375
Ind. Eng. Chem. Prod. Res. Dev. 1884, 23, 375-379
Statistics of Resols Anafia Vbquez, Humberto E. Adabbo, and Roberto J. J. Wlllla!ns* Institute of Materials Sclence and TechnorOgy (INTEMA), University of Mar del Plata and National Research Council, J.B. Just0 4302, (7600) Mar del Plata, Argentina
Statistical quantities such as gelation condltlon, number and weight average molecular weights, and distribution of molecular species are derived for resols, i.e., the alkali-catalyzedphenol-formaldehyde polymerization products. This model predlcts gelation for a formakJehyde/phenol molar ratio greater than or equal to 0.5. This Is in agreement with reported experimental results. Operating with high formaldehyde/phenol molar ratios always led to gelation due to the possibility of condensing methylols between themselves. A convenient formaldehyde/phenol molar ratio for preparing NaOH and MgO-catalyzed resols lies in the range 1.5-1.7. I n this range the model predicts the smallest contents of both free phenol and formaldehyde as well as an adequate balance between free sites and methylol groups.
Introduction The resols are phenol-formaldehyde resins obtained with a formaldehyde/phenol molar ratio greater than one under alkaline conditions. They are one-step resins and must therefore be discharged from the reactor before gelation takes place. The first step of the reaction, i.e., the formation of methylolated products, has been thoroughly studied by several investigators. A good review is available ( D r u ” and LeBlanc, 1972). However, the condensation reactions, which finally can lead to gelation, are much less understood. Moreover, commercial products show a broad range of formaldehyde/phenol molar ratios and are prepared using different catalysts. Although there is a considerable amount of empirical knowledge, there is a lack of fundamental study dealing with the influence of different formulations on the structure of the resulting resols. A statistical model for the formation of resols was derived by Pal et al. (1981). However, the results are of relative value because kinetic parameters were taken from the novolac chemistry which differs widely in the nature of reactions, reactivity ratios, and substitution effects. The aim of this paper is to derive a statistical model for resol formation by applying the available kinetic information (Dnurun and LeBlanc,1972) by using the recursive method developed in the previous paper (Aranguren et al., 1984). The major concern of this analysis will be to delineate the influence of the formaldehyde/phenol molar ratio and the nature of the alkaline catalyst on the gelation and structure of the resulting resol. Both different reactivity of sites and substitution effects will be taken into account. However, it will be assumed that no intramolecular reaction occurs in finite species (Aranguren et al., 1984). Kinetics Resol chemistry is characterized by a high ratio of the addition to condensation rates. Thus, mono-, di-, and trimethylolated species appear in the reaction mixture, making it necessary to take into account all the nineteen species shown in the previous paper (Aranguren et al., 1984). Figure 1 shows some possible condensation reactions between o-methylolphenol and p-methylolphenol. As the formation of ethers is not a significant reaction under strong alkaline conditions (Yeddanapalli and Francis, 1962),reaction I will be deleted from the following analysis. Reactions I11 to V compete with reaction 11, the degree of which depends upon the structure of the methylolphenols 0196-4321/84/ 7223-0375$01 .SO10
Table 1. Reactivity Ratios per Reaction Site reactivity ratio reaction with NaOH MgO r[2/1 I r[l(Mel’)/l] r[l(Me2)/11 r[ l(Me1’ ,Me2)/1] rI2(Me1)/21 r [ 2(Mel,Me1‘)/2] r[Me2/Mel] r[Mel(Mel’ )/Me11 r[Mel(MeZ)/Mel ] r[Mel(Mel‘,Mea)/Mel] r[Me2(Mel)/Me2] r[Me2(Mel,Mel’)/Me2] r[Mel/B] r [ Me 1 / B] r[Mel - M e l / l - B] r[i/el
B or Me B o r Me B or Me B or Me B or Me B or Me free site o r Me free site or Me free site or Me free site or Me free site or Me free site or Me free sites (except those of phenol) free sites of phenol
___------_---
B, Me, or free sites
1 2 1 1 1.5 3 1.5 2 1 1 1.5 3 1/8
1/2.7 8/2.7 1 3/2.7 2 4 1.5 2 1 1 1.5 3 l/S
1/16
1/16
1/16 1/2
1/16 1/2
involved and the reaction conditions (Knop and Scheib, 1979). It is also necessary to take into consideration that the condensation reactions between two methylolphenols proceed substantialIy faster than the reactions between methylolphenols and free phenol (Yeddanapalli and Francis, 1962). The selected alkaline catalysts are NaOH, one that is used most typically in the industrial practice, and MgO, which, as other divalent metal oxides do, favors o-substitution (Drumm and LeBlanc, 1972). The reactivity ratios of the methylolation reactions are well established for both catalysts (Zavitsas, 1968; Zavitsas et al., 1968). Table I shows the selected reactivity ratios. Several comments are in order. (i) Positions of the phenolic ring occupied with methylol groups are indicated between brackets, i.e., r [l(Mel’)/l] means the reactivity of an o position when there is a methylol in the o’position, with respect to the reactivity of the same site when there is no substitution in 0’. (ii) For NaOH-catalyzed reactions the substitution effect is assumed to be same for the reactivity of methylols and for free sites. The only exception is the reactivity of a p-Me with respect to an o-Me, which has been increased in 50%, r[Me2/Mel] = 1.5, according to Francis and Yeddanapalli (1969). (iii) For a MgO-catalyzed resol it is assumed that the divalent ion participates only in the addition reactions on 0
1984 American Chemical Society
376
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984
r
ylene glycol, CH,(OH),; ktl0 is the true specific rate constant, and klo is the apparent specific rate constant, which is assumed to remain constant along the reaction. Rate Equations Rate equations may be written in terms of a dimensionless time, t* = klo A. t , by taking into account all possible steps leading to or consuming the particular species. The notation will be the same as the one used in the previous paper (Aranguren et al., 1984). As an example, the rate equation for species T is shown below dT/dt* = R([a] [b]) P[c] BL - T([d] [e]) (4)
OH
+
L v i
HO~JYJ-CH~
*OH
+
~
2
0
CH,OH
Figure 1. Possible condensation reactions between o-methylolphenol and p-methylolphenol.
free sites. Reactivity ratios for condensation reactions were assumed to have the same values as in the case of NaOHcatalyzed reactions. (iv) As suggested by Minami and Ando (1956), the condensation/addition ratio was taken to be for free sites except for those of phenol; for phenol the ratio was taken to be accounting for its reported lower reactivity (Yeddanapalli and Francis, 1962). (v) By taking the reactivity r[Mel - M e l / l - B] = the self-condensation of 2,4,6-trimethylolphenol through the p position is characterized by a reaction rate lying in the range of addition reactions, as has been shown by Freeman (1967). Thus kMe2(Mel,Mel’)-Me2(Mel,Mel’)llt10
=
r[Mel - M e l / l - B] X r2[Me2(Mel,Mel’)/Me2] X r2(Me2/Mel) X (1/4) = 0.316
klo is the specific rate constant for the addition reaction of formaldehyde on an o position of phenol. The factor 1/4 in the above equation accounts for the o positions of phenol and the two reaction sites of formaldehyde. (vi) It is assumed that the reactivity of external sites is not affeded by the formation of a methylene bridge, except by the possible change in the substitution effect. However, when two methylene bridges are joined to a ring the remaining internal site is assumed to be less reactive; a factor l / z is arbitrarily selected. Although the accuracy of several of the selected reactivity ratios may be questionable, their values may be regarded as reasonable on an order-of-magnitude basis. The ratio of specific rate constants may be obtained from the reactivity ratios by taking into account the number of available reactive positions for each kind of reaction. These ratios may be expressed by taking kl0 as a reference value, as was illustrated above for the condensation of two methylols. It must be recognized, however, that the correct way of expressing the formaldehyde addition rate to an o position of phenol is (Zavitsas, 1968 Zavitsas et al., 1968; Drumm and LeBlanc, 1972) where This leads to [A-] is the relative concentration of ionized phenol over the initial phenol concentration, A,,; [B] is the relative concentration of total free formaldehyde over initial phenol concentration; y is the fraction of formaldehyde as meth-
+
+
+
T is produced by the condensation of Me2 of the species R with all other methylols ([a]) or with free reactive positions ([b]), by the condensation of the free reactive position of the species P with all methylols ([c]), and by formaldehyde addition to the free o position of the species L. On the other hand, T is consumed by condensation of its o-methylols with free reactive positions ([d]), or with all other methylols ([e]). For a NaOH-catalyzed resol, the factors defined in eq 4 take the following values [ a ] = 0.106(1 + K ) + O.O7O(J + L + M) 0.053N+ 0.0350 + 0.281(P T ) + 0.229(Q S) 0.914R [b] = 0.422(A + P) + 0.281(0 E I L) 0.070(F G) 0.492J 0.141(K Q) 0.211M [c] = 0.141(1 + K ) + O.O94(J L + M) + O.O7ON+ 0.0470 + 0.750P + 0.305(Q + S) 0.609R + 0.375T [d] = 0.375(A + P) + 0.250(0 E I L) 0.063(F G) + 0.438J 0.125(K Q) 0.188M
+ + + + + + + + + + + + + + + + + + + + + + [e] = 0.094(1 + K ) + O.O63(J + L + M) + 0.047N+ 0.0310 + 0.250P + 0.147(Q + S) + 0.406R + 0.500T
+
A set of 19 differential equations was thus obtained. A numerical solution was performed with a fourth-order Runge-Kutta method. The constancy of the number of phenolic rings and formaldehyde segments (including free formaldehyde,methylols, and methylene bridges) was used to check the numerical solution. The corresponding equations are A + D + E + F + G + H + I + J +K + L +M + N + 0 + P + Q + R S T = 1 (5) B+I+J+K+L+M+N+O+ 2(P + Q+ S + 7‘) + 3R + 2 = &/A0 (6)
+ +
where 2 = (1/2) [D + E
+ K + L + M + S + T + 2(F+ G
+ N + 0) + 3H]
is the relative concentration of methylene bridges. The formaldehyde conversion is calculated as PB = rPA where r = 3A0/2B0
(7)
is the stoichiometric ratio of functionalities, and pAis the conversion of reactive phenolic positions, given by PA = (1/3)(D + E + I + J) + (2/3)(F + G + K + L + M + P + Q) + ( H + N + 0 + R + S + r ) (9) Statistics Statistical parameters were generated from the distribution of molecular species, using the method developed in the previous paper (Aranguren et al., 1984). Number Average Molecular Weight M , = [94 + 30(Bo/Ao) - 182]/[B + 1 - 21 (10)
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984
377
Table 11. Gel Times of Different Resols Prepared at 65 "C
fh: B,/A,
cat.
cat./A,, mol/mol
0.5 0.5 0.6 0.7 1 1 2 2 3 3
NaOH MgO MgO NaOH NaOH MgO NaOH MgO NaOH MgO NaOH MgO
0.28 0.14 0.14 0.28 0.28 0.14 0.28 0.14 0.28 0.14 0.28 0.14
4 4
mol m-3
gel time, h
7.49 7.49 7.08 6.84 5.92 5.92 4.04 4.04 3.02 3.02 2.41 2.41
did not gel did not gel 720 60 32 36 22 25 21 27 20 37
Figure 3. Gel time under alkaline conditions as a function of the formaldehyde/phenol molar ratio for a MgO-catalyzed resol (points are experimental data).
-
k,, 4 2 IO-*
STAND NOV
F-S NW
SM
EXP
OONOV
m'
40
04
06
08
lo
Bo/Ao
Figure 4. Gelation limits for phenolic resins at full formaldehyde conversion (F-S: Flory-Stockmayer, S.M.: our structural model, Exp: experimental value).
0
1
2
3
4
&/A0
Figure 2. Gel time under alkaline conditions as a function of the formaldehyde/phenol molar ratio for a NaOH-catalyzed resol (points are experimental data).
This equation is exactly the same as the one obtained for novolacs. However, in this case a simplified approximation (Aranguren et al., 1984) cannot be used because of the high concentration of methylolated species. Weight Average Molecular Weight.
M , = (9OOB + (94 + 30[(&/Ao) - B] - 182)(94 + 30[(&/Ao) - B - z ] + 2Y2))/(94 + 30(&/Ao) - 182) (11)
where Y= (106 + (15/2)[K + L + M + 2(N 0 + S + T ) ] ) / (1- [F + G N + 0 + 3H]/Z} (12)
+
-
+
Gelation Condition. The resol gels when Y a),implying
H
= (1/3)(D + E
-
+ K + L + M + S + T)
aldehyde conversion. Also shown is the fact that NaOH led to gelation at shorter times than MgO. The comparison of model predictions with experimental results is shown in Figures 2 and 3. The ratio of the true specific rate constants for formaldehyde addition to an o position of phenol for both catalysts, has been reported by Zavitsas et al. (1968) k'10
(NaOH)/k'lo(MgO) = 1.138 (at 57 "C)
(14)
Figures 2 and 3 show that the model gives a reasonable fitting of experimental results, using klo as an adjustable parameter for each formulation with the restriction that their ratio must satisfy eq 14. The apparent specific rate constants, klo, are two orders of magnitude smaller than the true constants, k'lo (Zavitsas et al., 1968; Drumm and LeBlanc, 1972). This accounts for the product of the formaldehyde fraction as methylene glycol and the phenol fraction in anionic form (eq 2). The experimental limiting value leading to a gel at full formaldehyde conversion is Bo/Ao = 0.5. The theoretical prediction is exactly the same (M, m at p B = 1for Bo/Ao = 0.5). Thus, the set of selected reactivity ratios (Table I) shows to be adequate for the prediction of gelation conditions. We are now able to compare our theoretical predictions for the gelation conditions of novolacs and resols, with predictions of the classical Flory-Stockmayer statistical model (Flory, 1941; Stockmayer, 1943). For the condensation of a trifunctional reactant (phenol) with a difunctional reactant (formaldehyde), assuming: (i) equal reactivity of sites, (ii) absence of substitution effects, (iii) absence of intramolecular condensation, and (iv) absence of reaction between sites of the same reactant (phenol or formaldehyde), the classical theory of gelation predicts --+
(M, (13)
The polymer gels when the number of methylene bridges joined to trireacted phenolic rings (3H) equals the number of methylene bridges joined to terminal phenolic rings (D + E + K + L M + S + T).
+
Results and Discussion Gelation Condition. In order to compare model predictions with experimental results, several resols were prepared and kept at 65 "C until gelation took place. Table I1 shows the different formulations and the corresponding gel times. The ratio of catalyst concentrations was selected to give the same amount of hydroxyl equivalents per mole of phenol. Phenol was a technical grade reagent and the 37% by weight commercial solution was the formaldehyde source. Results showed that, for both catalysts, a formaldehyde/phenol molar ratio close to 0.5 was the limiting value to gel the system at full form-
PAPB
=
(15)
72
with the aid of eq 7 and 8, we get (2B&'/3Ao)
(16)
= 1/2
and, at full formaldehyde conversion @,/Ao) = 0.75
bB= 1) (17)
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 3, 1984
378
0.
BO/.IO
=
-
s
0 2 . i
Mw Mn
BoIAo 2 NaOH
2
NaOH Y
I
Q
f
-1 a
01
0
5
t‘
10
Figure 5. Evolution of several species for a NaOH-catalyzed resol: (-) methylolphenols; (- - -1 terminal units; (0) chain-extending units; ( 0 )branching units.
Figure 4 illustrates the experimental and predicted gelation conditions for phenolic resins. The deviations of experimental results from the Flory-Stockmayer prediction is due to the failure of hypotheses (i), (ii), and (iv) (functional sites have unequal reactivity, there are substitution effects, and methylols may condense between themselves). The fact that (iv) is not verified also accounts for the absence of an upper gelation limit for resols, when operating at high Bo/Aoratios. Novolacs catalyzed by strong acids gel somewhere in the range Bo/Ao= 0.85-0.90, as discussed by Drumm and LeBlanc (1972). Novolacs prepared in coonditions favoring 0,0’substitution will gel at higher Bo/Aoratios. In the limit of a pure 0,o’substitutions phenol acts as a difunctional reactant leading to gelation for Bo/Ao= 1,at full formaldehyde conversion. Species Distribution and Statistical Parameters. Figure 5 shows the evolution of the molecular species present in high concentrations, for a NaOH-catalyzed resol prepared with Bo/Ao= 2 (species are designed with the same notation as Aranguren et al., 1984). The concentrations of methylolphenols (I,J, P, Q, and R) increase rapidly at the beginning of the reaction, go through a maximum, and decrease continuously. At this time, a great increase in terminal units, e.g., S and T , with only one methylene bridge and two methylolated positions, may be noted. Also, a typical chain extender as 0, with two methylene bridges and a methylolated interior site, begins to increase rapidly, becoming the predominant fragment of the mixture. The continuous increase in the concentration of branching units, H, leads to gelation at a dimensionless time t* = 11.6. The trends depicted in Figure 5 are typical of NaOH-catalyzed resols. When MgO is used as catalyst, the favoring of the o-substitution results in a considerable increase in the concentration of species J,P, and N . The model predictions for the evolution of free phenol and formaldehyde along the polymerization, as well as the increase in number and weight average molecular weights, is shown in Figure 6. A rapid increase in M , is observed near the gel point. This corresponds to the viscosity runaway which would result if the resol were not discharged in due time from the reactor. A convenient residence time for this type of resol seems to be t* 5 . At this time the phenol concentration has been sufficiently depleted, the formaldehyde concentration varies at very low rate, and the gelation condition is sufficiently apart. The number average molecular weight is M , = 250 and the weight average molecular weight is M , = 475. The influence of the initial formaldehyde/ phenol molar ratio, Bo/Ao,upon the properties of the resulting resol is shown in Figures 7 and 8. The criterion for comparing
Figure 6. Evolution of free formaldehyde (B), phenol (A), and number (M,) and weight (M,)average molecular weights for the resol depicted in Figure 5. t*/Ao
Mw = 473
m’/md]
75Lr I’ A o
25 a Me
41 sites free
0 1
1.5
2
-3
3
2.5 Bo/Ao
Figure 7. Relative concentrations of methylols, free sites (0,o‘, and p ) , phenol, and formaldehyde (both with respect to their initial
concentration), and reaction time @*/A,) to reach the desired M,, as a function of the formaldehyde/phenol molar ratio, for a NaOHcatalyzed resol.
-
Figure 8. Relative concentrations of methylols, free sites ( 0 , o’, and p ) , phenol and formaldehyde (both with respect to their initial concentration), and reaction time (t*/Ao)to reach the desired M,, as a
function of the formaldehyde/phenol molar ratio, for a MgO-catalyzed resol.
Ind. Eng. Chem. Prod. Res. Dev., Vol.
Table 111. Resols Prepared for the Acid Condensation catalyst BJA,
T."C
A,, Ai/m3 cat./A, t*,d tgel, h
k,, x lo8 m3/(mols)
resol I
resol I1
NaOH
MgO 1.5 70 4730 0.06 17.7 19 5.5
1.5 70 4730 0.12 14.6 14 6.1
acid curing PH ( C W T,"C
k,, x lo8,m3/(mols)
-0 50 2.2
the different products is a constant M,,which for thermosetting systems may be correlated with the polymer viscosity (both go to infinite at the gel point). Thus, the comparison is also valid for resols discharged from the reaction vessel at more or less the same viscosity. An increase in the Bo/Ao molar ratio gives a corresponding increase in the amounts of free formaldehyde and the 0,of, and p phenolic positions occupied by methylols (% Me), but a decrease in the concentrations of free phenol and reactive positions of phenolic rings (9% free sites). The convenient Bo/Aomolar ratio for preparing a resol for acid cure depends on several requirements, two of which are: (i) minimum amounts of remaining free phenol and formaldehyde and (ii) fast final crosslinking rate. This last requirement may be achieved by providing a product with a similar amount of free sites and methylols because their reaction is very fast under acid conditions. According to these requirements, Figures 7 and 8 show that the convenient Bo/Aoratio, theoretically predicted, is in the range 1.5-1.7. The actual reaction time, t*/Ao,for producing the resol has a minimum at Bo/Aoclose to 2 but does not increase very much at Bo/Ao= 1.5-1.7. Both catalysts, NaOH and MgO, give similar results, although it is worthy of pointing out some differences. Thus, the model predicts that NaOH-catalyzed resols need shorter reaction times to reach the same M, (this agrees qualitatively with results shown in Table 11), and show less free formaldehyde but a higher free phenol than the MgO counterpart. Acid Condensation. In some commercial applications of resols, as in the production of phenolic foams, the resin is cured in an acid medium at room or moderate temperatures. It is useful then to analyze the capability of the model to simulate this process. Two resols were prepared according to the conditions shown in Table 111,using an experimental device described elsewhere (Rojas and Williams, 1979). Samples were extracted from the reactor at several intervals and cured at T = 50 "C with the addition of HC1 (pH -O), determining the gel time. The process could be simulated by using the reactivity ratios described in this paper for alkaline condensation until the sample was extracted from the reactor, and then changing the specific rate constants, using the values analyzed in the previous paper for an acid condensation (Aranguren et al., 1984). In this case it was necessary to define the reactivity ratio for the condensation of two methylols under acid conditions. It was assumed, arbitrarily, that this value was one half of the reactivity of a methylol with a free site. Thus r (Me1 - M e l / l - B) = (1/2)r(Mel/B) = 4
23, No. 3, 1984 379
NaOH Model--Exp.
M$
-
0
I
0
5
lo
tpolym.lhl
Figure 9. Gel time under acid conditions vs. polymerization time at 70 "C in an alkaline medium (points are experimental data).
Figure 9 shows predicted and experimental values of the gel time under acid conditions. The theoretical dimensionless times have been transformed into actual times by using the klo values, in alkaline and acid conditions, reported in Table 111. The value of klo in the alkaline medium was calculated from the experimental gel time in these conditions (note that the ratio between both klo is again in agreement with eq 14). The klo value under acid conditions was arbitrarily adjusted to set the theoretical curves in the actual experimental range. Figure 9 shows a reasonable fitting of the experimental data, without significant differences between both types of catalysts. Increasing the polymerization time decreases the gel time under acid conditions. However, after 5-7 h of alkaline reaction at 70 "C, the rate of change is relatively slow.
Conclusions Resols gel at a formaldehyde/phenol molar ratio equal to or greater than 0.5. There is not an upper limit for avoiding gelation at a great formaldehyde excess due to the possibility of condensing methylols between themselves. The model predicts that a convenient Bo/Aoratio for preparing NaOH and MgO-catalyzed resols lies between 1.5-1.7. This gives the smallest contents of both free phenol and formaldehyde as well as an adequate balance between free and methylolated phenolic sites. NaOHcatalyzed resols need shorter reaction times to reach the same weight average molecular weight (or gelation) and, according to calculations, contain less free formaldehyde but more free phenol than MgO-catalyzed resols. Using MgO or NaOH as alkaline catalysts does not result in any significant difference in the posterior curing under acid conditions. In this case, the model may be used to calculate the convenient polymerization time under alkaline conditions. Registry No. (Phenol).(formaldehyde)(copolymer), 9003-35-4. Literature Cited Aranguren, M. I.; Borrajo, J.; Willlams, R. J. J. Ind. Eng. Chem. prod. Res. D e v . 1984, preceding paper in thls Issue. Drumm, M. F.; LeBlanc, J. R. Klnet. Mech. Polym. 1972, 3 , 157. Flory, P. J. J . Am. Chem. Soc. 1941, 63, 3083, 3097. Francis, D. J.; Yeddanapalll, L. M. Makromo/. Chem. 1989, 725, 119. Freeman, J. H. Am. Chem. SOC. Dlv. Ora. - Coat. Plast. Chem. R e m . 1987, 2 7 , 84. Knop, A,; Schelb, W. "Chemistry and Applications of Phenolic Resins", Poiymers/Propetties and Appilcatlons 3, Springer-Verlag: Beriln-Heidelberg, 1979. r n A5 Mlnami, T.; Ando, T. K o ~ y oKagaku Zasshll956, 59, 668. Pal, P. K.; Kumar, A.; Gupta, S. K. Polymer 1981. 22, 1699. Rojas. A. J.; Williams, R. J. J. J . Appl. Polym. Sci. 1979, 23, 2083. Stockmayer, W. H. J . Chem. h y s . 1943, 7 1 , 45. Yeddenapaili, L. M.; Francls, D. J. Makromol. Chem. 1982, 55, 74. Zavltsas, A. A. J . Polym. Scl. Part A - 7 1988. 6 , 2533. Zavitsas, A. A.; Beaulleu, R. D.; LeBlanc, J. R. J . P w m . Sci. Part A- 7 1968, 6 , 2541. ' - . - I
Received for review October 14, 1983 Accepted February 10, 1984
