Energy & Fuels 1994,8, 1020-1023
1020
Calculating the Number-Averaged Molecular Weight (Mo) of Aromatic and Hydroaromatic Clusters in Coal Using Rubber Elasticity Theory Jean-Loup Faulon Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87185-0710 Received December 15, 1993. Revised Manuscript Received July 5, 1994
For the past 10 years, rubber elasticity theory has been applied t o coal to calculate the numberaveraged molecular weight between branch points (M,) and the number-averagedmolecular weight of aromatichydroaromatic clusters (MOL In the equations that have been proposed in the context of coal, the average number of statistical links between branch points (N) was always assumed to be equal t o the average number of clusters between branch points (N'). The present paper demonstrates that this assumption is invalid for highly cross-linked polymer networks. Modified equations are proposed for which the assumption N = N' is no longer considered valid. The main consequence of the modification is that the number-averaged molecular weight of coal clusters is greater than previously reported.
Introduction Rubber elasticity has been applied in the area of coal science to evaluate the cross-link densities of several coals using solvent swelling data. In the last decade, modified-Gaussianmodels (Kovac m ~ d e l ,and ~,~ Barr-Howell-Peppas model9 have been used with coal7,* to evaluate the number-averaged molecular weight between branch points (M,) and the numberaveraged molecular weight of aromatichydroaromatic clusters (Mo). More recently, Painter et aL9 applied for coal network a technique originally developed by Bastide et a1.,6 and they derived a relation between the average number of carbon atoms per aromatichydroaromatic cluster and the average number of carbon atoms between branch points. We have some criticisms regarding the uses made of these models. In the Kovac m ~ d e lany ~ , polymer ~ chain is composed of identical freely jointed repeat units (atoms) having a molecular weight Mo, connected by N links. The polymer network is assumed t o be formed by molecular chains cross-linked at branch points. Based on the classical rubber elasticity theory,'S2 the Kovac model assumes that a branch point is any location in the network linking more than two polymer chains. Kovac's equation4 relating Mc and N can be expressed in the form
~~
(1)Flory, P. J.; Rehner, J. J . Chem. Phys. 1943,11, 512. (2)Flory, P. J.; Rehner, J. J . Chem. Phys. 1943,11, 521. (3)Fixman, M.;Kovac, J. J . Chem. Phys. 1973,58,1564. (4)Kovac, J. Macromolecules 1978,1 1 , 362. (5)Barr-Howell, B. D.;Peppas, N. A. Polym. Bull. 1986,13, 91. (6)Bastide, J.;Picot, C.; Candau, S. J . Macromol. Sci. Phys. 1981, B19, 13. (7) Larsen, J. W.; Green, T. K.; Kovac, J. J. Org. Chem. 1985, 50, 4729. (8) Lucht, L. M.; Peppas, N. A.Fuel 1987,66, 803. (9)Painter P. C.; Graf, J.; Coleman M. Energy Fuels 1990,4,393. (10)Flory, P.J. J . Chem. Phys. 1977,66,5720.
where
r is defined by
r=
Vl Uln(1 - U J
(2)
+ u2 + p,21
where vz is the final equilibrium volume fraction of the swollen network, VI is the molar volume of the solvent, U is the specific volume of the polymer, and is the Flory solubility parameter. The x parameter depends on the temperature and the concentration of the polymer in the swollen gel. In the Barr-Howell-Peppas5 model the polymer network is composed of polymer chain each having N' = M,IMo repeat units. The number of links per polymer chain is defined as
x
N=/W
(3)
where 1 is the average number of links per repeat unit. The Barr-Howell-Peppas model explicitly introduced the average number of molecular chains per branch point ($1 in the calculation of M,. The q5 parameter is greater than 2, and is also called the average functionality of the network. With the notation used in eqs 1 and 2, the generalized Barr-Howell-Peppas equation corelating Mc and N is
where M , is the number-averaged molecular weight of the original polymer chains before cross-linking, and w is defined as w = 2I4 (5) According to Flory,lo A+ is a structural factor which varies as follows
1 > A, > 1 - 214 (6) When both models were applied to study the crosslink density of c0al,'3~the repeat units were considered
0887-0624/94/2508-1020$04.50/00 1994 American Chemical Society
Energy & Fuels, Vol. 8, No. 5, 1994 1021
Cross-Link Densities of Coal Branch point
Figure 1. Difference between average number of links (M and average number of repeat units (N') per polymer chains. (a) Watermelon polymer network composed of four polymer chains each having three links. The average number of links per chain is N = 3. The network is composed of 10 repeat units, the average number of repeat units per chain is therefore,N' = 1014 = 2.5. (b) Corresponding structural network as it could appear in the coal macromolecule. In the graph representing structure b, the vertices are the aromatichydroaromatic clusters and the edges are the ether or aliphatic bridges. This representation was first proposed by Painter et aL9 (cf. Figure 2, p 394) and agree with Kovac's equation. However, one may question if the graph representing structure b is the right interpretation of rubber elasticity theory. More precisely, one may reverse the graph of structureb to obtain a dual graph where the vertices are the ether or aliphaticbridges and the edges are the aromatichydroaromatic clusters. When drawing the dual graph, one may observe that this interpretation of the theory is not valid, because the branch points cannot be represented in the dual graph (in graph theory every edge links exactly two vertices). to be polycondensed aromatic and hydroaromatic clusters, and the links were considered to be ether or aliphatic bridges between these clusters. The Kovac equation was applied for coal7 in a form equivalent to eq 1. The Barr-Howell-Peppas equation was applied for coal8 in the form:
1 MC
r-
(7)
From the differences between eq 4 and eq 7, we can conclude that in eq 7 it was implicitly assumed that coal is a tetrafunctional network (4 = 4)and that A4 = 1. The most important point for both models when applied to coal is that it was assumed that N = N', and therefore, that MO is equal to the ratio Mc to N . For long polymer chains the difference between N and N' is small and can be neglected, and N' can be assumed to be equal to N . However, for short polymer chains as it may occur in coal, we think this assumption is invalid. Figure 1 clearly shows for an arbitrary network that the average number of links per chain (N) is not equal to the average number of units per chain (N'). We feel it is important to correct the calculation of Mo, since MOis a key parameter which characterized coal structure, furthermore, MO is used as an input in several kinetics models simulating the pyrolysis of coal. The ratio M , to N is not the number-averaged molecular weight of the coal clusters (Mo) but the numberaveraged molecular weight per link (M& In order to prove that MO is different than M L the relationship between these two quantities is developed as follows.
Table 1. Examples of Number-Average Molecular Weight of Coal Clusters (Mo) as a Function of the Functionality of the Coal Network Mob cross-linked densities for extracted coal,a 4 Mc = 550, Mi = 180 Mc = 640, M I = 320 M , = 780, MI = 620 Mc = 880, Mi = 880 cross-linked densities for 0-acetylated coal,a 4 M , = 910, Mi = 230 Mc = 1000, Mi = 330 Mc = 1160, Mi = 580 Mc = 1320, MI = 880
2.1
3.0
4.0
10.0
-
183 202 215 243 268 328 384 427 533 640 644 843 1029 1702 3022 ca 924 1320 1760 4400 2.1 3.0 4.0 10.0
-
233 335 594 880
251 371 696 1131
263 395 773 1320
288 448 967 1886
308 492 1160 2640
a The values of Mc and Mi = MJN have been taken from Larsen et a1.I0 and were obtained from swelling experiments of Illinois No. 6. MOwas calculated using eq 13. All the molecular weights
are expressed in amu.
Methodology The equations derived in this paper are based on the formula of Euler applied to polymer networks.ll Assuming a graph composed of V vertices and E edges, the number 5 of independent loops in the graph is determined using the formula of Euler V E - E = 1. When applied to a polymer network, V can be the number of atoms, the number of clusters, or the number of branched points. Depending on the definition taken for V, E is the total number of bonds between atoms, the number of links between clusters, or the number of polymer chains attached to the branched points. The number E, calculated using the formula of Euler, is the cycle rank of the network.
+
(11)Duplantier, B. J. Stat. Phys. 1989, 54, 581.
Faulon
1022 Energy & Fuels, Vol. 8, No. 5, 1994 35
-
I 50
-: b
5
T
--
20
10
0
0 0
20
0
60
40
20
40
60
Nc
Nc
a)
b)
Figure 2. Plots of n, the number of carbon atoms per link, and n,' the number of carbon atoms per cluster, versus N , the number of carbon atoms between branch points (4 = 3, and a, = 0.6): (a) random walk ( u = 0.5); (b) self-avoiding walk (v = 0.6). 60
40
I
0 4 0
100
50
150
a) $J
=
Let us assume a polymer network composed of X clusters, Y polymer chains, and xb branch points. The average functionality of the network is 4. Each branch point links 4 chains, and each chain is linked to two branch points, therefore (8)
The total number of links in the network is (YN), and according to the formula of Euler
YN = X + 6 - 1
(9)
where 5 is the cycle rank of the polymer network. j is calculated by applying the formula of Euler considering this time the relationship between xb and Y
E = Y-xb
+1
(10)
by replacing eq 10 into eq 9
by replacing eq 8 into eq 11and substituting X I Y by N'
N
= N' -F 1 - 214
100
b) 4,and a, = 0.5: (a) random walk ( u = 0.5); (b) self-avoiding walk ( u = 0.6).
Results and Discussion
Y = xb$/2
50
Nc
Nc
Figure 3. Plots of n, and n,' vs N , for
0
(12)
Equation 12 can easily be verified with Figure 1. Having the relationship between N and N', we finally obtained the following equation
(13)
The results of the application of eq 13 for different values of M I ,M,, and 4 are presented in Table 1. It is clear from Table 1that Mo is always greater than Mi. We can conclude that the number-averaged molecular weight of the clusters for coal calculated using the modified-Gaussianmodels and reported in the literature is underestimated. Furthermore, the most important consequence of eq 13 is that Mo cannot be calculated for coal without knowing the functionality of coal network. The model of Painter et aL9follows the calculation of Bastide et a1.F where an expression between N (number of statistical links) and u2 (volume fraction of the swollen network) is obtained considering the volume of the chains present in a sphere surrounding a particular branch point. Let R, be the root-mean-square "chain" end-to-end distance in the swollen network; hence, the volume of the sphere surrounding any branch point is (4/3)xRm,3. In the nonswollen network, according to Bastide et aL6 the volume of the chains attached to a given branch point is @Nul,where u1 is the volume of the statistical link. If the network is assumed to have a uniform distribution of branch points, the following relation is true: (14)
Cross-Link Densities of Coal
Energy & Fuels, Vol. 8, No. 5, 1994 1023
where R,, can be replace by the following expression derived by Flory:12 Rms
=a F
(15)
where a1 is an average virtual link length and the exponent v has values 0.5 for a chain following random walk and 0.6 for a self-avoiding walk. By substituting (15) into (14) and rearranging the expression, the following equation is obtained:
and by substituting eq 24 into eq 23 we finally obtain an expression between the average number of carbon atoms per aromatichydroaromatic clusters (n,') and the average number of carbon atoms between branch points (Nc):
(16) Painter et aL9 applied eq 16 and assumed in the case of coal network u1 = v,n,
(17)
N = NJn,
(18)
al = n,a,
(19)
and
where n, is the average number of carbon atoms per aromatichydroaromatic cluster, uc the molar volume of the (nonswollen) coal per carbon atom, N , the number of carbon atom between branch points, and a, the average virtual link length per carbon atom. By substituting eqs 17 to 19 into eq 16, the authors finally obtained (20)
It is clear from eqs 17 to 19 that the number of link N was assumed to be equal to the number of clusters between branch points (N'). As we discussed previously we think that this assumption is invalid for coal network. In the original article of Bastide et u Z . , ~ v1 is the volume of a statistical link, and according to Flory,12a1 is the length of a link. Therefore, in eqs 17-19, nc is the average number of carbon atoms per link and not the average number of carbon atom per aromatic/ hydroaromatic clusters as defined by Painter et aL9 Let n,' be the average number of carbon atom per aromatic/ hydroaromatic clusters; this number is simply obtained by dividing the number of carbon at.oms between branch point (N,)by the number of clusters (N'): n,' = NJW
+ 219; therefore = NJ(N - 1 + 214)
(21)
according to eq 12, N' = N - 1 n,'
(22)
by substituting N by Nclnc,where n, is the average number of carbon atoms per link, we obtain
Equation 25 was applied for different values of a,, 9, and u. The different values of n,' are presented in Figures 2 and 3 and compared with the n, corresponding It is clear from Figures 2 and values of Painter et 3 that n,' is always greater than n,. Therefore, the size of the aromatichydroaromatic clusters is underestimatd by the calculations of Painter et al. Recently, Painter and Shenoy13 developed a new model for swelling of polymer network using the c* theorem of de Gennes.14 The model has been applied for coal15 by the same authors and they derived an expression between N and the volume fraction of the swollen coal network. An important difference exists between this new model and the previous models. In the new model N does not represent the number of links of the molecular chain but the so-called number of segments per molecular chain. A segment is the smallest statistical element that is freely jointed. A link connects exactly two monomers (cf. Figure 11, but a segment can connect more than two monomers; therefore, eq 12 can no more be applied in the case of segment. Instead of eq 12, Painter and Shenoy13 propose another expression between N and N':
N = WIC,
(26)
where C, is the average number of monomers per segment. It is evident from eq 26 that the confusion that existed in the previous models between N and N' is eliminated with this new model. However, it is not clear how C , can be calculated for coal, and therefore, how MOcan be determined using the model of Painter and Shenoy. To summarize, rubber elasticity models when applied for coal have underestimated the number-averaged molecular weight of aromaticlhydroaromatic clusters (Mol. In this paper, modifications of the rubber elasticity equations are proposed that lead t o correct values for Mo. These modifications are derived from graph theory and hold true for any type of polymer network. In the context of coal, an important consequence of the modifications is that MO can no longer be calculated without knowing the functionality of the coal network.
(23)
Acknowledgment. Funding was provided by the U.S. Department of Energy at Sandia National Laboratories under contract DE-AC04-76DP00789.
(12)Flory, P. J. Statistical Mechanics of Chain Molecules; Hanser: Munich, 1969; p 8.
(13)Painter, P. C.; Shenoy, L. J . Chem. Phys. 1993,99,1409. (14)De Gennes, P. G . Scaling Concepts i n Polymer Physics; Cornel1 University: Ithaca, NY,1988. (15)Painter, P. C.; Shenoy, L. Prepr. Pap.-Am. Chem. Soc., Diu Fuel Chem. 1993,40,1304.
lln: = lln,
+ (214 - lYN,
eq 20 can be written is the form
