Effect of surface H₂ on molecular hydrogen formation on interstellar grains

Molecular hydrogen is the most abundant molecules in astronomical sources. Moreover, gas-phase H₂ is crucial for the subsequent formation of the molecular ion H${}^{+}_{3}$, which triggers the rich gas-phase interstellar chemistry (Williams, 1998; Herbst, 2001). Therefore, the H₂ formation mechanism has been an interesting and important research topic since its discovery. Although almost all interstellar molecules were believed to be formed in the gas phase in astrochemical models, H₂ formation in the gas phase has been found to be inefficient while interstellar dust grains can play an important role in increasing the efficiency with which H atoms convert to H₂ molecules (Gould & Salpeter, 1963).

However, it is not straightforward to explain H₂ formation in astronomical sources even when the catalytic roles of dust grains are introduced into models. Interstellar species are believed to be formed on cold grain surfaces via the so called Langmuir-Hinshelwood mechanism (Watson & Salpeter, 1972; Pickles & Williams, 1977; Hasegawa et al., 1992). To form H₂, H atoms accrete on dust grains and then bind weakly with surfaces, which is known as physisorption. They can overcome the diffusion barrier and move on the grain surfaces via quantum tunneling or thermal hopping. However, laboratory and theorectical studies showed that quantum tunneling does not contribute much to the mobility of H atoms on silicate, carbonaceous or solid amorphous water (ASW) surfaces (Pirronello et al., 1997, 1999; Katz et al., 1999; Nyman, 2021). If H atoms encounter other H atoms, then H₂ molecules are formed. But H atoms can also desorb and leave grain surfaces. A hydrogen atom must reside on a grain long enough to find a partner H atom to form H₂. As the dust temperature increases, the H atom desorption and diffusion rates also increase. So the temperature of grain surfaces must be sufficiently low so that a H atom can encounter another one before it desorbs. On the other hand, the temperature of grain surfaces must be high enough so that H atoms can diffuse on the grain surface. The parameter that measures how strongly species are to bound to grain surfaces is called desorption energy. It was found that if we assume a single H desorption energy in models, the dust temperature range over which efficient H₂ formation occurs is narrow (6-10 K for olivine grains) (Katz et al., 1999). Moreover, it was found that the highest dust temperature at which H₂ can be formed efficiently is less than 17 K (Katz et al., 1999). However, the grain surface temperature in the unshielded diffuse clouds, where hydrogen molecules are believed to be efficiently formed, is around 20 K (Li & Draine, 2001).

In order to make the dust temperature range over which efficient H₂ formation occurs wider, H₂ formation models adopted at least two types of sites, the stronger and weaker binding sites, on grain surfaces (Hollenbach & Salpeter, 1971; Chang et al., 2005; Cuppen & Herbst, 2005; Iqbal et al., 2012). In these models, the stronger binding sites are able to hold H atoms on grain surfaces long enough so that other H atoms can encounter them to form H₂ before they desorb even when the dust temperatures are high. On the other hand, H atoms in the weaker binding sites can diffuse to encounter other H atoms even when the dust temperatures are low. These models are able to significantly enlarge the dust temperature range over which efficient H₂ formation occurs. The upper limit of dust temperatures at which the efficient conversion from H to H₂ occurs depend on the strongest binding sites on grain surfaces. As the desorption energy of the strongest binding sites increases, this upper limit also increases. The recent review by Wakelam et al. (2017a) further summarizes H₂ formation models, experiments and observations.

Because H₂ molecules cannot react with H atoms on grain surfaces, the impact of the existence of surface H₂ was not considered in earlier H₂ formation models (Hollenbach & Salpeter, 1971; Biham et al., 2001; Chang et al., 2005; Cuppen & Herbst, 2005; Iqbal et al., 2012). However, studies using astrochemical models for the cold ($\sim 10$ K) and high density (H nucleus densities $>10^{12}$ cm^-3) sources showed that in order for the modeling results to be physical, the H₂ desorption energies in binding sites occupied by H₂ must be much lower than that on water ice (Garrod & Pauly, 2011; Hincelin et al., 2015). If a chemical model adopts a single H₂ desorption energy and the H₂ desorption energy on water ice is used in the model, then gas-phase hydrogen molecules would be heavily depleted onto dust grains in the chemical model (Hincelin et al., 2015), which has not been observed. To solve this problem, Hincelin et al. (2015) suggested the encounter desorption (ED) mechanism in their models. The ED mechanism assumes that when a hydrogen molecule hops into a site occupied by another H₂ molecule, the desorption energy for the hopping H₂ molecule becomes the desorption energy of H₂ on H₂ substrate, which is much lower than that on water ice surfaces. So, surface H₂ can desorb much more efficiently even around 10 K if the ED mechanism is included in models. As a result, models predict that the abundances of surface H₂ in cold and high density sources are only no more than a few monolayers (Hincelin et al., 2015). In a recent study, Chang et al. (2021) extended the H₂ ED to the H and H₂ ED, which assumes that when H atoms encounter H₂ on grain surface, their desorption energy becomes the H atom desorption energy on H₂ substrate, which is much lower than that on water ice surfaces. So, the H and H₂ ED mechanism enhances the desorption rate of surface H, thus, it becomes more difficult to hydrogenate CO on grain surfaces. So, the production of a few surface species such as methanol was strongly reduced if the model adopted the H and H₂ ED mechanism.

The desorption rate of H atoms should also increase if we consider H and H₂ ED mechanism in H₂ formation models because the desorption energy of H on H₂ substrate is much lower than that on bare dust grain surfaces. On the other hand, the existence of H₂ on grain surfaces may affect H₂ formation in another way. The efficient H₂ formation at higher temperatures relies on the existence of stronger binding sites on grain surfaces. If we consider physisorption only, the stronger binding sites are those where surface species can see more horizontal neighbors, which are the upward step edges on rough surfaces (Cuppen & Herbst, 2005). Therefore, a stronger H binding site must also be a stronger H₂ binding site if we consider physisorption only. After a hydrogen molecule occupies a stronger binding site, this site becomes a weaker binding site for two reasons. First, for the H atom in this site, the vertical neighbor below it becomes H₂. Secondly, the H atom may not see horizontal neighbors, thus loses lateral bonds with it neighbors because a H₂ molecule fills the vacancy. We refer to Cuppen & Herbst (2005) for details of how horizontal and vertical neighbors affect the desorption energies of surface species.

The impact of the existence of surface deuterium molecules on D₂ formation has been studied in laboratory (Gavilan et al., 2012). It was found that D₂ molecules formed more efficiently on silicate surfaces covered by D₂ than on bare silicate surfaces. Gavilan et al. (2012) argued that the reason was that D atoms diffuse more quickly on surfaces covered by D₂. However, the flux of D atoms in their experiments was more than seven orders of magnitude larger than the H atom flux in diffuse clouds (Biham et al., 2001). As the H atom accretion flux significantly decreases, the H atom population on grain surfaces also drops a lot. Because H₂ formation requires at least two H atoms on grain surfaces, it is not clear if the existence of surface H₂ can still help to increase H₂ formation efficiency on grain surfaces.

In this paper, for the first time, we investigate H₂ formation by the recombination of physisorbed H atoms on interstellar dust grains using models that consider the existence of H₂ on grain surfaces. Although our major purpose is to study H₂ formation in diffuse clouds, we also investigate H₂ formation in translucent clouds, where typically $25\%$ H nucleus are in the form of H atoms (Zuo et al., 2018). These H atoms will also be converted to H₂ as translucent clouds gradually become denser and form dense dark clouds (Zuo et al., 2018).

The organization of the paper are the follows. Our models are explained in Section 2 while the numerical method is introduced in Section 3. Our results are shown in Section 4. Finally, Section 5 is the section for the discussions and conclusions.

Based on the work by Biham et al. (2001), we assume in our models that the H nucleus density of diffuse clouds is 10 cm^-3 while the gas-phase temperature is 100 K. Although both olivine and carbon grains exist in diffuse clouds, we only study the H₂ formation efficiency on silicate-composed olivine grains in diffuse clouds. The effect of H₂ existence on molecular hydrogen formation on carbon grains will be briefly discussed in Section 5. Assuming the gas-phase species is composed of H atoms only, the H atom accretion flux is $f1=1.8\times 10^{-9}$ ML s^-1 in diffuse cloud models (Biham et al., 2001). Diffuse clouds are those in which more than $10\%$ H nuclei have been converted to H₂ (Wakelam et al., 2017a), so, we assume $75\%$ or $50\%$ H nuclei exist in the form of H atoms in diffuse cloud models. The purpose of this work is to study how the existence of surface H₂ may affect only the physisorbed H atom recombination efficiency. Therefore, H₂ formation scenarios involving chemisorption are ignored in models.

In translucent clouds, the gas temperature is 15-40 K, the H nucleus density is between 500 and 5000 cm^-3 while dust grains are believed to be covered by water ice (Wakelam et al., 2017a). In translucent cloud models, we therefore assume that the H nucleus density is 1000 cm^-3 and the gas temperature is 30 K. In a typical translucent cloud, $75\%$ of the H nuclei have been converted to H₂ (Zuo et al., 2018). The H atom accretion flux in these translucent clouds is found to be $f2=1.3\times 10^{-8}$ ML s^-1.

Diffusion barrier and desorption energy are the key parameters that determine the rates of surface species diffusion and desorption respectively. The desorption energy of H₂ on amorphous silicate surfaces, $E_{D1_{H_{2}}}$, was found to follow a continuous distribution based on experimental analysis (He et al., 2011). On the other hand, He et al. (2011) found the value of $E_{D1_{H_{2}}}$ can be represented by three discrete values, 35, 53 and 75 mev. The lowest value, 35 mev is about $40\%$ lower than the intermediate value, 53 mev while the highest value, 75 mev is about $40\%$ higher than the intermediate value. Hama et al. (2012) experimentally studied the diffusion barrier of H atom, $E_{b2_{H}}$ on ASW surfaces and found that $E_{b2_{H}}$ should be represented by at least three discrete values, $<18$ mev, 22 mev and $>30$ mev. The lowest value, $<18$ mev is more than $18\%$ lower than the intermediate value, 22 mev while the the highest value, $>30$ mev, is about $40\%$ higher than the intermediate value.

Since the aforementioned experimental findings suggest that three categories of binding sites can represent all the various binding sites on surfaces, we assume the desorption energies of a surface species on dust grain without H₂ existence always take three discrete values in models. The lowest value is $40\%$ lower than the intermediate one while the highest value is $40\%$ higher than the intermediate one. Hereafter, following nomenclature by Hama et al. (2012), the binding sites with the lowest, intermediate and highest desorption energy are called the shallow, middle and deep potential sites. For simplicity, the number of shallow, middle and deep potential sites are assumed to be equal on grain surfaces in our models. So, the desorption energy of the middle potential sites is the average desorption energy of all binding sites on grain surfaces. The three discrete desorption energies are taken from literature if they are available. For instance, we adopt the three discrete H₂ desorption energies on silicate surfaces, which were suggested by He et al. (2011) in our models. On the other hand, if only a single desorption energy for a surface species is available in literature, we assume that single value to be the desorption energy in the middle potential sites.

In a recent review (Wakelam et al., 2017a), the suggested desorption energies of H atoms on silicate and ASW surfaces are 44 and 50 mev respectively. These two values are adopted for the H atom desorption energies in the middle potential sites for the silicate and water ice surfaces respectively. The desorption energy of H₂ on water ice surfaces varies much in literature. We adopt the commonly used value of $E_{D2_{H_{2}}}=440$ K (Cuppen & Herbst, 2007). Moreover, in order to study the effect of $E_{D2_{H_{2}}}$ on recombination, we also adopt a much larger value, $E_{D2_{H_{2}}}^{{}^{\prime}}=800$ K (Wakelam et al., 2017b). These two values are assigned to the desorption energy of H₂ in the middle potential sites on water ice surfaces in translucent cloud models.

The suggested diffusion barrier of H atoms on the silicate surfaces is 35 mev (406 K) (Wakelam et al., 2017a). So, the ratio of the H atom diffusion barrier to desorption energy is $R=35/44\sim 0.8$. We assume R is fixed to be 0.8 for the shallow, middle and deep potential sites on silicate surfaces. Moreover, because the diffusion barrier of H₂, $E_{b1_{H_{2}}}$, on the silicate surfaces is not available in literature, we assume $E_{b1_{H_{2}}}=0.8E_{D1_{H_{2}}}$. Therefore, R is 0.8 for all species on silicate surfaces in our models.

Hama et al. (2012) found that diffusion barrier of H atoms in the middle potential sites on ASW surfaces is 22 mev (255 K). So, on the water ice middle potential sites, R is $22/50\sim 0.44$. We assume R=0.44 for both H and H₂ on all water ice binding sites in our models. The H atom diffusion barrier on the shallow water ice sites is $40\%$ lower than that in the middle potential sites. The adopted energy value is 13.2 mev (153 K), which is lower than its upper bound suggested by Hama et al. (2012). The adopted H atom diffusion barrier on the deep potential sites is 30.8 mev (357 K), which is larger than its lower bound suggested by Hama et al. (2012). The diffusion barrier of H₂ on water ice middle potential sites is, $E_{b2_{H_{2}}}=0.44E_{D2_{H_{2}}}$ or $E_{b2_{H_{2}}}^{{}^{\prime}}=0.44E_{D2_{H_{2}}}^{{}^{\prime}}$.

Table LABEL:table1 summarizes the desorption energies and diffusion barriers on the shallow, middle and deep potential sites, which are not occupied by H₂, in our models.

The desorption energies of H₂ and H on H₂ substrate are much lower than these on silicate or ASW surfaces. At 10 K, the desorption energy of H on H₂ substrate was calculated to be $E_{D_{HH_{2}}}=45$ K (Pierre et al., 1985; Vidali, 1991; Cuppen & Herbst, 2007). Cuppen & Herbst (2007) estimated that the desorption energy of H₂ on H₂ substrate is, $E_{D_{H_{2}H_{2}}}=23$ K. Recently, by performing quantum chemical calculations, Das et al. (2021) found that $E_{D_{H_{2}H_{2}}}$ is between 67 K and 79 K while $E_{D_{HH_{2}}}$ is between 23 K and 25 K. Moreover, if a site has been occupied by H₂, a H atom in this site may not be able to see horizontal neighbors, thus cannot form lateral bonds with its horizontal neighbors. So, the desorption energy of the surface species is further reduced when the existence of H₂ is considered. Fig. 1 illustrates lateral bonds formed between H atoms and their horizontal neighbors in different binding sites.

Not only H₂ molecules but also H atoms can enhance the desorption rates of H₂. Cuppen & Herbst (2007) estimated that the desorption energy of H₂ on H substrate is, $E_{D_{H_{2}H}}=30$ K, which is also significantly lower than that on silicate surfaces. In all our models, if H₂ molecules resides in sites occupied by H atoms, the desorption energy of H₂ on these sites become 30 K.

Models M01-M06 are used to study H₂ formation on the olivine grains in diffuse clouds. Model M01 is a reference model that is similar to earlier models (Chang et al., 2005; Cuppen & Herbst, 2005). Gas-phase H₂ are allowed to accrete on grain surfaces in models M02-M06, but not in model M01. When a exothermic chemical reaction occurs on grain surfaces, products may desorb immediately, which is known as chemical desorption (Garrod et al., 2007). The chemical desorption coefficient, $\mu$, measures the percentage of products that desorb immediately after a reaction fires. In the reference model M01, $\mu$ is set to be 1. In model M02, the desorption energies of H₂ and H on H₂ substrate are 23 K and 45 K respectively. Moreover we assume a surface species loses all lateral bonds with its horizontal neighbors if the surface species is in a site already occupied by H₂. The chemical desorption coefficient is set to be 1 in model M02. To study the effect of $\mu$ value on H₂ formation efficiency, we simulate model M03, which is similar to model M02, but $\mu$ is set to be 0.

Because both $E_{D_{H_{2}H_{2}}}$ and $E_{D_{HH_{2}}}$ are not well known yet, we simulate models M04 and M05 to study how the value of $E_{D_{H_{2}H_{2}}}$ and $E_{D_{HH_{2}}}$ may affect the H₂ formation efficiency. These two models are similar to model M02. The difference between models M02 and M04 is that the desorption energies of H and H₂ on H₂ substrate are set to be 24 K and 73 K respectively in model M04 based on the energy values suggested by Das et al. (2021). Since the residence time of H atoms on dust grains increases with $E_{D_{HH_{2}}}$, H₂ formation efficiency at higher temperatures should increase as $E_{D_{HH_{2}}}$ becomes larger. We would like to set the value of $E_{D_{HH_{2}}}$ to be as high as possible in model M05. However, to the best of our knowledge, the upper bound of $E_{D_{HH_{2}}}$ is not available in literature. We argue that the desorption energy of H atoms on the silicate shallow potential sites may be a upper bound for $E_{D_{HH_{2}}}$ for the following reasons. Hydrogen atoms in sites occupied by H₂ can hardly form lateral bonds with their horizontal neighbors in sites occupied by H₂ as shown in Fig. 1. So, even if the vertical bond formed between H atoms and H₂ substrate is as strong as that between H atoms and silicate surfaces, $E_{D_{HH_{2}}}$ cannot be higher than the desorption energy of H atoms on the silicate shallow potential sites. Similarly, the desorption energy of H₂ on the silicate shallow potential sites may be a upper bound for $E_{D_{H_{2}H_{2}}}$. In model M05, both $E_{D_{H_{2}H_{2}}}$ and $E_{D_{HH_{2}}}$ are set to be their upper bounds.

In models M02-M05, deep H atom potential sites must also be deep H₂ potential sites. This assumption is based on our current understanding about how stronger binding sites are formed (Cuppen & Herbst, 2005), which might be limited. Therefore, we also simulated model M06, in which deep potential sites for H atoms cannot be deep potential sites for H₂ molecules. The deep H atom potential site can be a middle or shallow potential site for H₂ molecules with equal probability. Similarly, it is equally likely that a deep H₂ molecule potential sites is a middle or shallow potential site for H atoms. In models M02-M06, the ratio of a surface species diffusion barrier to its desorption energy on sites occupied by H₂, R${}_{H_{2}}$ is fixed to be 0.8. Moreover, the ratio of H₂ diffusion barrier to its desorption energy on sites occupied by H atoms, R_H is also 0.8.

Models M11-M15 are used to study H₂ formation on dust grains in translucent clouds. In all these models, $\mu$ is set to be 1. The model M11 is a reference model in which H₂ cannot accrete while H₂ are allowed to accrete in models M12-M15. The desorption energies of H and H₂ on sites occupied by H₂ are 45 K and 23 K respectively in models M12 and M13. In model M12, R${}_{H_{2}}$ and R_H are fixed to be 0.44, which is the same as the value of R on water ice binding sites without H₂. These two ratios are 0.8 in model M13 to study how R${}_{H_{2}}$ and R_H may affect the recombination efficiency. The H₂ desorption energy on water ice is 440 K in models M11-M13. To study how the H₂ desorption energy on water ice may affect the recombination efficiency on grain surfaces, we simulated models M14 and M15. The H₂ desorption energy is 800 K in these two models. In model M14, R${}_{H_{2}}$ and R_H are 0.44 while in model M15, these two ratios are 0.8.

Our purpose is to make the reference models to be as close to the earlier models (Chang et al., 2005; Cuppen & Herbst, 2007) as possible for comparison, so, some of the assumptions in the earlier models are kept in our models. The H atom quantum tunneling effect was not considered in the earlier models because H atoms diffuse mainly by thermal hopping on silicate or ASW surfaces, so this effect is also ignored in our models. The rate of hopping is $\nu\exp(-E_{b}/T)$, where $E_{b}$ is the diffusion barrier, T is the dust grain temperature, the pre-exponential factor $\nu$ is $10^{12}$ s^-1. The rate of desorption is $\nu\exp(-E_{D}/T)$ where $E_{D}$ is the desorption energy. When H atoms land on empty sites or sites occupied by H₂, the sticking coefficient is 1. However, H atoms are not allowed to accrete on sites already occupied by H atoms. On the other hand, we assume the H₂ sticking coefficient is always 1 regardless of the landing sites.

Table LABEL:table2 summarizes models in this work.

In this work, we use the microscopic Monte Carlo (MMC) method to perform model simulations. The MMC method is a rigorous approach that can treat the finite size effect. In this section, we only briefly introduce this method and refer to Chang et al. (2005) and Chang & Herbst (2012) for details.

To perform MMC simulations, binding sites on grains are put on a $L\times L$ square lattice, where L is the square root of the total number of sites on the grain. Previous studies show that L=100 is already large enough to eliminate finite size effect (Chang et al., 2005), so in this work, L is fixed to be 100. Each site has four nearest neighbor sites on the square lattice. Gas-phase species can accrete and become surface species on the lattice. We keep track of the position of each surface species, which hops from one site to one of its nearest neighbor. A chemical reaction occurs when two reactive species are in the same site on the lattice. Other than hopping, surface species can also desorb. To decide whether a surface species hops or desorbs, we generate a random number, X, that is uniformly distributed between 0 and 1. Assuming the hopping and desorption rates of the surface species are $b_{1}$ and $b_{2}$ respectively, the species hops if $X<\frac{b_{1}}{b_{1}+b_{2}}$; otherwise, it desorbs. There are three types of events on the lattice, accretion, hopping and desorption. We keep track of the absolute time when each event occurs. The smallest absolute time is sorted out and the event that occurs at this time is executed.

The absolute time when each event happens are calculated as the follows. In MMC simulations, after an event occurs, the same event will occur again after a “waiting time”, $\tau=-ln(Y)/k$, where Y is another random number uniformly distributed within (0, 1) while k is the rate of the event. So, assuming a surface species hops from one site to another at time t, the absolute time it hops again or desorbs is calculated as, $t^{{}^{\prime}}=t-ln(Y)/(b_{1}+b_{2})$. Similarly, the absolute time when an accretion event of a gas-phase species occurs is, $t_{acc}=t_{acc}^{{}^{\prime}}-ln(X^{{}^{\prime}})/k_{acc}$, where $X^{{}^{\prime}}$ is a random number uniformly distributed within (0, 1), $t_{acc}^{{}^{\prime}}$ is the absolute time when last accretion event occurs (it is 0 in the beginning of simulations) while $k_{acc}$ is the accretion rate of the species.

In the beginning of model simulations, the population of surface H and H₂ fluctuates but also increases. After some time, the steady state condition is reached when both H and H₂ population fluctuates around the converged value. After steady states have been reached, we choose a time period $\tau^{,}$ to calculate the recombination efficiency, $\eta$, which measures how efficiently H₂ are formed on grain surfaces. We count the number of H₂ formed, $N_{H_{2}}$ and the number of H atom accretion events, $N_{H}$ during $\tau^{,}$. The recombination efficiency is calculated as, $\eta=\frac{2N_{H_{2}}}{N_{H}}$. We set $\tau^{,}$ large enough so that the value of $\eta$ converges.

Although H₂ molecules may account for a large fraction of gas phase species in the diffuse and translucent clouds, their existence on grain surfaces has been ignored by almost all previous H₂ formation models. To the best of our knowledge, this work is the first time that the detailed MMC method has been used to investigate how the existence of H₂ may affect the efficiency of molecular hydrogen formation on cold interstellar dust grains in diffuse and translucent clouds. We found that the impact of the existence of H₂ on recombination efficiency mainly depends on the diffusion barriers of H₂ on grain surfaces.

In the diffuse cloud models M02-M06, because of the high diffusion barriers on olivine grains, H₂ molecules can be trapped by the deep potential sites. Site occupied by H₂ become much weaker binding sites for H in these models. So, in models M02-M05, it becomes more difficult for H atoms to become trapped in these sites already occupied by H₂. Hydrogen atoms can also efficiently desorb when they encounter H₂ molecules in models M02-M05, leading to a lower predicted recombination efficiency in models M02-M05 than in the reference model M01, which does not allow H₂ to exist on grain surfaces. Moreover, models M02-M05 all fail to predict the minimum recombination efficiency (0.5) to explain observations.

The recombination efficiency increases with the value of $E_{D_{HH_{2}}}$, which is, however, not well known yet. The adopted value of $E_{D_{HH_{2}}}$ in models M02-M04 is much lower than the desorption energy of H atoms on silicate surfaces while model M05 adopts a upper bound of $E_{D_{HH_{2}}}$. Future more rigorous study might suggest higher values of $E_{D_{HH_{2}}}$ and $E_{D_{H_{2}H_{2}}}$ than these used in models M02-M04. However, because even model M05 predicts significant reduction of the recombination efficiency, we argue that even if these higher values of $E_{D_{HH_{2}}}$ and $E_{D_{H_{2}H_{2}}}$ are adopted by models M02-M04, these models may still predict lower recombination efficiency than the reference model M01 does.

In model M06, because H atoms and H₂ molecules do not share common stronger binding sites, the stronger H atom binding sites can still trap H atoms when H₂ molecules are trapped on grain surfaces. However, H atoms can still desorb efficiently when they encounter H₂. Model M06 results show that as long as H₂ are trapped by the strong binding sites and the desorption energy of H atoms on H₂ substrate is much lower than that on silicate grain surfaces, efficient H₂ formation ($\eta\geq 0.5$) occurs only when the dust surface temperature is around 11 K.

We used olivine grains as an example in the diffuse cloud models. The diffusion barrier of H₂ on carbon grain surfaces is even larger that on silicate surfaces. So, H₂ molecules could be more tightly trapped on grain surfaces. The tightly trapped H₂ molecules should also make H₂ formation on carbon grains less efficient.

In the translucent clouds models M12 and M13, the diffusion barriers of H₂ on grains covered by water ice are low enough so that H₂ molecules diffuse even faster than H atoms and cannot be trapped by the deep potential sites. Moreover, H₂ molecules can efficiently desorb when they encounter H atoms on grain surfaces. So H₂ population in these two models are very low so that the enhanced H atom desorption by encountering H₂ molecules is minimal. Therefore, the existence of H₂ on grain surfaces can hardly affect the recombination efficiency. On the other hand, in models M14 and M15, the diffusion barriers of H₂ are almost a factor of two larger than these in models M12 and M13, so H₂ can be trapped by the deep potential sites. In models M14 and M15, efficient H₂ formation is not possible when the temperatures are above 10 K.

Based on the translucent cloud model results, we can conclude that lower H₂ diffusion barriers on bare dust grains can enhance the recombination efficiency. This conclusion suggests that $\eta$ may also increase if lower H₂ diffusion barriers is used in diffuse models. In order to find out how lower H₂ diffusion barriers may affect diffuse cloud model results, we simulate a test diffuse cloud model M07 using reduced H₂ diffusion barriers, which are $50\%$ of the H₂ desorption energies on silicate surfaces. The H₂ diffusion barriers in the shallow, middle and deep silicate potential sites are 203 K, 308 K and 435 K respectively in model M07. Other than the H₂ diffusion barriers in silicate sites, parameters for model M07 are the same as these for model M02. Fig. 10 shows the recombination efficiency as a function of T predicted by the test model M07 and the reference model M01. We can see that H₂ can be efficiently synthesized ($\eta\geq 0.5$) when the grain temperatures are between 10 K and 14 K in model M07, which agrees well with model M01 predictions. So, lower H₂ diffusion barriers can also enhance recombination efficiency in diffuse clouds. However, to the best our knowledge, the H₂ diffusion barriers on silicate surfaces are poorly known. More work should be done to gauge this energy value in order to better understand H₂ formation in diffuse clouds.

Since the classical work by Hollenbach & Salpeter (1971), models that adopted grain surfaces whose binding sites have various desorption energies and diffusion barriers have been successfully used to enhance the H atom recombination efficiency at higher temperatures. The key to the success at higher temperatures is the existence of stronger H atom binding sites in these models. Ironically, these stronger sites may become much weaker H atom binding sites after H₂ molecules occupy them. Therefore, the success may not be as plausible as it seemed to be if we consider surface H₂ in models. Our model results show that H₂ molecules must diffuse fast enough on grain surfaces so that they are not trapped on grain surfaces in order that H₂ can be formed efficiently.

We consider physisorption only in our models while H atoms can also be chemisorbed on silicate or carbon surfaces (Cazaux & Tielens, 2004). It is not clear whether models that involve chemisorption can explain H₂ formation in diffuse clouds or not if the existence of H₂ on grain surface is considered. More study should be done in order to unveil the mystery of H₂ formation in diffuse clouds.

Hydrogen atoms can only be physisorbed on water ice surfaces (Wakelam et al., 2017a). Our translucent cloud model results further show that the conversion from physisorbed H atoms to H₂ molecules can be very slow on grain surfaces if the diffusion barrier of H₂ on water ice surface is large enough. Therefore, this diffusion barrier could be important for the molecular evolution of translucent clouds. However, because surface chemical reactions that involve H₂ typically have a large reaction barrier, this diffusion barrier is not well studied so far. More rigorous research should be done to gauge this energy value.

This work was funded by the National Natural Science Foundation of China (grant Nos. 11973099, 11973075 and 12173023). We thank our referee for constructive comments to improve the quality of the manuscript. We would like to thank Professor Laura Rowe for her help with proofreading and scientific English writing. We thank Di Li for helpful discussions.

No new data were generated or analysed in support of this research.