size distribution of 0.1 µm. Integrationof the recombination efficiency over the full size distribu-tion shows a less dramatic effect, since the small grains havea considerably larger surface area. It was found that Jura’srate could be reproduced as long as grain surfaces of at leastmoderate roughness were used.43

6.2 Ice chemistryThe astrochemical simulations presented in Section 6.1 arerelevant for environments where there is either too much ra-diation for ices to build up or it is too warm for ices to sus-tain. Dense molecular clouds, on the other hand, providea surrounding with low temperature, high density, high H2abundance and typically high values of AV . The latter isespecially ideal for the formation of icy mantles reachingthicknesses up to 100 ML. The high extinction shields theinner regions of the cloud from UV light and therefore, atomaddition reactions are the prevailing process, together withcosmic-ray induced mechanisms. Typical values for diffuseand dense clouds are summarized in Table Table 1.

Table 1: Typical physical conditions for interstellar clouds

AV Tgrain Tgas nH H/H2(mag) (K) (K) (cm−3)

Diffuse ≤ 1 15–20 60–100 ∼ 102 HDense ≥ 5 10–15 10–30 ∼ 104 H2

The main problem when simulating grain surface chem-istry in these dense conditions is that one leaves the sub-monolayer regime and ice layers build up. The grain istherefore constantly changing and the binding energies andhopping barriers change consequently. In diffuse clouds, thedesorption and diffusion behavior of the hydrogen atoms ismainly determined by the underlying grain and the interac-tion between absorbates occurs through reaction. Now, indense clouds newly formed species determine the bindingenergy, barriers and reaction possibilities of their neighbors.Many more interaction energies are thus needed to describethe system, because there is not only interaction between thedifferent species with the bare substrate, but also with allother adsorbed species. The problem is that many of these in-teraction energies are unknown or at best poorly constrained.We are aware of 12 KMC studies of interstellar ices, treatingthe ice at least in terms of layers, but often with more mi-croscopic detail. A short summary of these studies is givenin Table Table 2. We will discuss them separately in the re-mainder of this section.

The first paper to simulate interstellar ices using micro-scopic KMC studied the desorption of these ices by cosmicray heating of grains.22 In dense clouds, the UV photon fluxis substantially reduced with respect to diffuse clouds andcosmic ray heating becomes an important desorption mech-

anism. Since, especially the Fe cosmic rays, can lead to atemperature increase of several hundreds of Kelvins, cosmicray hits will be able to lead to substantial desorption of grainmantles. The applied heating model is similar to the onereported in Ref. 43, but the deposited energy per hit is dif-ferent and the grains do not only cool radiatively but also bydesorption, which extracts energy from the grain. Every des-orption the grain loses Ebind in energy. N2, CO and H2O icesof different thicknesses and underlying substrates were stud-ied. A single monolayer of water was found to efficientlydesorb for all but the largest grains studied as well did themonolayers of CO and N2. Desorption of five monolayers ofwater ice was found to proceed because of the strong bondsbetween water molecules.

One of the first papers to use microscopic KMC to studyice build-up was by Cuppen and Herbst 147 . The paper con-centrated on the formation of water ice and it aimed to un-derstand the threshold value of infrared water ice detection atAV = 3.3±0.1 mag.152 A range of different conditions wassimulated. Density, grain and gas temperature and visualextinction were varied; representative for diffuse, translu-cent and dense interstellar clouds. Water ice was formedunder all circumstances, but in diffuse clouds the amountstayed within the submonolayer regime, well below the de-tection limit. The threshold value for detection was repro-duced, taking into account this detection limit. In translu-cent clouds, the ice was heavily processed by UV radiationand the abundances of oxygen-rich species, like O2 and O3,were found to increase at later times due to the loss of hydro-gen by photodesorption. The route to water formation alsochanged depending on the conditions – atomic or molecu-lar gas. In diffuse clouds OH+H−−→ H2O was dominantwhereas in dense clouds OH + H2 −−→ H2O + H but alsoH2O2 +H−−→ H2O+OH became more important. Hydro-gen peroxide in the latter route is formed through O2 hydro-genation and the finding that this route can account for 20%of the water production initiated the experimental study ofthis formation route.79,130,131

Another early model focusing on ice formation was byChang et al. 46 . They used a coupled gas-grain code wherethe gas was described by rate equations and the grain chem-istry by microscopic Monte Carlo simulations to simulatethe evolution of a grain. Since the main aim of the paper wasto show the coupling between the gas and grain chemistry,only a small network of 12 reactions was employed, lead-ing to the formation of CH3OH, H2O and CO2. The resultswere compared to a master equation model. Rough and flatsurfaces were used at T = 10 and 15 K and lateral interac-tions were switched either on or off. A reasonable agreementwith ice observations was obtained, better than for the mas-ter equation model, and this agreement improved with thelateral interactions switched on. The formation of methanolwas found to be less efficient at 15 K than at to 10 K, inagreement with experiments,68 because of a competition be-tween thermal hopping and tunneling for reaction. CO2 wasfound to be formed through the reaction CO+OH with theOH formed at the site next to the CO. Starting from a roughsurface, the grain was found to be slowly smoothed duringthe build up of the ice layer. This is caused by the lateral in-

24

Table 2: Overview of KMC ice studies with some microscopic detail

Reference # reactionsa reaction routes gas D/H? photo- commentsphase?b processes?

Herbst and Cuppen 22 0 – N N N CR desorptionChang et al. 46 12 CH3OH,H2O,CO2 Y N N

Cuppen and Herbst 147 10 H2O2,H2O,O3 N N YMuralidharan et al. 148 0 – N N N water chemisorption

modelFuchs et al. 68 5 CH3OH N N N sequential accretion of CO

and HCuppen et al. 48 5(12) CH3OH, (CO2,H2O) N N N one simulation with larger

networkCazaux et al. 149 101 H2O2,H2O,O3 N Y Y

Das et al. 85 10 CH3OH,H2O,CO2 N N NDas and Chakrabarti 86 11 CH3OH,H2O,CO2 N N Y

Marseille and Cazaux 150 0 – I N N water accretion timescaleChang and Herbst 88 29 CH3OH,H2O,CO2,CH4 Y N Y only photodesorption is in-

cludedVasyunin and Herbst 125 full gas-grain network Y N Y only layers

Meijerink et al. 151 101 H2O2,H2O,O3 I Y Y included in XDR model

a

Photodissociation reactions are not included in this count.b “I” stands for Indireclty if the results are later fed to another simulation program treating the gas phase.

teractions, which lead to filling the strong binding sites, i.e.,the valleys and holes.

Section 5.2 discussed a CO hydrogenation KMC modelwhich was compared against experiments.68 In this paper,also a few simulations on CO hydrogenation under inter-stellar conditions were performed. In these simulations, aformed CO ice was exposed to hydrogen atoms. The simula-tions showed that the abundances do not scale with flux, go-ing from the experimental flux to interstellar fluxes over allthese orders of magnitude. This can be seen in Figure Fig-ure 16 where the simulated experimental and interstellar iceevolution is plotted as a function of H atom fluence. One rea-son for this discrepancy is the high surface H abundance inthe experimental conditions, leading to more H+H reactionsas compared to the CO hydrogenation channel. A secondeffect is the difference in sticking probably: for hydrogenatoms this is rather temperature-dependent and it changesfrom roughly 0.1 for room temperature H atoms (laboratory)to 1 for molecular cloud conditions.61 This example showsthe need for applying models to bridge the gap between ex-perimental and interstellar conditions.

The final H2CO/CH3OH ratio that was obtained in the in-terstellar simulations showed only a reasonable agreementwith observations. The authors argued that a proper modelof interstellar CH3OH formation should have simultaneousdeposition of CO and H. These type of simulations were per-formed in a follow-up paper where the model was used tosimulate the surface formation of H2CO and CH3OH underdifferent conditions, including density and temperature.48

The applied H flux was constant in time, whereas the COflux gradually diminished because of CO freeze-out. Theobtained CO/CH3OH and H2CO/CH3OH abundance ratioswere found to be predominantly determined by the H/COflux ratio and the surface temperature. This resulted in a lay-ered structure of simulated grain mantle with CH3OH on top.The species CO and H2CO were found to exist mainly in thelower layers of ice mantles, which are formed in early timeswhen the CO flux was relatively high and these layers are

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Figure 16: KMC simulations of CO-ice hydrogenation at12.0 (top) and 16.5 K (bottom) as a function of fluence.The thick lines are at a constant atomic hydrogen gas phasedensity of 10 cm3 and a gas temperature of 20 K. The thinlines represent experimental conditions with a particle fluxof 5×1013 atoms cm−2 at room temperature. The results areshown in terms of column density and the figure is adaptedfrom Ref. 68. Credit: Fuchs, Cuppen, Ioppolo, Romanzin,Bisschop, Andersson, van Dishoeck, Linnartz, Astron. As-trophys., 505, 629, 2009 adapted with permission c© ESO

25

not available for hydrogenation at late times. This findingdiscredits many gas-grain models, which do not take into ac-count the layering of the ice. The extend of effect of layeringcan be seen in Figure Figure 17. The right panel shows theevolution of the CO, H2CO, and CH3OH ice abundance fora dense cloud of density of nH = 105 cm−3 and grain surfacetemperature of 12.5 K simulated with a microscopic KMCsimulation. The left panel shows the abundance evolution forthe same species obtained with a rate equation model withsimilar input parameters and conditions. One can clearly seethat whereas the initial ice composition is very similar, atlater times the ice composition starts to deviate, since in therate equation model also the lower layers are still availablefor hydrogenation, whereas in reality they are blocked bythe top layers, which is included in the microscopic KMCmodel.

The obtained model results in Ref. 48 agree reason-ably well with observations of solid H2CO/CH3OH andCO/CH3OH abundance ratios in the outer envelopes of anassortment of young stellar objects. The results suggest thatthe large range of CH3OH/H2O observed abundance ratios isdue to different evolutionary stages of the objects. Extensionof the limited network to include competing reactions thatalso lead to the formation of H2O did not alter the overallconclusions concerning H2CO and CH3OH formation.

1×103 1×104 1×105

microsc. KMC

1×103 1×104 1×10510-3

10-2

10-1

100

101

102

rate equations

CO

H2CO

CH3OH

N(t)[1015moleculescm

-2]

Time [years]

CO

CH3OH

H2CO

Figure 17: The evolution of the CO, H2CO, and CH3OH iceabundance for a dense cloud of density of nH = 105 cm−3

and grain surface temperature of 12.5 K obtained by (left)microscopic KMC simulations and (right) a rate equationmodel. Adapted from Ref. 48. Credit: Cuppen, vanDishoeck, Herbst, Tielens, Astron. Astrophys., 508, 275,2009 reproduced with permission c© ESO

Cazaux et al. 149 developed a model to form water and itsdeuterated forms including many different reaction routes.This model simulated the initial stages of water ice forma-tion on carbonaceous grains in the submonolayer regime andbuilds on earlier models for H2 formation.143 The modeldoes not allow for a second layer to build up and reactionproducts can desorb using the excess energy from the reac-tion. The aim of this model was to study the effect of surfacechemistry on the gas phase composition. Different physicalconditions were simulated using a microscopic KMC model

to study this effect. Return to the gas phase occurs throughphotodesorption and chemical desorption. To model theseenergetic processes Molecular Dynamics would probably bebetter suited, although the required timescales cannot bereached. At a low dust temperature of T = 10 K, hydrogena-tion reactions leading to species like H2 and H2O are foundto dominate; for higher temperatures (T = 30 K) oxygena-tion starts to take over, leading to the formation of O2 and O3.Different water formation routes were found to be importantin different conditions: in diffuse clouds, formation is mainlythrough H+O−−→ H2O in agreement with Ref. 147. Sincethis is a highly exothermic reaction and the chemical desorp-tion probability is assumed to scale with the exothermicity, alarge fraction of the formed water molecules is released intothe gas phase. In dense clouds H2 +O−−→ OH+H domi-nates (this reaction was not included in Ref. 147, and thereis no experimental evidence for it to occur153) and since thisreaction is less exothermic (it is in fact endothermic154), asmaller fraction desorbs into the gas phase. In photon domi-nated regions, more oxygenated species are released and alsothe water formation routes through O2 +H and O3 +H in-crease in importance.

Das et al. 85 developed a discrete-time, random-walk KMCmodel to look at the formation of CH3OH, H2O and CO2under different physical conditions. In discrete-time simula-tions, time is advanced using

∆t =1

ktot(xi)(51)

instead of Eq. (23) and the time steps do not follow a Poissondistribution. Very thick ices were obtained in this way (60–500 MLs) and a range of physical conditions was determinedwhere observations were best reproduced: the so-called fa-vorable zone. Ice abundances of CO2 were underproduced inall conditions, probably since the OH+CO reaction was notincluded in their model. Only a small network containing tenreactions was used; reactions involving H2 are the most strik-ing omission, especially since both Cuppen and Herbst 147

and Cazaux et al. 149 concluded that reactions with H2 are themost important formation route for the production of H2O indense cloud conditions. Also the H+O2 route is omitted,whereas large amounts of O2 are formed on the grain. Ina follow-up paper, the reaction network was extended withsix more addition reactions, now also including the H+O2,H+O3 and H2 +OH reaction channels,86 as well as witheleven photodissociation reactions. Molecular hydrogen wasstill not allowed to land on the surface – probably to keep thesimulation time within reasonable limits – and the H2 +OHreaction could only proceed through Eley-Rideal reactions,which seems a strict limitation. The aim of the paper was tostudy the effect of photodissociation on the evolution of thegrain mantle. Photodesorption is only allowed to occur inthe topmost layer; photodissociation is allowed deeper intothe mantle. Physical conditions within the favorable zonewere chosen. The percentage of photodissociated moleculesas a function of AV and number of UV-exposed layers wasstudied. A high percentage was found for low AV , as couldbe expected, and this percentage increases with number of

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UV-exposed layers, until it saturates beyond 50 monolayers.The mantle was found to be more oxygenated for low vi-sual extinction, in agreement with the work of Cuppen andHerbst 147 .

Several microscopic KMC studies investigate the so-called snow line. This is the line around a protostar thatseparates the inner area where bare grains exist from theouter area where grains are ice covered. Earth has formedwithin the snow line and should therefore be rather dry.Several hypotheses exist on how water arrived in the innerSolar System. One of these hypotheses is put forward byDrake 155 and Stimpfl et al. 156 who showed that chemisorp-tion sites exist on forsterite grains where water can absorb.Grains can retain a submonolayer of water well beyond thesnow line in this way. Muralidharan et al. 148 build on thiswork by applying a lattice-gas KMC simulation. The ad-sorption/desorption energies were mapped on three differentfosterite surfaces (100, 010, and 110) by constructinga fine grid, placing one water molecule at a time on a gridpoint, and performing a geometry optimization to obtain theminimum energy. The resulting map for the 100 surfaceis shown in Figure Figure 18. This map was then fed to aKMC model that determined the surface coverage of a grainas a function of grain temperature. Hopping rates were ob-tained from the diffusion constant that was used as an inputparameter. Due to the many assumptions used in the model,the authors argue that the results only give a conservativelower limit to the amount of water that can be absorbed ontograins prior to accretion into planets. Their results show thatthis amount can indeed be substantial.

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Figure 18: Surface energy potential map for water on the100 surface of fosterite. Color scale represents energy inkJ/mol. Figure reproduced with permission from Ref. 156.Copyright 2006 Elsevier

Marseille and Cazaux 150 constructed a water-accretionmodel where water molecules are physisorbed on car-bonaceous grains and can cluster together, leading to astronger binding, with a maximum of six water neighbors.Timescales for condensation of water ice below 100 K weredetermined in this way and these were found to be ratherlong compared to timescales of grain mixing, caused by tur-bulence and infall. This results in a snow border, rather thana snow line, a region where bare and icy grains can coexist.

This region can be significant compared to the size of the ob-ject. For massive dense core this region can take up a fourthof the total system size.

7 New directionsThis section discusses the new directions towards which thefield of grain surface chemistry modeling is moving. Be-cause of the increased computer power, one can see an in-crease in complexity of the Monte Carlo models that arebeing reported. On the one hand more molecular detail isintroduced. This was already visible in the H on graphitesimulations where more and more structural and system spe-cific information was introduced. This evolves even furtherin off-lattice KMC models. On the other hand, models aredeveloped in which the complexity of the chemical networkis increased or in which gas and grain surface chemistry arecombined. In the latter case, one can both notice existinggas-grain routines to get more detail in their grain surfacetreatment as well as surface models being extended with agas phase treatment.

7.1 Unified Gas-Grain KMC modelsSeveral unified gas-grain KMC models have been recentlydeveloped.88,125,157 The grain surface is treated at differentlevels of detail in these models. A first attempt to unify amicroscopic KMC model of the grain surface with a methodthat treats the gas phase was by Chang et al. 46 . They usedrate equations to calculate the time-dependent chemical evo-lution of the gas phase including accretion onto grains, butno surface chemistry or evaporation was included. From thisinformation, temporal fluxes were determined to use as in-put in a microscopic KMC model to treat the surface chem-istry. Desorption of species was then included again in therate equation treatment of the gas phase chemistry and inthis way the chemical evolution of both gas and grain man-tle was obtained iteratively. For dense clouds, convergencewas reached in two iterations. The success of this methodcomes from the fact that for the conditions chosen in theirpaper, dense cloud conditions, there is very little return ofgrain surface species into the gas phase and the influenceof the grain surface chemistry on the gas phase is thereforelimited. Under these conditions, only atomic hydrogen evap-orates to a significant extent. Although it has little effect onother gas-phase species, the evaporation of atomic hydrogenchanges its gas-phase abundance, which in turn changes theflux of atomic hydrogen onto grains. For astronomical ob-jects where more grain mantle material is evacuated into thegas phase, like hot cores or protoplanetary disks, this influ-ence becomes clearly larger and many more iterations willbe needed to reach convergence.

Another approach to couple gas phase chemistry and grainsurface chemistry is to use the KMC approach for bothphases. Early attempts did not result in the required reso-lution of the gas phase abundances of rare species,158 butVasyunin et al. 6 managed a way to circumvent this by cu-mulative averaging of the gas phase abundance in time. They

27

demonstrated this by simulating molecular clouds at differ-ent temperatures and densities and comparing this to rateequation and master equation results for a large gas-grainchemistry network of roughly 6000 gas-phase and 200 sur-face reactions.

Recently, a new unified KMC model was introducedin which macroscopic KMC is used to describe the gasphase chemistry whereas microscopic KMC treats the sur-face chemistry.88 A medium-size surface network of 29 reac-tions was used, leading to the formation of stable moleculeslike CO2, CH4, CH3OH, and H2O. Although CH4 is over-produced in the simulations, the agreement with ice obser-vations159 for other carbon-bearing ices like CO, CO2 andCH3OH is much more reasonable. In the analysis of thesimulated ice mantles, an anti-correlation between CO andCO2 was found, in agreement with the observations. Theobserved CO-rich non-polar ice segment is however not re-produced. In a few of the models, CO is formed early in theevolution of the grain mantle by reaction of C with O. In-crease in the initial H/H2 ratio, destroys this route however,since then most of atomic oxygen reacts with H to form OH.Part of the simulated CO2-ice abundance is obtained througha “chain reaction” in which OH is formed by hydrogenationof O in the vicinity of a CO molecule. CO and OH can thenreact without much diffusion involved. In this way, the mo-bility leading to the formation of CO2 is done by the H atomthat performed the initial reaction. Detailed mechanisms likethis show the value of microscopic models when simulatinggrain chemistry.

Macroscopic models include more and more “micro-scopic” information in their models. This is achieved byincluding layering into the macroscopic model, where thecomposition of each layer is followed in time. Other mod-els only distinguish between the surface and the bulk of theice mantle: the so-called three-phase model.160 Examplesof gas-grain models that treat the ice mantle in at least twophases can be found in Refs 124,125,160–162. Since Ref.125 is the only model that makes use of KMC we will re-strict ourselves to the discussion of this model.

This macroscopic model tracks the species per layer. Itis assumed that the formation of a new monolayer startswhen the previous layer is completely filled. Chemical re-actions and desorption are only allowed to occur in the topmonolayers. The exact number of chemically active layerswas calibrated against desorption experiments as discussedin Section 5.2. Here the assumption is made that chemi-cal reactions only occur in the same layer as where desorp-tion occurs. This requirement is probably too strict. HighVacuum experimental studies of energetic processing clearlyshow that reactions in the bulk can occur. The model wasemployed to study the chemical evolution starting from amolecular cloud through the collapse and warm-up phasesleading to a hot core/corino. The main motivation to use alayer-by-layer model was that it treats the desorption cor-rectly, which is important in the warm-up phase. In gen-eral, a good agreement with ice and gas phase observationsis obtained for all stages, better than when using a two-phasemodel: in particular, the formation of polar and apolar icefractions in the cold stage and the observed jump abundances

in H2CO in the hot corino phase. On the other hand, specieslike HCOOCH3 which have photoproducts as precursors areunderproduced. This is most likely because photochemistryand bulk diffusion should be active in the inner layers aswell.

7.2 Combining different modelsIntroducing more microscopic-detail into macroscopic mod-els can also be done in a more pragmatic way, where ded-icated, detailed microscopic simulation programs are usedto obtain information about reaction efficiencies/rates as afunction of density, temperature etc.. This is then given tothe macroscopic model in terms of lookup tables or fittedanalytical expressions. Chang et al. 163 give such lookup ta-bles for H2 formation rates for different H fluxes, grain struc-ture – rough vs. flat surface and olivine vs. carbonaceous –and surface temperatures. Separate tables are provided forthe use in pure gas phase models and gas-grain models. Inthe latter case, effective evaporation rates and diffusion ratesare provided. Ref. 147 provided an analytical expression forthe H2 formation rate on graphitic grains, resembling PAHs,with the density and gas temperature as input parameters.The dust temperature was kept constant at 15 K.

Something similar was done for the water formation in X-ray exposed environments (XDRs).151 Here a rate equationmodel was constructed to determine the formation rates ofOH and H2O and the release into the gas phase and analyticalexpressions for these quantities were then obtained. Thesewere then included in a XDR model. KMC simulations usingthe model presented in Ref. 149 were used as a consistencycheck for the rate equation model. At dust temperatures be-low 35–40 K, the KMC and rate equation method yield thesame results within the uncertainties. Since the simulationsstay within the submonolayer regime, rate equations are rel-atively simple to construct and layering does not need to beconsidered. At higher dust temperatures, the system entersthe stochastic regime and the rate equation method systemat-ically obtains too high efficiencies. OH formation on grainsand release to the gas phase is dominated by O+H−−→ OHfor low temperatures and by O3 +H− > OH+O2 at highertemperatures, consistent with the earlier findings in Ref. 149.H2O formation on grains and release to gas phase is domi-nated by OH + H−−→ H2O. The fraction of H2O desorb-ing into the gas phase is further increased under the influ-ence of strong UV radiation fields by formation-dissociation-formation loops. These additional formation routes for warmgas phase water could help explain unusually strong waterlines observed for Mrk 231, which has an accreting super-massive black hole with X-ray luminosity LX ∼ 1044erg s1.Typical starforming environments, without an X-ray source,do not show these strong water lines.

7.3 Off-lattice modelsOne of the main restrictions of traditional KMC methods isthe confinement of the system to a predefined lattice. Therestriction of the atomic coordinates limits the amount ofphysical detail contained by the simulation and makes the

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use of realistic interaction potentials far from straightfor-ward. In the solid state, the lattice approximation is oftenwell-justified for simple crystalline systems, but the situa-tion already becomes more complicated for molecular crys-tals which often show some degree of disorder, like manyhydrates and the hydrogen bond network in H2O ices. Smallsite-to-site differences in these systems affect the rate con-stants and make the generation of the table of events a te-dious task. For amorphous systems, the situation obviouslybecomes even worse and lattice-based KMC simulations ofthese systems should be more seen as toy models. Needlessto say, the same is true for simulations of systems in the liq-uid phase.

A second problem with traditional KMC simulations is theneed to define the table of events before the start of the simu-lation. As with the limitation to a lattice, this can be done forsimple crystalline systems but quickly becomes more chal-lenging as the complexity of the system increases and thetransition mechanisms become less intuitive. A prime ex-ample of such a mechanism is the diffusion of H atoms intoa CO ice, which proceeds through an exchange mechanismrather than the H atom taking up interstitial positions.68 Ide-ally, one would like to evaluate every process individually, asaccurately as possible. This can be done in off-lattice KMCby applying forcefields83,164–166 or even electronic structuremethods167 to calculate individual process rates. As everyprocess is considered to be unique, this evaluation needs tobe done on-the-fly, which considerably increases the com-putational costs. The advantage of not having any a prioriassumptions about processes in the system is, however, of-ten worth this additional effort.

In the remainder of this section we will describe twodifferent approaches which take KMC simulations beyondthe lattice approximation. The continuum Kinetic MonteCarlo Method168 is a particularly interesting off-lattice KMCmethod which incorporates diffusion of particles analytic-cally into a continuous time KMC scheme. Reaction ratesand diffusion constants still need to be specified beforethe simulation starts though. The Adaptive Kinetic MonteCarlo169 method is an on-the-fly, off-lattice, KMC methodwhich allows for very high atomistic detail, requiring noother input than an interatomic interaction potential.

7.3.1 Continuum Kinetic Monte Carlo

The Continuum Kinetic Monte Carlo method168 was de-signed for simulating reactions in solution, a typical casewhere one would like to go beyond the lattice approximation.The challenge lies in the fact that conformational changes,reorientations and individual diffusion steps often occur onmuch shorter timescales than the chemistry itself. In con-tinuum KMC, this problem is solved by coarse graining themaster equation, which allows for diffusion to be treated inan analytic way whereas the chemical reactions are simu-lated in a variation on the CTRW algorithm.

A continuum KMC cycle starts, just like CTRW KMC,by constructing a list of possible reactions and their reactionrates. In CTRW the event time is then determined for eachparticle following Eq. (23). In continuum KMC the event

time for each possible reaction is calculated. This calcula-tion of reaction times though, is somewhat more involvedthan in the case of a lattice-gas. The rate constants depend,besides on the reaction barrier, also on the diffusion con-stants of the reactants, their initial positions and on time it-self. Once the list is complete, the first reaction is executedand the positions of the reacting particles are updated. Thesenew positions are determined by assuming a Gaussian dis-tribution for the diffusing particles by generating a randomnumber from the appropriate probability distribution. Afterthis, the reaction list is updated and a new cycle starts.

The advantage of the continuum KMC method lies in thefact that it treats diffusion analytically, enabling the simu-lation of rather large systems for long timescales. Prede-fined reaction rates and diffusion constants keep the compu-tational effort down to a very moderate level but are at thesame time an assumption in the model. For dilute systems,where the reactant diffusion is limited by the inert solvent,this is likely well-justified but at higher concentrations thiscould become a problem. Until now, the applications of themethod remain limited168,170 but the approach certainly of-fers appealing features which may possibly also be useful forsimulating astrochemically relevant processes like reactionsbetween minority species in ice mantles in molecular clouds.

7.3.2 Adaptive Kinetic Monte Carlo

Predefined event tables (process rates) in KMC simations arenot only severe assumptions in the model, they can also bevery tedious to construct. During its construction, one shouldalways ensure that all important, high rate, processes havebeen included, because a missing process in the event tablewill not only close certain evolution routes, it will also causethe simulation time to progress too slowly. For complex sys-tems it is therefore desirable to have an unbiased method forfinding transition mechanisms, eradicating possible humanerrors. This is the essence of the Adaptive Kinetic MonteCarlo method.169 With this method, it is possible to runoff-lattice Kinetic Monte Carlo simulations without havingto define the event table beforehand. The two most severelimitations of traditional, lattice-gas KMC are therefore notpresent when using AKMC.

Since the AKMC method is an off-lattice method, a stateis defined as a local minimum in the potential energy sur-face. At the start of a simulation, only one of these minimais known, the initial state. As a first step, the AKMC pro-gram needs to determine the table of events for all processesleaving this initial state. This requires the exploration of thePES in the vicinity of the current state to search for other lo-cal minima that can be reached via a transition state and todetermine the corresponding rate for this process. To locatethe relevant transition states out of this minimum, single-ended search methods are used. Double-ended methods likeNEB cannot be applied to find transition states, since they re-quire knowledge of both the initial and final states, whereasthe AKMC simulation has only information about the ini-tial state. Several single-ended methods exist,171,172 but theminimum-mode-following method173 is the most frequentlyused. This method can be used when the transition state is

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assumed to be a single point on the potential energy surface(a first order saddle point). Olsen et al. 171 have performeda comparative study of different methods for finding saddlepoints without knowledge of the final states. They concludethat minimum-mode following methods are to be preferred ifa more complete mapping of all existing (low-energy) saddlepoints is desired for large systems, which is typically the casefor AKMC simulations. The method requires the lowest-eigenvalue eigenvector of the Hessian which can be calcu-lated exactly or approximated by a method such as the Dimermethod,173 Lagrange Multipliers174 or Lanczos method.175

As soon as a transition state is found, the rate of the cor-responding process is calculated. Until now, harmonic tran-sition state theory is used for the rate estimation, but anyother method may also be used to account for effects likequantum tunneling and zero point motion. By performing aseries of transition state searches, the table of events of theinitial state is filled up until there is sufficient confidence thatall low-lying saddle points have been found.167 Once this isthe case, time is advanced and the system proceeds to a newstate according to the n-fold way algorithm.

The need to perform transition state searches every time astate is entered for the first time makes the AKMC methodmore demanding from a computational recourses point ofview then traditional lattice KMC. However, since the indi-vidual searches are fully independent, they can be distributedover a large number of computational nodes in a straightfor-ward manner.176 This feature is currently implemented in theEON code.177 Naturally, when a state is entered which hasbeen visited before, no new searches need to be performedas the table of events is already known for that particularstate. For systems with only a limited number of thermallyaccessible states, this means that eventually the relevant partof the PES will be fully explored and then the KMC simu-lation can progress just as quickly as a lattice gas simulationwith a predefined event table. This situation was for exampleencountered in the simulation of CO diffusion of hexagonalice, which will be discussed below. In this case, due to thelow temperatures, only the weakly bound CO molecule wasfree to move and when the surface binding sites were all ex-plored, long time kinetic Monte Carlo trajectories could berun to extract the surface diffusion constant. Notice that inthis case, the AKMC simulation essentially reduces to a nor-mal KMC simulation, with a predefined event table whichhas been constructed in an unbiased way, i.e., without anypossible bias following from human intuition. Moreover, therates are determined from a realistic interaction potential andare not estimated based on experimental information.

The aforementioned study of CO diffusion on hexagonalwater ice, the first application of AKMC in an astrochemicalcontext,83 also showed the importance of an unbiased wayto construct the table of events. Despite the near-crystallinenature of the water ice substrate, the dangling bond patternon the surface leads to a large variation in both the CO ori-entations and the binding energies. This is illustrated in Fig-ure Figure 19, where both the configurations of the bind-ing sites and a sketch of the potential energy landscape areshown. Since the typical binding energy of the CO moleculeis much lower than the cohesive energy of the ice substrate,

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Figure 19: The binding sites and potential energy surface ofCO on the basal plane of hexagonal ice, investigated with theAKMC method.83 The left panel shows the binding sites onthe substrate. The large variation in CO orientation on thesurface is an example of the importance of having an unbi-ased method for finding potential energy minima and transi-tion states. The right panel shows the potential energy sur-face. Again, due to the proton disorder in the substrate, alarge variation in binding energies is observed. Figure re-produced from Ref. 83 by permission of the PCCP OwnerSocieties.

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all the surface binding sites could be explored without thesubstrate itself evolving. Long time kinetic Monte Carlo tra-jectories could therefore be obtained which enabled the ex-traction of the surface diffusion constant.

The required resources for AKMC simulations depend,besides on the obvious system size, heavily on the methodby which interactions are calculated and the number of stateswhich has to be explored. For small systems with onlya limited amount of states, the method permits the use ofdensity functional theory as was demonstrated by Xu et al.,who calculated the dynamics of methanol decomposition onCu(100)178 and Pd island formation on MgO(100).167 Forlarger systems or when many states need to be explored, sim-pler interaction models, like effective pair potentials can beused.83,166

Several techniques have been developed to improve theefficiency of the transition state searches and thereby reducethe computational cost of AKMC simulation. One of thesetechniques is the recycling of previously found saddle points,a method proposed by Xu and Henkelman 167 . In this case,the knowledge of the transition state configurations from pre-vious states is used as an initial guess for transition statesin unexplored states. This method is particularly effectivewhen the system is still rather ordered and only a few atomsare significantly involved in the considered process. Anothermethod which can speed up the simulations is the cancella-tion of transition state searches, if they reach a point wherethere is a high certainty that the saddle point which will befound was already identified during a previous search.179

The power of the AKMC method lies in its ability to per-form long-timescale simulations of thermally driven rare-event systems whilst retaining information of all atomic co-ordinates as the system evolves in time. This makes for apowerful tool when studying kinetic processes in grain man-tles. Of course, the high amount of structural detail limitsthe system size and simulation time compared to traditionalKMC. However, individual processes can be systematicallyinvestigated at high accuracy. Dynamics of important pro-cesses like diffusion, segregation and even chemical reac-tions (given an interaction method capable of treating reac-tive processes) can be studied using AKMC and the resultsmay then be fed to larger scale models like on-lattice KMCor rate-equation methods.

7.4 Coarse grainingA common problem in all KMC simulations occurs whenthe rates in the table of events differ from each other byseveral orders of magnitude. This situation can for exam-ple be encountered when energy barriers are very differentor when the temperature is low. When simulating interstellarices, both cases apply and the KMC simulation may becomestuck in a certain subset of states, wasting many iterationswithout interesting dynamics. The problem is illustrated inFigure Figure 20. Due to the low barrier (high rate) betweenstates 1 and 2, the KMC simulation will repeatedly pick theprocess between these states without evolving to the otherstates, whereby wasting valuable computational recourses.

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Figure 20: Schematic potential energy surface with KMCstates. Without coarse graining, the simulation will be stuckin states 1 and 2 for a long time, due to the low barrier.Coarse graining can solve the problem by merging states intocomposite states.

Novotny 180 provided a solution to this problem by us-ing the theory of absorbing Markov chains. Using this the-ory, the states which are joined by the high-rate process aremerged together to form a composite state. This compos-ite state can in principle contain any number of ‘real’ states.When the simulation visits a composite state, dynamics arenot treated by the usual KMC theory that dictates the choiceof the leaving state and the time evolution but rather by us-ing Absorbing Markov chains. This gives, in one step, theprocess through which the simulation will leave the compos-ite state and the corresponding time it spends in the com-posite state. Hence, as Figure Figure 20 illustrates, the pro-cesses between states inside the composite state are effec-tively removed from the event table allowing the simulationto progress much more efficiently. It is important to notethat the dynamics generated in a coarse-grained simulationfollow exactly the same statistics as in a normal KMC sim-ulation. However, the exact evolution inside the compositestates is lost.

The decision to merge states into a composite can be basedon several criteria. One such algorithm was proposed byPedersen et al.181 Here, each state is assigned a fictitiousenergy level, initially equal to the total energy of the state.This level is then raised by each subsequent visit to the stateand once it becomes higher than the transition state energyto a neighboring state, the states are merged into a compos-ite. The application of coarse graining can accelerate KMCsimulations by many orders of magnitude in terms of bothCPU-time and simulation time per iteration. Figure Fig-ure 21 shows the speed-up factor of AKMC simulations ofthe aforementioned CO diffusion study on top of crystallinewater ice as a function of temperature. Speed-up is here de-fined as the factor by which the simulated time increases withrespect to a simulation without coarse graining using thesame number of iterations. Since the difference in rates in-creases with decreasing temperature, for thermally activatedprocesses, the coarse graining algorithm is clearly more ad-

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vantageous at lower temperature where speed-ups of 13 or-ders of magnitude can be achieved. The figure further showsthat the obtained speed-up depends on the amount of statesinside the composites, the larger the composite states grow,the higher the speed-up. For most physical applications how-ever, one should not let the composite state size grow in-definitely because the exact dynamics inside the compositesis not retained in the Absorbing Markov chain theory. It istherefore advisable to put a limit on the number of states in-side each composite. This limit should be chosen such thatthere is a good balance between the obtained speed-up andthe chemical detail in the simulations.83

The application of coarse graining can significantly extendthe timescales which can be reached by KMC simulations. Itis particularly at low temperatures, when small differences inbarrier heights lead to huge differences in process rates, thatthe method is at its most useful. This makes the method par-ticularly applicable to in KMC simulations of astrochemicalsystems. Since the underlying dynamics remain unaltered,the physical quantities extracted from the simulation remainthe same, only the time over which statistics can be gatheredbecomes longer and the temperatures may be lower.

8 Conclusions and outlookKinetic Monte Carlo was initially introduced in surface as-trochemistry to circumvent the accretion limit, in whichclassical rate equations break down and obtain abundanceswhich are off by many orders of magnitude. The technique,although still not generally applied, has come a long waysince then. Since the method has no real restriction on howmolecules are represented: in terms of number densities or infull atomistic detail, it has proven its additional strength byallowing layering and more detail in the simulations. We

have discussed several examples in the previous sections.Mostly lattice KMC models are applied with the aim of (i)gaining more insight in the physico-chemical mechanismsbehind astrochemically relevant experiments or of (ii) mod-eling grain surface chemistry with a relatively high level ofdetail. As discussed in the earlier sections, several of theinterstellar ice studies have revealed that bulk diffusion isstill an ill-understood phenomenon that cannot be well mod-eled using the lattice-gas KMC models. It appears how-ever to be the limiting factor for obtaining better agreementbetween models and experiment. This agreement is a re-quirement for realistic models that can be applied to predictthe chemical evolution of different astrophysical environ-ments/objects. We therefore see the field moving in two di-rections: on one hand, gas-grain models include more micro-scopic detail and layering in order to introduce some distinc-tion between surface and bulk diffusion – bulk diffusion isusually switched off entirely in these models –, on the otherhand we see the move to a more atomistic description of thesystem, which allows for detailed investigation of the under-lying mechanisms behind bulk diffusion. Unfortunately, theapplication of these atomistic models remains limited due totheir high computational load. The real progress in the fieldwill be made, if these two approaches can be combined byfeeding the results of the atomistic models to the gas-grainmodels.

AcknowledgementH.M.C. is grateful for support from the VIDI research pro-gram 700.10.427, which is financed by The NetherlandsOrganisation for Scientific Research (NWO). H.M.C. andL.J.K. acknowledge the European Research Council (ERC-2010-StG, Grant Agreement no. 259510-KISMOL) for fi-nancial support. T.L. is supported by the Dutch Astro-chemistry Network financed by The Netherlands Organisa-tion for Scientific Research (NWO). The authors would liketo thank Stefan Andersson and Eduardo Monfardini Pen-teado for general discussions concerning this review and EricHerbst, Stephanie Cazaux, Qiang Chang, Kinsuk Acharyya,and Jochiam Krug for fruitful discussions regarding micro-scopic reversibility in particular.

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BiographiesHerma Cuppen studied Chemistry at the Radboud UniversityNijmegen (then named University of Nijmegen), the Nether-lands, where she earned her M.Sc. degree in 2000 and herPh.D. in 2005 (with highest honor) on theory and simula-tions of crystal growth. During this time, she was introducedto the Kinetic Monte Carlo technique. She then moved to theOhio State University as a postdoctoral fellow in the groupof Eric Herbst where she applied KMC simulations to astro-chemically relevant surface processes. In 2006, she movedback to the Netherlands to join the Sackler Laboratory of As-trophysics in Leiden where she lead the theoretical interpre-tation of experimental data and bridged the gap between lab-oratory and space conditions. She was offered an assistantprofessorship in the Theoretical Chemistry at the RadboudUniversity Nijmegen in 2011, where studies both interstellarices and molecular crystals using a range of different compu-tational chemistry techniques, including KMC and Molecu-lar Dynamics. Herma Cuppen is recipient of a NWO VENIgrant (2006-2009), an ERC Starting Grant (2011-2016), aNWO VIDI grant (2011-2016) and the 2005 Crystal GrowthAward for the best Doctoral Thesis from the Dutch CrystalGrowth Society.

Leendertjan Karssemeijer was born in the Netherlands andobtained his B.Sc and M.Sc degrees in Physics at the Rad-boud University of Nijmegen. He is currently a Ph.D candi-date under supervision of Dr. Herma Cuppen at the Instituteof Molecules and Materials of the Radboud University of Ni-jmegen. His project involves atomistic computational mod-eling, applying mostly AKMC and Molecular Dynamics, oflong-timescale kinetic processes, like diffusion and segrega-tion, in interstellar ice mantles.

Thanja Lamberts studied Chemistry at the Radboud Uni-versity of Nijmegen, where she obtained her B.Sc and M.Scdegrees (with honors). She is currently a Ph.D candidate

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with a shared position between the Institute of Moleculesand Materials at the Radboud University of Nijmegen un-der supervision of Dr. Herma Cuppen and Leiden Obser-vatory at Leiden University under the supervision of Prof.Dr. Harold Linnartz. Her project involves the simulation ofsurface chemistry of water formation in interstellar medium,both experimental on grain analogs (Leiden) and by grainsurface models (Nijmegen). The latter aims at understand-ing the relevant experiments as well as modeling chemistryin astrochemical environments using lattice-gas KMC.

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