Adsorption and Mobility on Amorphous Surfaces. Application to

J. Phys. Chem. 1983, 8 7 , 4229-4233

Downloaded by WEST VIRGINIA UNIV on September 11, 2015 | http://pubs.acs.org Publication Date: October 1, 1983 | doi: 10.1021/j100244a050

too small, or absorp in frequency regions blocked by the earth's atmosphere to make them defdctible with present day techniques. However, it is expected that with improved signal-to-noise ratio, with higher frequency resolution, and with observations from above the atmosphere even the weaker interstellar features should become measurable. These observations are expected to yield a wealth of information on the chemical history of the grains and the gas. 6. Concluding Remarks We have studied in the laboratory the infrared spectra of mixtures of H 2 0 and other molecules at low temperatures. Dilution of H 2 0 with non-hydrogen-bonding molecules gives rise to considerable changes in the H 2 0 spectral features. The changes include (1)the appearance of a new feature at 3700 cm-l and a shoulder to the 3250-cm-' absorption at 3220 cm-', (2) a shift to the blue and an increase of the width of the 3250-cm-' absorption, and (3) an increasing relative intensity of the 1650-cm-' peak. Observations of these features in interstellar spectra allow an estimate of the degree of dilution. The general shape of the HzO features is maintained for H 2 0 concentrations as low as 33%. At concentrations below 10% the broad absorptions of solid H20 are replaced by a number of sharp and weak lines. Dilution of H20with hydrogen-bonding molecules alters ita spectrum considerably. Simple empirical relations are established between the basic (or acidic) properties of the dilutant and its influence on the spectra. Strong bases

422s

induce a low-frequency wing to the 3250-cm-' absorption in solid mixtures through hydrogen bond formation with H20. Such a wing can also be caused by CH-stretching vibrations of diluting molecules. At low resolution these absorption features might smear out into a wing on the 3250-cm-' feature. Theoretical calculations of the chemical composition of interstellar grain mantles have been presented. The calculated mixtures consist mainly of H20, H2C0, N2, CO, 02,C02,and NH3in varying ratios. The exact composition reflects the physical conditions in the gas phase. Mantles accreted at a density of 103-104 will contaifi large concentrations of H 2 0 (-60%) and consequently show a broad 3250-cm-' absorption band. We attribute the lowfrequency wing observed on the 3250-cm-' feature in interstellar spectra to absorption by CH-stretching vibrations. Molecules containing CH groups have a concentration of about 25% in the mixtures calculated for densities of 103-104 ~ m - ~ The . concentration of strong bases is generally low (-2%) in the calculated mixtures. The contribution to the low-frequency wing by absorption of H 2 0 molecules hydrogen bonded to strong bases is therefore expected to be small. Acknowledgment. We thank Dr. Allamandola for his assistance in many stages of the project and Drs. Erickson, Hollenbach, and Werner for critical reading of an earlier version of this manuscript. Registry No. H20,7732-18-5; H2C0,50-00-0;NH3, 7664-41-7; N2, 7727-31-9; 02, 1782-44-7; CO, 630-08-0; C02, 124-38-9.

Adsorption and Mobility on Amorphous Surfaces. Application to Astrophysical Problems R. Smoluchowskl Departments of Physics and Astronomy, The University of Texas, Austin, Texas 78712 ( R e c e M : September 23, 1982; I n Final Form: November 23, 1982)

Adsorption of atoms and molecules on crystalline surfaces and their mobility have been studied in detail for many years. Much less effort has been devoted to similar phenomena on amorphous surfaces except for rather simple experimental investigations related to amorphous catalysts. The latter are usually complicated and not amenable to theoretical analysis. Thus a detailed treatment of adsorption and mobility of hydrogen atoms on the surface of amorphous ice has been made since it appears to play an important role in interstellar space where hydrogen molecules are formed on various, mostly icy, grains. Using experimental data concerning the structure of amorphous ice, one can evaluate the adsorption of a hydrogen atom with a suitable Lennard-Jones potential. One obtains a broad spectrum of adsorption energies which indicates that about 10% of the sites or traps are deep, while the others are more shallow. Since the process occurs below 30 K quantum mechanical treatment is necessary. It appears that a newly adsorbed hydrogen atom is quickly localized in a deep trap so that its wave function has a limited spacial extent. Thus in general the interaction time with another hydrogen atom adsorbed elsewhere is slow as compared to the residence time unless the two atoms happen to be localized near each other. This phenomenon reduces the rate of Hz formation by several orders of magnitude when compared to the situation on crystalline surfaces. This result seems to have a general applicability not limited to H on H20 ice. At higher temperatures in the classical region the mobility is being studied by use of molecular dynamics. The results so far available throw light on the importance of the deep traps.

Introduction The enormous success of quantum mechanics in explaining and predicting all kinds of basic properties of solids under the assumption that they were made of perfect or nearly perfect crystals delayed similarly rigorous attempts to understand noncrystalline or amorphous solids and especially their surfaces. It is well-known that various reactions taking place on amorphous surfaces play a very 0022-365418312087-4229$0 1.50/0

important role in catalysis. Unfortunately the systems of practical interest for catalysis are usually very complicated as far as the solid and the readants are concerned and for this reason a quantitative theoretical understanding of these phenomena is largely lacking. In contrast the formation of H2 molecules from hydrogen adatoms on ice, both crystalline and amorphous, presents a much simpler situation. In the following we describe briefly the 0 1983 American Chemical Soclety

4230

The ,kwm/ of Pnyshl Wwmfstry, Vd. 87, No. 21, 1983

smduchowski

5

M

4

3

2 h

h Y

0

0 E

1


Flgm 1. An amaptm~ tetragonalry c x " t e d sl".(Caurtw of Professor William Paul.)

amorphous ice and the interaction of a single hydrogen atom with a water molecule, and then discuss in detail, for the first time, the derivation of the spectrum of energies of hydrogen atoms adsorbed on amorphous ice and their mobility calculated with quantum-mechanical concepts. Finally, we mention very briefly the rate of H2 formation on amorphous ice as compared to those obtained for ideal or nealy ideal crystalline ice surface.

Amorphous Ice By an amorphous network, see Figure 1, one understands a two-or three-dimensional array of atoms or molecules in which the distances between the nearest neighbors and the angles between the bonds leading to these neighbors are close to those in a perfect crystal but there is no translational symmetry. Much of the difficulty in understanding these structures came from the unsettled disputes Concerning experimental diffraction results which because of inherent limitations were unable to distinguish unambiguously between very fine-grained or strained crystalline solids and truly amorphous structures. Even now many of these questions seem to be rather controversial and thus we fmd it convenient to use an operational definition' that in an amorphous monatomic solid a structural order, that is, a translational symmetry, does not extend beyond distances of about 10-20 A. This criterion is roughly applicable to molecular solids if the molecules are small and simple but clearly has to be modified when the elementary units are larger. Since the two hydrogens of a water molecule produce only a minor perturbation of the spherical electronic charge distribution in the oxygen the structure of crystalline or amorphous ices can be described, to a good approximation, by the arrangement of the oxygens only. The location, correlation, and motion of hydrogens, or of their vacancies, in crystalline ice is a well-known and much-explored topic.2 There seems to remain, however, the question of the location of hydrogens of the water molecules, i.e., of the dipoles, on the surface of ice. Although the dipole moment of a water molecule is small, its presence undoubtedly has some effect on adsorption and mobility of adatoms. To the author's knowledge few if any systematic studies have been made of these dipole effects for crystalline ices and certainly none for amorphous ices. For this reason and in view of other uncertainties concerning the surfaces of

0

-1

0

-1

2

0

Flgwe 2. Oxygen atom pak conelation functions for polycrysta"3 hexagonal Ice (top) and for amorphous Ice (bottom) prepared and studied at 77 K. (Reprinted wlth permission of the American Instftue of Physics. Copyright American Instttute of Physics, 1976).

amorphous ice, in the following the possible role of these dipoles will be neglected. The usual hexagonal crystalline ice is hydrogen bonded, solid germanium has covalent bonds, and SiO, is partly ionic and partly covalent, but the structures of these solids are tetrahedrally coordinated and are quite similar. Experiment shows3that this applies also to amorphous germanium, vitreous silica, and amorphous ice in which interbond angles deviate from the ideal 109.5'. Upon rapid cooling of water vapor amorphous ice forms in two phases, one denser (-1.2) and rarer near 10 K and the other less dense (-0.93) usually studied near 77 K. The upper limit for the formation and stability of the denser ice, which appears to have some interstitial H20molecules, is not known and thus the present work is limited to the normal less dense form which is stable up to about 150 K. The reader is referred for details to the extensive studies made by Rice and his collaborators and summarized by Rice.' It is possible to account for the X-ray, density, and other data of amorphous ice without assuming so-called dangling (or broken) bonds. Such bonds are well-known from the study of electrical properties of amorphous semiconductors. One should not conclude, however, that such bonds do not exist in amorphous ice, but rather that they are not numerous. From experimental X-ray diffraction data one obtains' a pair correlation function g ( r ) for the oxygen atoms. Such a function (Figure 2) indicates the probability that an oxygen atom, thus in ice a water molecule, exists in a small volume at a distance r from a given oxygen. This ~~~~~~~

(1) Konnert, J. H.; Karle, J. Acta Crystaffogr.,Sect. A 1973,2!l, 702. (2) Hobb, P. V. 'Ice Physics"; Clarendon Press: Oxford, 1974.

(3) Narten, A. H.; Venkatesh, C. G.; Rice, S. A. J. Chem. Phys. 1976, 64,1106. (4) Rice, S. A. Top. Current Chem. 1976, SO, 109.

The Journal of Physical Chemistry, Vol. 87, No. 21, 1983 4231

Adsorption and Mobility on Amorphous Surfaces

1

00

05

10

15

A

20

Flgure 3. Repulsive potential between a neon and a hydrogen atom.


b

function is the only quantitative information about amorphous ice used in the subsequent calculations.

Interaction of a Hydrogen Atom with the Surface of Ice Unfortunately there are no experimental or theoretical average potential curves available for the hydrogen-water interaction. Those published either dwell upon the effect of anisotropy of the water molecule on the interaction or are concerned with the various excited states, the formation of H,O molecules or ions, etc. For this reason it was decided to use the usual Lennard-Jones potential E(r) = 4t[(u/r)12 - (a/rI6]

(1)

based on the attractive polarization interaction calculated from the well-known properties of a hydrogen atom and a water molecule 4093 = ~

+

H p o ~ H I ~ l ~ ~ ( I H , ~

(2)

where e and u are the usual Lennard-Jones parameters and a and I are the polarizabilities and ionization potentials of water and of hydrogen, respectively. The repulsive part of the potential was assumed to be the same as that experimentally determined (Figure 3) between a hydrogen and a neon atom5because neon is isoelectronicwith a water molecule and of the same size. The best fit was obtained with t = 1.6 X eV and u = 2.84 A yielding an equilibrium distance 3.18 A. A hydrogen adatom at rest on a single crystal of ice is a nearest neighbor of three water molecules on a basal plane and of three or four molecules on prismatic planes parallel to the c axis. The binding of an H atom derived from the directly measureda value for H2 for crystalline ice for T between 4 and 40 K is about 6 X eV which is in good agreement with binding energies of 5.5 X 10-2-8.8 X eV or 637-790 K calculated for various sites by using the value of e given above. The contributions of the second and third neighbors have been included. The total binding is somewhat reduced by the zero point energy of the light hydrogen atom. As discussed later a minor change in the value of t does not affect the conclusions of the paper. The situation is much more complicated when the binding of a hydrogen atom to amorphous ice surface is of interest. One assumes that the hydrogen atom is nearest neighbor of three water molecules of the surface. These ( 5 ) Mason, E. A.; Vanderslice, J. T . J. Chem. Phys. 1958, 28, 1070. (6)Lee, T.J. Astroph. Space Sci. 1975, 34, 123.

ro A 80

Flgure 4. Position of a hydrogen atom above the surface of amorphous ice: H, hydrogen atom; (H,O), “virtual” water molecule (see text). (a) The hydrogen atom is nearest neighbor of three water molecules which lie in the plane of the surface of the ice; (b) the same but the plane of the three water molecules is inclined 25’.

three molecules form a plane which lies either in the surface plane of the solid, Figure 4a, or is tilted with respect to it by less than 20-25O,Figure 4b. For higher inclinations another set of three water molecules determines the geometry of the configuration. It would be, however, erroneous to assume that the pair correlation function between two water molecules applies to the hydrogen atom directly because the equilibrium distance of a hydrogen atom from the surface, as determined by its equilibrium distance to three water molecules in the surface, is different from the distance at which a “virtualn water molecule, shown in parentheses in Figure 4, would be located if it were above the surface and had the same three molecules as nearest neighbors. Thus the binding of a hydrogen atom is a sum of two contributions: the bonding due to the three nearest water molecules at 3.18 A and a contribution of other more distant molecules are determined by the pair correlation function centered on the “virtual” water molecule. The first is constant, the second varies from site to site. The calculation of the latter is the crux of the problem. Since E(r) drops off rapidly with increasing r the calculation was limited to values for which ro is larger than 3.18 8, but smaller than 6.5 A. Within this range of ro a certain number of water molecules are present, as required by the density of the solid. The location of these molecules, which

4232

The Journal of Physical Chemistry, Vol. 87, No. 21, 1983

Smoluchowski

atom adsorbed on a solid at temperatures of 10-30 K has 8 to be described in terms of the behavior of its wave

I

n

.6

3

5

4

6

E(IO-2eV)


Flgure 5. Number N E of adsorption sites of hydrogen atoms on amorphous ice per unit area of the surface and per unR energy as a function of binding energy E . The total number of sites is 1014-1015 per cm2. (Reprinted with permission of Reidel Publishing Co. Copyright Reidel Publishing Co.)

varies from site to site, was obtained by choosing a suitable set of random values of ro and calculating the corresponding E(r). Clearly among these molecules only those contribute to the binding for which the angle P in Figure 4a is smaller than P,,. Also as seen from Figure 4a, for a given r and ro there is a ring of radius ro sin Bo r sin P and of cross section roAtloAro rApAr in which the probability of finding a molecule is g(ro). Thus the probability of finding a ,molecule in the solid at a particular distance r from the hydrogen atom is p ( r ) = 2arOsin 8,g(ro)roA80Aro 2ar sin Pg(ro)rA@Ar (3) For small ro and r the binding is strong but the volume of the ring is small; the opposite is true for large ro and r. Adding up these contributions for a set of random ro one obtains the binding energy due to more distant neighbors at a given site. Repeating this calculation for a large number of sets and adding the constant contribution of the three nearest-neighbor molecules, one obtains the number of sites for which the total binding energy lies within certain limits. The result is shown in Figure 5. An analogous calculation made for the situation illustrated in Figure 4b leads to a curve which is very similar to that in Figure 5 except that it is displaced toward higher binding energies by up to 10%. The conclusions remain, however, the same. As intuitively expected, but previously never demonstrated, there is a broad range of binding energies which contrasts strongly with the four discrete states possible on crystalline ice. about 10% of the sites are quite deep while the remainder are more shallow. The deepest sites are those in which the hydrogen atom has another water molecule neighbor which is about as far from it as the three water molecules in the surface. These characteristic few high binding energy sites play an essential role in determining the mobility of atoms adsorbed on amorphous surfaces, in our case ice, as discussed further below.

-

-

-

Mobility of Hydrogen Atoms on Amorphous Ice There were only a few theoretical studies7 made concerning the motion of adatoms on amorphous surfaces in the classical high-temperature region and none in the quantum mechanical low-temperature region. As pointed out by Hollenbach and Salpetels the motion of a hydrogen (7) Mruzik, M. R.;Pound, M. G. J. Phys. F. 1981, 11, 1403. (8) Hollenbach, D.J.; Salpeter, E. E. J . Chem. Phys. 1970, 53, 79.

function because for such a light atom and low velocities its de Broglie wavelength is comparable to the interatomic distance of the solid. On a crystalline surface which has a periodic structure the wave packet representing the atom spreads over it almost instantaneously and therefore the probability of finding the adatom is the same at every point of the surface. This phenomenon is clearly impossible on the surface of an amorphous solid because, by definition, the surface is not periodic and the wave packet becomes localized, as f i s t discussed by Ander~on.~ From the point of view of quantum mechanics the situation is analogous to the motion of electrons in two- and three-dimensional semiconducltors studied extensively for many There, the essential feature is that nonperiodic lattices at low temperatures are nonconductive below the so-called mobility gap. Similarly here, as shown below, the hydrogen adatom can become immobilized for long periods of time. There are various theories and ways to treat this localization. The simplest and most direct way seems to be that of Mott.12 In his notation the wave function $ ( r ) of a particle localized at a site decreases in all directions as exp(-cur), that is 01-l is the distance at which #(r) drops to e-' of its value at its maximum. In two dimensions, the jump or rather tunneling frequency of the atom over a distance d is then given by p = u exp[-2ad - (ad2NElzT)-']

(4)

where u = 1012-1013s-l is the frequency of vibration of the atom, which should be comparable to that of the solid, and NE is the number of sites (cm-2eV-') at energy E. In two dimensions d2 = li(aNEkT)-' so that p

u

eXp[-4a(aNEkT)-l/2]

(5)

The average size of an adsorption site on amorphous ice should be about the same as on crystalline ice so that putting 01r = 1one has CY-='10-15 A. The values of NE can be obtained from Figure 5, keeping in mind that E is the positive difference between the highest binding energy eV and E b . Equation 5 combined Eb(max) = 5.85 X with Figure 5 is thus the solution of the problem of mobility of a hydrogen atom adsorbed on amorphous ice. It is clear that, while eq 5 has general validity, the density of states vs. binding energy curves, analogous to Figure 5, have to be calculated for each kind of amorphous surface and of adsorbed atom separately. The curve in Figure 5 is, however, a good approximation for most amorphous tetragonally coordinated lattices and small adatoms. The form of eq 5 indicates that, as expected, the probability of a jump is higher for small 01 than it is for less sharply localized atoms, for larger density of states between which the atom can tunnel, and at higher temperatures. In other words with increasing temperature more paths and a larger area of amorphous surfaces become available for motion. Substituting in eq 5 01-1 = 10 A, T = 20 K, and E b smaller eV, that is in the range of shallow traps, than 4.8 X leads to 3 X 10sjumps per second of which nallog n = 10s jumps will be made to different sites.13 On the other hand, for the case 01-l = 15 8, the result is 5 X lo9 s-l and lo9. As (9) Anderson, P. W. Phys. Rev. 1958,109, 1492. (IO) Mott, N. F.; Davia, E. A. "Electronic Process in Non-Crystalline Materials"; Clarendon Press: Oxford, 1971. (11)Tauc, J. "Amorphous and Liquid Semiconductors"; Plenum Press: New York, 1974. (12) Mott, N. F.Phil. Mag. 1969, 19, 835. (13) Montroll, E.W.;Weiss, G. H.J . Math. Phys. 1965, 6, 167.

The Journal of Physical Chemistry, Vol. 87, No. 27, 1983 4233

Adsorption and Mobility on Amorphous Surfaces

mentioned before, Figure 5 shows that about 10% of the eV, and for traps are deep, with Ebgreater than 4.8 X those the jump frequency is 4 X lo* and 2.7 s-l for the two values of a,respectively. There is of course the possibility that an atom moving initially among shallow traps will fall into a deep trap. A recent theory of Rosenst~ck’~ indicates that this will happen on the average after

[I - C - log

(TQ)I(.Q)-~

(6)


jumps, where Q is the concentration of deep traps and C is the Euler number, 0.5772. For the values given above the average number of jumps of the hydrogen atom before it becomes trapped is five which happens in 5 X lo-* to s. I t follows that the mobility or diffusion of the hydrogen atom on a surface of amorphous ice is controlled by the probability of its tunneling between deep sites which is a very slow process. If Mott’s formulae are used the effective diffusion coefficient on an amorphous surface can be shown to be

D = [ 2 / ( ~ N a k T ) ] v~exp(-[11.5~u~/(NdT)]~’~) /~ (7) where N = NE in the high binding energy part of Figure 5. What will happen if the amorphous solid is not really amorphous but consists of a collection of small, say 100 A diameter, rather perfect grains? Clearly within such a grain the wave packet describing the hydrogen atom will spread instantaneously to its boundary, but a t the boundary it will become localized in the usual way on a deep site. It is known from the theory of large- and small-angle grain boundaries that such sites abound along such boundaries because of the inherent lattice mismatch. The resulting diffusion will be thus limited to slow motion along the boundaries.

Astrophysical Applications As discussed elsewhereI5 in considerable detail an important application of the very slow mobility of hydrogen (14)Rosenstock, H. B. J. Math. Phys. 1980,21, 1643.

atoms on amorphous surfaces is the rate of Hzformation on interstellar grains. In the classical treatment it is assumed that the grains are perfect, or nearly perfect, single crystals so that the mobility of adsorbed hydrogens is high and the probability of their encountering each other before either one escapes from the crystal is also high. Actually the rate itself is then first order in the concentration of hydrogen in the gaseous phase. On the amorphous crystals, on the other hand, the situation is quite different. The hydrogen atoms become rapidly localized and, in general, it will require a very long time for two of them to diffuse close enough to each other to form an H, molecule which escapes. This time may turn out to be larger than the mean lifetime of these atoms on the surface before evaporation. An H2molecule can be then formed only if upon arrival on the grain, two hydrogen atoms accidentally become localized at nearby sites. The probability of such an event requires, of course, a high density of hydrogen atoms in the surrounding vapor. The resulting rate of Hz formation is second order in the gas density and at least lo3 times lower than on crystalline grains. The explanation of the observed high concentration of hydrogen molecules in space requires an explanation of either why ice grains in space are rather perfect crystals and not amorphous in spite of being formed undoubtedly at low temperatures, or why the density of hydrogen atoms in areas of space where the grains are present is so much higher than it is usually assumed. Neither one of these questions has as yet been answered in a satisfactory manner.

Acknowledgment. The most helpful discussions with Professor P. W. Anderson of the localization problems are gratefully acknowledged. This research was supported by NASA Grant NSG Sup. 4. Registry No. Atomic hydrogen, 12385-13-6;water,7732-18-5; hydrogen, 1333-74-0.

(15) Smoluchowski, R. Astroph. Space Sci. 1981, 75, 353.

Adsorption and Mobility on Amorphous Surfaces. Application to

Recommend Documents