874
Ind. Eng. Chem. Prod. Res. Dev. 1981, 20, 674-679
any deterioration due to continuous exposure to contaminants such as vapors from kitchen oils, soy sauce, or dust, thereby guaranteeing trouble-free operation for more than 20000 h. The ceramic humidity sensor has already been utilized in the automatic microwave oven in which it controls the cooking time or microwave power supply by detecting the humidity given off by the food during cooking. This advanced automatic microwave oven consistently results in perfectly cooked food as well as allowing greatly simplified operation since housewives no longer have to predetermine the cooking time. Similarly, the ceramic humidity sensor has great potential for application to a wide range of devices which require accurate and reliable humidity control. Finally, both the newly developed ceramic humidity sensor and the automatic microwave oven controlled by
humidity detection are shown in Figure 12. The new ceramic humidity sensor has received an I.R.lOO Award in 1980. Acknowledgment
The author expresses sincere thanks to Dr. S. Hayakawa, Director of the Corporate Engineering Division, for continuous encouragement. L i t e r a t u r e Cited "a,T.; Hayakawa, S. IEEE Trans. Components. HybMs, Menuf. Techtwl. 1980, CHMT-3, 237-243. "a,T.; Terada, 2.; Hayakawa, S. U.S. Patent 4086556, 1978a; US. Patent 4080564, 1978b; U.S. Patent 4210894, 1980a. "a, T.; Terada, Z.; Hayakawa, S. J. Am. Ceram. Soc. 1980b, 63, 295-300.
Receiued for review December 10, 1980 Accepted July 21,1981
Cross-Linking Chemistry and Network Structure in High Solids Acrylic-Melamine Coatings David R. Bauer" and Gregory F. Budde Engineering and Research Staff, Ford Motor Company, Dearborn, Michigan 48 12 1
The cross-linking chemistry of typical high sotiis acrylic-melamine coatings has been measured as a function of bake temperature and catalyst level. Using a network structure model, a network parameter has been determined which correlates well with the state of cure of these coatings. Using this parameter, the effects on network formation of the acrylic molecular weight, the use of low functionality small molecules, the melamine cross-linker type, and the bake temperature have been investigated. The type of melamine used determines in large part the cross-linking chemistry. Lowering the resin molecular weight or adding mono- or difunctional materials requires a small but significant Increase in the cure temperature of a given coating. Lowering the resin molecular weight also increases the gel point of the coating and has a large effect on the rheological properties of the coating in the pre-gel region.
Introduction
One approach to achieving low emissions coatings is through the use of high solids acrylic-melamine enamels. In these coatings it is necessary that both the resin and cross-linker have a low molecular weight. In conventional coatings, resin molecular weights are around 10000 while in high solids acrylics molecular weights range from 1000 to 3000. Similarly, the highly polymerized butylated melamines currently used in conventional acrylics are not desirable in high solids acrylics. There are basically two classes of melamines commercially available for use in high solids acrylics: monomeric fully alkylated melamines and slightly polymeric partially alkylated melamines. During the bake, the polymer and cross-linker react to form a network. The structure of this network influences many coating properties. In this paper, means for characterizing network structure in high solids coatings are described and the effects of formulation changes on network formation are determined. In previous reports (Bauer and Dickie, 1980a,b) the cross-linking chemistry and network formation in both conventional solvent-based and water-based acrylics were studied as a function of melamine cross-linker type, bake temperature and time, and catalyst type. It was found that the cross-linking chemistry was determined predominantly 0196-432118 111220-0674$01.25/0
by functionality of the melamine cross-linker. The reactions of melamine-based cross-linkers have been studied by Blank and Hensley (1974) and recently reviewed by Blank (1979). In our systems, two reactions were found to dominate the cross-linking chemistry. The first was the condensation of acrylic hydroxy and carboxy groups with alkoxy groups on the melamine to form acrylic-melamine cross-links. The second was the condensation of methylol groups on different melamines to form melamine-melamine cross-links. It should be noted that the reaction mechanism of the alkoxy reaction depends on whether or not the melamine is fully alkylated or partially alkylated (Blank, 1979). The cross-linking reaction of the alkoxy group in fully alkylated melamines requires catalysis by strong acids (e.g., p-toluenesulfonic acid). In partially alkylated melamines, the reaction is catalyzed by weak acids (e.g., acrylic acid). In this paper we have determined the amounts of the different catalysts necessary to achieve a given extent of chemical reaction for typical high solids acrylics under different process conditions. The final network structure depends not only on the extent of reaction but also on other formulation variables such as the acrylic copolymer molecular weight. Using a statistical network model based on the work of Miller and Macosko (1976),we have found 0 1981 American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 4, 1981 675 Table I. Characterization of Acrylic Resins
a
resin no.
%
%
HEAa
AAb
1 2 3 4 5 6 7 8 9
30 30 35 30 20 25 20 20 20
02 -_
__ __ __ -__
Hydroxyethylacrylate.
'I6 L
M,
MJM,
TG,"C
2300 2300 1100 4000 3900 4100 4600 5900 8200
1.8 2.0 1.4 2.0 2.0 2.3 2.0 2.2 2.2
20 13 ?
15 25 --32 56 19 18
Acrylic acid.
a network parameter, the elastically effective cross-link density, which correlates well with film solvent resistance in these coatings. Using this parameter, we have determined the effects of lowering the acrylic resin molecular weight on network formation. Since high solids coatings often employ low functionality materials to increase solids, we have determined the effects of the addition of these materials on network formation. Experimental Section Several acrylic copolymers were prepared for this study. They are characterized in Table I. All resins except no. 3 were prepared by conventional free radical polymerization. Composition heterogeneities which can occur in the polymerization of multicomponent monomer systems were minimized in these polymerizations by adding the monomer mixture dropwise into the reaction flask. The molecular weights and molecular weight distributions were determined by gel permeation chromatography (GPC). The GPC results were calibrated using vapor phase osmometry measurements. With the exception of resin no. 3 which was a commercial acrylic obtained from Rohm and Haas, the polymers had values of Mw/M,, equal to 2 (f15 %) consistent with the free radical polymerization technique used. Glass transition temperatures were calculated from the comonomer composition (Brandup and Immergut, 1974). Two nonacrylic oligomers were also used in this study. One was bisphenol A while the other was the diadduct of EPON 1001 and acetic acid. The two melamine cross-linkers used in this study were characterized by 'H NMR (Christensen, 1970). One was a fully methylated melamine (Mel-A of Bauer and Dickie, 1980) and was found to contain 98% methoxy groups. The other was a partially methylated melamine (Mel-D). It was found to contain 47% methoxy functionality, 13% methylol functionality, 36% amine functionality, and 4% melaminemelamine bonds. The weight average molecular weight of Mel-D was around 450. The extent of reaction of the methylol bonds during the polymerization of this melamine was found to be 0.40. The cure chemistry of this melamine was similar to that of Mel-B (Bauer and Dickie, 1980). Two strong acid catalysts were used in this work: p-toluenesulfonic acid (PTSA)and phenyl acid phosphate (PAP). PAP was found to be similar in acid strength to butyl phosphate. Films were cast and cured, and infrared spectra were measured as described previously (Bauer and Dickie, 1980a). Cure times were 20 min. The formation of acrylic-melamine bonds was monitored by measuring the disappearance of the acrylic OH band. The formation of melamine-melamine bonds was monitored by measuring the disappearance of the melamine OH band after correcting for the NH band and the extent of polymerization of the melamine. Film solvent resistance was determined using the methyl ethyl ketone spot test previously described (Bauer and Dickie, 1980a). Films which did not
" = 1000
M
.IO
g .08 .06
.04
.02
0
0
5
IO
I5
20
25
30
n
Figure 1. Weight fraction of polymer with functionality n vs. n for acrylic resin containing 25% hydroxyethylacrylate.
noticeably soften after a 10-s exposure to MEK were considered to pass. Network Structure Calculations In previous work (Bauer and Dickie, 1980a), we developed a model based on the approach of Miller and Macosko (1976) to calculate post-gel properties of the networks formed by acrylic melamine coatings. The model determines the various post-gel properties by first calculating the probability that a particular functional group on the polymer or cross-linker looks out to the infinite network. This probability is a function of the different extents of reaction and of the number of functional groups on the different species. In order to more accurately treat network formation in high solids coatings, it was necessary to make some modifications to the model. These are described below. The first modification was to incorporate the molecular weight distribution of the resin into the model. The earlier model assumed that the functionality of the resin was monodisperse. I t was known that this was not the case, but, for high molecular weight resins for which the average functionality was greater than 10 it was shown that the exact value of the resin functionality and the distribution of that functionality was not important and the network could be well described using an average functionality. For low molecular weight resins, on the other hand, the distribution of functionality becomes important. Miller and Macosko (1976) have given expressions for calculating network properties when the distribution functions are known. In the Appendix, modifications to our earlier model are described which incorporate the distribution of functional groups for our acrylic copolymers calculated assuming a most probable molecular weight distribution (consistent with the observed Mw/M,,ratios) and random copolymerization (consistent with the method of polymerization). The weight fraction of material with a given functionality, n, is shown vs. n in Figure 1 for different resin number average molecular weights. The way in which the effective cross-link density is calculated has also been modified. In Bauer and Dickie, 1980a, a cross-link was counted as effective whenever it looked out to the infinite network. This effective cross-link density was found to correlate well with film solvent resistance in both water base and conventional solvent-based paints. The acrylic resins in these coatings all had molecular weights around 10000. For the low molecular weight resins used in this study, it was found (see Results and Discussion section) that the minimum effective
676
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 4, 1981
cross-link density necessary to achieve solvent resistance increased with decreasing molecular weight. The reason for this can be seen by extrapolation to the limiting case of a resin with two functional groups. In this case any bond that is formed by the resin will only cause chain extension. The resin molecule cannot form an elastically effective joint since such a joint requires three independent paths out to the infinite network. However, by our previous method of calculation such a bond would have been counted as effective if it led out to the infinite network. In order to correct for this deficiency we have modified the calculation of the effective cross-link density. Instead of counting every bond that leads out to the infinite network, we only count those bonds which lead out to the infinite network and for which there are two other paths out to the infinite network from the resin or cross-linker into which the bond is looking. Thus we require that the resin or cross-linker have at least three bonds out to the infinite network before any of them can be counted as elastically effective. The equation for the elastically effective cross-link density is given in the Appendix. As we shall see in the Results and Discussion section, this method of calculating elastically effective cross-link density gives cross-link densities which correlate well with film solvent resistance independent of resin molecular weight. The last modification that has been made is to include the presence of small low functionality materials in the formulation. While this could have been accomplished simply by adjusting the distribution of functional groups to include these materials, we chose to treat these materials as separate species so that we could allow the reactivity of these materials to be different from that of the functional groups of the resin. One of the goals of our network structure calculations has been to find a parameter which correlates well with cure (i.e,, the elastically effective cross-link density). For this reason we have up to now limited our calculations to the post-gel region. It is also possible to calculate the increase of the weight average molecular weight (M,) in the initial stages of the cross-linking reaction before the gel point (Macosko and Miller, 1976). Since the coating viscosity is related to the molecular weight, the initial rate of molecular weight increase will influence the viscosity stability. Also, the time dependence of the molecular weight increase will affect coating rheology. We have incorporated the pre-gel molecular weight buildup into our network model for the case of the random acrylic copolymer cross-linked with fully alkylated melamine. This case has been treated by Macosko and Miller (1976), and we have used their expressions directly in our model (eq 39). One of the critical parameters that is calculated is the extent of reaction at the gel point. It is given by the following expression (1) Pgel = r / ( ( . f e- l)(g where r is the ratio of melamine alkoxy groups to acrylic functional groups, g is the melamine functionality (g = 6 for a fully alkylated melamine), and fe is the functionality average functionality of the acrylic resin and is defined by CffLAf CffAf where A f is the number of moles of polymer with functionality f. Results and Discussion Cross-LinkingChemistry. The extent of reaction of the hydroxy functionality of resin no. 1 cross-linked with fe
=-
1.0 r
50
70 90 110 I30 150 170 CURE TEMPERATURE *C
Figure 2. Extent of reaction of acrylic hydroxy groups of resin no. 1 cross-linked with 20% Mel-A catalyzed with 0.1% PTSA vs. cure temperature. 1.0
2l / 50
90 110 130 150 170 CURE TEMPERATURE .C
70
Figure 3. Extent of reaction of acrylic hydroxy groups of resin no. 2 cross-linked with 35% Mel-D (-) and extent of reaction of Mel-D methylol groups (- - - - -) vs. cure temperature.
Mel-A (20%) and catalyzed with 0.1% PTSA is shown as a function of temperature in Figure 2. The extent of reaction rises rapidly with the bake temperature and goes nearly to completion at high temperature. The minimum cure temperature for this formulation as defined by film solvent resistance is 120 "C. The extent of reaction and the minimum cure temperature are strong functions of the amount of PTSA catalyst that is used (Bauer and Budde, unpublished). For example, addition of 0.5% PTSA is sufficient to lower the minimum cure temperature to 90 "C. Since Mel-A does not contain much methylol functionality, melamine self-condensation does not occur under normal bake conditions. The extent of reaction of acrylic melamine bond formation (and thus the resulting network structure) necessary to achieve film solvent resistance was independent of catalyst level. The extent of reaction of the hydroxy groups of resin no. 2 cross-linked with Mel-D (30%)is shown as a function of temperature in Figure 3. Also shown is the extent of reaction of the melamine-melamine self-condensation reaction. The acrylic-melamine extent of reaction rises to a plateau value of 0.75-0.8. The melamine-melamine extent of reaction increases slowly with temperature. The minimum cure temperature of this formulation as defined by film solvent resistance is 130 "C. Lowering the cure temperature further to 90 "C requires a strong acid. It was found that 5% PAP or 3% PTSA was required to give a solvent resistance film for a bake temperature of 90 "C.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 4, 1981 877 x
Table 11. Effect of TG on Minimum Elastically Effective Cross-Link Density
a
B g 3.0 i
p
2.5
-
0
2.0-
6
-32 15 25 56
5
1
7
5 l.5X
$
TG,"C
4
fn
Y
resin
X
cross-link density l o 3 mol cross-links/g
* 0.15 * 0.15 0.99 f 0 . 1 5 0.94 * 0.15 1.33 1.05
e9 1.6 -
I.0-
z
0
Y
*
2
1.4
-
1.2
-
X
d 0
400
1000
10000
d
Mn
Figure 4. Minimum elastically effective cross-link density (0) and vs. number average mominimum effective cross-link density (0) lecular weight of resin. The bars at M, = loo00 indicate the range of values for water base and conventional solvent based formulations studied previously (Bauer and Dickie, 1980). The vertical bars indicate values for the low molecular weight oligomers (see text).
Thus much larger amounts of strong acid are necessary to achieve a 90 "C cure when partially alkylated melamines are used than when fully alkylated melamines are used. Also, strong acid catalysis has been found (Bauer and Dickie, 1980b) to enhance melamine-melamine bond formation over acrylic-melamine bond formation. For the partially alkylated melamine used in this study, it was found that when 5% PAP is added, the extent of reaction of acrylic hydroxy groups increases from 0.8 to 0.85 at the minimum cure temperature while the extent of reaction of the melamine methylol groups increases from 0.6 to 0.95. In addition to changes in cross-linking chemistry, the addition of acid also affects the resulting network structure a t minimum cure conditions. In the above system, the minimum elastically effective cross-link density necessary for cure is 40% higher when 5% PAP is used. This may be due to the large amount of highly polar unreacted material present in the catalyzed coating. Network Structure. In order to determine the effect of acrylic resin molecular weight on cure properties, it is necessary to determine the effect of molecular weight on cross-linking chemistry. We have found that as long as the catalyst level and bake temperature are constant, the extents of reaction are simply a function of stoichiometry of the resin and cross-linker functionality and not the resin molecular weight. We have also determined the lowest extent of reaction that is necessary to achieve good solvent resistance in acrylic resins cross-linked with Mel-A as a function of resin molecular weight and glass transition temperature. The cross-linking chemistry of Mel-A with two low molecular weight oligomers (bisphenol A and the EPON 1001 diadduct) was also studied. From the lowest extent of reaction and the known molecular weight or functionality of the resin, we can use our network structure model to calculate the minimum effective cross-linkdensity necessary to pass the methyl ethyl ketone solvent resistance test. In Figure 4 we show both the minimum effective cross-link density and the minimum elastically effective cross-link density as a function of resin molecular weight. While the minimum effective cross-link density increases with decreasing molecular weight, the minimum elastically effective cross-link density is nearly independent of resin molecular weight or functionality and has a value of 1.0 f 0.15 X mol of crosslinks/g. The elastically effective cross-link density also correlates well with solvent resistance for formulations with different melamines. Values of the minimum elastically effective cross-link density for the water base and conventional solvent base acrylics also
z
E x 1.0 E
I: 8 8 ' $ .e-
k
.4-
3 -1 4
.e -
fd
50
0
70
90 110 120 I50 CURE TEMPERATURE *C
I70
Figure 5. Calculated minimum elastically effective cross-link density vs. cure temperature for acrylic resin containing 25% hydroxyethylacrylate cross-linked with 20% Mel-A as a function of the number average molecular weight of the resin. The catalyst concentration is 0.1% PTSA.
fall within the above range. As the molecular weight of the resin becomes large, the difference between the two cross-link densities becomes relatively small since there are vanishingly small amounts of very low functionality polymer when the average molecular weight is large. Thus all bonds which lead out to the infinite network are also elastically effective since the probability is high that a t least two other bonds will also lead out to the infinite network. The elastically effective cross-link density correlates well with film solvent resistance independent of resin molecular weight or melamine type. The minimum elastically effective cross-link density is a weak function of the glass transition temperature of the resin, however. As shown in Table 11, the minimum elastically effective cross-link density decreases with increasing glass transition temperature. This is apparently due to the fact that networks formed from low glass transition temperature resins are softer than those formed from resins with high glass transition temperatures. All of the acrylic resins shown in Figure 4 had glass transition temperatures around 20 "C, typical of practical coatings. As shown in Figure 4,the elastically effective cross-link density correlates well with the state of cure independent of molecular weight of the resin. Thus, we have a single parameter which can be used to assess the state of cure as a function of a variety of formulation variables. Calculation of the elastically effective cross-link density requires only a determination of the extent of reaction, the resin molecular weight, and the functionality of the melamine. Since we know the extent of reaction of the acrylic hydroxy group as a function of cure temperature (see Figures 2 and 3), we can construct a plot of the elastically effective cross-link density as a function of cure temperature for different formulations. Shown in Figures 5 and 6 are the elastically effective cross-link densities vs. cure temperature for acrylic resins cross-linked with Mel-A and Mel-D as a function of resin molecular weight. The resin
678
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 4, 1981 0
X DILUENT
AluL.-2
I!
50
70 90 110 130 160 C U R E T E M P E R A T U R E 'C
I
I70
Figure 6. Calculated minimum elastically effective cross-link density vs. cure temperature for acrylic resin no. 2 crw-linked with 30% Mel-D as a function of the number average molecular weight of the resin.
in Figure 5 contained 25% hydroxyethylacrylate while the resin in Figure 6 had a composition identical to that of resin no. 2. For both melamines the cure temperature necessary to achieve solvent resistance increases when the molecular weight of the resin decreases. The effect is significant but not large. Going from a conventional acrylic to a typical high solids acrylic while holding everything else constant requires an increase in the cure temperature of around 6 "C for both melamine types. Equivalently, the cure temperature could be held constant and the cure time increased. The performance of these resins degrades rapidly as the molecular weight goes below 1000. The reason for this can be seen in Figure 1. When M,,= 1000, over 35% by weight of the polymer chains have 2 or less functional groups. These polymers cannot form elastically effective cross-links. Such a low molecular weight resin would require a higher average functionality to achieve cure. Although the effect of molecular weight on cure temperature is relatively small, variations in molecular weight, especially unexpected decreases in resin molecular weight, could lead to incomplete cure. Based on the above results, we might expect that the addition of low functionality small molecules would also have an effect on network formation. Since such materials are often added to high solids formulations to increase solids, it is important to determine the magnitude of the effect. We have calculated the dependence of the elastically effective cross-link density on bake temperature as a function of the amount of added mono and difunctional material for the different melamine cross-linkers. For the purpose of this calculation we have assumed that the reactivities of the functional groups on the polymer and on the small molecule are the same. The equivalent weight of the functionality of the small molecule is assumed to be 264 g, which is typical for such materials. As expected, the effect of adding these materials to a given formulation is to raise the minimum cure temperature (see Figures 7 and 8). Incorporation of 10% of the monofunctional material increases the cure temperature around 6 "C for the Mel-A formulation and 8 "Cfor the Mel-D formulation. Incorporation of 10% of the difunctional material increases the cure temperature by only 2-3 "C. Another important parameter that can be calculated by the network model is the amount of material that is extractable from the coating after cure (i.e., the amount of material that is not connected to the infinite network; 15-209'0 of the monofunctional material is extractable at
u 5 w
.2
50
70
90 ll0 130 150 C U R E T E M P E R A T U R E *C
170
Figure 7. Calculated minimum elastically effective cross-link density vs. cure temperature for acrylic resin containing 25% hydroxyethylacrylate cross-linked with 20% Mel-A as a function of amount of added monofunctional (-) and difunctional material (- - - - -). The catalyst concentration is 0.1% PTSA.
% DILUENT
Figure 8. Calculated minimum elastically effective cross-link density vs. cure temperature for resin no. 2 cross-linkedwith 30% Mel-D as a function of amount of added monofunctional (-) and difunc-
tional (- - - - -) material.
the minimum cure temperature in the Mel-A formulation compared with 3540% in the Mel-D formulation. For the difunctional material, only 5% of the material is extractable in the Mel-A formulation while 15% of the material is extractable in the Mel-D formulation. Pre-Gel Behavior. In addition to achieving a cure which yields acceptable physical properties, a coating must also have good appearance. This depends to a large part on coating rheology. The viscosity of a coating depends on the molecular weight of the resin, the amount of solvent in the coating, and the coating temperature. All of these variables change as the coating enters the bake oven; the solvent evaporates, the coating temperature increases, and the molecular weight of resin increases as cross-linking begins. While a complete description of coating rheology is outside the scope of the present paper, it is possible to calculate one important factor, namely the increase of weight average molecular weight with extent of crosslinking. It is also possible to calculate the gel point (i.e., that extent of reaction for which M , = m). The gel point is important because it is the point at which flow ceases. From eq 1 it can be seen that the gel point is inversely proportional to the square root of the initial molecular weight of the polymer. The increase in M , with extent of
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 4, 1981 670
assume an average functionality for both the alkoxy denoted by (g) and the methylol denoted by ( k ) functionalities. Including the acrylic copolymer distribution into eq 9 of Bauer and Dickie (1980a) yields the following equations
P(FAoUt) = 1 - PA
+ ~AP(FB""t)g-'P(FCoUt)h (Al)
P(FC""t) = 1 - pc
+ pcP(Fc""t)gP(Fc""t)h-'(A3)
x = 2 IXn(I)
(A4)
z=1
1
.05
I
I
.I ,I5 .2 .25 EXTENT OF R E A C T I O N
,
.3
I
,35
Figure 9. Weight average molecular weight of formulation crosslinked with Mel-A w. extent of reaction as a function of the number average molecular weight of the acrylic resin. The vertical lines indicate the gel points.
reaction is shown in Figure 9 for different values of the initial polymer molecular weight when Mel-A is used as the cross-linker. It is clear that as the initial polymer molecular weight decreases, the coating will have a greater propensity to flow in the bake oven. This flow typically leads to such problems as sagging and in the case of formulations containing aluminum flake, aluminum reorientation. Conclusion We have measured the cross-linking chemistry for typical high solids acrylic melamine coatings. Using these data and a network structure model, we have found a parameter, the elastically effective cross-link density, which correlates well with the solvent resistance of these coatings. We have used this parameter to show that lowering the acrylic resin molecular weight or adding low functionality materials to a given formulation requires an increase in cure temperature to achieve the same state of cure. Lowering the resin molecular weight also increases the extent of reaction at the gel point and greatly affects the buildup of molecular weight in the pre-gel region. Appendix. Calculation of Elastically Effective Cross-Link Density The approach used by Miller and Macosko (1976) requires knowledge of the distribution of the number of functional groups on the polymer and cross-linker. For the acrylic copolymer, a most probable molecular weight distribution was used. In addition, it was assumed that the distribution of functional groups on a given chain was random. The validity of these assumptions is discussed above. With these assumptions it is an easy matter to , number fraction of polymer chains calculate ( X n ( I ) the with I reactive groups, and Wn(l),the weight fraction of such chains, as a function of M, and the percent of functional groups. Such a plot is shown in Figure 1. With the exception of Mel-A the distributions of the different functionalities on the cross-linker are not known. We
where P(FxoUt) is the probability that looking out from group X is a finite chain and px is the extent of reaction of that group. Subscript A denotes the acrylic hydroxy, B the melamine alkoxy, and C the melamine methylol. As before, we have assumed that A and B react with one another and that C reacts with itself. Equations A1-A3 can be solved using Newton's method. The effective cross-linker density is given by
c,ff=
4
([I - P(FA""~)] + ALK[1 - ~ ( F B " "+~ M[1 )] -
P(FcoUt)l 1 (A5) where Q is the initial moles of acrylic hydroxy per gram of solids, ALK is the ratio of alkoxy to hydroxy, and M is the ratio of melamine methylol to hydroxy. The quantity 1- P(FAout) is the probability that a single group leads out to the infinite network. To calculate the elastically effective cross-linking density, we also need to calculate the probability that there are two other paths out to the infinite network ce1
=
1
g(l - P(F~""t))(P(F~""t)~-' X P(FC""t)h-')])(A6)
Literature Cited Bauer, D. R.; Budde, G. F. Ford Motor Company, unpubllshed data, 1980. Bauer, D. R.; Dicke, R. A. J . folym. Sci., folym. fhys. 1080. 18, 1977a. Bauer, D. R.; Dickie, R. A. J. folym. Sci., folym. fhys. 1080, 78, 2015b. Blank, W. J. J . Coat. Techno/. 1870. 51, 61. Blank, W. J.; Hensley, W. L. J . faint Techno/. 1074, 46, 48. Brandup, J.; Immergut, E., Ed. "PolymerHandbook";Wlley: New York, 1974; 111-61. Christensen, G. frog. Org. Coat. 1070, 5 , 255. Macosko, C. W.; Miller, D. R. Macromo/ecu/es1976, 9 , 199. Miller, D. R.; Macosko, C . W. Macromolecules 1976, 9 , 206.
Received for review March 5 , 1981 Accepted July 30, 1981
