Excess Electron Energy in Rare Gas-Propane Systems
(17) Y. A. Attia and J. Rubio, Br. Polym. J . , 7,135 (1975). (18) Th. F. Tadros and J. Lvklema. J. Nectroanai. Chem.. 17. 267 (19681. ilQj W.M. Heston: Jr.,-Rli R,). Namely
+
E,'(ls)
+ E,'(ls)
= [E,'(ls) + Eml(ls)lp+
[E,'(ls) + Eml(ls)Im (3)
where subscripts p and m indicate pure propane and a liquid rare gas-propane mixture, respectively, f l s is a self-consistent field potential which is obtained from Poisson's equation,3.13c is the optical dielectric constant, and Q1, is a hydrogenic Is wave function. The optical dielectric constant in the liquid rare gas-propane mixture region outside the nth propane layer is calculated assuming that the molar polarization of a liquid rare gas-propane mixture can be approximated by the sum of the product of the molar polarization and the mole fraction of pure propane and that of a pure liquid rare gas. (2) The short-range interaction of an excess electron with medium electrons which includes repulsive interactions is approximated by
E , = Vo,c[Cis(R)- Cis(R&l+ Vo,p[Ci,(R,)- Cl,(R)I + Vo[1 - C1,(R,)] (6) where Vo,c,Vo,p,and Voare the quasifree electron energy in the first coordination layer of propane (R, Ir IR, R, = R - Rd),that in a propane cluster containing the second
quasifree electron energy in a medium is a function of the density of the medium, Vo,cwas calculated by the Wigner-Seitz method21 taking the density of the first coordination layer instead of the bulk medium density. The quasifree electron energy in the liquid rare gas-propane mixture region, Vo, was assumed to be given by
Vo = VopX + V0,Jl - x )
(7)
where Vo,pand Vo,rare the quasifree electron energy in pure propane and a pure liquid rare gas, respectively, and x is the mole fraction of propane. It has been shown experimentally that a relation such as (7) holds for hydrocarbon mixtures.20 Values of the physical constants used in the present calculation are listed in Table I.
Results and Discussion The results of the calculation are given in Figures 2-4 and Tables 11-IV. In the calculation it is assumed that the concentration of excess electrons is so low that the number of propane molecules in propane clusters containing excess electrons is negligibly small compared with that outside the propane clusters for x L 0.1. Therefore, the propane concentration in the region outside the propane clusters practically represents the propane concentration in the total system when x 20.1. A criterion for electron localization is expressed in terms of the energy difference between the localized electron state, Els,and the quasifree electron state, V0;l2 the excess electron is localized for El, < Toand delocalized for El, 5 Vo. Values of El, - Vo for excess electrons in propane clusters (N = 4 and n = 1-6) are plotted against the propane concentration in Figures 2-4 for xenon-propane, krypton-propane, and argon-propane systems, respectively. In Figure 2, El, - Vo for n = 1is negative for the propane concentration x = 0 to 1indicating that the excess electron in the propane cluster which has only a single layer is localized over _the entire propane concentration range. However, El, - Vofor n = 2 is positive up to x = 0.5 and then suddenly becomes negative. Therefore the excess electron can be localized in the propane cluster with n = 2 in the propane concentration range, x 1 0.6. The localization of excess electrons in propane clusters becomes less favorable with increasing n. In propane clusters with n = 5 or 6 excess electrons can be localized only when x is greater than 0.9.
TABLE 11: Charge Distribution of Excess Electrons in Liquid Xenon-Propane Systems at 170 K (N= 4 )
1
0
C,s(Rn1
x n = 0
0.1
-0.6))
'
' ' 0.2
' 0.4
' ' 0.6 ' 0.8:.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
J
- Po)for
systems at 170 K ( N =
1
2
3
4
5
6
0.162 0.212 0.290 0.395 0.527 0.641 0.757 0.796 0.821 0.839
0.028 0.034 0.043 0.057 0.086 0.147 0.996 0.996 0.996 0.996
0.012 0.015 0.021 0.024 0.032 0.046 0.081 0.184
0.007 0.010 0.013 0.017 0.021 0.025 0.043
0.006 0.006 0.009 0.009 0.013
0.003 0.006 0.006 0.006
1.000 1.000
1.000
0.029 0,044 0.120
1.000
1.000
0.081
0.018
0.010 0.015 0.021 0.036
0.081
1.000
1.0
( m o l e fraction 1
X
Flgure 2. (E
I
82,No. 20, 1978 2217
The Journal of Physical Chemistry, Vol.
Excess Electron Energy in Rare Gas-Propane Systems
excess electrons in liquid xenon-propane 4) as functions of x and n .
TABLE 111: Charge Distribution of Excess Electrons in Liquid Krypton-Propane Systems at 1 2 3 K (N= 4 ) C,,(Rn) x n=
0 0.1 0.2
0
0.3 0.4 0.5 0.6 0.7
0.8 0.9
1
2
3
4
5
6
0.255 0.385 0.646 0.667 0.775 0.809 0.830 0.846 0.859 0.868
0.022 0.030
0.011
0.007 0,009 0,009 0.01 6
0.006 0.006
0.003 0.006 0.006 0.006 0,009 0.014
0.041 0.063 0.107 0.997 0.997 0.997 0.997 0.997
0.014 0.017 0.023 0.035 0.064 0.098
0.020
0.008 0.008 0.012
1,000
0.029 0.047 0.095
0.017 0.028 0.059
1.000 1,000
1,000 1.000
0.182
0.020 0.034 0.104
1,000 1.000
TABLE IV: Charge Distribution of Excess Electrons in Liquid Argon-Propane Systems at 87.5 K ( N = 4 ) C,S(R, 1 1
2
3
4
6
6
0.006 0.006 0,009 0.015 0.019 0.028 0.062 0.130
0.003 0.005
0.003
0.008 0.008
0.8
0.881 0.903
1.000 1.000
1.000 1.000
0.005 0.005 0,009 0.013 0.025 0.041 0.130
0.9
0.019 0.029 0.043 0.077 0,999 0.999 0.999 0.999 0,999 0.999
0.009
0.6 0.7
0.568 0.695 0.793 0.821 0.840 0.854 0.865 0.874
x n=
0 -0.8;"
" 0". 4 '
0.2
0.8
I
i
1.0
(mole fraction)
X
Figure 3. (El,
0.6
- Po)for excess electrons in liquid krypton-propane
systems at 123 K (N = 4) as functions of x and
n.
0.1 0.2 0.3 0.4
0.6
0.011 0.016 0.022 0.038 0.063 0.139
1.000 1.000 1.000
0.012 0.021 0.033 0.066
0.003
1.000
centrations than in xenon-propane systems, namely, El, - Vo for n = 2 in xenon-propane systems (Figure 2)
X
( m o l e fraction)
Figure 4. (€ls - Po)for excess electrons in liquid argon-propane systems at 87.5 K (N = 4) as functions of x and n.
For krypton-propane systems (Figure 3), abrupt changes in El, - Toare seen for n = 2-6, and El, - Vo tends to change from positive to negative a t lower propane con-
changes from positive to negative at a propane concentration of x = 0.5-0.6 but in krypton-propane systems the change occurs at a propane concentration of x = 0.4-0.5. The abrupt change in El, - Uo for n = 3 occurs a t a lower propane concentration in krypton-propane systems than in xenon-propane systems. For argon-propane systems (Figure 4), El, - Vofor n = 2 changes abruptly from positive to negative at a still lower propane concentrations than for krypton-propane systems. In Tables II-IV, the charge distribution of excess electrons for N = 4 is shown for xenon-propane, krypton-propane, and argon-propane systems, respectively. For example, for the 0.1 mole fraction argon system (Table IV), the excess electron charge enclosed within R,, Cls(R,), is 0.695 for n = 1 indicating that the excess electron is mainly localized in the cluster, and it decreases abruptly to 0.029 for n = 2 indicating that the excess electron distribution is very spread out. For the argon-0.5 mole fraction propane system, 99.9% of the excess electron charge is enclosed within R, for n = 2, but only 6.3% for n = 3. The abrupt change in El, - i;6 in Figures 2-4 is associated with that in the charge distribution in Tables II-IV; the value of Cl,(R,) is high in the case where the excess electron is localized, while it is very low whenever
2218
The Journal of Physical Chemistry, Vol. 82, No. 20, 1978
the excess electron is delocalized. The ground state energies in xenon-propane, krypton-propane, and argon-propane systems for N = 6 and 8 were also calculated and similar results were obtained except that El, increases with increasing N. Thus electron localization tends to become slightly unfavorable with increasing N. The difference in electron localization among N = 4,6, and 8 appears only for n = 1; excess electrons are localized in propane clusters with n = 1 in xenonpropane systems in propane concentration ranges of 0-1, 0.2-1, and 0.4-1 mole fraction for N = 4,6, and 8, respectively, in krypton-propane systems at 0-1,O.l-1, and 0.2-1 mole fraction propane concentrations for N = 4,6, and 8, respectively, and in argon-propane systems at 0-1, 0-1, and 0.1-1 mole fraction propane concentrations for N = 4, 6, and 8, respectively. The results of the calculation can be summarized as follows: (1)Excess electrons are mostly localized in these systems for n = 1. In the case of N = 4, the electron localization occurs over the entire range of x for all systems (Figures 2-4). The cases where electrons me not localized in propane clusters with n = 1 are N = 6 at 0 I x < 0.2 and N = 8 a t 0 I x < 0.4 in xenon-propane systems, N = 6 at 0 I x < 0.1 and N = 8 a t 0 Ix < 0.1 in krypx < 0.1 in arton-propane systems, and N = 8 at 0 I gon-propane systems. ( 2 ) Electron localization is progressively favorable in the order of N = 8,6, and 4. This is because El, becomes lower in this order. (3) A rather high concentration of propane is required to localize excess electrons in propane clusters with n 1 2. For n = 2, 60, 50, and 40 mol % propane concentrations are required for electron localization in xenon-propane, krypton-propane, and argon-propane systems, respectively. The propane concentration required to localize excess electrons in propane clusters with n = 3 increases to 80,70, and 70 mol % for xenon-propane, krypton-propane, and argonpropane systems, respectively, and they do not change very much for n = 4-6. (4) Generally, electron localization is progressively favorable in the order of xenon-propane, krypton-propane, and argon-propane systems. We may say that an excess electron exists temporarily in the localized state if its mobility is
