41
J. Phys. Chern. 1980, 84, 41-50
A Model for Fluorine Atom Recombination on a Nickel Surface Eric J. Jumper, Department of Aeronautics, United States Air Force Academy, USAFA, Colorado 80840
Casper J. Ultee, United Technologies Research Center, East Hartford, Connecticut 06 108
and Ernest A. Dorko" Department of Aeronautics and Astronautics, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio 45433 (Received July 27, 1978; Revised Manuscript Received July 16, 1979) Publication costs assisted by the United States Air Force Academy
A detailed steady-statemodel for fluorine atom recombination on a nickel catalytic surface was developed based on the Langmuir-Rideal heterogeneous reaction mechanism. The model was compared to fluorine wall recombination data for a dilute mixture of atomic and diatomic fluorine in an inert diluent. The mixture flowed in a nickel tube. The model is able to duplicate the character of the data and to predict fluorine recombination coefficients as a function of temperature and other influencing conditions.
Introduction The goal of modeling the heterogeneous recombination of atomic fluorine is to arrive at a recombination coefficient as a function of the various influencing parameters including temperature, other gas-phase particles (atoms and molecules), and the surface in question. Because of the complexities of the problem, detailed theoretical analysis has, for the most part, been neglected; however, in order to understand the changing character of the reaction with temperature, it is essential to have a detailed mathematical model to properly interpret experimental results. The model proposed herein is capable of qualitatively explaining the nature of steady-state heterogeneous fluorine recombination on a nickel surface. Further, the treatment presented should give new insights into the modeling of heterogeneous chemical kinetics. In order to test the applicability of any model, experimental data are needed to use as a metric. Experimental results are scarce but not totally lacking for the fluorine/ nickel recombination system. Previous experimental work includes the work of Nordine and LeGrangel and Arutyunov and Chaikin.2 Nordine and LeGrange's work represents a single, well-documented, room-temperature point, while Arutyunov and Chaikin's represents a wide temperature range but lacks adequate description of the experimental conditions to be of use in the detailed analysis which follows. Included in this paper is a new set of data, which represents well-known experimental conditions for initial and final fluorine concentrations after exposure of a diluted atomic fluorine mixture to a nickel surface. Included in the known conditions are the surface temperature distributions. Even this data should be treated with some caution, but, as it compares favorably in the mean with prior results, we feel that the qualitative behavior of the recombination reaction is properly portrayed. These data appear primarily to demonstrate the applicability of the model. While the paper is not meant to be an experimental one, the experimental section is detailed enough to stand on its own, since the data do represent a new set for the fluorine/nickel recombination problem. 0022-3654/80/2084-0041$0 1.OO/O
Fluorine Heterogeneous Recombination Model The physical picture of the chemical reaction itself is assumed to be that of the Langmuir-Rideal m e ~ h a n i s m ~ - ~ for recombination of atoms at a surface. This reaction is a two-step process. A gas-phase atom, A, must first collide with the surface and stick at an empty surface site, s. Another gas-phase atom may strike the adhered atom and recombine, leaving the site again empty. The mechanism may be written as follows: (11
A + --+A I
s
A t A-A, I
S
1
s
t
I
(2)
S
Reaction 1 shows the necessary direction for the Langmuir-Rideal mechanism to proceed; the reaction is further complicated, however, by the fact that the reverse reaction does exist and cannot be neglected at elevated temperatures. The choice of the surface site of either being empty or filled with a reacting atom is a naive one in dealing with fluorine recombination since any recombination of fluorine leads to a concentration of molecular fluorine near the wall. McKinleya sets a precedent for a fast chemically bonded surface concentration of molecular fluorine at room temperature. We must, therefore, deal with the possibility of surface sites being filled with other than the reacting atoms. Finally, the type of bonding process should also be addressed in forming the most general set of rate equations. The rate equations, then, can be written in component form in much the same way as Brennen and Shuman:' did for their simplified single temperature, nonreversing, single-boriding model. Rate Equation Components. In order to properly write the rate equations, a cursory component overview is given here. Some of the components, or at least parts of the components, will need further comment and will be treated in detail in subsequent sections. Species. We will limit ourselves to three possible gasphase species, atomic fluorine, F, molecular fluorine, F2, and an inert diluent, I. Each of these species will be al0 1980 American Chemical Society
42
The Journal of Physical Chemistty, Vol. 84, No. 1, 1980
Jumper, URee, and Doko
lowed to occupy a surface site. The fraction of the total surface sites available taken up by a given species will be given by 6 with the appropriate subscript to denote the type species occupying that fraction. The surface will be clean nickel. Surface Bonding. The surface bonding can take one of two forms,6,8physical bonding due to van der Waals forces, and chemical bonding requiring a certain activation energy in the collision process. Both forms of bonding will be permitted for the F and F2species but, because the diluent is assumed inert, only physical bonding will be allowed for the diluent. The fractional surface concentration, 6, will be further subscripted p for physical and c for chemical on the F and Fz species to indicate the type of surface bonding. Adsorption Equation. The equation for an adsorption (physical or chemical) of a given species can be represented as follows: adsorption rate = NS
(3)
where fl is the rate of surface impingement per unit area and S is the sticking coefficient, a function of temperature and surface-site coverage. The rate of impingement is given by the well-known kinetic equationg
N = n-c
4
(5)
(6)
where 6 is the thermal desorption rate per unit area for a given species, and 6 the fractional surface concentration of that species. The thermal desorption rate is given by Glasstone, Laidler, and Eyring" as 6=
C,xexp(-D/kT) kT
where nnF is the surface concentration of F per unit area. All the F subscripts refer to atomic fluorine, and the p subscripts to physical bonding. Similar rate equations can be written for chemically surface-bonded atomic fluorine, physically surface-bonded and chemically surface-bonded molecular fluorine, subscripted F2,and the physically surface-bonded inert diluent, subscripted I. (dnnF/dt), = (SOF),$F(~ - 0,) - ( 6 ~ ) ~ ( -6 P ~ )N~F ( ~ F ) ~ (10)
(so~l)~N~~(l - 0,) - ( 6 ~ ~ ) ~ ((11) 6 ~ ~ ) ~
(dnsF,/dt), = (SOF~)C~TF,(~ - 6,) - ( 6 ~ ~ ) ~ ( (12) 6 ~ ~ ) ~ dnSI/dt = sO1NI(l - 6,) - 6161
(13)
Equations 11-13 reflect only one depletion term, that of thermal desorption. The c subscripts indicate chemical bonding. For the case under consideration, inert diluent and molecular fluorine are present in addition to atomic fluorine, and 6, is the total fractional surface concentration which is limited by 0 I6T I 1.0. 6T
= 6FP
+ 6F, + dFZp+ 6F2, + 61
(14)
Substituting eq 14 into eq 9-13 and solving these equations at steady state for the various fractional surface concentrations, we obtain 1 - 01 - 6F, - $Fa - 6Fh $FP
=
(6F)p
P
(sOF)~NF
(SoF)p
1+-+-
(15)
1 - 61 - 6FP - 6Flp - 6 F h $F,
=
(6F)c
P
CSoF>&F
(SOF)c
1+-+-
eF2p
--
(16)
1 - 61 - 6Fp - dF, - 6 F k (17) 1+
(6F,)p
(SOF,)~NF~
1 - $1 - OFp - 8F, -
(18)
$F% =
1+
(8)
where P is a steric factor. The steric factor is less than 1.0 and takes into account the effectiveness for reaction of the gas-phase/surface-adhered atom collisions. Since the reaction is an atom-atom recombination, P may be reasonably assumed to be independent of temperature.
6 ~ ) f -i ~( 6 ~ ) ~ ( 6-~p)N~ ~ ( 6 , ) ~ (9)
(7)
where C, is the number of surface sites per unit area (for a monolayer of adhered atoms this is taken to be approximately equal to the metal surface atom packing;8any experimental surface will undoubtedly be polycrystalline but, for the purpose of analysis, it will be taken to be a face centered cubic12),k the Boltzmann constant, h the Planck constant, and D the desorption energy (Dis taken to be the well depth for the particular bonding process; well depths will be addressed later). Recombination Desorption Equation. The desorption by recombination is applicable only for the surface-bonded atomic fluorine and is given by the kinetic equation recombination desorption rate = p N F 6 F
(SOF)p(l -
(d&F,/dt), =
where the subscript T on 6 indicates the total fractional surface concentration (i.e., the s u m of the fractional surface concentrations of all species present). Thermal Desorption Equation. The equation for thermal desorption is given by8 thermal desorption rate = 60
(dnSF/dt), =
(4)
where n is the number density of particles in the gas and E is the average velocity of the particles. If the clean surface or initial sticking coefficient is given by So, a function of temperature, T , the form of S, as experimentally verified by Christmann et al.,'O is given by
S(T9'9T)= (1 - 6T)so(V
(The possibility of a temperature dependence does, of course, exist because of the added complexity of the surface, but for the purpose of simplicity is not addressed in this paper.) Note further that eq 8 assumes that the rate-determining step is the recombination rate of the newly formed F2;this is in keeping with the suggestion by Laidlera5 Rate Equations. The rate equation for the change in surface concentration of atomic fluorine bonded in physical wells under the assumption of the Langmuir-Rideal mechanism in light of the above is given by
er =
(6Fz)c
(SOFJ C~TF,
1 - OFP - OF, - $Fzp - $Fa 61,
1+SOIN*
(19)
The Journal of Physical Chemistry, Vol. 84, No. 1, 1980 43
FI Atom Recombination on a Ni Surface
sulting in a recombination to the total number of fluorine atom wall collisions, is then given by
TABLE I : Arguments for Eq 25 diluent
Fp
FC
F2p
Fzc
The following definitions are used for isolated fractional surface concentrations: 1 (20) (6F)p P 1+( S ~ F ) ~(SOF)~ F
+-
1 (6Fc)o
(6F)c
P
(SoFIC&
(SOF)c
l+-+-
(21)
(The fractional surface concentrations are given the subscript o because they correspond to the actual surface coverage of the particular species in the absence of the others.) Substituting the definitions for the isolated fractional surface concentrations, eq 20-24, into eq 15-19 and simultaneously solving these, we obtain the explicit equation for the fractional surface concentrations
H = T/B
(25)
where
T = L[(1+ (&&3L4L5) - (L2L3L4) - (L2L3L5) (LZL4L5) -- (L3L4L5) + (L2L3) + (LZL4) + (L2L5) + (L3L4) + (L3L5) + (L4L5) - L, - L3 - L4 - L5) B = 1 + 4(LlL2L&&5) - ~ ( ( L I L ~ L+~ (LL~I L ) ~ L ~ L+, ~ ) (L1L21d4L5)-k (L1L&4L5) + (L!&3L4L5)) + 2((Ll&L4) + (LlL2L4) + (LlLZL5) + (L1L3L4) + (LlL3L5) + (11'1L4L5) + (L2L3L4) + (LZL3L5) + (L2L4L5) + (L3L4L5))((LlL2) + (LlL3) + (LlL4) + (LlL6) + (LZL3) + &&4) + (L2L5) + (L3L4) + (L3L5) + (L4L5)) and the arguments for H (i.e., L1, Lz, L3, L4, and L5) are given in Table I. Recombination Coefficient. Having solved for the surface coverage of atomic fluorine, using eq 25, we can calculate the recombination rate, uR, from UR = PNF(BFp+ 6,) (26) The recombination coefficient, y,defined as two times the ratio of the number of fluorine atom wall collisions re-
If suitable values for sticking coefficients, thermal desorption rates, and steric factor can be either theoretically or experimentally determined, then the recombination coefficient can be found as a function of temperature, species present, and surface. These many parameters ultimately depend on only three quantities which must be determined experimentally: P, EOF, EeF2 (where EOF and EoFz are the well depths for chemical surface bonding for F and F2, respectively); all other quantities are either known or theoretically determinable. Sticking Coefficients To determine the sticking coefficients, the atom-wall inelastic collision process must be addressed. This will not only involve modeling of the collision process but also require some knowledge of the well depths associated with the various types of particle-surface bonds: (F-Ni)p, (FNi)c, (F2-Ni!p, (F2-NiIc,and (1-Ni). Physical bonding; will take place with all the species although analysis given here will show that this type bonding is only important at temperatures below about 70 K. We have already mentioned the justification for including chemically surfacebonded Fz and the need for chemically surface-bonded F is implicit in the Langmuir-Rideal mechanism at room temperature and above. Neither the well depth for physical or chemical surface bonding is known for fluorine. Physical Surface Bonding. A simple model for physical surface bonding can be constructed for the potential well near a surface based upon an adsorption (attractive) potential and a repulsive potential as suggested by Lennard-Jones.13 The construction is described by example. The closest distance of approach of nickel atoms in metal is 2,48 A, while that for fluorine atoms is 1.28 A. The equilibrium distance of approach, RE,for nickel and fluorine can be taken as the mean of these two distances or 1.88 A. At this equilibrium distance the repulsive forces are from 1/3 to 1/2 that due to the attractive forces. Thus, by letting the repulsive force be 40% that of the attractive force, a well depth can be determined by algebraically summing the two potentials. Thus, an attractive potential, assumed to be W(r),where r is the distance of separation, will yield a well depth, AE,, at the equilibrium position, RE, of AE, O.~W(RE) (28) No reference is made to an activation energy and in fact no activation energy is experimentally observed for physical adsorption.8 Several authors have described the functional relationship of the attractive potential for physical bonding, W(r). The earliest of these descriptions13was the simplest, and yielded fair results. As reported by several authors,1P16 this relationship served as an upper limit on the physical attractive force. In a more recent paper, Mavroyanni~l~ derived a formula for W(r)and compared the results of his relationship with four previously derived relationships, those of Lennard-Jones,13Bardeen,14Margenau and Pollard,15and Prosen and Sachs.16 The results of this comparison showed that the Mavroyannis formulation compared as well as, or better than, the previous relationships when compared with data for several atom-surface systems. In addition, the Mavroyannis relationship makes use of readily available material properties. For these reasons the Mavroyannis potential was adopted here for
44
The Journal of Physical Chemistry, Vol. 84, No. 1, 7980
.ll~wper,Ultee, and Dorko
TABLE 11: Computational Parameters ~~
P
0.018
A Q F
1.571 X 1.261 X
EOF
AEWF2 EOF2 AEWAr
AEWHe
Cil(k/h)
~~
erg/atom erg/atom
erg/molecule 1.328 x 1.215 X lo-'' erg/molecule 1.448 X erg/atom 7.317 X erg/atom 3.3878 X loz5particles/(Ks cm2)
use in approximating the attractive potential for physical surface bonding. The formula for this potential is given by (42) h w,/21/2 W(r) = -(29) 12r33N + AW,/21/2 (q2) where ( g 2 ) is the sum of the electronic charge of each electron in the adsorbed particle times the expectation value of its orbital radius, A W, is the work function of the metal surface, and N is the number of electrons in the adsorbed atom. The results obtained by use of eq 28 and 29 are listed in Table I1 for those atom-surface systems of interest. The values of these well depths are consistent with those to be expected for van der Waals adsorption.* Chemical Surface Bonding. The well depth in chemical surface bonding is not as easily modeled as that for physical surface bonding; this parameter is usually experimentally determined. The values for chemical bonding, E,, given in Table I1 were obtained by an optimization process discussed in the Appendix with the data presented in the results portion of the Experimental Section. Atom-Wall Inelastic Collision. In order to model the inelastic collision process, the soft cube model by Logan and Keck was used.ls Some results of this model can be found in the 1 i t e r a t ~ r e . lAs ~ ~dictated ~~ by the soft cube model, the surface of the wall is pictured as made up of wall atoms vibrating at a frequency consistent with the bulk Debye temperature (i.e., at the Debye frequency, 0,) in a direction normal to the plane of the surface. During a surface interaction only the velocity component of the gas-phase atom normal to the surface was considered. The use of the normal velocity component enabled data correlation which was not able to be made by using the total velocity in the case of low incident angle particle-surface collisions. The gas-phase atom was further assumed to interact with only one surface atom (i.e., each surface atom was treated as a surface site). The various frequencies and potentials were then modeled as shown in Figure 1. The shape of the well beyond the interaction position was of no importance; the incoming particle was simply given additional velocity consistent with the well depth above its Boltzmann-predicted gas-phase velocity. In the case of physical surface bonding, this presented no problem since no activation energy is required to enter the well. Pagni21 makes no distinction between chemical and physical bonding except for the well depth. It is, however, possible to formulate a cutoff energy (activation energy) required to enter the chemical-bonding well and to incorporate it into Logan's model. (Very little is available on theoretical or experimental determination of the activation energies required for the present application. A reasonable estimate of the activation energies for chemical bonding was taken to be 10% of the well depth. Based on this estimate, sticking coefficient curves were obtained which resembled the expected shapes presented by EmmetL8) A collision of a gas-phase atom (or molecule) with
Figure 1. Gas-wall collision model.
the wall can have only one of three possible outcomes, adherence to the wall in a physical well (physical adsorption), adherence to the wall in a chemical well (chemical adsorption), or avoidance of trapping (scattering). By including an activation energy barrier, we allowed gasphase atoms (or molecules) below the barrier to interact solely with a physical well and those above the barrier to interact with only a chemical well. (If the particle never overcomes the activation barrier it cannot be chemically bonded, and if the energy of the reflected particle is sufficient to overcome the chemical well plus the activation barrier, it could not be expected to be trapped in the somewhat smaller physical well.) Having entered the appropriate well, the atom (or molecule) encountered an interacting spring of spring constant k . The spring, following the discussion of Modak and Pagn? represented an exponential repulsive potential, which led to a spring constant of
k g = 0.2D1.17/~2
(30)
where D is the well depth (AEwfor the physical and E, for the chemical well), and K the distance over which the interaction takes place. The spring constant for the surface atom of mass m, was
k , = wa2rns (31) A collision began when a gas phase particle of mass mg came in contact with the moving end of the spring with constant k,. From this point until the collision ended, the system was treated as a simple undamped two mass system. The collision, then, took place with a gas-phase particle of velocity u u = V , + (2D/mJ1l2 (32) representing a combination of the Boltzmann-predicted random surface-normal component of velocity, V3,the probability for which is given by9
2 v,
where T is the gas temperature at the wall, and the additional kinetic velocity gained from entering the well. The position of the spring at the moment of the collision, Y(O),
The Journal of Physical Chemistry, Vol. 84, No. 1, 1980
FI Atom Recombination on a Ni Surface
m=
.bI \
1
1
45
1 i
I 0
100
200
400
300
500
600
700
800
0
100
200
TEMPERATURE I K I
was also random but equal in magnitude to the position of the wall atom at that moment, Z(0). The initial conditions were Z(0) = Y(0) (34)
800
I
I
I
I
I I
I I
I
I
I I
(36)
(or D plus the activation energy in the case of chemical surface bonding) the gas particle was trapped, and if (37)
(or D plus the activation energy in the case of chemical surface bonding) the gas particle escaped the well and avoided trapping. Referring to Figure 1,the equations of motion for the collision process are m,Z + ( k g k,)Z - k,Y = 0 (38)
+
m , Y - kgZ + k,Y = 0
700
I I
Equation 35 is interpreted to mean that if the velocity, u, were less than Y(O),there would have been no collision at that initial condition. The collision ended at time t, when the magnitude of colliding-particle position, Y(t,),again equaled the magnitude of the wall-atom position, Z(t,). At that point, if
)/2m~Y(t,)~1 2D
800
7
(35)
Xmg[Y(t,)zI< D
500
Figure 3. Initial sticking coefficient for F, on Ni as a function of temperature. The curves show computed values for physical bonding, chemical bonding, and a composite of the two.
T
> Y(0)
400
TEMPERATURE I K I
Flgure 2. Initial sticking coefficient for F on Ni as a function of temperature. The curves show computed values for physical bonding, chemical bonding, and a composite of the two.
u
300
(39)
The solution for eq 38 and 39 then gave analytic expressions for Y(t),Z(t),and Y ( t )which could be used to find t,. Having t,, the collision criteria, eq 36 and 37, were used to make decisions of either a stick (trapped) or no stick (escape) for each set of initial conditions. Since the model assumed that the surface site represented by the surface atom was clean, the result of the solutions of eq 38 and 39 was used to calculate the initial sticking coefficient of eq 5 for the specific set of initial conditions. Initial Sticking Coefficient. The stick or no stick decisions from above were made for 40 initial conditions of surface-atom phase angle for each of 26 initial nondimensional velocity conditions for each temperature. These in turn were weighted by the probability of that velocity condition occurring according to the probability density function, eq 33. This was done for 20 temperatures between 0 and 800 K for each of the gas-metal systems of interest. The result, of this process yielded the initial or clean surface sticking coefficient, S,(T), defined as the statistical probability that any given collision of a gas-phase particle with the clean surface at the temperature T would cause the particle to be trapped by the surface. The gas and surface were assumed to be in thermal equilibrium.
\ 0
100
200
300
400
500
600
700
0
TEMPERATURE i K I
Figure 4. Initial sticking coefficient for Ar on Ni and He on NI as a function of temperature (physical bonding only).
The initial sticking coefficients were calculated for both physical surface bonding and, where applicable, for chemical surface bonding. The result of the calculations for atomic and molecular fluorine, and for argon and helium diluents on a nickel surface, are given in Figures 2-4. The constants for intermediate calculations of the physical well depths by use of eq 29 were taken from the l i t e r a t ~ r e ~ ~ ~ ~ ~ for F, Ar, and He. In the case of Fz, (q 2 ) was obtained from ab initio calculationsz4which solved the time-independent Schrodinger equation by using a configuration-interaction method.
Computational Results As previously mentioned, there are three parameters which had to be experimentally determined: P, ET,and EOFz. The experiment used to optimize these parameters will be presented in the next section and the optimization process used to determine these parameters is discussed in the Appendix. The results of the optimization appear in Table 11. In order to maintain the emphasis on modeling the fluorine recombination process at the wall, we will assume, at this point, that the results of our optimization are correct and demonstrate the use of the model. First we will examine the isolated surface concentrations represented by eq 20-24. It is clear from the presence of the impingement rate in these equations that these isolated surface concentrations are dependent on the gas-phase number densities of the various species being calculated; therefore, as a representative case, Figure 5 shows the
The Journal of Physical Chemistty, Vol. 84, No.
46
I, 1980
Jumper, Ultee, and Dorko
PHYSICAL
BONDING
10- 1
CHEMICAL BONDING
I
1I
7--f 1.0
-
*---_-----_
~
1-
1 I I
/
,\ \
1 I 1
1 6 ~ 1
\ \
------ !6
10-2
Fpi
! 0 ARL 1
