Anisotropic translational energy distribution due to gas-phase collisions in rapid desorption of molecules from surfaces

Surface Science 200 (1988) 113-134 North-Holland, Amsterdam

113

ANISOTROPIC TRANSLATIONAL ENERGY DISTRIBUTION DUE TO GAS-PHASE COLLISIONS IN RAPID DESORP'HON OF MOLECULES FROM SURFACES I. NOORBATCHA, Robert R. LUCCHESE Department of Chemistry, Texas A&M University, College Station, Texas 77843, USA

and

Yehuda ZEIRI Physics Department, Nuclear Research Center-Negev, P.O. Box 9001, Beer-Sheva, Israel Received 5 November 1987; accepted for publication 17 February 1988

The velocity distribution of NO molecules rapidly desorbing from an LiF(100) surface, in the presence of gas-phase collisions, is calculated using a direct Monte Carlo simulation procedure. The gas-phase collisions are found to transform the initial MaxweU-Boltzmann like distribution into an ellipsoidal Boltzmann distribution. In this respect the rapid desorption process is found to be similar to the supersonic expansion process in which case the gas-phase collisions also convert the random thermal motion of the gas molecules and the rotational energy into directed motion along the beam axis. Even though the total velocity distributions show non-equilibrium behavior, the velocity distribution of the molecules desorbing near the normal direction in the few gas-phase collisions limit is four,.l to be reasonably well represented by a Maxwell-Boltzmann distribution if the temperature is allowed to vary as a function of desorption angle. However, the ellipsoidal Boltzmann distribution function characterized by two temperatures is found to describe the velocity distributions of the molecules desorbing into specific desorption angles as well as the total velocity distribution of the molecules desorbing in all tile directions very well. The translational energy corresponding to the peak of the time-of-flight distribution calculated using an ellipsoidal distribution is found to be greater than the value predicted by the Knudsen layer model. The probable reasons for this discrepancy are analyzed. We find that under the conditions at which postdesorption collisions are important, the mean translational energy of the desorbed molecules need not necessarily be directly related to the surface temperature. The implications of these results for the existing mechanisms for rapid desorption are discussed.

1. Xntroducfion

Recently there has been active interest in understanding the factors influencing the velocity distribution of the molecules desorbed rapidly from surfaces. In experiments such as laser-induced desorptions, heavy ion induced desorptions, electron stimulated desorptions or flash desorptions, the heating (;039-6028/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

114

I. NoorBatcha et ai. / Velocity distribution of NO desorbing from LiF(IO0)

rate (101°-1012 K s -1) and the desorption rate are usually very high and a significant number of adsorbates are desorbed in a time on the order of 10-6-10 -8 s. Under these conditions the dynamics of the desorptien process need not be same as that observed in conventional thermal desorption studies. An important characteristic, which could lead to better understanding of the desorption process, is the velocity distribution of the desorbed molecules. Although there has been significant progress made in characterizing velocity distributions of the rapidly desorbed molecules [1-17], there are basic questions which still need to be resolved. The velocity distributions of desorbed molecules are usually analyzed in terms of Maxwell-Boltzmann distributions. The velocity distributions of the molecules at a fixed desorption angle obtained by laser-induced desorption [1-8] and flash resorption [9] are found to be described by Maxwell-Boltzmann distributions. However, in a detailed study of the laser-induced desorption of D~ from tungsten, Cowin et al. [10] found that the velocity distribution of the molecules desorbing at low desorption fluxes at a given desorption angle 0 (measured from the surface normal) could be fit to a Maxwell-Boltzmann distribution where the characteristic temperature was found to depend on 0. A~ high resorption fluxes ( - 1 monolayer/laser-pulse) the velocity distribution deviates significantly from a Maxwell-Boltzmann distributions. Similar deviations at high desorption fluxes have also been found in the laser-induced desorption of 0 2 and CO physisorbed on polycrystaUine copper [12], Xe on a copper film [13], and in the laser ablation of CuCI [14], Al203 [15], CdS [16], and GaAs [17]. Namiki and co-workers [16,17] have investigated the angular dependence of the translational energy distribution of the neutral particles desorbed by laser irradiation of CdS and GaAs. They have found that at low laser power the velocity distribution of the desorbed particles follow a Maxwell-Beltzmann distribution throughout the range of angles (0 °-60 °) they have investigated, whereas at high laser powers Maxwell-Boltzmann distributions were found only in the narrow cone around the surface normal. The mean translational energy of the desorbed molecules was found to decrease with increasing resorption angle and the rate of decrease increased with the laser power [16,17]. Similar angular dependence of the translational energy was also obtained by Taborek [9] and Cowin et al. [10]. Cowin et al. [10,11] have explained the deviations from Maxwell-Boltzmann u~u~uuuvu~ ~t ~u~t desootion ttuxe~ on the "---' oas~s of the gas-phase " collisions between the desorbed molecules. Recently, we have exa,rfined [18] the possibility of gas-phase collisions in rapid desorption experiments by simulating the laser-induced desorption of D z from tungsten, using direct Monte Carlo simulation procedures [19]. We found that at high surface coverages the desorbates make about 3 collisions per molecules which decreases to zero collisions at low coverages. Although the postdesorption ~11: . , ~ . . ~ " 1 1 . . . , . " . . . .

~

L ; ~ L

k'll . . . . . .

L NoorBatcha et al. / Velocity distribution of NO desorbingfrom LiF(IO0)

115

collisions play a major role in influencing the asymptotic distributions of the desorbed molecules, we found that the non-equilibrium distributions at low coverages have to be included in the Monte Carlo simulations to obtain good agreement with the experimental results of Cowin et al. [10]. We have studied the effects of postdesorption collisions on the internal energy of the rapidly desorbing molecules by simulating the desorption from LiF(100) of NO molecules [20], which have a large probability for rotational energy transfer [21,22]. The mean rotational energy of the NO molecules was found to decrease with increasing surface coverage due to the increase in the number of gas-phase collisions experienced by the desorbing molecules [20]. Additionally, the rotational energies of the NO molecules were found to decrease with increasing desorption angle 0 implying that molecules with large translational energy would also have large rotational energy. The angular dependence of the rotational energy distribution of the rapidly desorbed molecules has not been studied experimentally. However, Dreyfus et al. [15] have found that rotational excitation of AIO molecules obtained by laser ablation of A1203 is larger for the molecules with larger velocities which is in good agreement with the Monte Carlo simulation results [20]. Kelly and Dreyfus [23] have analyzed the effects of postdesorption collisions on the time-of-flight (TOF) measurements, in terms of the formation of a Knudsen layer above the surface. They have assumed that the velocity distribution of the desorbed particles reacl-.ing the TOF detector is given by: fK (Vx, Vy, V~) dv x dvy dv: 0c v: exp -

m v~+Vy . 2kTK

- c ~
dv x dvy dv:, (1)

where UK is the center-of-mass velocity, T K is the time-of-flight temperature, m is the mass of the desorbed particles, k is the Boltzmann constant, Vx and Vy are the velocity components parallel to tile surface and v~ is the velocity component perpendicular to the surface. They have shown [23] that for measurements along the surface normal the translational energy Ep, corresponding to the peak of the TOF distribution is related to the surface temperature T~, by the relation: Ep = ~KkT~, (2) where

+ [4
[{'1' + 16) ' / 2 - "t"/212( J + 4) 2

S = [ ( j + 5 ) / 2 ( j + 3)] '/2, y = ( j + 5 ) / ( j + 3),

(3) ' (4) (5)

L NoorBatcha et ai. / Velocity distribution of NO desorbingfrom LiF(IO0)

116

and j is the number of internal degrees of freedom of the desorbed particles. According to eqs. (2)-(5), Ep depends only on j and Ts and, if confirmed experimentally, eq. (2) would be a powerful tool for predicting Ep under conditions in which gas-phase collisions are important. The value of ~? has been found to range from 2.53 for monatomic species to 3.28 for polyatomic species with many internal degrees of freedom [23]. These relations qualitatively explains the experimentally observed increase in Ep relative to the surface temperature, though the present simulations indicate that eqs. (2)-(5), probably underestimate the influence of gas-phase collisions on Ep (see section 3). The non-equ'dibrium velocity distributions observed in the laser irradiation of compound semiconductors have been explained on the basis of the nonradiative recombination of electron-hole pair, which could energize surface species and induce desorption [14,24,25]. This mechanism explains the very high translational energy of the molecules desorbing along the surface normal. However, the angular dependence of the translational energy of the desorbing molecules is not addressed by this mechanism. The similarity between the non-equilibrium distributions observed in several different types of experiments [9-17] suggest that a common mechanism is responsible for the nonMaxwellian velocity distributions and the angular dependence of the translational energy. The results of the Monte Carlo simulations [18,20,26] suggest that the postdesorption collisions can account for the angular dependence of the translational energy of the desorbing molecules. Moreover b.y simulating the rapid desorption of single [18,20] and mixtures of gases [26,27], we found that the translational energy of the particles desorbing along the surface normal are greatly enhanced relative to that of the surface temperature, depending on the mass and compos;tion of the desorbing mixture, and the desorption flux. As a part of our continuing efforts to understand the effects of postdesorption collisions, in this paper, we present a detailed characterization of the velocity distribution of the NO molecules rapidly desorbed from an LiF(100) surface in the preseace of the gas-phase collisions. Lucchese and Tully [28] have studied the dynamics of laser-induced desorption of NO from LiF(100), by means of stochastic trajectory methods [29]. Using the distribution of the NO molecules obtained from stochastic trajectory calculations [28] as the input, we have included the effects of gas-phase collisions using direct Monte It""

t~sJ

•

We find that the asymptotic total velocity distributions as well as the velocity distributions of the particles desorbed at specific angles can be well represented by ellipsoidal Boltzmann distributions. The ellipsoidal Boltzmann distribution for the velocities is characterized by two temperatures, one for the distribution c,f the velocities in the direc,don normal to the surface (higher than the surface temperau.re) and the other for the distribution of velocities in the

L NoorBatcha et al. / Velocity distribution of NO desorbingfrom LiF(IO0)

117

direction parallel to the surface (lower than the surface temperature). The simulations show that these two temperatures are very different and reflect the inherent anisotropy of the rapid desorption process. We also find that the internal degrees of freedom can also be characterized by a temperature (lower than the surface temperature) which is distinct from the translational temperatures although its variation with surface coverage is very similar to the translational temperature in the direction parallel to the surface. An analysis of the angle-resolved velocity distributions reveals that the velocity distribution of the molecules desorbing into a narrow cone around the surface normal at low surface coverages can be described in terms MaxweU-Boltzmann distributions by allowing the temperature to vary as a function of desorption angles, even though the total velocity distributions deviate from MaxwellBoltzmann distributions.

2. Method The details of the direct Monte Carlo calculations have been described before [18-20]. In this section, we describe only the details relevant to the present calculations. The molecules are assumed to be rigid-rotors and hard spheres, and desorbed into the system as a function of time according to a first-order rate law. The mean residence time ¢, of the molecules on the surface is assumed to be [20,28] 3.43 × 10 -12 s. We have made use of 66660 particles in this simulation and the appropriate surface coverage O, is obtained by varying the surface area. We have performed simulations at three surface coverages, O = 0.1, 1.0, and 3.0 monolayers. For simulation purposes the region above the surface is divided into a network of cells and the molecules are desorbed into these cells as a function of time with appropriate distribution functions. A major simplification in our simulation is that the positions of the desorbed molecules are characterized by the distance above the surface, the z coordinate, but the velocity components are treated for all the three dimensions. This is valid as long as the rotational energy and velocity components are independent of x and y directions. This is equivalent to assuming infinite size for the laser spot (effective heating area from which most of the desorption occurs) and uniform temperature across the laser spot. As a result the present simulation is strictly valid only when the clinrnotor n f the laser ~nnt i,~ lnroer t h a n Ihe clistanc.e travelled b y the m o l e c u l e s

in the x - y plane before velocity segregation occurs in the z-direction. At 500 K, the mean speed of the desorbed NO molecules in the x - y plane is 5.3 x 104 c m / s . Hence in 10 - 7 S, during which time more than 80% of ga3-phase collisions occur, the molecules will travel -- 0.5 mm in the x - y plane. Most of the laser induced desorption experiments have laser spot size which is larger than this value.

118

L NoorBatcha et al. / Velocity distribution of NO desorbing from LiF(IO0)

The motion of the molecules was followed for 5 x 10 -6 s. By the end of the simulation, the density has decreased to such an extent that collisions at later times are not significant. To increase the efficiency of sampling, the propagation incremental time and the cell size are changed several times during the simulation. The correct collision frequency is maintained by a modified time counter method as described in ref. [18]. A time counter is placed in each cell and after each collision it is advanced by a time increment Ate, equal to [19] N2

At c =

N2 + 12

2

(6)

oc~nNm'

where Nm is the number of molecules in a given cell, n is the number density in that cell, c~ is the relative velocity between the colliding pair of molecules, and o is the collision cross section. An empirical correction factor of N2m/(N2m + 12) has been included to avoid statistical scatter at low values of Nm. Sufficient collisions are calculated in each cell to keep the time counter concurrent with the overall time of the system. If for a given collision the updated cell time exceeded the overall flowtime, then the possibility of such a collision is decided using the acceptance-rejection method on the basis of the ratio of the remaining sampling time in a cell to that of the collisional time Ate. This method of performing the last collision in each cell was found to be essential for properly describing the system when the cell sizes were changed

[ 81. We have assumed zero sticking probability for the molecules after desorption. As a result if a desorbed molecule strikes the surface during the simulation, such a molecule is reflected back into the system. At the end of the Mo~nte Carlo simula~ior~ --- 2~ o~" ~he .molecules are found Io have v. < O. If we had further continued our simulation for sufficiently longer time, these molecules would have collided ~,~:~L,,.,,~..~,,,,. , ~urface and reflected back mto~ the system with the same I o~l. As the number of gas-phase collisions has already become very small these molecules would reach the asymptotic distribution with the same i v~l. Hence~ in our analysis of final translational energy distributions we have considered o.~iy the magnitude of v~ for the particles moving towards the surface at the end of the simulation and all other velocity components are treated as such. Neglecting those particles with - v : does not significantly affect any conclusions of our analysis. The initial distributions of the desorbea molecules are obtained by fitting the stochastic trajectory data [281 to appropriate distribution functions. The initial distributions used in the present calculations correspond to simulation D of ref. [281. The angular distribution of the desorbed molecules is described by cos~940 and the translational energy distributions are described by:

fMB(V, 0)

dO d r =

½(m/kT(O))2v3 exp(-mv2/2kT(O))

sin t9 dO d r ,

(7)

where v is file velocity, m is the mas.,; of the desorbed molecules and the

L NoorBatcha et al. / Velocity distribution of NO desorbingfrom LiF(IO0)

119

translational temperature T(#) is allowed to vary as a function of desorption angle. For the initial translational energy distributions we have taken T(O)--A - B02, where the parameters A and B are found to be 562.1 K and 114.2 K / r a d 2, respectively [20,28]. The rotational temperature of the desorbed molecules is taken to be [28] 562 K at all values of 0. The collision number for rotational relaxation is assumed to be unity [21,22]. The main objective of the present calculation is to obtain the asymptotic energy and angular distribution of the NO molecules rapidly desorbed from an LiF surface using lasers. As a result we have used ~-= 3.43 x 10 -12 s for NO on LiF and the distribution of NO molecules obtained from stochastic trajectory calculations [28] as the nascent distribution of the desorbed N O molecules. It can be expected that the results of the Monte Carlo simulations would depend on these initial conditions. However the results of the simulations of the model systems [26] involving single and mixtures of gases, with ,r---- 10 -9 S and thermal initial distributions for the desorbed molecules, show that the qualitative features of the effects of the gas-phase collisions as discussed above, persists even with longer ~- and if the nascent desorbed molecules have thermal distributions. From the observed trend [26] in the decrease in the gas-phase collisions with increasing I" values, we predict that the gas-phase collisions will be less significant for ~-> 10 -8 s.

3. Results and disc!~ssion Preliminary results of the direct Monte Carlo simulation of rapid desorption of NO from an LiF smface have been published elsewhere [20]. Here we direct our attention to the characterization of the translational energy distribution of the NO molecules. The velocity distributions of the particles desorbed from surfaces are usually described by Maxwell-Boltzmann distribution functions. Hence we have attempted to described the calculated velocity distributions of the NO molecules using the Maxwell-Boltzmann distribution given in eq. (7), which was used to describe the nascent distributions. The velocities of the NO molecules desorbed within a range of 10 ° are fitted to eq. (7) and for discussion purposes we consider the midpoint in this range as the angle at which desorption occurs. The fitted curves obtained at the desorption angles 0=5°

and 55°

at 6)=13.1 are s h o w n in fig.% 1 a n d 9, n n d :~t 6 ) = ' ~ f ) :~re .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

--

. ~

~ _

_

shown in figs. 3 and 4. Similar plots are obtained at O = 1.0 and at different desorption angles. Even though the velocity distributions at 6 = 5 o and at low surface coverages are found to be adequately described by Maxwell-Boltzmann distribution, the velocity distribution at O= 55 ° is found to deviate significantly from the equilibrium distribution. The extent of these deviations increases with ,h,, surface coverage. Similar deviations from the

120

L NoorBatcha et aL / Velocity distribution of NO desorbing from LiF(IO0)

.]

l 0=0.1 #=5" eQ

*= /

#

/

k\

f

\-

// .,¢¢ 04

i

0.0

"V', "~'--.. I

i

I

0.5

1.0

i

I

i

1.5

I ~''" i

2.0

2.5

v(lOs (:ms -t) Fig. 1. Velocity distribution of the NO molecules at the desorption angle, 8 = 5 o and surface coverage e = 0.1. The points refer to the Monte Carlo results and the continuou3 smooth curve is the fit obtained using Maxwell-Boltzmann distribution (eq. (3)). The fitted temperature is 895 + 25 K. The dashed curve is obtained using eq. (5) with the parameters given in table 1. The values of do and dO are 5 x 103 c m / s and 0.1745 radians, respectively.

1.2-

e=0.1 8=55" 0.8-

~

0.4

l

7]

~"

0.0"

0,0

0,5

1,0 V (105

1,5

2.0

2,5

Cm S- t )

Fi_~. 2. Same as fi~. 1 for 0 = 55 o. The temperature obtained using eq. (3) is 296 + 14 K.

I. NoorBatcha et al. / Velocity distribution of NO desorbing from LiF(IO0)

121

0.03 1 ~ = 3.0 0=5 ° 0.02

°° "i/ fl

0.01"

"%. V~

'!

0.00

'

0.0

~,\ .',,

I

0.5

'

I

'

1.0

I

i

1.5

I

2.0

'

2.5

v (105 Fig. 3. Same as fig. 1 for O = 3.0. The temperature obtained using eq. (3) is 1148 + 57 K.

Maxwell-Boltzmann distributions, particularly at broad desorption angles and large desorption fluxes, have been observed in the laser induced desorption of D E from tungsten [10], and in the laser ablation of CdS [16] and GaAs [17]. The fitted temperatures for 0 = 5 ° (859 _ 25 K for @ = 0.1 and 1148 _+ 57 K for O = 3.0) are found to be greater than the surface temperature (500 K) as well as the temperatures obtained by fitting the velocity distribution at 8 =

~ = 3.0 e=55 ° u

I O

~.~4"

.

. 0.0

0,5

1.0

v (105 crn s -T) Fig. 4. Same as fig. 2 for O = 3.0. The temperature using eq. (3) is 36 _+2 K.

L NoorBatchaet al. / Velocitydistributionof NO desorbingfrom LiF(IO0)

122

0 = 55 ° (296 + 14 K for O=0.1 and 36 _+ 2 K for O = 3.0). The error limits correspond to a 957o confidence interval. The fitted temperatures at O = 1.0 and at other resorption angles (00-90 °) are found to be similar to the translational temperatures reported in fig. 2 of ref. [20], where we have expressed the mean translational energy in terms of ( E ) / 2 k . The fitted translational temperatures are found to decrease with increasing desorption angle and the translational temperature along the surface normal increases with increasing surface coverage. These effects are found to correlate with the average number of collisions experienced by the desorbing molecules [18,20]. The different temperatures obtained at different desorption angles suggest that the translational energy of the rapidly desorbing molecules, under the conditions in which gas-phase collisions can occur, is not simply related to the surface temperature. The fact that the velocity distributions of the molecules desorbing along surface normal obey Maxwell-Boltzrnann distribution does not signify thermal equilibrium with the surface. It is also clear that there is no thermal equilibrium between the particles desorbing in different directions. To our knowledge, no attempt has been made to characterize the overall velocity distribution of the molecules desorbing in all the directions. In the following discussion we show that the complete velocity distribution of the rapidly desorbed particles can be described by just three parameters: temperatures Txy and T~ for the distributions of the velocity in the plane parallel to the surface and in the direction perpendicular to the surface, respectively, and v*, the most probable speed of the molecules in the z-directions. In contrast to eq. (7) which seems to represent the "local equilibrium" in the translational energy distribution of the rapidly desorbing molecules along the normal direction only, the ellipsoidal Boltzmann distribution function provides a global representation of the translation energy distribution of all the desorbed molecules. We find that the translational energy distribution of the molecules desorbing in all the directions can be well represented by an ellipsoidal Boltzmann distribution function of the form:

(8a)

f(Oxy, oz) ---f,(Vz)f2(Oxy), f l ( ~ ) dvz= (a/cl) exp[-m(Vz-V~*)2]2kT~ dye, f2(Vxy) dv.,,y= (1/c2)V,,y exp

my)

2kTxy

d°xy '

v~ > O.

> O,

(8b)

(8c)

where (~*)2m cI -

~

c2 = ( k r . . / m ) .

erf-

2kT:

1/2

} +1

,

(Sd) (Be)

I. NoorBatcha et al. / Velocity distribution of NO desorbingfrom LiF(IO0)

123

Vxy and v~ are the speeds in the xy plane and z-direction, respectively, k is the Boltzmann constant, and m is the mass of the desorbed molecules. The normalization constants cl and c2 are obtained by integrating the distribution functions corresponding to vxy and v~, within the limits of 0 and oo. It should be noted that the parameter v* is not equal to the mean center-of-mass velocity of the desorbed molecules in the z-direction, as it can be seen that the integration of ozf(oxy, Oz) within the limits of 0 < v~, vxy < oo does not result in v*. Hence v* is the most probable speed of the desorbed molecules in the z-direction and not the mean speed in that direction. The ellipsoidal model for translational energy distribution allows characterization of the velocity distributions in terms of separate and uncoupled Maxwellian distribution fuactions. This model has been used in many theoretical and experimental studies of free jet expansions [30-33]. The existence of such distributions for rapidly desorbing molecules indicates the similarity in the effects due to gas-phase collisions in rapid desorption processes and free jet expansions [34,35]. The analogy between the jet expansions and rapid desorpt ~on processes ca~ be explained as follows. Just as in the case of jet expansions, the molecules escaping from the surface also have relatively large velocity components in the direction normal to the surface. The gas-phase collisions that occur just above the surface (close to the orifice in the case of jet expansions) further increase the normal velocity components at the expense of the velocity components parallel to the surface and possibly the rotational energy of the desorbed molecules also. These collisional effects will be manifested in the final energy distributions by the shift in the peak of the velocity distribution along the normal direction, to higher velocities and the shifting of the peak of the velocity distribution along the directions parallel to the surface, to lower velocities. In the case of polyatomic molecuYes there will be rotational energy cooling also. The above similarities also bring out the limitations of comparing jet expansions with rapid desorption processes. Firstly, the normal velocity component of the desorbing molecules depends very much on the dynamics of the d~sorption processes and invariably there is a significant fraction of the molecules desorbing in the broad angles. Secondly, the number of gas-phase collisions that occur in the rapid resorption process is less than the gas-phase collisions that occur in and downstream of the orifice of the expanding beam. As a result the increase in the normal velocity components or the decrease in the parallel velocity components and the accompanying narrowing of the velocity distributions in the case of rapid desorption processes cannot be as much as expected in jet expansion processes. The translational energy distribution of the desorbed molecules along xy and z directions, and the fitted curves obtained using eq. (8) for O = 0.1 are shown in figs. 5 and 6, and similar plots for O = 3.0 are shown in figs. 7 and 8. The calculated velocity distributions are found to be in very good agreement

124

L NoorBatcha et aL / Velocity distribution of NO desorbing from LiF(IO0)

.~ i

e = 0.1

T"o" 0.4

,,, 0.0

0.2

0.4

0.6

0.8

1.0

Vxy (105 cm s -1) Fig. 5. Velocity distribution of desorbed N O molecules in directions parallel to the surface, f~r O = 0.1. The points refer to the Monte Carlo results and the smooth curve is the fit obtained using eq. (4). The value of dv.~v is 2.5 × 10 3 c m / s .

05 a®®~ m

,~

0.4 "

® = 0.1

0.1 t 0.0

i

0.0

|

0.5

,

i

i

I

1.0 1.5 Vz 005 crn s -t)

i

m 2,0

2.5

Fig. 6. Same as fig. 5 for the velocity distribution in direction normal to the surface. The value of dr: is 5 x 103 cm/s.

!. NoorBatcha et aL / Velocity distribution of NO desorbingfrom LiF(IO0)

125

1.2 "

0=3.0

1o0 " A

TO0.8"

,~0.6

"

0.4"

0.2-

0.0

i

=

0.0

I

i

i

0.2

i

l

0.4

'

I

0.6

i

0.8

1.0

vxy (105 cm s-l) Fig. 7. Same as fig. 5 for e = 3.0.

with eq. (8). Similar levels of agreement between the calculated translational energy distributions and the fitted curves are also obtained fer O = 1.0. The distribution of the density of the particles as a function of V,+y and v: are shown in figs. 9-11. Fig. 9 shows the initial distribution and fig. 10 shows the

0=3.0 6@

0.4"

!

0.3"

~~>NN0.2 -

0,1

0.0

"

" - " ' T ~ -- ]i 0.0

0.5

i

i 1.0

~

I

i

1.5

':z ( 105 ¢rn S -1)

Fig. 8. Same as fig. 6 for @ = 3.0.

i 2.0

+

2.5

126

L NoorBatcha et aL / Velocity distribution of NO desorbing from LiF(IO0)

1

®=3.0

O.58

Dens~

2O

C 6

3

#

0.19 ) °'°°o.~

O.

~'0

.~

zo

Fig. 9. r~nitial density distribution of the desorbed molecules as a function of Ox.fl and oZ values, for O = 3.0 monolayers. The contour levels are in the units of 2 x 10-11 s2//cm2.

final distribution obtained in the Monte Carlo simulations. The transformation from an initial Maxwell-Boltzmann like distribution to an ellipsoidal Boltzmann distribution is clear from these figures. The ellipsoidal Boltzmann distribution calculated using eq. (8) with the fitted temperatures at O = 3.0 monolayers is shown in fig. 11. The simulated final distribution (fig. 10) is very

0.75 O= 3.0 Flnd

0"SB 1

only

I

I0

o

0.37!

0.0

0.5

L5 ~

1.0

2.o

(lo5 ~ s-~

Fig. iG. Final density distribution of the desorbed molecules as a function of vxy and v: values, for O = 3.0 monolayers. The contour levels are in the units of 10-10 s2/cm 2.

L NoorBatcha et at,. / Velocity distribution of NO desorbingfrom LiF(IO0) 0.75

127

¸

®=25.0 Fllled

oenstty

0.58

I

M

fi

0.37

0.19

0.00

o.o

o~s

~o

~

z0

vz 0~s ~ s-b Fig. 11. Density distribution calculated using eq. (4) with the temperatures obtained by fitting the final distribution for O = 3.0 monolayers.

similar to the ellipsoidal Boltzmann distribution (fig. 11) except for the presence of a high intensity peak at small values of Vxy and ~ (innerraost contour line in fig. 10) in the simulated distribution. This peak is absent in fig. 11. This implies that the ellipsoidal Boltzmann distribution underestimates the intensity of the particles for small values of V~y and o~. The perpendicular temperature, T~, and parallel temperature, Txy, obtained by fitting the calculated asymptotic velocity distribution of desorbed molecules along with the rotational temperatures and the average number of collisions experienced by a molecule ~, are given in table 1. The perpendicular temperatures of the desorbed molecules are very much larger than the parallel temperatures and this difference between the values of T~ and Txy increases rapidly 'c'ith increasing surface coverage as a result of increase in the gas-phase Table 1 Results for NO desorption from an LiF surface a) O

T~ (K)

T~ (K)

v* (10 4 c m / s )

Trot (K)

0.1 !.0 3.0 1.0 t,)

294 + 3 !00_+ 2 41 ± 1 35±1

1055 + 70 99,~0 + !5! 2565 _+134 1086+ 58

3.60 + 0.20 2,!5 ± 0.39 1.90+0.33 2.66+0.18

a00 + 4 131 + 1 53 + 1 55S:t:3

1.90 12.91 32.62 11.12

a) The error bars correspond to 95% confidence intervals. Temperatures are obtained by fitting the appropriate distributions from the Monte Carlo simulations to tile distributions given in eq. (8). b~ Results obtai,~ed using 104 as the collision number for rotational energy transfer.

/. NoorBatcha et al. / Velocity distribution of NO desorbing from LiF(IO0)

128

collisions between the desorbed molecules. However the most probable speed of the desorbed molecules in the z-direction v.*, decreases with increasing surface coverage. It should be noted that the velocity distribution of the rapidly desorbed molecules along the z-direction is broader than the distribution observed in the expansion through a nozzle. As explained before, this can be due to the difference in the number of gas-phase collisions occurring in the rapid desorption ~nd jet expansion processes. Additionally, in rapid desorption from a surface there is a relatively large fraction of molecules desorbed in broad angles and gas-phase collisions occur between particles desorbed in different orientations with respect to the surface. These collisions scatter the particles in all directions leading to a broad range of velocities along the z-direction rather than a narrow distribution as observed in the nozzle beams. Encouraged by the success in fitting the distributions of vxy and v. using eq. (8), we have examined the velocity distributions in specific angles by rewriting eq. (8) to include the 0 dependence as:

f(v, 0) -

dO

v2 s i n 0 e x p -

2kT~y

CIC2

exp -

2kT~

d~, d0,

(9)

where the symbols have same meaning as i~n eq. (8). I:~owing the values of T~y, T~ and v* the velocity distributions of the mole,-ules of the particles desorbed into a range of desorption angles can be obtained by numerically ;,,t,~,,r~,n,, eq. ~¢~ ,~vor a, for sr~ecific velocities. We have already multiplied the distribution given in eq. (9) by the appropriate solid angle (note the sin 0 term) so that the angular integration is performed directly on eq. (9). The velocity distributions obtained in this manner are shown in figs. 1-4. The simulated distributions are found to be in very good agreement with the ellipsoidal Boltzmann distributions, except in the case of O = 0.1 and 0 = 5 o. In this case (fig. 1) the velocity distribution of the particles with high velocity are found to deviate from the ellipsoidal Boltzmann distributions. These deviations at low surface coverage and small desorption angles can be attributed to the small number of gas-phase .::ollisions between the desorbed particles and as a result the velocity distrib, ution of the fast moving part,;cle~ does not completely evolve into an ellipsoidal Boltzmann distribution. The total velocity distribution of the particles desorbed in all orientations can be obtained by integrating eq. (9) over 0 ~n the range of 0 ° to 90 °. The aaA ~,~.

La 1.Jr a a ~

~k/ /

.

.

.

.

.

.

.

A

and O = 3.0 monolayers, respectively For comparison purposes we: have also shown the "best fit" obtained using eq. (7) with a single tempera~l,re. Clearly the velocity distributions of the desorbed particles obey ellipsoida| Boltzmann distribution rather than Maxwell-Boltzmann distributions. It should be noted that eq. (8) reduces to a single temperature fit with an additional parameter

I. NoorBatcha et aL / Velocity distributioh of NO desorbing from LiF( IO0)

129

1~0 -

(}=0.1 0.8

"

I I I

,,w~ k # t I t

t t

t

TOO.6" [ ' \ \ ,~.,~ 0.4 0.2 0.0

,=,=

0.0

0.5

1.0

1.5

2,0

2.5

vOe =,, =-') Fig. 12. Velocity distribution of the particles desorbing in all the orientations, for 19 = 0.l. The continuous smooth curve is obtained using eq. (5) with the parameters given in table 1. The dashed curve is the fit obtained using eq. (3). The.fitted temperature is 492 ___40 K.

0.6 "

t~

!",,

ik

i

"

_,..,0.4 I N :

'°

"

I 0.2-

!\\

1/

0.0-I-"

,,= 3.o

\ ',,,

-

ooo

~,,,,I~

'

|

o.s

'

I

, ....

'

i

I

~.o

1.s

2.o

v Oe

='. ~-')

"'

I

=I-=~='T

2.s

3.0

Fig. 13 Velocity distribution of the particles desorbing in all the orientations, for @ = 3.0. The smooth curve is obtained using eq. (5) with the parameters given in table 1. The dashed curve is the fit obtained using eq. (3). The fitted temperature is 1084 + 256 K.

130

I. NoorBatcha et aL / Velocity distribution of NO desorbingfrom LiF(IO0)

1.5

1 -~

~ = 0.1

|

1

0.0 0

2

4

6

8

10

Rotational Energy (KJ/mol) Fig. 14. Rotational energy distribution of desorbed NO molecules for O = 0.1. The points refer to the Monte Carlo results and the smooth curve is the fit obtained using the classical Boltzmann distribution function for internal energy. The value of (i Ero t is 0.5 kJ/tool.

4*, if Txy = T.. However, the large difference in the T~ and Txy values suggest that a single temperature fit is not sufficient to represent the global velocity distribution of the desorbed molecules. Due to the directed .flow of the desorbed molecules, we expect that the condition Try = T z will be rarely met in rapid desorption processes with significant gas-phase collisions. We have obtained the rotational temperatures of the desorbed molecules by fitting the calculated rotational distributions to classical Boltzmann distribution functions for rotational energy, and a typical plot for O = 0.1 is shown in fig. 14. The fitted rotational temperatures are in good agreement with the mean rotational energies reported in ref. [20]. A comparison of Tz, Txy and T~ot given in table 1 suggests that the additional translational energy along the surface normal is due to the flow of energy, not only from the rotational modes but also from the parallel components of the translational energy. The postdesorption collisions convert the random motions parallel to the surface into directed motions along the surface normal and this transformation is enhanced by the flow of energy from rotational modes. To study the effects of rotational modes on the anisotropy of the translational energy distribution we have simulated desorption of NO from LiF at a surface coverage of unity, ,~ith a very low collisional probability for rotational to translational energy transfer equal to 1 0 - 4 . The results obtained from this simulation are shown in table 1. It is significant to note that the anisotropy in the translational distribution persists even in the absence of energy flow from

L i,~,~^.a_..~. II[JlJl OI~I~IILI|II~I~ . . .Ilffi . lag.i

/

tz..p....~,., Iv ~]~qttrilg~ Ik61t.llll m . , . a , . . , ; , , . , a f ~ r ~ ,t,~,~,h~,,g~,,,,,, ~I, . . ilJ~614J14

~

•

I

~

~ o ~ -

•

•

~

t;.b"(lO0!

. . . . . . .

t

131

rotational modes. Additionally, there is a decrease ir the parallel and perpendicular temperatures compared to the case where the probability for rotational energy transfer is unity. This is due to the absence of energy transfer from internal degrees of freedom. The large difference observed in the perpendicular temperatures suggests that a major portion of the rotational energy is converted into perpendicular components of translational energy during the postdesorption collisions. When there is a less probability for rotational energy transfer, there is a drop in the average number of collisions experienced by each molecule. This is due to relatively fewer inelastic collisions which restrict the flow of energy from rotational modes leading to relatively low translational energy in the desorbed molecules and the presence of these "slow" molecules results in fewer collisions (elastic and w,elastic) compared to the case in which all collisions lead to rotational energy transfer. The presence of internal energy is found to broaden the velocity distribution in xy and z-directions, as can be seen by the increase in Txy and ~ . This is due to the increase in ~ values and the increased flow of energy from rotational to translational modes. Finally we compare the ~IK values given by eq. (7) with that predicted by the eUipsoidal Boltzmann distribution (eq. (6)). Following the treatment given by Kelly and Dreyfus [23], the TOF signal for the molecules desorbing along the surface normal with the eUipsoidal Boltzmann velocity distribution is: signal oc zt -3 e x p [ - m ( z - o * t t2 )2

(10)

,

where z is the perpendicular distance from the surface to the detector and t is the flight time. The velocity co~responding to the peak of this TOF distribution is given by: op= [o*fl~ + ((o*)2fl: + 6)11:] /2B~,

(11)

where ~2 = m / 2 k T ~ .

(12)

Hence, if the desorbed the molecules follow the ellipsoidal Boltzmann distribution then the relation between Ep and T~ is given by: Ep = ~lekT~ (13) where -\1/21 2

rle = m v~*fl~ + ((v~*)2f12 + 0)

2

] /8kTsfl; .

t'!4~,,

Eq. (13) is not as useful as ¢q. (2) in Fred;,ctir,.g E o as it requi.res the values of T~ and 07 to calculate tie. However, a comparison of tie and rIK will be helpful in evaluating the Knudsen layer model [23] against the results obtained using Monte Carlo simulations. The r~ values obtained using eqs. (3) and (14)

132

I. NoorBatcha et al. / Veloct~ distribution of NO desorbing from LiF(IO0)

Table 2 Comparison of ~ values for desorption of NO from an LiF surface ~K 0.10 1.00 3.00

562 562 562

4.11 7.11 7.79

2.80 2.80 2.80

a) The initial temperature of the molecules desorbed along the surface normal (see section 2). b) We values calculated from eq. (14). o ~K vahes calculated from eq. (3) with j ffi 2 (ref. [23]).

are given in table 2. The ~K values are found to be significantly different from the ~e values for all three surface coverages. The ~e values increase with surface coverage whereas ~K values are independent of O values. The large differences between the two ~ values is both due to the different manner in which the gas-phase coUisional effects are incorporated in the Knudsen layer model [23] and in the Monte Carlo simulations, and due to the different asymptotic distribution functions. The relative insensitivity of the ~z values to j values (~K changes from 2.8 for j = 2 to 3.1 for j = 10) suggests that the Knudsen model probably underestimates the influence of j on the gas-phase collisional effects. Under the conditions at which postdesorption collisions are important the energy distributions of the rapidly desorbed molecules are mainly determined by the gas-phase dynamics rather than the desorpt:.on dynamics and the final energy distribution of the desorbed molecules would depend on many factors in addition to the surface temperature and internal degree of freedom. Our Monte Carlo simulations [18,20] as well as several experimental [10,16,17] results suggest that the translational energy of the rapidly desorbed particles is a function of several variables like surface coverage, laser powei, and desorption angle in addition to the surface temperature mid internal degrees of freedom, On this basis we conclude that the Knudsen layer model, given by eqs. (1)-(5), only qualitatively represent the effects of postdesorption collisions along the normal direction and cannot describe high coverage results without the inclusion of the effects of adiabatic expansion [36].

4. Conclusion

By performing Monte Carlo simulations of the rapid desorption of NO molecules from an LiF surface, we have shown that the postdesorption collisions can lead to an ellipsoidal Boltzmann dis,~ribution for the velocity distribution of the desorbed molecules. In contrast to the usual practice of using Maxweil-Boltzmann distributions with several teraperatures at different desorption angles to describe (poorly) the velocity distributions of the rapidly

I. NoorBatcha et al. / Velocity distribution of NO desorbing from LiF(IO0)

133

desorbed particles, the ellipsoidal Boltzmann distribution with just three parameters pro~des an accurate description of the complete velocity distribution of the desorbed particles. The fact that the velocity msmuu~v,~o' ~ "'.... :""~ " of ,1,,~..,. rapidly desorbed particles can be well represented by ellipsoidal Boltzmann distributions suggests that postdesorption collisions transform random motions parallel to the surface into directed motions perpendicular to the surface. The anisotropy in the translational energy distribution persists even in the absence of any significant energy transfer from internal degrees of freedom. During inelastic collisions a major portion of the internal energy is converted into translational energy in the directions normal to the surface resulting in the rotational cooling of the desorbed molecules. The gas-phase collisions occurring during the rapid desorption process are found to cause large increases in the translational temperatures along the surface normal an0 result in the angular-dependent translational energy. Conclusions about the mechanism of the desorption process, derived exclusively on the basis of the measurements made along the surface normal, under the conditions at which postdesorption collisions are important, can be ~is!eading and need to be supported by the measurements made at different desorption angles. The Knudsen layer model [23] is found to yield smaller increases in the translation energy of the desorbed molecules compared to the present calculations. We attribute this difference between the two approaches to explain the gas-phase collision effects in the rapid desorption processes, to the relative insensitivity of the Knudsen model to the internal degrees of freedom of the desorbed particles and to the absence of any surface coverage dependent effects in the Knudsen layer model. Although several experimental studies on rapid desorptions have shown non-MaxweU-Boltzmann behavior for the translation er.~rgy distributions, no experimental evidence is available for eUipsoidal Boltzmatm distribution in desorption experiments. However, ellipsoidal Boltzmann distributions have been ol:served in experimental as well as theoretical studies of free jet expansion [30-33]. This indicates close analogy between the rapid desorption process and the expansion through nozzle sources. Such a similarity brings out the exciting possibility that by rapidly desorbing "seeded" adsorbed layers, the translational energy of the desorbed molecules can be varied over a wide range.

Acknowledgements Acknowledgement is made to the Monsanto Company and to the Celanese Chemical Company for partial support of this research. This research is based upon work in part supported by the National Science Foundatio~ 11nder Grant

I. No~rBatcha et al. / Ve:~city distribution of NO desorbing from LiF(I O0)

134

CHE-8351414. In addition this work has been supported by a grant from the Office of International Coordination of Texas A&M University.

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