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Calculation of time-dependent electron velocity distributions and rate coefficients for electron attachment to chlorofluoroethanes
CHEMICAL PHYSICS LFXIERS
Volume 118. number 6
-16 August 1985
CALCULA-I-ION OF TlMEDEPE; TDEN-I- ELECl-RON VELOCITY AND RATE COEFFICIENTS FOR ELECI-RON A-l-l-ACHMENT l-0 CHLOROFLUOROETHAN R. SCHMIDT
DISTRIBU’T’IONS
and W_ STILLER
Cenrral fnsrrrure of fsorope and Radrarron Research. DDR -7050 Lmrpz,g. Germon Denrocrarrc Repubtrc
Academy
of Scrences of rhe GDR,
Received 8 May 1985
For electron allachmcm IO l,l.l-C,CI,F,. 1,1,2X,C!,F,. 1.1~C$ZI,F_,. and 1,2-C,CI,F, (carrier gas Ar) Lhe time-dependent electron velocity dislnbuuon runc~~ons are calculaled wa the Bollunarm equatron. rsulting in time-dependen altachment rate coelklems smaller than the equihbnum rate coclflcienk
1. Introduction Ln recent years there has been much interest in electron attachment to halogenated hydrocarbons. These compounds, characterized by Iarge electron attachment cross sections, are of interest because of their potential use as high-voltage gaseous insulators. An electron swarm method has been applied [14] to measure the electron attachment rate coefficients k as a function of the density-reduced electric field E/N for a large number of halogenated hydrocarbons. From the set of rate coefficients k(E/N) the electron attachment cross sections aR(e) have been calculated as a cunctlon of the electron energy by the swarm unfolding technique. The electron attachment reaction has been
studied
for very small
amounts
of hydrocarbon
m an mert earner gas (N2 or Ar). In this low-concentration case the electron energy distribution function which has ‘ro be known for the unfolding procedure is assumed to be undisturbed by the attachment reaction_ Then the electron distribution is characteristic of the earner gas and may be considered as a mven function for all E/N applied. In this paper we investigate the influence of the attachment reaction on the electron velocity distribution for a higher concentration of the electron attaching gas. Considering the large attachment cross sections [llri, this influence is expected to be quite strong. 564
For the electron attachment to chlorofluoroethanes (CFE) (carrier gas Ar,E/N = 0), the time-dependent electron velocity distributions are calculated and in the next step the time-dependent kinetic electron temperatures and attachment rate coefficients are derived.
2. Reaction model and cross sections The following three-component system is considered: electrons (number density n,) and one of the four CFEs 1 ,1,l-C2C13 F3 (1 ,1,1 -trich.lorotrifluoroethane), I ,I ,2-C2C13F3 (1 ,1,2-tnchlorotrifluoroethane), 1 ,l-C2C12F4 (1 ,l -dichlorotetrafluoroethane), or 1 ,2-CZCIZFq (1,2-dichlorotetrafluoroethane) (number density nm) are diluteIy dispersed in an inert carrier gas (Ar, number density nc). An electric field is not applied. The number denslties fulfil the condition n,
enc.
(1)
We are interested in the time evolution of the electron velocity distribution function f(u,f) which in our case is governed by two kinds of collisions: elastic e-Ar collisiorts and reactive e-molecule (attachment) collisions. The influence of other collisions or of reaction products onf(u,t) is neglected. At tune r = 0, the distribution functions of all components are Maxwell 0 009-2614/85/S (North-Holland
03.30 0 Elsevier Smence Publishers B.V. Physics Publishing Division)
CHEhIlCALPHYSICSL~~
Volume 118,numbP,r6 distributions
(MD) v&h temperature
T = T, (tempera-
ture of the tamer gas T, = 298 K). For t > 0, only the electron distribution will r&ange whereas the distributions of the other components are assumed to be undisturbed MD. For the elastic e-Ar collisions the cross section u given by Milloy et al. [S] is used in our calculations. The cross sections on for the electron attachment to the CFE are taken from McCorkle et al. [l] . In the subthermal energy range we approximated oR(c) by a linear function of c to give the correct thermal attachment rate coefficient k,, [l] when OR 1s applied to
ktha
=
(2/m,)“2$ •“~u~(Elfhl,,(~) de,
(2)
0
give the order of magnitude now reads
of u(x) and on(x).
Eq. (3)
afed -=$+=(*,(;~+xf)] a7
- -p+,(x)f,
(4)
cot f(x,O) = (4/n’!‘) F(O,7) = F(~,T)
exp(-x2) = 0
(initial
(boundary
condition),
conditions),
where F(x,T) - x2f(x,7).
F(x,T) is connected with the number electrons
relative to the initial number
density of the density by
wherefMD(c) is the MD of electron energy and me and E are the electron mass and energy, respectively.
3. calculations
and with the probability tron speed by
The velocity distnbutron functions of a multicomponent system can be calculated from a system of coupled Boltzmann equations [6,7] _As mentioned above, for our model only the velocity distribution of the electronsf, =f(u.t) has to be studied. The MD of both the other components will be undisturbed by the very small number of electrons present in the system. Because me em, (me,mc: mass of electrons and Ar, respectively) the elastic colhsion term may be simplified, replacmg the integral term by a differential term [g] _The resulting partial differential equation for
fbt) is
distribution
P(x,T) of the elec-
From the electron distribution functions calculated by eq. (4) the time-dependent macroscopic quantities we are interested rn can be derived: ICinetrc electron temperature reduced to the temperature T, of the carrier gasT&)/T,
=$ i 0
x’P(x,l)
dx.
(7)
Non-equilibrium departure Arc(~) of the electron attachment rate coefficient k(7)from the thermal (equilibrium) attachment rate coefficrent k(O): -
q.p$(vlf.
(3)
It is convenient to change from electron speed u and time t to drmensionless variables x = v/v0 (reduced electron speed) and r = t/to (reduced time), where u. = Sc = 2nz,/na,. (2’CnTc/me)1’2 and to = (~Qcv~u~)-I, The elastic cross section u(x) and the reaction cross section uR(x) are replaced by dimensionless cross sectiOJISE(X) = O(X)/U~ and CR(X) = UR(JC)/UR~ where uo and one are constant cross sections suitably chosen to
Ak(7) = (k(T) - k(‘))/k(‘) I
(8)
k(T) = uoiRo J xE,(x)P(x,T) dx,
(9)
0 k(O) = (4/n’/‘)”
0u RO 1 ZR(*)x3 exp(-x2) 0
dx. (10)
Eq. (4) is solved by a standard numerical method (difference method) which is based on replacing the 565
Volume
CHEMICAI.
118, number 6
PHYSICS LEITIERS
parabobc differential equation (4) by a system of difference equations treated by an effiaent numerical procedure (see, e.g., refs. [9, IO]). The accuracy of the calculated electron distribution F(x,T) is tested by inserting F(x,T) into the balance equations of number density and kinetic energy of the electrons. The error of the numerical computations is < 1% for aU t > 0.
I
4. Results
and discussion
The electron speed probabihty distnbution P(x,T) (electron energy E = 0_02ti2 eV, time t = 1.4 X 1067s) has been calculated for electron attachment to the CFEs 1,l ,I-C,CI,F, , 1 ,1.2-C,Cl,F,, 1 .l-C2C1,F,, and 1 ,2-C2C12F, in the carrier gas Ar (number density n, = 2.687 X 10lg cmm3) for a CFE/Ar ratio of n,/n, = 10B5. For tlus relatively large CFE concentration a distinct departure of P(x ,T) from its initially assumed MD is expected to result. Fig. 1 shows the time evolution
I,5 i
of the electron
dis-
16 August1985
tributionP(x,T) for attachment to 1 ,lil-C&F3 _ After a very short time T = O.C!l the initial MJ3 is largely depleted in the low-energy part (range of strong reactivity, as indicated by the function xaR(x)) and shifted to higher energy. At T = 05 the similarity ofP(@,T) . with a MD disappears completely_ P(x ,T) shows two peaks: the first peak is at lower energy (X 5 1.7) and a second peak is built up in the high-energy range (JC= 3.4)_ At T = 1 the distribution consists of a single highenergy peak (x = 3.4) and remains unchanged in shape for T > 1. It can be seen that in such strongIy reacting systems the explicitly time-dependent calculation of P(x,T) 1s required because the departure ofP(x,T) from a MD is quite large (see, e.g., the double-peaked distribution at 7 = 05). Replacing thrs method of calculation by a simpler approach, like the method described by Keizer [l l] using for P(Jc,T) a MD with time-depen-
dent temperature,
is not expected
to be successful.
In
the case of the other CFE studied in this paper the dis-
tribution functions P(x,T) are disturbed by the attachment process to a lesser extent. In contrast to the
-
7
O,!5-
: L
Fig. 1. Time evoluuon of the election speed probabMty Ar, “m/n, = lo-). Dotted curve. r(x) - XUR(X)-
566
disbribution function P(x,T) for attachment to l,l,
l-C2QF3
(carrier gas
Volume 118, number 6
CHEMICAL PHYSICS LETI-ERS
16 August 1985
7
0
Fig 2. Reduced kinetic eIection temperature Te(r)/Tc for electron attachment to chlorofluoroethanes (CFE) (carrier gas Ar, nm/nc = 10-5). 1 ,1 ,l C2C1,F, case the mean kinetic energy of the electron distributions becomes lower for r > 0. Fig. 2 shows the variation of the reduced electron
temperature T,(r)/T, with tune r (eq. (7)) Depending on the details of the cross sections for the 4 CFE a different behaviour of T,(r)/T, may be found. In the case of 1,l ,l-C$l,F, the electron temperature increases from the initial value T,/T, = 1 to a high-temperature steady state (TJT, = 8 2), whereas for the other CFE the electron temperature decreases to a lowtemperature steady state (T,/T, = 0.82 for 1,1,2CzC13F3, TJT, = 095 for 1 ,I-C2C12F, and 1,2C2%F4)-
Fig. 3 shows the time-dependent departure NC(T) (eqs. (8)-(10)) of the electron attachment rate coefficient k(7) from the thermal rate coefficient IJo) (Hl = k,, in ref. [I], table V). The deviation ofP(x,r) from a MD under the influence of the attachment reaction results in a considerable decrease of the rate coefficient . For 1,l ,1 -C2C1, F, , the nonequrlibrium departure AJc(r) of the rate coefficient goes up within a time interval AT = 1 from zero (T = 0) to a stationary value of ]AJc I = 93%. The other CFEs are characterized by smaller non-equilibrium effects: the stationary deviations ]ak] attained within AT = 1 are 13% (1,1,2C,Cl,F,), 7% (1 ,l-C,Cl,F,), and 3% (1 .2-CzC12F4). It should be mentioned that-calculation for the lowconcentration case (n,/n, = 10S7-10A) have shown a negligibly small decrease (
1
2
3
T
Fig. 3. Non-equilibrium departure u(r) of the electron attachment rate coefficient k from the thermal value k(O) for attachment to CFE (carrier gas Ar. nm/nc = 1Oj).
An exception is 1,I , 1-C,Cl, F, where n,.,.,& has to be as small as 10mg for IA7c I to be
References [l] D.L. McCorkle, I. Szarnrej and LX. Christophorou, J. Chem. Phys. 77 (1982) 5542. [2] D.L. McCorkle, A-A. Christodoulides. L G. Christophorou and I. Szamrej. J. Chem. Phys. 72 (1980) 4049. (31 AA. Christodoulides, L G. Christophorou, R.Y. Pai and CM Drug. J. Chem. Phyn 70 (1979) 1156. [4] R.Y. Pai, L.G. Christophorou and A-A. Christodoulides, J. Chem. Phyr 70 (1979) 1169. [5] HE. MJJloy, R.W. Crompton, J.A. Rees and A.G. Robertson, Autielian J. Phys. 30 (1977) 61. [6] J_ Ross and P. Mazur. J. Chem. Phys. 35 (1961) 19. [7] B. ShizgeJ and M. Karphrs, J. Chem. Phys. 54 (1971) 4345.43s7. [B] EA. Desloge and S.W. Mathysse, Am. J. Phys. 28 (1960) 1. [9] D-U. v. Rosenberg, Methods for the numerical solution of partial differential equations (Elszvier, Amsterdam, 1969). [lo] J. Wilhelm and R. Winkler, Conhib. Plasrnaphys 16 (1976) 287. [ll] J. Keizer, J. Chem. Phys. 58 (1973) 4524.
567
Volume 118. number 6
-16 August 1985
CALCULA-I-ION OF TlMEDEPE; TDEN-I- ELECl-RON VELOCITY AND RATE COEFFICIENTS FOR ELECI-RON A-l-l-ACHMENT l-0 CHLOROFLUOROETHAN R. SCHMIDT
DISTRIBU’T’IONS
and W_ STILLER
Cenrral fnsrrrure of fsorope and Radrarron Research. DDR -7050 Lmrpz,g. Germon Denrocrarrc Repubtrc
Academy
of Scrences of rhe GDR,
Received 8 May 1985
For electron allachmcm IO l,l.l-C,CI,F,. 1,1,2X,C!,F,. 1.1~C$ZI,F_,. and 1,2-C,CI,F, (carrier gas Ar) Lhe time-dependent electron velocity dislnbuuon runc~~ons are calculaled wa the Bollunarm equatron. rsulting in time-dependen altachment rate coelklems smaller than the equihbnum rate coclflcienk
1. Introduction Ln recent years there has been much interest in electron attachment to halogenated hydrocarbons. These compounds, characterized by Iarge electron attachment cross sections, are of interest because of their potential use as high-voltage gaseous insulators. An electron swarm method has been applied [14] to measure the electron attachment rate coefficients k as a function of the density-reduced electric field E/N for a large number of halogenated hydrocarbons. From the set of rate coefficients k(E/N) the electron attachment cross sections aR(e) have been calculated as a cunctlon of the electron energy by the swarm unfolding technique. The electron attachment reaction has been
studied
for very small
amounts
of hydrocarbon
m an mert earner gas (N2 or Ar). In this low-concentration case the electron energy distribution function which has ‘ro be known for the unfolding procedure is assumed to be undisturbed by the attachment reaction_ Then the electron distribution is characteristic of the earner gas and may be considered as a mven function for all E/N applied. In this paper we investigate the influence of the attachment reaction on the electron velocity distribution for a higher concentration of the electron attaching gas. Considering the large attachment cross sections [llri, this influence is expected to be quite strong. 564
For the electron attachment to chlorofluoroethanes (CFE) (carrier gas Ar,E/N = 0), the time-dependent electron velocity distributions are calculated and in the next step the time-dependent kinetic electron temperatures and attachment rate coefficients are derived.
2. Reaction model and cross sections The following three-component system is considered: electrons (number density n,) and one of the four CFEs 1 ,1,l-C2C13 F3 (1 ,1,1 -trich.lorotrifluoroethane), I ,I ,2-C2C13F3 (1 ,1,2-tnchlorotrifluoroethane), 1 ,l-C2C12F4 (1 ,l -dichlorotetrafluoroethane), or 1 ,2-CZCIZFq (1,2-dichlorotetrafluoroethane) (number density nm) are diluteIy dispersed in an inert carrier gas (Ar, number density nc). An electric field is not applied. The number denslties fulfil the condition n,
enc.
(1)
We are interested in the time evolution of the electron velocity distribution function f(u,f) which in our case is governed by two kinds of collisions: elastic e-Ar collisiorts and reactive e-molecule (attachment) collisions. The influence of other collisions or of reaction products onf(u,t) is neglected. At tune r = 0, the distribution functions of all components are Maxwell 0 009-2614/85/S (North-Holland
03.30 0 Elsevier Smence Publishers B.V. Physics Publishing Division)
CHEhIlCALPHYSICSL~~
Volume 118,numbP,r6 distributions
(MD) v&h temperature
T = T, (tempera-
ture of the tamer gas T, = 298 K). For t > 0, only the electron distribution will r&ange whereas the distributions of the other components are assumed to be undisturbed MD. For the elastic e-Ar collisions the cross section u given by Milloy et al. [S] is used in our calculations. The cross sections on for the electron attachment to the CFE are taken from McCorkle et al. [l] . In the subthermal energy range we approximated oR(c) by a linear function of c to give the correct thermal attachment rate coefficient k,, [l] when OR 1s applied to
ktha
=
(2/m,)“2$ •“~u~(Elfhl,,(~) de,
(2)
0
give the order of magnitude now reads
of u(x) and on(x).
Eq. (3)
afed -=$+=(*,(;~+xf)] a7
- -p+,(x)f,
(4)
cot f(x,O) = (4/n’!‘) F(O,7) = F(~,T)
exp(-x2) = 0
(initial
(boundary
condition),
conditions),
where F(x,T) - x2f(x,7).
F(x,T) is connected with the number electrons
relative to the initial number
density of the density by
wherefMD(c) is the MD of electron energy and me and E are the electron mass and energy, respectively.
3. calculations
and with the probability tron speed by
The velocity distnbutron functions of a multicomponent system can be calculated from a system of coupled Boltzmann equations [6,7] _As mentioned above, for our model only the velocity distribution of the electronsf, =f(u.t) has to be studied. The MD of both the other components will be undisturbed by the very small number of electrons present in the system. Because me em, (me,mc: mass of electrons and Ar, respectively) the elastic colhsion term may be simplified, replacmg the integral term by a differential term [g] _The resulting partial differential equation for
fbt) is
distribution
P(x,T) of the elec-
From the electron distribution functions calculated by eq. (4) the time-dependent macroscopic quantities we are interested rn can be derived: ICinetrc electron temperature reduced to the temperature T, of the carrier gasT&)/T,
=$ i 0
x’P(x,l)
dx.
(7)
Non-equilibrium departure Arc(~) of the electron attachment rate coefficient k(7)from the thermal (equilibrium) attachment rate coefficrent k(O): -
q.p$(vlf.
(3)
It is convenient to change from electron speed u and time t to drmensionless variables x = v/v0 (reduced electron speed) and r = t/to (reduced time), where u. = Sc = 2nz,/na,. (2’CnTc/me)1’2 and to = (~Qcv~u~)-I, The elastic cross section u(x) and the reaction cross section uR(x) are replaced by dimensionless cross sectiOJISE(X) = O(X)/U~ and CR(X) = UR(JC)/UR~ where uo and one are constant cross sections suitably chosen to
Ak(7) = (k(T) - k(‘))/k(‘) I
(8)
k(T) = uoiRo J xE,(x)P(x,T) dx,
(9)
0 k(O) = (4/n’/‘)”
0u RO 1 ZR(*)x3 exp(-x2) 0
dx. (10)
Eq. (4) is solved by a standard numerical method (difference method) which is based on replacing the 565
Volume
CHEMICAI.
118, number 6
PHYSICS LEITIERS
parabobc differential equation (4) by a system of difference equations treated by an effiaent numerical procedure (see, e.g., refs. [9, IO]). The accuracy of the calculated electron distribution F(x,T) is tested by inserting F(x,T) into the balance equations of number density and kinetic energy of the electrons. The error of the numerical computations is < 1% for aU t > 0.
I
4. Results
and discussion
The electron speed probabihty distnbution P(x,T) (electron energy E = 0_02ti2 eV, time t = 1.4 X 1067s) has been calculated for electron attachment to the CFEs 1,l ,I-C,CI,F, , 1 ,1.2-C,Cl,F,, 1 .l-C2C1,F,, and 1 ,2-C2C12F, in the carrier gas Ar (number density n, = 2.687 X 10lg cmm3) for a CFE/Ar ratio of n,/n, = 10B5. For tlus relatively large CFE concentration a distinct departure of P(x ,T) from its initially assumed MD is expected to result. Fig. 1 shows the time evolution
I,5 i
of the electron
dis-
16 August1985
tributionP(x,T) for attachment to 1 ,lil-C&F3 _ After a very short time T = O.C!l the initial MJ3 is largely depleted in the low-energy part (range of strong reactivity, as indicated by the function xaR(x)) and shifted to higher energy. At T = 05 the similarity ofP(@,T) . with a MD disappears completely_ P(x ,T) shows two peaks: the first peak is at lower energy (X 5 1.7) and a second peak is built up in the high-energy range (JC= 3.4)_ At T = 1 the distribution consists of a single highenergy peak (x = 3.4) and remains unchanged in shape for T > 1. It can be seen that in such strongIy reacting systems the explicitly time-dependent calculation of P(x,T) 1s required because the departure ofP(x,T) from a MD is quite large (see, e.g., the double-peaked distribution at 7 = 05). Replacing thrs method of calculation by a simpler approach, like the method described by Keizer [l l] using for P(Jc,T) a MD with time-depen-
dent temperature,
is not expected
to be successful.
In
the case of the other CFE studied in this paper the dis-
tribution functions P(x,T) are disturbed by the attachment process to a lesser extent. In contrast to the
-
7
O,!5-
: L
Fig. 1. Time evoluuon of the election speed probabMty Ar, “m/n, = lo-). Dotted curve. r(x) - XUR(X)-
566
disbribution function P(x,T) for attachment to l,l,
l-C2QF3
(carrier gas
Volume 118, number 6
CHEMICAL PHYSICS LETI-ERS
16 August 1985
7
0
Fig 2. Reduced kinetic eIection temperature Te(r)/Tc for electron attachment to chlorofluoroethanes (CFE) (carrier gas Ar, nm/nc = 10-5). 1 ,1 ,l C2C1,F, case the mean kinetic energy of the electron distributions becomes lower for r > 0. Fig. 2 shows the variation of the reduced electron
temperature T,(r)/T, with tune r (eq. (7)) Depending on the details of the cross sections for the 4 CFE a different behaviour of T,(r)/T, may be found. In the case of 1,l ,l-C$l,F, the electron temperature increases from the initial value T,/T, = 1 to a high-temperature steady state (TJT, = 8 2), whereas for the other CFE the electron temperature decreases to a lowtemperature steady state (T,/T, = 0.82 for 1,1,2CzC13F3, TJT, = 095 for 1 ,I-C2C12F, and 1,2C2%F4)-
Fig. 3 shows the time-dependent departure NC(T) (eqs. (8)-(10)) of the electron attachment rate coefficient k(7) from the thermal rate coefficient IJo) (Hl = k,, in ref. [I], table V). The deviation ofP(x,r) from a MD under the influence of the attachment reaction results in a considerable decrease of the rate coefficient . For 1,l ,1 -C2C1, F, , the nonequrlibrium departure AJc(r) of the rate coefficient goes up within a time interval AT = 1 from zero (T = 0) to a stationary value of ]AJc I = 93%. The other CFEs are characterized by smaller non-equilibrium effects: the stationary deviations ]ak] attained within AT = 1 are 13% (1,1,2C,Cl,F,), 7% (1 ,l-C,Cl,F,), and 3% (1 .2-CzC12F4). It should be mentioned that-calculation for the lowconcentration case (n,/n, = 10S7-10A) have shown a negligibly small decrease (
1
2
3
T
Fig. 3. Non-equilibrium departure u(r) of the electron attachment rate coefficient k from the thermal value k(O) for attachment to CFE (carrier gas Ar. nm/nc = 1Oj).
An exception is 1,I , 1-C,Cl, F, where n,.,.,& has to be as small as 10mg for IA7c I to be
References [l] D.L. McCorkle, I. Szarnrej and LX. Christophorou, J. Chem. Phys. 77 (1982) 5542. [2] D.L. McCorkle, A-A. Christodoulides. L G. Christophorou and I. Szamrej. J. Chem. Phys. 72 (1980) 4049. (31 AA. Christodoulides, L G. Christophorou, R.Y. Pai and CM Drug. J. Chem. Phyn 70 (1979) 1156. [4] R.Y. Pai, L.G. Christophorou and A-A. Christodoulides, J. Chem. Phyr 70 (1979) 1169. [5] HE. MJJloy, R.W. Crompton, J.A. Rees and A.G. Robertson, Autielian J. Phys. 30 (1977) 61. [6] J_ Ross and P. Mazur. J. Chem. Phys. 35 (1961) 19. [7] B. ShizgeJ and M. Karphrs, J. Chem. Phys. 54 (1971) 4345.43s7. [B] EA. Desloge and S.W. Mathysse, Am. J. Phys. 28 (1960) 1. [9] D-U. v. Rosenberg, Methods for the numerical solution of partial differential equations (Elszvier, Amsterdam, 1969). [lo] J. Wilhelm and R. Winkler, Conhib. Plasrnaphys 16 (1976) 287. [ll] J. Keizer, J. Chem. Phys. 58 (1973) 4524.
567