it is obtained
keeping in mind that p1 and p z is approximately equal to
P. Equation 14 explains why it is important to let zero signal on the recorder correspond to an empty oxygen reservoir. In that case VOwill have its least value and so the factor before ap2/pZ will be minimum. In the present application, it was not possible to obtain VOequal to zero due to gas trapped in the vessel 7 . This circumstance can account for the estimated error in An. The error in the capacitance measurements was about 1% of full scale, and the same is true for the pressure measurements. Due to the amplifying factor before the error of the pressure in eq 14, the estimated value 3% of the error of A n is quite likely.
Safety Considerations. When the apparatus is used for oxygen, care should be taken that the steel vessel 7 cannot be filled with oxygen from the reservoir. It would be dangerous with oxygen of such a high pressure in contact with the rubber membrane. Acknowledgment Thanks are due to Olle Lindstrom for reviewing the manuscript and encouraging its publication and to Henrik Backstrom for many valuable suggestions. Literature Cited Bolland, J. L . . Proc. Roy. SOC.,Ser. A , 186, 218 (1946). Pauling, L., "The Nature of the Chemical Bond", p 607, Cornell University Press, Ithaca. N.Y., 1960.
ReceiGed for reuieu M a y 22, 1975 Accepted F e b r u a r y 28, 1975
A Model and Apparatus for Electrical Discharge Experiments Daniel L. Flamm Department o i Chemical Engineering Texas A & M University College Station Texas
An apparatus used to maintain radiofrequency electrodeless discharges is described a n d modeled. T h e voltage across the plasma gas, c u r r e n t through t h e discharge, and t h e capacitance between t h e plasm a and t h e reactor walls are obtained from t h i s model.
This paper describes a n apparatus which has been used to maintain radiofrequency electrodeless discharges over a wide range of pressure. The apparatus has been employed to investigate the reaction of hydrogen chloride with oxygen (Flamm, 1973) and the reaction of oxygen with organic material in synthetic polluted water. The measurement of the voltage, current, power, and the calculation of V,, the voltage across, and i, the current actually passing through the plasma gas are also described. These parameters may then be used to calculate
which are among the similarity variables governing the behavior of a discharge (Bell, 1971; Flamm, 1973). Under appropriate conditions the electronic energy and density may also be derived from this information. The essential parts of the experimental apparatus are first described in detail and then a model is presented which has been successfully used to calculate the above parameters. The conditions under which the model applies are indicated and justified both theoretically and experimentally. The theoretical discussion indicates how the model can be extended to other conditions. The overall apparatus used to react hydrogen chloride with oxygen has already been described (Flamm, 1973). A schematic of the electrical connections is given in Figure 1. Power from the radiofrequency source, a transmitter, is coupled into a coaxial line, an RF wattmeter, and into the helical resonator. High voltage from the helical resonator is then directly applied to the discharge electrodes. Ex-
perimentally measured variables include the peak voltage applied to the discharge electrodes, the power input to the resonator, total current through the discharge reactor, and optionally, the RF waveform. The helical resonator functions as a matching network which is capable of boosting the output of the radiofrequency source to a very high voltage. Furthermore, because of its inherently high Q, it can do so with negligible power loss in the resonator itself. A t microwave frequencies these properties can be attained using resonant cavities. However, the inductorcapacitor combinations which are generally used for impedance transformation a t radiofrequencies (2-30 MHz) can only achieve a maximum Q on the order of 100; as shown below, this leads to a very substantial power loss when operating at high voltage. Since the resonator dissipates negligible power, the measured power fed into the resonator can be taken to equal that which is dissipated in the discharge. Helical resonators also have other desirable characteristics such as large internal intra-resonator element spacing (for insulation) and high power capability. Design equations for these resonators are available in the literature (Westman, 1956). Macalpine and Schildknecht (1959) present a nomograph which eliminates the need for iterative calculations. The helical resonator used in the research on HC1 at 6 MHz is shown in Figure 2. The resonator was designed for operation at 6 MHz when connected to the capacitance of the discharge. The theoretical design equations predict a Q of 1500 and a characteristic impedance 20 of 1300 ohms Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
263
Table I. Parameters for a 14-MHz Helical Resonator (Design Frequency is 19 MHz) HELICAL RESONATOR
VOLTMETER
-Figure 1 . Schematic of apparatus
k
TEFLON TUBING
COIL TS
Figure 2. Helical resonator for 6 MHz (design frequency is 7
MHz).
L f
f
C
R
Figure 3. Equivalent resonator circuit.
a t its nominal self-resonant frequency of 7 MHz. Large values for these parameters are advantageous as shown in the theoretical discussion. The internal supports must be constructed of a n insulating material which has a small diefectric dissipation factor a t the frequency of operation. Teflon and polystyrene are suitable. Nylon screws are used to fasten internal members, but should be placed away from the central conductor a t the top, high-field region of the resonator. The large copper can for this resonator was rolled from a large sheet of copper and the seam was silver soldered. Silver solder is essential since in operation there are large circulating currents around the circumference. The dimensions of another resonator used a t 14 MHz are given in Table I. The “can” for this resonator was made of 5-in. copper pipe. In operation, the copper tubing projecting from the top of the resonator is connected to the “hot” discharge electrode using a short lead which is kept away from surrounding objects. The RF source is first adjusted to the resonant frequency of the resonator-reactor system. The coaxial jack just above the lower end of the helical coil connects to a clip with a piece of braid. This clip fastens over one turn of the helical winding. The clip is moved u p the winding, about l/z turn a t a time from the bottom, 264
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
5 in. 0.22 in./turn 4.6 in.
Can inside diameter Helical conductor pitch Coil length Helical conductor wire diameter Helical coil diameter Overall resonator length Characteristic impendance 2, at 19 MHz Plasma reactor capacitance
‘/8
in.
2.5 in. (center to center) 7 in. 1050 ohms 5.6 pF
until a satisfactory match to the discharge is obtained as indicated by a low standing-wave ratio in the coaxial feedline. Provision was also made for kerosine to be circulated through the resonator coil as a coolant as shown in the figure. However, this was never needed because a t the voltage used (about 8000 V maximum) dissipation was always completely negligible. If the voltage is high enough for dissipation to be measurable, the kerosine temperature rise can be used to make a calorimetric correction to the total power indicated by the RF wattmeter. The power absorbed by the elements in the transmission line following the RF wattmeter is readily estimated as follows. About 3 ft of RG l l 5 / U Teflon dielectric coaxial line joined the meter to the helical resonator. Losses for this line lie between 0.25 and 0.65 db/100 ft of cable a t the 6 MHz frequency of operation. The maximum nominal line loss is thus about 0.02 db. A standing wave loss factor, which must be added to the nominal attenuation may be determined from published tables (Westman, 1956b). Assuming the worst case in which half the forward power is reflected back from the resonator, the standing wave ratio becomes slightly less than 6:1, but even a t a n SWR of lO:l, this additional attenuation loss is somewhat less than 0.1 db. Close to the natural frequency of the helical resonator, the power loss within the resonator and frequency of operation may be estimated from the equivalent circuit in Figure 3. The fictitious voltage source E replaces the action of the RF power supply in so far as the internal resonator power loss is concerned and
where wo is the resonant frequency
1 =
JLC
( 3)
and 20 is a parameter called the “characteristic impedance” of the resonator. For the large resonator of Figure 2 this was 1300 ohms. RI,,, represents a n internal resonator resistance and CH is the added capacitance due to the discharge, a voltmeter and stray coupling. The unloaded resonator Qu (before connecting the discharge reactor to the resonator) is given by Q,
50Df’’2
(4)
At 7 MHz Q u is about 1500 so that Rloss = 2 x l o 6 ohms
( 5)
If the effect of C,, is momentarily ignored, the equivalent circuit gives
(power into d i s c h a r g e ) (power into r e s o n a t o r )
I
The effect of the added capacitance, C f < ,is to lower the resonant frequenc of the circuit. If L and C were physical circuit components as drawn, the new resonance would occur a t w = [LfC
+
C,)]-”?
A 5 6 4 2
Figure 4. Vacuum diode voltage probe
(7)
In fact this is a good approximation providing
c, >
1
(30)
-
which is valid in many RF discharges, the distribution al0. As CY increases the elecways approaches eq 27 as CY tron concentration becomes larger at the ends relative to the center and approaches a rectangular distribution as ( t
-
m.
Now the plasma conductivity, and is given by
u,
is proportional to n
This may be rewritten as (3
and ro is the radius, d is the distance between electrodes, and 5 = 2z/d (see Figure 8). ( is zero midway between the plates. In this expression, the capacitive admittance of the parallel plates (neglecting fringing) in the absence of a discharge
=
(32)
OOf(S)
where uo is the plasma conductivity midway between the plates and f(() is the axial distribution of electrons in the discharge. Then for a given f([) the discharge admittance is a function of only U O . These relations may be written as
3=
770e-j
ctn"
ii
(33)
WQO
has been subtracted from the total. Allis and Brown (1951) have investigated the variation of u as a function of z, the axial distance from the midpoint of the discharge, for diffusion control. Their original concepts are extended in the following discussion. The balance of electrons is determined by the steadystate diffusion equation d2n
+
,Z(Ui
-
VJ
= 0
where vi - v a is the net ionization rate per electron. Two additional assumptions are made: (1) the electron density is sufficiently high that L), may be taken as the ambipolar diffusion coefficient and (2) as the electric field in the discharge varies about a certain value Eo, the variations in the parameters in eq 21 are given by
where the subscript (ave) denotes conditions for the average field E,,,. D, may always be taken to be the coefficient of ambipolar diffusion providing that
s o that (34) is the ratio of electron current to displacement current at the center of the discharge and
a
(35)
= v/w
(which are identical with the parameters in eq 24) i h a measure of the dissipation. Thus the discharge admittance will be determined by q o , d, and ( \ . At fixed pressure these variables correspond roughly t o current. lrequency and electric field. Everhart and Brown (1949) present the discharge admittance computed for the cosine distribution (eq 25) at small ~0 (their original figure appears to have been mislabeled): the curves have been recalculated and a more a c curate plot is presented in Figure 9. However, for a radiofrequency discharge. 0 0 is apt t o be much larger than those values Everhart and Brown encountered in a microwave discharge. This may be seen 103-104 tor from Table IV. A t 6 MHz and 10 Torr $ both HCI and 0 2 . For an ambipolar glow plasma n
-
Ind. Eng. Chem., Fundam.. Vol. 14, No. 3, 1975
-
267
104
C
-t 1.0
I
I 0
I
I
I
I
10.2 10.1
1.5
1.0
.5 Gpd/nr2 (reo
Figure 9. Dimensionless discharge admittance. Calculations are for f(> 1 as is the case here. T O is in general also a function of /3 as per eq 24. The discharge admittance for f ( < ) = cos n u 2 (36)
for a wide range of 7 0 is presented along with the admittance for (37) f ( 5 ) = ( 1 - 591’2 in Figures 10 and 11. These numerical evaluations of the integral in eq 19 are detailed in Appendix 111. From the figures it may be seen that either for small 70 (either high frequency or low electron density) or small N , when f(> 1. However for @ >> 1 with 7 0 large and a 2 so that the circle distribution, (or one even more nearly approaching a “box”, for N > 2 ) applies, the plasma impedc x e is effectively resistive.
Lv..Ld,:Lion oftv The remaining uncertainty in stating that the discharge impedance is real is the question of whether a N 2 (or greater). Allis and Brown (1951) present values of N for an ambipolar discharge in helium. For large 8, N is a function of pod alone (or E l p o ) . For lengths characteristic of RF 1 cm) and moderate pressures discharge apparatus ( d ( p -v 10 Torr) N = 1.7 for helium. This value increases rapidly with p or d
-
268
-70-
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
-
-go----------cl--i
10
-
100
101
-
i
102
.
-
I -I .+-J
1 o3
lo4
17.
Figure 11. Phase of the dimensionless discharge admittance, arg [ Y p d / x ~ ~ w c ovs] , 7 0 .
Experimentally, N may be determined from the operating characteristic of a discharge where the electron density is kept low enough (00 small) so that f ( < ) = cos 77.512 (38) and the electric field in the discharge is uniform. A plot of
(39) has a slope of 2a - 2 (Allis and Brown, 1951). Allis and Brown used this to determine cy as a function of E/po or pod for helium. Unfortunately such data are not generally available in the literature. (There are some data on ( v l - va)/DeE’ against E/po for breakdown conditions, not to be confused with ( V I - va)/DaE2.) The parameter a can also be calculated from other basic data. In a diffusion-controlled discharge, measurements of the breakdown field at constant pressure give ( v l - va)/DeE2vs. E / p from measurement of E, p and the dimensions of the discharge reactor. This is true because
where A = d/n
2.0 Y
--Y
,
I
D,E~ 1 .o
0.8 3.6 5.4
3.2
10
20
30 40
I. 1 60 80 100
EiPO
Figure 12. N and
(uI
- u,)/D,EZ in oxygen,
under these conditions. In fact the measurements can be performed in a discharge without parallel electrode geometry providing that .i is replaced by the appropriate “diffusion length” for the particular geometry involved (McDaniel, 1964). The parameter (vi - va)/DeE2 is then multiplied by De/Da to obtain ( v I - va)/DaE2. De/Da can be related to commonly tabulated parameters
Da
N
kT, e
P+
(42)
s o that (43) the ratio of electron and positive ion drift velocities. Thus (44) When breakdown data for eq 44 are not available, may be estimated from data obtained at DC
N
e Po Equations 44 and 39 have been used to compute CY for oxygen. The results are presented in Figure 12. Values of ( v I - va)/DeE2 were obtained from Brown (1956), Ue is given by Hake and Phelps (1967), and u t has been tabulated by Varney (Loeb, 1961). Brown (1959) obtained good agreement between his experimental data for ( v l va)/DeE2 and computations based on eq 39. Hake and Phelps (1967) present and analyze more recent data on ( l y I - a a ) / p o and u e . Thus the calculations show that, at least for oxygen, CY 1 2 for a range of E/po of interest. Since this condition was also generally applicable in helium it might be expected to apply in other gases as well. Unfortunately there are not yet data available to investigate this question for many other gases of interest. In the following discussion it will be assumed that N = 2 so that the admittance curves for [f(E) = (1 - t2)’ 2] apply. The admittance parameters 7 and /3 in Table IV were calculated from drift velocity data given by Healey and Reed (1941) and Hake and Phelps (1967). At low w or at DC, the drift velocity i s related to the collision frequency, v , by eE
2’e
=
117,V
(46)
0
100
300 400 POWER ( W A T T S )
200
500
Figure 13. Series wall capacitance vs. power in reactor P1.
In the investigation at 6 MHz described by Flamm (1973), a mixture of HCl and 0 2 was reacted in a discharge between parallel plates. E / p o was generally in the range of 5 to 60 or more and p was between 10 and 160 Torr. For these conditions Table IV indicates
p 2 lo3 (4 7) If n is taken as 1012, the upper limit for a “cold” plasma, then from Table IV
-
77 103 (48) Figure 10b shows that under these conditions the electron distribution (1 - t2)’ will produce a negligible imaginary admittance component (BP 0); even for a cosine distribution the susceptance is small (arg[ Ypd/.lrr2wco] = lo” or B,/G, 0.2). Thus the imaginary part of the plasma conductance can be neglected as was assumed in deriving the model of Figure 6c.
Experimental Data The model may be tested by comparing C,, as calculated from the model, with theoretical predictions. The procedure for calculating experimental values of C, is presented in Appendix 11. At high power in the investigation at 6 MHz described by Flamm (1973), the plasma occupied most of the crosssectional area of the reactors close to their electrode surfaces. Under this circumstance the capacitance C, ought to correspond to the physical geometrical capacitance across the reactor wall. The reactors used are shown in Figure 7. Figures 13 and 14 present typical smoothed capacitance versus power data for the operating discharge. Table V compares the asymptotic high power capacitance in the figures with the geometric capacitance calculated from the known dimensions of each reactor. These values are in close agreement. The experimentally calculated capacities displayed considerably more scatter at low than at high power. This is explained by a decrease in plasma resistance from several thousand ohms at low power to about 500 ohms o r less at high power. Since 10 p F (see Table V ) represents a reactive impedance of only 2650 ohms at 6 MHz, i t is apparent that the determination of R,, was relatively less sensitive to wall capacity at low power. It is felt that the upward bow of part of the low power data in the figures is an artifact of correlated error due to the insensitivity of calculations to C, at low power. Ind. Eng. Chem.. Fundam., Vol. 14, No. 3, 1975
269
I-
,a
~
CM. 2b
'
are in common use [however, the above does not hold for peak reading wattmeters which always give a signal proportional to the peak power due to a longer integration time (Bird, 1967)l. The true power, on the other hand, is proportional to the RMS voltage squared. These meters are calibrated for the ratio of these quantities when a pure sinusoidal waveform is present. Therefore, in the presence of modulation the true power is related to the indicated power by
W
-
r r r 9
z
a 2
4
a.
a
vzRMS,CW
u
J
For 100%audio modulation of the applied voltage T
0 400 500 POWER (WATTS) Figure 14. Capacitance vs. power in the cylindrical reactors.
0
100
200
300
V =
I
l +v o k (1 + k c o s oat)cos
wt
(50)
where k , the fraction of audio applied is unity; one obtains from eq 49
Table V. Theoretical and Experimental C a p a c i t y Reactor P1 P2 P3
Asymptotic capacity, pF
The o r et ic a1 capacity, pF
10 10 22.5
10.1 11.3 22.6 1.1 11.7
PM 1.6 CM 13 c3 5.7 5.3 Pillbox capacities were calculated using Kirchoff's formula (NP, 1963). Capacity for each cylindrical reactor was first computed as 0.2416LK/log ( r l / r z ) where L is the electrode length ( 2 cm here), K is the dielectric constant of quartz (taken as 3.78) and r l , r2 are the inner and outer radii of the cylinder. This formula neglects and effects. If the end effects for an equivalent disk capacitor of radius equal to the length of the cylinder are computed, capacity is increased by the factor of 1.094. Capacities from the above formula were multiplied by this factor to obtain the entries (I
in the table.
The decrease of capacitance with decreasing power from a high power asymptote agrees with expectations based on visual observations. The core of the discharge was more contracted at high pressure and likewise the capacitance was lower at each power level. The cross-sectional area of the discharge increases with power as the capacitance also increases. The flow rate of feed had little effect on capacity; this was also true of contraction. Measurements Using a Modulated Waveform In a previous article (Flamm, 1973) it was shown that audio modulation of the applied radiofrequency power can be used to test certain postulated reaction mechanisms. The measurement and calculation of voltage, power, and current must be performed somewhat differently when modulation is present. The indication of many RF wattmeters is proportional to the squared mean value of the radiofrequency carrier voltage (Flamm, 1970; Bird, 1964). This occurs because the RF waveform is rectified and integrated by the wattmeter over a time that is long compared to a cycle of the RF carrier but short compared to a cycle of audio modulation. The indicating meter then averages the signal applied to it, which will vary at an audio rate. In particular this is true of many CW radiofrequency wattmeters which 270
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
For two-tone modulation V = V, s i n w t sin wat = 0.81 Pactual
'rea
(52)
It should be noted that the calculation of plasma voltage, current, and resistance in Appendix I1 also involves RMS quantities. The total impedance of the circuit in Figure 6c should be taken as
z=
(53) However, if the instantaneous impedance is not sensitive to current (on the average) VRMs/iRMS
Tr
(54) may be an acceptable approximation. The RMS current is read from the thermocouple RF ammeter and the peak current is measured using the indication of a peak reading voltmeter connected to the coaxial resistor (see Figure 1 ) . The RMS voltage used in the calculations of Appendix I1 is then taken as \e&*
vR,,
iwak
(55)
Conclusions Through the use of a high-Q resonator, plasma chemistry experiments can be conducted at high voltage and moderate pressure. An equivalent circuit has been presented which can be used to calculate the basic parameters of the discharge, to determine discharge current and power from experimental data, and to design for a particular frequency of operation. Appendix I. Capacitance Approximation The approximation CshI >> Csp,II is examined here for reactor P1. The gap to radius ratio
-
d/r 1 (A- 1) so that using Kirchoff s formula fur a parallel disk vacuum capacitor (AIP, 1963)
c
= q,($
+
y [ l 16" n T - l]}
(A-2)
-
It is apparent that with r/d 1, the fringe capacitance (second term) is about equal to that of the volume shunt (first term). However in the reactor, the fringing field travels partly through quartz of dielectric constant 3.78 and partly through silicone oil (the coolant in which the reactor is immersed) of dielectric constant 2.2. Moreover, quartz occupies part of the gap as well and will focus some of the field around the gas volume. This is confirmed experimentally. The theoretical air gap capacitance without fringing is
R , is taken as the independent variable and a NewtonRaphson iteration is performed
where Rp* is an estimate of R, and
( A- 3)
The effect of focussing ought to diminish this contribution. The measured capacitance C s h (determined in the absence of a discharge) was 2.13 p F so that ( A- 4)
at least. Finally it was shown from Figures 10 and 11 that Y,d/sr%co, the dimensionless plasma admittance, is large. Thus the current through Csh" will be negligible compared to the current through the plasma gas. However, the current through C,h* is not necessarily also negligible because the potential across this capacitance is larger by an amount equal to the voltage across C.,
+ -az - az
(A- 14)
R,* = V 2R M S / P - 2
(A- 15)
dz az -=d ~ p aR,
az,
aR,
A good initial guess for R , is
The iteration, eq A-12, is repeated until the experimental impedance 20 and that calculated from eq A-12 are sufficiently close. C, is calculated from eq A-8 and A-9. Appendix 111. Numerical Evaluation of the Discharge Admittance Following Everhart (1948), the plasma admittance, eq 19, is expressed in terms of the two integrals
Appendix 11. Calculation of P l a s m a Voltage a n d Current While the resistance of the plasma does not necessarily stay perfectly constant in time, an average value, R,, can be defined as (power input to d i s c h a r g e ) f A- 5) R, = i2,S
and the RMS voltage across the discharge gas is calculated from
V~,RM= S
~RM&
( A- 6)
Let V be the RMS voltage applied to the circuit of Figure 6c and let i be the total current. The power dissipated in the series coaxial resistor of this circuit is subtracted from the total input power as measured on the RF wattmeter. Let P be this corrected power input to the discharge. Define 2, =
v/i
( A- 7)
to be the total experimental impedance of the network. Then
where 2, =
1
wcw
( A- 9)
and
where @ =
so that G,d w2we0
= ;2+ Q,
Re[o/weo] =
+
B,d Qi
& -
W'WE~
-8
Qr2
(A- 18)
Qi2
- 1 (A- 19)
When q(1 + 1/62) is large, the major contribution to these integrals occurs in a small interval near 5 = 1 when f(() approaches zero and *(I + l/p2)1/2f(() decreases below unity. For the cosine distribution this ocl/[*(l + l/d2)l 2] while with the circle curs when 4 distribution 5 1/[Q(1 + 1/@2)] at this point. An accurate numerical evaluation of these integrals therefore cannot be obtained with a reasonable (e.g., 1000-10,000) number of equally spaced intervals over ( 0 , l ) . The problem was overcome by using Simpson's rule with 600 divisions over each of the intervals
--
Then Z, = (R,VZRMs/P- R,?"'
(A- 10)
If
(A- 20) where
ZSh = -5-
(A-21)
wcsh
for the cosine distribution, or
1
= @ ( l+
then the total impedance of the parallel circuit in Figure 6c, may be expressed as Z = (Re2
+
Z2)'/'
(A- 12)
1/67
( A- 22)
for the circle distribution when 'k(1 + 1/b2) was large. For the cosine distribution the numerical value of QI for @ = m can be compared with the exact answer lor this case Ind. Eng. Chem.,
Fundam., Vol.
14, No. 3, 1975
271
= permittivity of a vacuum, 8.85 X
to
F/m
7 = dimensionless ratio of electron to displacement current, a/weo
