Energy & Fuels 1996,9,364-371
364
A New Model for the Swelling of Coal Paul Painter" and Suresh Shenoy Polymer Science Program, Penn State University, University Park, Pennsylvania 16802 Received September 26, 1994. Revised Manuscript Received December 13, 1994@
The Flory-Rehner approach to the swelling of polymer networks is modified so as to abandon the affine deformation assumption, instead using the assumptions of the c* model of de Gennes. The resulting model, in which the mixing term is also modified to account for hydrogen bonding, is applied to coal networks. The results indicate that the number of statistical segments between junction points is small and in treating the swelling of coal networks energetic terms will probably have to be taken into account.
Introduction Practically all attempts1-6 to relate coal structural parameters (such as the molecular weight between cross-link points) to the degree of swelling in a good solvent have taken as a starting point the Flory-Rehner t h e ~ r for y ~the ~ ~swelling of macromolecular networks. Modifications have been made to account for short chain length^^,^ and, in recent work in our laboratory, the effect of hydrogen bonding i n t e r a c t i o n ~ . ~This J ~ is also the general approach that has been used in many studies of polymer gels and rests on the fundamental assumption that the contribution of elastic and mixing terms to the free energy can be treated separately and additively. Although this model continues to be widely used, various studies reported over the last 15 years (e.g. refs 11-15) have now made it clear that there is a fundamental problem with this approach. Some data have been interpreted so as to bring into question the assumption of the separability of the elastic and mixing free energies,16J7but the most significant deficiency of Abstract published in Advance ACS Abstracts, February 1, 1995. (1)Sanada, Y.; Honda, H. Fuel 1966,45,295. (2) Nelson, J. R. Fuel 1983,62, 112. (3) Kirov, N. Y.; OShea, J. M.; Sergeant, G. D. Fuel 1968,47,415. (4) Larsen, J. W.; Green, T. K.; Kovac, J. J . Org. Chem. 1985,50, 4729. (5) Lucht, L. M.; Peppas, N. A. Fuel 1987,66,803. (6) Green, T.; Kovac, J.;Brenner, D.; Larsen, J. W. In Coal Structure; Meyers, R. A., Ed.; Academic Press: New York, 1982. (7) Flory, P. J.; Rehner, J. J . Chem. Phys. 1943,11, 512 and 521. (8)Flory, P. J. Principles of Polymer Chemistry; Cornel1 University Press: Ithaca, NY,1953. (9) Painter, P. C.; Park, Y.; Coleman, M. M. Energy Fuels 1988,2, 693. (10) Painter, P. C.; Park, Y.; Sobkowiak, M.; Coleman, M. M. Energy Fuels 1990,4,384. (11)Bastide, J.; Candau, S.; Leibler, L. Macromolecules 1981,14, 719. (12)Bastide, J.; Picot, C.; Candau, S. J . Macromol. Sei. B 1981,19, 13. (13) Bastide, J.; Duplessix, R.; Picot, C.; Candau, S. Macromolecules 1984,17,83. (14) Horkay, F.; Zrinyi, M. In Biological and Synthetic Polymer Networks; Kramer, O., Ed.; Elsevier: Amsterdam, 1988. (15)Geissler, E.; Hecht, M.; Horkay, F.; Zrinyi, M. In Springer Proceedings in Physics; Baungartner, A,, Picot, C. E. Eds.; Springer: New York, 1989; Vol. 42. (16) Nauberger, N. A,; Eichinger, B. E. Macromolecules 1988,21, @
mfin
(17) McKenna, G. B.; Flynn, K. M.; Chen, Y. H. Polym. Commun. 1988,29,272.
the Flory-Rehner theory involves the assumption that the deformation of the elementary chains of the network is in some fashion affine with the macroscopic deformation (swelling) of the sample. The neutron scattering work of Bastide et al.11-13has demonstrated that the network elementary chains have approximately the same dimensions as equivalent non-cross-linked chains in solutions of the same concentration, which is much less than would be expected on the basis of an affine deformation model. These results are in good agreement with the c* theorem of de Gennes,18however, who proposed that in a good solvent, the swollen coils of the network largely exclude one another from a volume that is (more or less) defined by their radius of gyration, but because the chains are forced into contact a t their crosslink points, the gel is analogous to the situation a t the overlap threshold in a semidilute solution. Accordingly, Bastide et a1.12 proposed that the swelling of a gel proceeds by a process of topological rearrangement or disinterspersion of the cross-link points and demonstrated that an analysis based on a scaling approach is in good agreement with experimental observations. We have taken these ideas and proposed a simple modification to the Flory-Rehner approachlg that abandons the affine assumption and instead uses the packing conditions that are a consequence of the c* theorem in order to obtain the necessary relationship between the degree of swelling and chain extension. The model appears to provide a good description of the swelling and deswelling behavior of model polymer networks and provides an explanation for various anomalous results, in particular those that had previously been interpreted in terms of a breakdown of the assumption that the elastic and mixing free energies can be separated. Here we will extend and combine this model with our treatment of hydrogen-bonding interactions20 in order to describe the swelling of coal networks. A preliminary and incomplete account of this work was presented at a recent ACS Meeting.21 (18) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornel1 University: Ithaca, NY,1979. (19) Painter, P. C.; Shenoy, S. L. J . Chem. Phys. 1993,99,1409. (20)Coleman, M. M.; Graf, J.; Painter, P. C. Specific Interactions and the Miscibility of Polymer Blends; Technomic: Lancaster, PA 1991. (21) Painter, P. C.; Shenoy, S. L. Prepr. Pap.-Am. Chem. SOC.,Diu. Fuel Chem. 1993,38( 4 ) , 1304.
0 1995 American Chemical Society
A New Model for the Swelling of Coal
Energy & Fuels, Vol. 9, No. 2, 1995 365
“UNOCCUPIED “ VOLUME
DANGLING END Figure 1. Schematic representation of the de Gennes “blob” model of a swollen network.
Theory This model starts with a description of the equilibrium swelling of a “perfect” network (i.e., no dangling ends, etc.) of very long chains, each made up of N statistical units or segments and joined through their ends by cross-link units of functionality f . The effect of imperfections, short chain lengths, etc., will be considered as modifications below. Three basic assumptions are made: 1. The free energy of the gel can be written as the sum of two separate components, describing the elastic free energy and mixing free energy, respectively. 2. These components of the free energy can be expressed in terms of the classic elastic free energy and the Flory-Huggins theory, modified to account for hydrogen bonding. 3. Following de Gennes, we assume that ut equilibrium in a good solvent the chains expand to the extent that they would in a dilute solution of the same solvent. The cross-link points rearrange their positions or disintersperse to the extent that the gel is a collection of spheres or “blobs” of individual network chains that as far as possible exclude segments of other chains from their volume, but are forced into contact at their crosslink points, as illustrated in Figure 1. Using the first two assumptions, the equations for the free energy can be written down immediately, while the final assumption provides the essential connection between the volume fraction of polymer segments (i.e., the degree of swelling) and the chain expansion factor.lg The free energy is given by
- u -(a E E2 =
1)- In a
31
-
In @c
+ n, In0 - 4J +
where n, is the number of moles of solvent, & and & are the volume fractions of solvent and coal, respectively, E is the cycle rank of the network, u is the number of moles of chains, x is the Flory-Huggins interaction parameter and a is the chain expansion factor. The A& term represents the contribution to the free energy from hydrogen bonding or other strong specific interac-
tions. The exact form of this term will be discussed in more detail below, but it should be recognized that this is not simply an empirical modification to the FloryHuggins equation. This result can be obtained by using a lattice model to describe the distribution of hydrogenbonded species,22and this model has been successfully applied to a description of the phase behavior of a wide range of polymer mixtures.20 There are a number of points concerning eq 1 that are important. First, apart from the AGH term, this equation is simply the Flory-Rehner result. Differences will arise later when we consider the relationship between & (=l/Q, where Q is the extent of swelling), the chain expansion factor, a , and packing conditions. Second, the inclusion of a logarithmic term, 5 In &, is controversial and is absent from the phantom network treatment. We agree with B r ~ c h a r dthat ~ ~ this term must be included, as in our view it accounts for the entropy of disinterspersion of the cross-link junctions over the volume of the network. Third, this treatment neglects the effect of dangling ends or pendant chains (those chains connected to the network at only one point), and physical entanglements. In the c* model, however, pendant chains will still swell and exclude other chains from their sphere of influence (see below), so that the effect of a limited number of such chains should be small, as the experimental results of Bastide et a1.12 demonstrate. The role of physical entanglements is more complex and determines the extent to which the junction points can unfold or disintersperse. For coal, however, we are presumably dealing with chains that are relatively short and probably below the entanglement limit. Neglecting this factor probably does not involve serious errors, but this comes at the price of the complication of dealing with short chains. The next step in the treatment is to obtain the chemical potential of the solvent in the network by differentiating eq 1and putting the result equal to zero, the chemical potential of the pure solvent in contact with the swollen coal gel:
The remaining tasks are to obtain expressions for the variation of a with &, and the contribution of hydrogen bonding to the chemical potential, A ~ H . Elastic Deformation Term. In models that assume an affine deformation, an expression for ada& is obtained immediately from the condition $c = l/a3. In using the c* model we can obtain a similar relationship from the packing condition, but there are some subtleties concerning the definition of concentration. We have discussed these in detail in our initial treatment,lg but we now need to consider how these arguments apply to a densely cross-linked network such as coal. If the individual chains in a gel act as swollen coils, they effectively exclude neighbors from their volume (apart from the overlap where they are forced into (22) Painter, P. C.; Graf, J.; Coleman, M. M. J. Chem. Phys. 1990, 92,6166. (23) Brochard, F.J.Phys. (Paris) 1981,42,505.
366 Energy & Fuels, Vol. 9, No. 2, 1995
Painter and Shenoy
contact a t their junction points). It would then seem necessary to use two concentration term @c and @g. The first of these reflects the overall or average concentration of coal segments within the spheres occupied by each of the swollen coils and is simply given by
(3) where we have used the end-to-end distance of the chains as a measure of coil diameter and the chains are assume to consist of spherical beads of diameter 1. The second concentration variable q5g reflects the average concentration of segments in the gel as a whole and will depend on how the swollen coils pack (see Figure 1). For example, if the cross-link density or cross-link functionality f is such that the swollen coils are arranged in a manner similar to random close packing of spheres, then we would expect that @g would be less than @p by a factor of about 0.637, the “filling factor” or fraction of space occupied by such a collection of spheres. We considered this point in some detail in our initial development of the modellg and noted that in general we could write
4g = W C
(4)
where P is a geometric or packing factor that should depend on the cross-link functionality f and the number of network “defects”, but should be largely independent of a. However, we found that in applying the model t o osmotic deswelling data reported in the literature P approached a value of 1 for values of swelling, Q, less than about 5. Because the maximum swelling of coal is of the order of 2-3, we will assume in this initial treatment that P = 1. We have obtained an approximate expression for P in terms of the network functionality, f, however, and this can be used in future work, if necessary. The chemical potential can now be obtained using
and substituting in eq 2 to give
Substituting for a
25 RT (7) This is very similar in form t o the Flory result, differing only in the first term, where the assumption of an affine deformation results in a @c1131N term. Nevertheless, a t large degrees of swelling the contribution of the elastic component of the free energy is very different in the two models.
Hydrogen-Bonding Term. The contribution of hydrogen bonding to the chemical potential follows from our previous work,9JoJ9~20 but there are some new subtleties that should be mentioned. We have recently found in our experimental studies of polymer blends that there are correlation effects due to the connectivity of the polymer chains.24 (We do not form as many unlike contacts as we would in a mixture of unconnected segments.) We have formulated a local composition model to describe these correlation^,^^ but this does not lend itself to simple calculations. Fortunately, a semiempirical entropy “correction term” compensates for this effect and has allowed us to successfully calculate the phase behavior of a wide range of polymerlpolymer mixtures.20 We used this term in our previous calculations of the phase behavior of coal solvent mixtures, but recent experimental results have shown this to be incorrect. We have determined that the equilibrium constants describing polymerlsolvent hydrogen-bonding interactions are within error exactly the same as those for equivalent low molecular weight model compounds.24 Without going into details, this means that correlation effects are apparently not as significant in polymer solutions (as opposed to mixtures of two polymers) and we do not have t o use any “correction term”, but simply apply the model in its “uncorrected” and more rigorous form. These points are all discussed in detail in a recent note to which the interested reader is referred.26 However, it should be noted that these recent results are also important from the point of view that equilibrium constants obtained from appropriate model compounds can be applied to polymer solutions and gels, within the error of determining these quantities. We will consider the effect of errors on calculated phase behavior in more detail later in this paper. The equations describing the stoichiometry of hydrogen bonding and their contributions to the free energy have been described in detail in previous work9Jo,20and will not be reproduced here. The elimination of the “entropy correction’’ factor affects only one term in the expression for the free energy and, as mentioned above, this is described and discussed in ref 26. The crucial point to keep in mind is that the number of hydrogen bonds at any given concentration in a solution or gel and their contribution to the free energy can be calculated from equilibrium constants. The equilibrium constants we used to describe self-association, OH-OH and OH-ether hydrogen bonds between coal segments or units, are the same as we employed previously, but we obtained new values for phenolic OH-pyridine hydrogen bonds from recent experimental work conducted in our laboratory.27 The details of this work will be reported in a future paper, but the relevant parameters are listed in Table 1. We use the symbols K2 and KB to describe coal OH-OH hydrogen bonds, KE to describe coal OH-ether interactions, and KAto describe coal OH-solvent interactions. Calculations Contribution of Hydrogen Bonding to the Free Energy. It is useful to first consider the number of coal/ (24) Coleman, M. M.; Xu, Y.; Painter, P. C. Macromolecules 1994, 27, 127.
(25)Veytsman, B.; Painter, P. C. J . Chem. Phys. 1993, 99, 9272. (26) Painter, P. C.; Veytsman, B.; Coleman, M. M. J . Pol. Sei. A,, Pol. Chem. (Rapid Commun.) 1994,32, 1189. (27) Hong, H. Unpublished work.
Energy & Fuels, Vol. 9, No. 2, 1995 367
A New Model for the Swelling of Coal Table 1. Parameters for Illinois No. 6 Coal at 25 “C cmVmol 6, (caYcm3)o5
212 12 8.95 5.6 28.6 5.2 20.9 5.0
Vg,a
KZ h2 (kcavmol) KB h g (kcavmol) KE h~ (kcallmol)
Molar volume of an “average” repeat unit.
Table 2. Solvent Parameters solvent pyridine THF a
Vs (cm3/mol) KA 219 81 74.3 42a
h~ (kcallmol) 8.9 5.8
6s ( c a l / ~ m ~ ) l ’ ~ 10.6 9.9
0
PYRIDINE
A THF
Dimensionless units normalized to a molar volume of VB =
212
0
A 0
PYRIDINE THF “INERTSOLVENT
U.0
0.2
0.4
0.6
0.8
1.0
@c Figure 3. Fraction of coal segments that are hydrogen bonded to solvent plotted as a function of coal concentration (&).
”.”
U.0
0.2
0.4
0.6
0.8
1.0
Figure 2. Fraction of coal segments that are hydrogen bonded to one another (self-associated) plotted as a function of the volume fraction of coal (qQ in the solvent-swollen gel.
coal and codsolvent hydrogen bonds in various systems and their contribution to the free energy of mixing. We will use Illinois No. 6 coal as our example and consider the extent of hydrogen bonding with three different solvents: pyridine, which forms very strong hydrogen bonds with phenolic OH groups; THF, which forms somewhat weaker hydrogen bonds; and a hypothetical non-hydrogen-bonded solvent. The data necessary for these calculations are the concentration of phenolic OH groups present in the coal, the equilibrium constants describing phenolic OWOH (and phenolic OWcoal ether) hydrogen bonds, and finally, equilibrium constants describing coal-solvent interactions. Although there are presumably some coal OHhasic nitrogen interactions within coal, we do not know the concentration of
the latter and will assume that it is small and can be neglected. Accordingly, the first two sets of data are combined to obtain values of the self-association (i.e., coallcoal) parameters, Kz, KB,and KE,which are listed in Table 2. The constants KZ and KB describe the formation of OWOH dimers and “h-mers” ( h > 21, respectively, while KE describes coal OWcoal ether hydrogen bonds. The constant KAdescribes coaYsolvent hydrogen bonds and values of these constants (from model compound studiesZ7)are listed in Table 2, together with the solvent solubility parameters. Figure 2 shows the fraction of coal segments that are hydrogen bonded to one another as a function of coal concentration in various coallsolvent mixtures. For the purposes of these calculations we arbitrarily defined a coal segment as the average unit containing one phenolic OH group. This definition merely serves to define a molar volume that is used to scale the free energy contributions to a common reference volume and is not important (see ref 10 for a more complete discussion of this point). The implicit assumption that is important is that the coal OH groups are randomly distributed throughout the sample. As might be expected, the fraction of coaVcoa1 hydrogen bonds decreases dramatically upon the addition of just small amounts (10-20%) of pyridine, while the fraction of coal segments hydrogen bonded to solvent, shown in Figure 3, increases dramatically. THF does not hydrogen bond as strongly to phenolic OH groups, so that the fraction of coallcoal hydrogen bonds is larger for swelling with equivalent amounts of this solvent. For Illinois No. 6 coal with a swelling ratio Q 2.5 (4c 0.4) only about 3-4% of the coal segments remain
-
-
Painter and Shenoy
368 Energy & Fuels, Vol. 9, No. 2, 1995 1
G O Y
-
9
-1
g
-2
* 0 z w
w w
.3
ez;
k .A
’
-5 0.0
1 0.2
0.4
0.6
0.8
1.0
VOLUME FRACTION COAL
Figure 4. Contribution of hydrogen bonding to the free energy of mixing coal with various solvents.
hydrogen bonded to one another in the presence of pyridine, but about 13% or so of the segments remain hydrogen bonded to one another in the presence of an equivalent volume of THF. The effect of an “inert” (i.e. non hydrogen bonded) solvent can be modeled by simply putting KA = 0 in the calculations and this result is also shown in Figure 2. As might be expected, the number of coaWcoal hydrogen bonds decreases only slowly with increasing solvent concentration and the number of coaIJcoa1 hydrogen bonds only starts to decrease dramatically when the volume fraction of solvent exceeds about 80%. Nonhydrogen-bonding solvents barely swell coals whose OH groups are unreacted, of course, and this brings us to one of the points we wish to make concerning how changes in hydrogen bonding contribute to the free energy of mixing. Figure 4 shows the calculated contribution of hydrogen bonding to the free energy of mixing (the AGH term in eq 1). As might be expected, mixing with pyridine results in a large negative (favorable to mixing) value of AGH, while the free energy change for mixing with THF is somewhat less. Mixing with a non-hydrogenbonding solvent results in an unfavorable (positive) value of AGH, because favorable intramolecular hydrogen bonds are “broken” upon dilution but no new favorable intermolecular interactions are formed. It has been proposed that hydrogen bonds act as crosslinks in coals and that these interactions must be “broken”if coal is to swell. If a bond is to act as a crosslink it will affect the free energy through both the elastic and mixing terms and will also confer certain dynamic properties on the network. It is our view that in the liquid state hydrogen bond “lifetimes” are far too short, (-10-5-10-11 s) to affect dynamic properties (in the glassy state segments are largely frozen in position and it is irrelevant weather or not they are cross-linked). Instead, we believe that Figures 2 and 4 provide the answer to the role of hydrogen bonds in swelling. If there were sufficiently favorable contributions to the free energy from combinational entropy and if “physical” (non hydrogen bonding) interactions were small (i.e., x 01, then coal would mix with non-hydrogen-bonded solvents, even though their segments hydrogen bond to one another to a significant degree, just as phenol mixes with cyclohexane. Because coal is a network, however, its combinatorial entropy of mixing with solvents is
-
much smaller than the mixing of two low molecular weight materials and is insufficient to overcome the unfavorable contribution of the A& term to the overall free energy of mixing. However, when coal OH groups are acetylated or methylated the contribution of the A& term to the free energy of mixing is removed and reacted coals swell significantly in the appropriate non-hydrogenbonding solvent^.^ Coals do not swell in non-hydrogenbonding solvents because the free energy changes are unfavorable, not because the hydrogen bonds are crosslinks. Hydrogen bonds are an important factor in the swelling of coal, of course, but their contribution manifests itself in the mixing terms, not the elastic deformation terms. Other Interactions. The complex heterogeneous nature of coal is such that there must be a variety of interactions that occur between segments of the network. We account for dispersion forces (which includes z-ztype interactions) and weak polar forces in our calculation of solubility parameters, while hydrogen bonds are handled explicitly, as described above. This leaves charge transfer or other strong specific interactions presently unaccounted for. The evidence for the existence of these interactions is not direct, however, and must be inferred from other measurements. Until methods are devised to “count” the number of these interactions our only choice is to neglect them. Calculations of the Degree of Swelling. By equating eq 7 for the chemical potential of the solvent to zero, the degree of swelling (Q = l/c,bc)for a network of known N (number of statistical segments between cross-link points) can be calculated, or alternatively N can be determined from swelling measurements. This presumes, of course, that the parameters describing mixing ( A ~ H and x) and the network functionality f a r e known. The factor [/v in eq 8 is related to f by 2
-=5 V
and in our calculations we will start by making the arbitrary assumption that f = 4. The parameters describing hydrogen bonding have been described above. We usually calculate the value of x from solubility parameters determined from atomic group contributions, which gives a value in the range 11.7-11.9 (caw cm3)lI2for Illinois No. 6 coal. In this study, however, we used the experimental value of 12.0 determined by Green et a1.28 We will consider the effect of errors and variations in the values of these parameters shortly. Figures 5 and 6 show calculated values of the chemical potential for the swelling of Illinois No. 6 coal in pyridine and THF, respectively. There are four curves corresponding to values of N = 1 , 2 , 5 , and 10 and each is plotted as a function of coal volume fraction, &. The zero value at the lefbhand edge of the plots corresponds to the chemical potential of the pure solvent, while the value of @c at which the calculated value of the chemical potential again becomes zero corresponds t o the equilibrium swelling value. It can be seen that for values of N = 5 or 10 we calculate that the degree of swelling would be greater than 10 (i.e., Q =- 10, @c < 0.1). The observed swelling in pyridine (& 0.4) corresponds to
-
(28) Green, T. K.; Chamberlin, J. M.; Lopez-Froedge, L. Prepr. Pap.-Am. Chem. Soc., Diu. Fuel Chem. 1989,34 (31, 759.
A New Model for the Swelling of Coal
Energy & Fuels, Vol. 9, No. 2, 1995 369
0.11
1
0.09
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ci
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0.03
-
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.
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.
‘
*
‘
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0.2
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.
0
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’
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Figure 5. Calculated solvent chemical potential of a pyridineswollen coal network plotted as a function of &.
Figure 7. Calculated solvent chemical potential of a pyridineswollen coal network plotted as a function of values of the cross-link functionality, f.
+c
for various
a calculated value of N = 1. We do calculate the observed trend in that we predict that this coal should swell less in THF (for N = 1,the calculated value of & is about 0.55). In addition, the small degree of swelling of many coals (8 2-3) in “good” solvents would lead one to expect that there are only a small number of statistical units between junction points. But a calculated value of N = 1 makes no sense in the context of the assumptions underlying these calculations. Although the number of “average chemical repeat units” or aromatic clusters in each statistical unit is probably of the order of 5-10, the classical theory of rubber elasticity is based on an entropic deformation of chains of statistical (freely hinged and rotating) segments. We calculate values of a (chain expansion) that are greater than 1 even for N = 1, because in the de Gennes blob model
-
-
1 a =1.36 d/3N1’6
U.0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 6. Calculated solvent chemical potential of a THFswollen coal network plotted as a function of &.
(9)
In other words, the statistical unit needs t o be “stretched” in order to satisfy the equations. One possibility is that we are calculating a value of N = 1 because the equations are sensitive to variations in the parameters. Alternatively, there could be some fundamental flaw in the assumption we have made. We will consider the former possibility first. Effect of Errors in the Parameters. The only parameter used in the calculation of the swelling of Illinois No. 6 coal that is not based on some experimen-
Painter and Shenoy
370 Energy & Fuels, Vol. 9, No. 2, 1995
0.12
0*14
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i
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B
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c
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but with no success. However, this suggested another approach which we belive provides an explanation and could prove useful. It has been recognized €or many years that the relatively small degree of swelling of coal in “good” solvents is by itself strong evidence that the number of statistical units between junction points must be small. However, a statistical unit consists of a sufficient number of “chemical units” that their collective behavior is equivalent to that of a hypothetical freely hinged and (29) Flory,P. J. Statistical Mechanics of Chain Molecules, reprinted ed.; Hanser Publishers: New York, 1988;Chapter 8.
A New Model for the Swelling of Coal
Energy & Fuels, Vol. 9, No. 2, 1995 371
Using the relationship between a and q+ required for the disinterspersion model, the contribution to this energetic deformation term to the solvent chemical potential is given by
-".-
1
0 10
0.30
0.20
0.50
0.40
0.60
@c
Figure 10. Calculated chemical potential for a model that includes energetic as well as entropic contributions to the elastic free energy. rotating segment. The "chemical units" in coal can be thought of as aromatic clusters linked by methylene and ether bridges (see ref 4 and citations therein). These are relatively stiff, so a statistical unit would presumably consist of something of the order of 5-10 such chemical units (we based this rough estimate on the calculations reported in Flory's book, ref 29, for various synthetic polymers). If the number of statistical units between junction points is small, however, then the strong osmotic forces that come into play during the swelling of the chains may be enough to provide an energetic as well as entropic force opposing swelling, through an additional redistribution of bond rotational angles. This is allowed for in the thermodynamics of rubber elasticity, of course, where the force/extension relationship is given as the sum of internal energy and entropic contributions
au
f=---
ai
Tas ai
If we assume that the modulus of the coal chains is simply the sum of entropic and energetic terms (and this may be a big assumption) we can simply add eq 15 to eq 8. We define
B' = BlNkT
and use a value of B = J, taken from the order of magnitude estimate of the modulus for stiff chains mentioned above, to obtain an estimate of B' 250lN per mole of chains. This value of B' is an estimate for very stiff chains, so we performed calculations with values of B1 = 50,100, and 300 for various values of N . The calculated chemical potential obtained with a value of N = 3 is shown in Figure 10. It can be seen that including an energetic term gives a degree of swelling that is now more reasonable, by which we mean the model is now self-consistent in that we calculate the observed degree of swelling with values of N > 1 and values of B' that are of the right order of magnitude. This result suggests that thermoelastic experiments would provide some interesting insight into the mechanism of coal swelling.
-
Conclusions
(11)
and TrelodO reports that for long flexible rubber chains the internal energy contribution is of the order of 20%. The contribution of internal energy to the thermodynamics of coal swelling could well be much larger than this and we believe this is something that needs to be explored in experimental work. At this point we can include a simple modification to the model, however, by adapting an approach described by Jones and Marques?l who considered the bending of a chain treated as a frozen random walk. The bending was assumed to occur through bond rotations and the following expression for the modulus G,was obtained
where B is an elastic force constant and an order of magnitude estimate for G, was determined to be lo6Pa. The elastic free energy is given by (30)Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd ed.; Clarendon Press: Oxford, U.K, 1975;Chapter 13. (31)Jones, J. L.;Marques, C. M. J.Phys. Fr. 1990, 51, 1113.
(15)
We are proposing a model where the swelling of coal is assumed to proceed by a process of chain disinterspersion. The c* theorem of de Gennes is used to obtain a relationship between chain expansion and the degree of swelling, while hydrogen-bonding interactions are described by an association model developed in this laboratory. The results demonstrate that this approach cannot describe coal swelling, in that a fundamental assumption of the model is violated by the results. These results also suggest that the principal difficulty is that the number of statistical segments between cross-link junctions is very small. However, an approach where an energetic as well as an entropic contribution t o the elastic free energy is included does result in a self-consistent model and may prove to be a viable approach in future work.
Acknowledgment. We gratefully acknowledge the support of the office of Basic Energy Sciences, Department of Energy, under Grant No. DE-FGO2-86ER13537. EF940182G
