Monte Carlo simulations of H2 formation on stochastically heated grains

0.35 MB·English

by  H. M. Cuppen

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
Monte Carlo simulations of H formation on stochastically
2
heated grains
H. M. Cuppen1, O. Morata1 and Eric Herbst2
1 Department of Physics, The Ohio State University,Columbus, OH 43210, USA
2 Departments of Physics, Astronomy, and Chemistry, The Ohio State University,Columbus, OH 43210, USA
6 5February2008
0
0
2
ABSTRACT

n Continuous-time, random-walk Monte Carlo simulations of H2 formation on grains
a have been performed for surfaces that are stochastically heated by photons. We have
J assumeddiffusecloudconditionsandusedavarietyofgrainsofvaryingroughnessand

4 size based on olivine. The simulations were performed at different optical depths. We
2 confirmed that small grains (r 60.02 µm) have low modal temperatures with strong
fluctuations, which have a large effect on the efficiency of the formation of molecular
1
hydrogen. The grain size distribution highly favours small grains and therefore H2
v
formation on these particles makes a large contribution to the overall formation rate
4
5 for all but the roughest surfaces. We find that at AV = 0 only the roughest surfaces
5 can produce the required amount of molecular hydrogen, but by AV = 1, smoother
1 surfaces are possible alternatives. Use of a larger value for the evaporation energy of
0 atomic hydrogen, but one still consistent with experiment, allows smoother surfaces
6 to produce more H2.
0
/ Key words: ISM:molecules - molecular processes
h
p
-
o
r
t 1 INTRODUCTION character of the grain or to the topological structure of the
s grain;i.e.,surfaceroughness.Theseresultsareinagreement
a
with the findings of Cazaux & Tielens (2004), who intro-
: Although molecular hydrogen is abundant in diffuse and
v duced extra chemisorption sites and made a distinction be-
i dense regions of theneutral interstellar medium (ISM), the
X tween H and D atoms. They found a larger temperature
detailed mechanism of its formation is still not completely
range dependingon thewidth and theheight of thebarrier
r clear.Atthelowgas-phasetemperaturestypicaloftheinter-
a between sites of physisorption and chemisorption.
stellar medium, a gas-phase formation is not possible, and
H2 is therefore formed on the surfaces of grains. An inter-
pretation of experimental studies(Pirronello et al. 1997a,b; In this paper, we will study the formation of molec-
Pirronello et al. 1999) indicates that the surface tempera- ular hydrogen using similar continuous-time, random-
ture range over which efficient H2 formation occurs is very walk Monte Carlo simulations as in our previous papers
small for olivine (6-10K) and for amorphouscarbon (13-17 (Chang et al. 2005; Cuppen & Herbst 2005) while grains
K) (Katzet al. 1999) under diffuse-cloud conditions. The are heated by the interstellar radiation field. The photons
grain temperature for midsize grains in unshielded regions hitting a grain give it a short heat impulse, resulting in
is however around 20 K. This means that molecular hydro- temperature fluctuations especially for small grains. These
gen cannot be formed efficiently on such grains assuming temperature fluctuations have received considerable atten-
that the olivine surfaces used in the laboratory are a re- tionintheliterature(Greenberg & Hong1974;Purcell1976;
alistic representation of interstellar grains. The analysis of Aannestad & Kenyon1979;Tabak1987;Draine& Li2001),
Katz et al.(1999)considered,however,onlyasinglebarrier where they are discussed with a varying degree of detail.
for diffusion between bindingsites and a single low binding While most of the studies simply use the heat capacity
energybetweenphysisorbedHatomsandthesurface.Inpre- to calculate the temperature changes, Draine & Li (2001)
vious papers (Chang et al. 2005; Cuppen & Herbst 2005), looked at the heating and cooling of small silicon clusters
weshowedthattheintroductionofsitesofdifferentbinding andPAHsbyconsideringtheinfraredemissionofthegrains,
energy leads to an increase of thetemperaturerange for ef- takingintoaccounttheinfraredbandsofthegrainmaterial.
ficient H2 formation. The different sites are the result of a Since our main goal is to study molecular hydrogen forma-
differenceinlocalenvironmenteitherduetotheamorphous tion, we will use a relatively simple model for the heating
(cid:13)c 0000RAS
2 H. M. Cuppen, O. Morata and Eric Herbst
and cooling. As we will show, this model is capable of re- dependence of Q becomes different. Here we used a set
abs
producing thedetailed method toa large extent. of empirical expressions to approximate the absorption co-
Since both the heating and the recombination of H efficients found by Draine & Lee (1984) and Draine (1985).
atoms into H2 are stochastic processes, the Monte Carlo Theyshowedtheabsorptioncoefficientsforsilicatesandcar-
technique is an ideal method to study how both processes bonaceous material to be significantly different over a large
influence each other. During the simulations, the individ- energy range. Weonly used thesilicate results here.
ual H atoms are followed as they land on thegrain surface, For the shielding of the radiation field, we followed
undergotheirrandomwalk,andevaporate.Iftwoatomsar- Draine & Bertoldi (1996), who derived the attenuation fac-
rive at the same position on the grain, they react to form tor, D :
λ
molecular hydrogen. As these processes occur, photons are
A
hitting the grain, leading to a pulsed temperature increase Dλ =exp −3.7 λ AV , (4)
and subsequent decrease for the smaller grains. The radia- (cid:18) A1000˚A (cid:19)
tion field and the absorption and emission coefficients used basedontheattenuationat1000˚A.Sincewepreferredtouse
iwnhtichhe stihmeuplaotsisoinbsleaerveeenxtpsla-inheodppiningS,ecetviaopno2r.atTiohne,taimndesdaet- AV instead of A1000˚A we took A1A00V0˚A =4.70 from Whittet
(2003) toarrive at the alternative expression
position of H atoms, and heating of the grain - occur are
randomlydeterminedusingtheratesoftheprocessesanda Dλ =exp −0.8AλAV . (5)
random number. The details of the Monte Carlo procedure (cid:16) AV (cid:17)
are discussed in Chang et al. (2005). We briefly summarise We then used the table of Aλ values in Mathis (1990) and
theminSection3,whichalsoexplainstheheatingandcool- AV
Whittet (2003).
ingofthegraininmoredetail.Section4containsourresults
TheleftpanelofFigure1givesthetotalfluxofabsorbed
for the fluctuations in temperature, while Section 5 shows
photonsP inphotonspersecondforgrainsoftwodifferent
their effect on the formation efficiency of H2 as well as the λ
sizes and at different visual extinctions as a function of en-
effects of grain temperature vs. grain size and immediate
ergy. The shape of thecurves for different grain sizes is the
grain desorption following formation of H2. Our results are sameforlowenergiesbecauseoftheconstantr3dependence
put into the context of the ISM in Section 6. Finally, our
stemming from the linear dependence of Q (λ). This re-
conclusions are discussed in Section 7. abs
lationship no longer holds in the high energy regime where
the curve shapes are different for different grain sizes. This
regime can be divided in two ranges: P depends on more
λ
2 THE PHOTON FLUX thanr3 fromapproximately1-7eVwhilethedependencyis
Therateofphotonsabsorbed byagrain dependsonitspo- lessstrongforenergieshigherthan7eV.Thegraphfurther
sition inthecloud andonitsopticalproperties. Weassume shows that dust attenuation mainly affects the high energy
the distribution of photons absorbed by the grain per sec- regime.
ond per wavelength interval as a function of wavelength to
begiven by
3 THE SURFACES AND THE MODEL
2
P =πr I Q (λ)D , (1)
λ λ abs λ
In our previous paper (Cuppen & Herbst 2005), we per-
with r the radius of the grain, I the interstellar radiation
λ formedourMonteCarlosimulationsoftherecombinationof
field in photons per unit area per second per unit wave-
hydrogenon differentsurfaces ofvaryingsurfaceroughness.
length, Q the wavelength-dependent absorption coeffi-
abs ThesesurfacesweregeneratedusingaseparateMonteCarlo
cient, and D the reduction factor due to attenuation of
λ simulationprogramstartingfromaflatsurfacewithasquare
theradiation field by dust.
gridtorepresentbindingsitesonagrain.Theresultingsur-
For the interstellar radiation field I we use different
λ faces were characterised by a matrix indicating the topo-
expressions in different regimes. In the high photon energy
logicalroughnessofeachlatticegrid.Weusedfourdifferent
regime,withwavelengths91.2-250nm,theapproximationof
surfaces:surface(a)iscompletelyflatwhilesurfaces(b),(c),
Sternberg (1988) is used. The lower photon energy regime,
and (d) have protrusions and ”holes”. In addition, surface
withwavelengths250nm-1cm,isdividedintofourdifferent
(c) has several features known as islands and surface (d)
regions using theexpressions from Zucconi et al. (2001).
is very rough with height differences of several monolayers
Atlowphotonenergies,theabsorptioncoefficient,Q ,
abs andnodistinctshapefortheprotrusions.Ifahydrogenatom
is approximated using
landsontheirregularsurface,itslocalhoppingandevapora-
8πr m2−1 tionenergyarecalculatedbasedonitsnumberofhorizontal
Qabs(λ)≈ λ Im(cid:18)m2+2(cid:19) (2) ‘grain’neighbours,i(seeFig.2inCuppen & Herbst(2005)).
Foreveryhorizontalneighbour,thereisanextrainteraction
(Bohren & Huffman 1983), where m is the complex refrac- energy EL. This method effectively results in surfaces with
tiveindex: amaximumoffivedifferentbindingsites.Thepresentpaper
uses the same concept to construct the three surfaces used
m≡n+ki. (3)
in the new simulations. Surface I has only one type of site
For the optical constants n and k, we used tabulated val- andistherefore flat.SurfaceIIhasaroughnesscomparable
ues for olivine from Ossenkopf et al. (1992). Since Eq. (2) to surface (c) in Cuppen & Herbst (2005), but we only dis-
is only valid for 2πr ≪ 1 we only used this approximation tinguish two types of sites. The first type of site has zero
λ
for Q for λ > 1000 nm. For smaller wavelengths, the r (i=0) horizontal neighbours while the second type, which
abs
(cid:13)c 0000RAS,MNRAS000,000–000
Monte Carlo simulations of H formation on stochastically heated grains 3
2
2
10
100 Av=0; a=0.1 µm
A=0; a=0.01 µm
v
A = 0.5; a=0.1 µm
v
-3 A=0.5; a=0.01 µm
-1m) 10 eV) 100 Avv = 1; a=0.1 µm
µ y ( Av=1; a=0.1 µm
-1s 10-6 rg
( e
P λ En 10-2
-9
10
-12 -4
10 10
-4 -2 0 0 0.2 0.4 0.6 0.8 1
10 10 10
X
Energy (eV)
Figure 1.(left)Theabsorbed photon fluxinphotons persecond per wavelength interval and (right)the relationbetween the random
numberandthephotonenergyfordifferentgrainsizes,r=0.01and0.1µm,andatdifferentvisualextinctions, AV =0,0.5and1(see
legend,whichappliestobothpanels).
canhaveoneormorehorizontalneighboursaccordingtothe
methodusedtogeneratethesurface,isstillassumedtohave
only oneneighbour(i=1).Surface IIIderivesfrom surface
(d) in Cuppen & Herbst (2005) and has fivedifferent types
of sites, with 0 to 4 horizontal neighbours.
Figure 2 shows representations of the non-flat sur-
faces II and III. Note that the colours used in this figure
indicate the different sites, in contrast with Figure 1 in
Cuppen & Herbst (2005) where the colour coding is used
to indicate the different heights. For surface II, the black
sitescorrespondtoi=0andthewhitesitestoi=1,whilst
forSurfaceIII,thelighterthecolour,thegreaterthenumber
of horizontal neighbours i. The hopping (E ) and evapora-
b
tion (ED) energies of these sites, labelled by thenumber of
neighbours i, are given by
Eb,i=Eb+iEL, (6)
ED,i=ED+iEL, (7)
respectively. The energies Eb, ED, and EL used here for H
and H2 are given in Table 1 for the different surfaces along
with the total number of different sites, itot and the dis-
tribution of these different sites ni. The lateral bond, EL,
is chosen to be 40% of the evaporation barrier, ED. Our
previous paper (Cuppen & Herbst 2005) showed that the
recombination efficiency for surface III is close to unity for
temperatures around 20 K. With this choice of the lateral
bond,wecandistinguishthreecases:zero,low,andhigheffi-
ciencyforthesurfacetemperaturesofinterestintheabsence
of fluctuations. Figure 2. The distribution of the olivine surface sites for (top)
Each MonteCarlo cyclestartsbycheckingwhichevent itot=2and(bottom)itot=5.Theblackareaindicatesthesites
will occur next. This is done by comparing the times at withi=0andthelightercoloursareforincreasingi.
which all future events occur: heating of the grain, cool-
ing of the grain, deposition of an H atom, or hopping or
evaporation of a particle on the surface. All these events
andtheircorrespondingtimesarediscussedinthefollowing
3.1 Desorption, hopping, and evaporation of H
threesections.Westartbyexplainingtheeventsinvolvedin
and H2
therecombination ofmolecular hydrogenwhile Sections3.2
and 3.3 contain discussions of the heating and the cooling The deposition rate of H atoms (s−1 site−1) is calculated
of the grain, respectively. from thegas abundanceusing
(cid:13)c 0000RAS,MNRAS000,000–000
4 H. M. Cuppen, O. Morata and Eric Herbst
Iftheparticle(H)landsontopofanotherparticle(Hor
Table 1. The parameters characterising the different surfaces
used. H2), it is checked if a reaction between the landing species
andspeciesalreadytherecanoccur.Ifso,theoldparticleis
I II III replacedbythenewmolecule,otherwisetheincomingparti-
cleremainsontopoftheoldonewithhoppingandevapora-
EEEbbD,,HH,H2((K(KK))1)12 232874732 232874732 232874732 ttiimoneroaftetsh,eRnhoepx,t0eavnednRt efovar,0t,hreesnpeewctoivrelnye.wInlybfootrhmceadseps,artthie-
ED,H2 (K)1 315 315 315 cleisdetermined.Weassumeinitially thatallH2 molecules
EL,H (K)3 149 149 thatareformed stayon thesurfaceuponreaction andlater
EL,H2 (K)3 126 126 evaporate. This corresponds to the parameter choice µ=1
ν (s−1)4 1012 1012 1012 introducedbyKatz et al.(1999),whereµindicatesthefrac-
itot 1 2 5 tion of molecules that remains on the surface immediately
n0 1 0.910 0.540 following reaction.
n1 0.090 0.258
n2 0.126
n3 0.052 3.2 Heating of the grain
n4 0.024
We consider photons in the wavelength range 91.2 nm-10
1 takenfromKatzetal.(1999). mm. The rate (s−1) of photons absorbed bya grain is
2 0.77ED,H2, where 0.77 is the ratio between Eb,H and ED,H 10mm
(Ruffle&Herbst2000).
R = P dλ (14)
34 EBiLha=m0e.4tEaDl.(2001) photon Z91.2nm λ
from which thetimebetween twophotonhitscan bedeter-
minedinamannersimilartoEqs.(10)and(11).Ifthegrain
vn(H)
R = (8) absorbsaphoton,theenergyE ofthephotonispickedfrom
dep
4ρ
thedistributioninEq.(1)usingarandomnumber.Theright
wheren(H)istheconcentrationofatomichydrogen,ρisthe panelinFigure1indicatestherelationbetweentherandom
surface site density, and v is the average velocity of the H number,Xλ,andthephotonenergy.Thisfigureisobtained
atoms in thegas: using therelation
λ
P dλ
v=r8kπTmgas, (9) Xλ = R91R.2pnhmotoλn (15)
withTgasthetemperatureofthegasandmtheatomicmass. which gives avaluebetween 0 and 1for a given wavelength
We use n(H) = 102 cm−3, Tgas = 60 K, and ρ = 2×1014 λ. This value is different for different grain sizes and dif-
cm−2(Biham et al.2001).Thetimebetweentwodeposition ferent visual extinctions. The r dependence is due to the r
events is determined using a random number X between 0 dependencein theabsorption coefficient.
The new temperature is then obtained bysolving
and 1, and thedeposition rate:
Tnew
ln(X)
∆t =− , (10) E = c(T)dT (16)
dep RdepNs ZTold
where Ns is the numberof sites on thegrain surface. for Tnew, where c(T) is the heat capacity. According
When a deposition event occurs, the deposition site is to Aannestad & Kenyon (1979), Draine & Li (2001), and
determinedusinganotherrandomnumber.Iftheatomlands Purcell(1976),olivinecanbeapproximatedbyaDebyesolid
onabaresiteofthegrain,itsimplylandsthereandthetime with a Debye temperature of 500 K. The low temperature
at which the particle will do its next event is determined. heat capacity in theDebyemodel is
This is donein a way similar todetermining ∆tdep: 12π4 T 3
c(T →0)= Nk , (17)
tevent =− ln(X) +t, (11) 5 (cid:16)TD(cid:17)
Rhop+Reva with N the number of atoms in the grain. If we assume
a density of 3.32 g cm−3 and a molecular mass of 153.3 g
where t is the current time in the simulation and R and
hop mol−1, theheat capacity in eV K−1 becomes
Reva arerespectively,thehoppingrateandtheevaporation
rate of the particle. These two rates are given by c(T)=61.38r3T3, (18)
R =νexp −Eb,i (12) with r in µm.
hop,i
(cid:16) T (cid:17) Every time the grain temperature changes, the time of
and the next event for each particle on the surface needs to be
adjusted from told totnew according tothenewtemper-
Reva,i =νexp −ED,i (13) ature. This proceevdenutre isedvoennteusing the equation
T
(cid:16) (cid:17)
respectively, where T is the temperature of the dust and ν tnew =(told −t) Rhop(Told)+Reva(Told) +t. (19)
is a trial frequency here chosen to be 1012 s−1 (Table 1). A event event Rhop(Tnew)+Reva(Tnew)
second random number is used to determine which event, SincetheratiobetweenRhopandRevaalsochanges,thetype
hoppingor evaporation, will occur at that time. of event that occurs at tnew must also be redetermined.
event
(cid:13)c 0000RAS,MNRAS000,000–000
Monte Carlo simulations of H formation on stochastically heated grains 5
2
50
0.005 µm 0.01 µm
40
30
20
10
K)
e (40 0.02 µm 0.05 µm
ur30
at
er20
p
m10
e
T 0.1 µm 0.2 µm
40
30
20
10
0
0 5×104 5×104 1×105
Time (s)
0 10 20 30 40
Temperature (K)
Figure3.Thetemperaturefluctuationsasafunctionoftimefor
differentgrainsizesandAV =0. Figure 4. The temperature distribution over time for AV = 0
and r = 0.01 µm. The lines indicate the modal temperature
(solid) and the two 99% levels (dashed). See the text for more
3.3 Cooling of the grain detail.
Thetemperatureofthegrainisrecalculatedatcertaintime
intervals using 40
35 A = 0 A = 0.5
V V
Told c(T) 30
∆t=− dT, (20)
Z dE 25
Tnew dt ) 20
K
15
where ddEt is given by ure ( 10
ddEt =4π πa2 Z ∞Qem(λ)Bλ(T)dλ (21) mperat 355 AV = 1 0.01 0.1
(cid:0) (cid:1) 0 Te 30
withQemtheemissioncoefficientandBthePlanckfunction. 25
We found that ∆t(s) = max 10r2/R ,30 with r in 20
photon
15
µm,isagoodtimeinterval,bec(cid:0)auseittakesinto(cid:1)accountthe
10
difference in photon flux and the temperaturefluctuations.
5
We integrate Eq. (21) numerically over the range 91.2 0
nm < λ 6 10 mm using Qem(λ) = Qabs(λ) and solve 0.01 r(µm) 0.1
Eq. (20) to obtain thenew temperature.
Figure5.Modaltemperatureofthegrainasafunctionofgrain
sizefordifferentvisualextinctions. Thebarsindicate99% confi-
4 TEMPERATURE FLUCTUATIONS dencelevels.
Before studying the formation of molecular hydrogen, we
first discuss the influence of the incoming photons on the fixed intervals chosen such that there were several photon
grain temperature. Figure 3 shows how the grain tempera- hits. Figure 4 shows a histogram of the temperature distri-
turefluctuatesasafunction of timefordifferent grain sizes bution so obtained for AV = 0 and r = 0.01 µm. The his-
at AV = 0. It clearly shows that although smaller grains togramclearlyhasamaximum,atwhichthetemperatureof
have fewer incoming photons per given time interval, the thegrain occursmostfrequently.Thismodeisindicatedby
temperature fluctuations are much larger, which is a con- asolid line, which is left of thecentreof thedistribution as
sequence of the r3 term in the heat capacity. For a grain couldbeexpectedconsideringtheshapeofthetemperature
size (radius) of 0.005 µm,fluctuations of up to 30 K occur, vs. time peaks. Figure 5 gives these modal grain temper-
whileforagrainsizeof0.05µm,analmostconstanttemper- atures as a function of grain size for AV = 0, 0.5 and 1.
ature is obtained. Furthermore, it can be noticed that the The bars indicate the 99% boundaries; i. e. the tempera-
minimumtemperaturereachedbeforeeachphotonhitisal- tures between which 99% of the determinations fall. These
most independentof theheight of thetemperatureincrease boundaries are indicated in Figure 4 by dashed lines and
given the rapid initial cooling. Draine (2003) shows similar are a measure of the amplitude of the fluctuations. Figure
graphsforcarbonaceousgrains,inwhichthetemperaturefor 5clearly showsthatthetemperaturefluctuationsarelarger
thegrainshassimilarfluctuationsbothinamplitudeandin for smaller grain sizes. All three temperature profiles show
modal temperature. a maximum, which is at rm = 0.02 µm for AV = 0 and
To obtain themodal grain temperatureand theampli- gradually moves to larger r for higher AV. The maximum
tude of the fluctuations, we determined the temperature at is sharper at AV ∼ 0, which indicates that the radiation
(cid:13)c 0000RAS,MNRAS000,000–000
6 H. M. Cuppen, O. Morata and Eric Herbst
40 1
0.005 µm 0.02 µm
Surface I
30 Surface II
Surface III
0.8
20
K)
ure (10 ency0.6
at ci
mper 0.1 µm 0 0.5 AV 1 1.5 effi0.4
e30
T
20 0.2
10
0
0 2 4 6 8 10 12 14 16 18 20 22 24
0 0 0.5 1 1.5 Temperature (K)
A
V
Figure7.Therecombinationefficiencyasafunctionoftempera-
Figure 6. Modal temperature of the grain as a function of AV tureforthethreedifferentsurfacesintheabsenceoffluctuations.
fordifferentgrainsizes. The closed markers all indicate simulations with µ = 1, while
forSurfaceIII,additionalsimulationsarerunanddesignated by
openmarkersforµ=0(uppercurve)andµ=0.3(lowercurve).
causing this peak is in the UV since this region is strongly
shielded at higher extinction. The effect is due to the r de-
pendence of the absorption coefficient in the high energy The definition of the recombination efficiency, η, used
range, as discussed in Section 2. in this paper is
Li & Draine(2001)showasimilargraph,wheretheav- η= 2NH2, (22)
erage temperatures of both graphite and silicate grains are NH
plotted for different radiation field intensities. Comparison
where NH is the number of hydrogen atoms that approach
lbeefttwpeaennetlhoefχFMigMurPe5=s1hocwursvtehfarotmthtehsahtagpreapohfbanotdhthcueruvpepseisr the surface in a given time interval and NH2 is the number
ofmolecules thatleavethesurface inthatinterval.Usually,
thesame,butthatthetemperaturesinFigure5areslightly
η is definedas
higher(1K).Thisisagoodagreementconsideringthediffer-
enceindetailusedinthemethods.Notethatinthelimited η= 2RH2, (23)
regime depicted by Li & Draine (2001) (r > 0.01 µm) the F
modal and average temperatures coincide. where RH2 is the rate with which H2 comes off the surface
Figure6givesthemodalgraintemperaturebutnowas and F is the flux of incoming hydrogen atoms. Since the
a function of AV for three different grain sizes. The three Monte Carlo method uses discrete numbers and not rates,
curves show very different behaviour. The modal temper- asinrateequationmethods,theevaporationratecannotbe
ature of the smallest grain remains constant as a function determined directly. We therefore use Eq. (22) which gives
of extinction but the amplitude of the fluctuations clearly thesame results as Eq. (23) in steady-state conditions.
decreases. The intermediate grain of radius 0.02 µm has a We first consider the efficiencies as a function of the
gradually decreasing modal temperature but theamplitude temperature for the three different surfaces introduced in
remains constant. It is interesting how the position of the Section 3. The results are shown in Figure 7 and were ob-
modeinthedistributionchangesfromthecentreforAv =0 tained with a simulation in which temperature fluctuations
to the lower temperatures for Av = 1.5. Finally, the 0.1 wereomitted.Thesizeofthegrainsislargeenoughthatthe
µm grain has almost no temperature fluctuations and the efficiencyisroughlyindependentofit(Chang et al.2005),a
curveshowsagradualdecreaseofthetemperatureforgrains regimeknownastheaccretion limit.Thesegraphswillhelp
deeper into the cloud. Notice the crossing of the curves for us to interpret the results we obtain from the simulations
0.02µmand0.1µmaroundAV =1,whichisaconsequence with temperaturefluctuations. The efficiency curvefor sur-
ofthegradualmovingofrmtolargervaluesforhighervisual face I, flat olivine, has one narrow peak at 8 K, surface II
extinction. (olivine with islands) has one broad peak (7-11 K) punctu-
ated by one dip and surface III (very rough surface) has a
very broad peak (6-18 K) with three dips. These dips are
due to hydrogen molecules, which occupy the high energy
5 EFFECT OF THE TEMPERATURE ON THE
sites (i > 0) and prevent hydrogen atoms from using these
EFFICIENCY
sitestomakenewmolecules.Atslightlyhighertemperatures
The previous section showed that not only the amplitude the formed molecules evaporate and the sites are available
of the temperature fluctuations is grain-size dependent but again. Since we use µ = 1 all molecules remain on the sur-
alsothemodaltemperature.Thismeansthatweshouldnot face after formation until they evaporate. In our previous
limit our study to sizes where fluctuations play a role but papers(Chang et al.2005;Cuppen & Herbst2005)weused
study a wider range of grain sizes in order to obtain the µ = 0 and only one peak was found in all cases. The open
complete size dependenceof therecombination efficiency. markers in Figure 7 give simulation results for µ = 0 and
(cid:13)c 0000RAS,MNRAS000,000–000
Monte Carlo simulations of H formation on stochastically heated grains 7
2
0.12 0.12
0.1 surface I surface II 0.1 surface I surface II
0.08 0.08
0.06 0.06
A = 0
0.04 Av = 0.5 0.04 0.005 µm
v 0.02 µm
cy 0.02 Av = 1 cy 0.02 0.1 µm
n n
efficie 0.8 surface III 0.01 0.1 efficie 0.8 surface III 0.5 1 1.5
0.6 0.6
0.4 0.4
0.2 0.2
0
0 0 0.5 1 1.5
0.01 0.1
r(µm) A
V
Figure8.Therecombinationefficiencyasafunctionofthegrain Figure 9. The recombination efficiency as a function of AV for
sizefordifferentsurfacesandvisualextinctions. differentsurfacesandgrainsizes.
µ = 0.3 on Surface III, where the µ dependence is signifi-
cant.Thevalueof0.3ischosensinceitisthevaluefoundby If we look at the larger grains, r > 0.05 µm, the fluc-
Katz et al. (1999) for H2 formation on olivine. Notice that tuations become even less salient and the efficiencies are as
for certain temperatures the difference in efficiency can be could be expected combining Figures 5 and 7. A grain of
aslargeasafactorof10.Thesedifferencesarehoweverlim- sizer>0.1µmatAV =1forexamplehasatemperatureof
itedtoverysmalltemperaturerangessothatthevalueofµ 14 K which would result in efficiencies of approximately 0,
only influences the average efficiency over the range within 0, and 0.65 for surfaces I, II, and III, respectively, without
a much smaller factor between thetwo extremes µ=0 and temperaturefluctuations.Figure8showsthatthisisindeed
µ=1.Theefficiencycurvesforµ=0.3andµ=1aremuch thecase.
closer. Weuse µ=1 for now. Figure 9 shows theefficiency again, but now as a func-
Figure 8 shows the recombination efficiency as a func- tion of visual extinction for three grain sizes, and should
tionofgrainsizewiththetemperaturefluctuationsdepicted be studied in combination with Figure 6. All graphs show
as in Figure 5 for the three different surfaces. With our as- an increase for increasing AV, except the 0.1 µm curve for
sumed surface site density of 2×1014 cm−2, we use array surfaceIII.Theseincreasesinefficiencyareduetothetem-
sizes of 25×25, 50×50, and 100×100 to represent the peraturedecreaseasafunctionofvisualextinctionforgrains
surfaces of grains of radii r = 0.005, 0.01, and 0.02 µm, re- of0.02µmandduetoadecreaseinfluctuationsforgrainsof
spectively. The efficiency at a given temperature for larger 0.005µm.Forgrainsof0.02µmwithsurfaceIII,thefluctu-
grains is in all cases independent of size. So we can use a ations gradually move inside the temperature range where
100×100 array for such grains as long as the temperature a non-zero efficiency is obtained. This occurs at AV = 0.7,
fluctuationsare headed correctly. which is the reason that for higher extinction the efficiency
Consider first the results for the smallest grains: r = hardly changes. The minimum in the 0.1 µm curve for sur-
0.005 µm. From Figures 5 and 6 we know that these grains faceIIIcorrespondstothedipintheefficiencycurve(Figure
haveamodalgraintemperatureof8K.AccordingtoFigure 7) around 15 K, since the temperature decreases from 17.5
7theefficienciesatthistemperatureareapproximately1.0, K to 13 K going from AV =0 to 1.5.
0.3 and 0.1 for surfaces I, II, and III, respectively, without Figure 10 shows the efficiency as a function of AV for
temperature fluctuations. Figure 8 shows however that for µ = 0 and 0.3 for surface III; it can be compared with the
thesesmallgrainstherecombinationefficiencyissimilar for lower left panel of Figure 9, which gives results for µ = 1.
all surfaces and rather low. The temperature fluctuations, We have showed already that the value of µ can have a
which occur frequently compared with the H-atom deposi- strong effect on the efficiency at specific small temperature
tioninterval,increasetheevaporationratewhichcausesthe ranges, but have also mentioned that on average the effect
averageresidencetimeofatomsonthesurfacetobereduced is much smaller. The low efficiency ranges with µ = 1 are
to such an extent that the efficiency is much lower and the around 8 and 11 K, at which newly formed H2 molecules
atomsarenolongeravailablewhenthegraincoolsdown.As interfere with subsequent reactions. Although the grains of
thegrainsizebecomessomewhat larger,themodaltemper- 0.005µmhaveamodaltemperatureof8K,thefluctuations
ature rises and theefficiency actually decreases for surfaces causeH2toevaporateeasily,resultinginaloweffectofµon
I and II,showing therise in themodal temperatureto out- the efficiency. For the other two grain sizes we see a visible
weigh the decrease in fluctuations. but still small effect due to variations in µ. This could be
(cid:13)c 0000RAS,MNRAS000,000–000
8 H. M. Cuppen, O. Morata and Eric Herbst
1 canbeassumedtobeclosetoone(Buch & Czerminski1991;
µ = 0 µ = 0.3
Al-Halabi et al. 2002). Eq. (25) can besimplified to
0.8
ency0.6 R= nαHrT3g0a0s, (26)
i
fic0.4 0.005 µm where
ef 0.02 µm
0.2 0.1 µm α=c 600πk rmaxη(r)r−1.5dr. (27)
r m Z
0 rmin
0 0.5 1 0.5 1 1.5
A Note that in the standard treatment where all grains are
V 0.1 µm in size and η = 1, α = 5×10−17 cm3 s−1. This is
in agreement with the estimate of R = 10−17 cm3 s−1 for
Figure 10. The efficiency as a function of the visual extinction diffuse-cloud conditions given by Jura (1974), which corre-
fordifferentvaluesofµforsurfaceIIIanddifferentgrainsizes. spondswith α≈2×10−17 cm3 s−1.
Figure 11 shows η(r)r−1.5 as a function of the grain
expected considering that the modal temperatures (12-20 size using the data from Figure 8. For surfaces I and II
K) do not fall in thelow efficiency dips. the maxima for the efficiency and r−1.5 coincide at rmin,
which obviously leads to a maximum for the smallest grain
sizesused.SurfaceIII,however,hasitslargestefficienciesfor
the largest grains. This results in a minimum for η(r)r−1.5
6 CONSEQUENCES FOR THE DIFFUSE ISM
around r =0.01 µm and a maximum at intermediate grain
In the previous sections, we showed that the amplitude of sizes, depending on AV. Figure 11 clearly shows that the
thetemperaturefluctuationsandthemodaltemperatureof higherefficiencyforlargegrainsisofalmostnoimportance.
grainsarestronglysizedependent,andthatbothquantities For flat grains and those with a minimal roughness, the
havea strong effect on theefficiency of molecular hydrogen overall H2 formation is mainly determined on grains with
formation on olivine. While for the large grains the type of r 6 0.01 µm, while for grains with a high roughness, the
surface also has a strong effect on the efficiency, the rough- intermediate range of 0.01 6 r 6 0.08 µm is most impor-
nessofthesurfaceseemstohavelittleeffectforthesmallest tant. Since surfaces I and II have their maximum value for
grains,sinceforthosegrainsthereisenoughthermalenergy η(r)r−1.5 atrmin,thechoiceforthelowerlimitingrainsize
for atoms to hop and evaporate even from the high energy distribution influences the value for α. Generally the min-
sites because of thestrong temperaturefluctuations. imum size for olivine grains is considered to be r = 0.005
Thissectiondiscussestheconsequencesofthesefindings µm (Mathis et al. 1977; Draine & Lee 1984), which is the
for molecular hydrogen formation in the diffuse interstellar valuewetake.Li & Draine(2001)concluded,however,that
medium. Usually, formation rates for molecular hydrogen up to ∼ 10% of interstellar Si could be in r 6 15 ˚A sili-
arebasedontheassumptionthatallgrainshaveastandard cate grains. We did not consider these ultra-small particles
size of 0.1 µm. In reality, there is a wide range of differ- inourcalculations because,inourview,suchparticlesneed
ent sizes in the ISM. The distribution of these sizes highly to be treated as a special case for several reasons: (i) they
favours small grains and can be approximated by a power consist of up to only a few hundred molecular units and
law (Mathis et al. 1977): therefore may not retain their bulk properties, such as the
ng(r)=cr−3.5, (24) heatcapacitythatweusetocalculatetheheatingandcool-
ing; (ii) the small number of units on the surface means
whereng(r)dristheconcentrationofgrainswithradiiinthe that our smooth and rough topologies are imprecise, (iii)
range(r,r+drincm−3).Draine& Lee(1984)foundavalue theefficiency of stickingbyweakly boundadsorbates isun-
c=1.51×10−25nH cm−21 for the proportionality constant likely to be high (Herbst & Dunbar 1991) and it is likely
using rmin = 0.005 µm and rmax = 0.25 µm (Mathis et al. that chemisorption is needed for high sticking efficiencies;
1977)fortheminimumandmaximumradii.Thegas-to-dust and(iv)strongadsorbate-substratebondsmaybenecessary
numberratioisfoundbyintegratingEq.(24)overtherange forHatomstowithstandtheverylargetemperaturepulses
and dividing by nH to be 2.91×109, which is significantly that photons will cause. Such considerations will also be
lowerthantheusually-assumedvalueof1012,anumberthat necessarytotreatmolecularhydrogenformationonPAH’s.
onlyconsiders0.1µmgrainsandthereforeneglectsthelarge Table2showsournumericallyintegratedαcoefficients.
portionofsmallgrains.Asomewhathighervalueof6.0×109 Note that the efficiencies used for the integration are only
was obtained by Lipshtat et al. (2004) who only considered determinedforonegas temperatureandonegasconcentra-
grains down to 0.04 µm in size. tionofH.Wedidnotcheckforanadditionaldependenceof
From the grain size distribution we can calculate the the efficiency on these two quantities. Considering that the
ratecoefficient Rincm3 s−1 formolecularhydrogenforma- efficiencies used to obtain the values in Table 2 are much
tion: lower than unity,the α coefficients are surprisingly close to
R= rrmmianx 21ng(r)vη(r)ξπr2dr, (25) 5nu×m1b0e−r1o7fcsmm3alsl−g1r.aBinustgwiveesmausvterreymleamrgbeesrutrhfaactethareealarfgoer
R nH reaction. Indeed, if we consider a constant high efficiency
whereξ is thestickingcoefficient of theatoms to thegrain. of one over the whole grain size range, an α coefficient of
For the low temperatures used here the sticking coefficient 1.53×10−16 cm3 s−1 isfound.Ofcourseourcomputedval-
(cid:13)c 0000RAS,MNRAS000,000–000
Monte Carlo simulations of H formation on stochastically heated grains 9
2
2.0×108 Table 3.αcoefficientincm3 s−1 ataconstanttemperature.
surface I surface II
1.5×108 AV =0 AV =0.5 AV =1
surfaceI <1.72×10−22 <1.72×10−22 <3.82×10−22
1.0×108 A = 0 surfaceII <2.85×10−22 3.28×10−20 1.41×10−18
Av = 0.5 surfaceIII 1.03×10−16 1.04×10−16 9.75×10−17
-1.5m)5.0×107 Avv = 1
c 17.6, 15.3, and 13.9 K for AV = 0, 0.5 and 1, respectively.
-1.5r ( surface III 0.01 0.1 Itnemthpiesrwatauyr,ewaencdanstosecphaarsatitcehoueatttinhgeeoffnecsmtsaolfltghreailnosw. NmoodHa2l
)
η(r2.0×108 was formed within our simulation time for the completely
flat surface I and slightly rough surface II at AV = 0; we
can therefore only give an upper limit for these cases. The
1.0×108 highervaluesinthecaseof thestochasticheating(Table2)
stemfromthelowermodaltemperatureforthesmallgrains.
Surface III, on the other hand, has a small increase in the
formation rate at constant temperature due to the higher
0.01 0.1 efficiency for small grains in the absence of the fluctuation
r(µm)
in temperature. Here the increase in modal temperature is
notasimportantbecauseofthelargetemperaturerangefor
Figure 11. The recombination efficiency × r−1.5 (or integrand high efficiency.
in Eq. (27)) as a function of the grain size for different surfaces
andvisualextinctions.
7 CONCLUSIONS
Table 2.αcoefficientincm3 s−1. Previouscalculations oftherateofmolecular hydrogenfor-
mation on interstellar grain surfaces in diffuse clouds have
AV =0 AV =0.5 AV =1 ignored the subtle role played by radiation in determin-
ing the surface temperature and its fluctuations as func-
surfaceI 6.50×10−19 1.99×10−18 2.91×10−18
surfaceII 5.08×10−19 1.51×10−18 3.20×10−18 tions of grain size. As shown by earlier investigators (e.g.
surfaceIII 3.12×10−17 4.75×10−17 5.62×10−17 Draine & Li(2001))andconfirmedhere,thetemperatureof
surfaceII1 1.32×10−18 5.42×10−18 2.74×10−17 grains fluctuates strongly for smaller grains (r < 0.02 µm)
but not for larger ones, and the modal temperature peaks
1 ED isincreasedby35K. for grains in the middle range of size (0.02 - 0.05 µm). We
have shown that the efficiency of formation of H2 depends
uesarelower.FromTable2,wecanseethatonlyourrough bothonthemodalsurfacetemperatureanditsfluctuations.
surfaceIIIisabletoproduceasufficientamountofmolecular According to the type of material and its smoothness or
hydrogen at AV = 0, while the other surfaces are possible roughness, each surface has a range of temperature where
alternatives, with α a factor of six below Jura’s value, at the efficiency is high. For olivine grains, which are consid-
AV =1, which pertains at more internal regions for diffuse ered here, a smooth surface has only a very small range of
clouds.Thisdifferenceiscausedbytheintermediategrains, surface temperature for high efficiency 6-9 K - while a very
which have a very low efficiency for surface I and II, but rough surface, with half of its binding sites different from
haveappreciable valuesfor surface III. the norm, allows efficient H2 production up to about 20
The low efficiency of surfaces I and II for intermediate K. For small grains, photons pulse the temperature above
grainsisdueultimatelytothelowevaporationenergyforH the range of efficiency, thus reducing it, although the low
given by Katzet al. (1999). In their paper they do indicate modal temperature aids H2 production, especially for flat
thattheirstated energy ismore precisely only alower limit andmoderatelyrougholivinegrains.Thenetresultforthese
within a range of 3 meV. Using a higher value results in twotypesofsurfacesisanenhancementintheefficiencyfor
larger αcoefficients for surfaces I andII sinceH evaporates smaller vs. larger grains although the overall efficiency re-
more slowly, and makes these surfaces more suitable candi- mainslowatallsizes.Therelativelyhighefficiencyforsmall
dates for efficient molecular hydrogen formation, especially grains is augmented by the high overall surface area given
at non-zero extinction. The last row in Table 2 gives the α thedistribution of grain sizes, so that theproduction of H2
coefficients for surface II if the value of ED is increased by is dominated by the smaller grains. For very rough grains,
theextra3meV(35K,anincreaseof9%.)Weseethatthere on theother hand, theefficiency is greater for larger grains
isaclearincreaseintheαcoefficient ofafactorofafewfor sincefluctuationsareminimalandthemodaltemperatureis
low AV that gets larger as the extinction increases, so that stillwithintherangeofhighefficiency,whileitisabovethis
for AV >0.5 the rate coefficient is high enough to produce range for smoother surfaces. Here the countervailing larger
asufficient amount ofmolecular hydrogenin diffuseclouds. surface area for smaller grains results in a peak H2 produc-
Table 3 gives values of the α coefficient for simulations tion for grains of intermediate sizes.
performedataconstanttemperature.Wechoosethemodal Acalculation oftheratecoefficient Rforhydrogenfor-
temperaturesofthestandardgrainofr=0.1µm,whichare mation in diffuse clouds for which grains from a minimum
(cid:13)c 0000RAS,MNRAS000,000–000
10 H. M. Cuppen, O. Morata and Eric Herbst
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8 ACKNOWLEDGEMENTS
E. H. wishes to thank the National Science Foundation (U.
S.)forsupportofhisresearchprogrammeinastrochemistry.
Theauthorswishtothanktherefereeforhiscarefulreading
of the manuscript.
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