Statistical Mechanics of Coal Treated as a Densely Cross-Linked

1250

Energy & Fuels 1997, 11, 1250-1256

Statistical Mechanics of Coal Treated as a Densely Cross-Linked Network Boris Veytsman and Paul Painter* Department of Materials Science and Engineering, Penn State University, University Park, Pennsylvania 16801 Received May 1, 1997. Revised Manuscript Received July 22, 1997X

An analysis of the statistical mechanics of a densely cross-linked system is presented. It is argued that for such systems fluctuations of the cross-link points will be severely inhibited and the assumption of affine deformation should be reasonable. It is first demonstrated that the usual result obtained by classical theories can be obtained by an alternative method without assuming Gaussian statistics or the Langevin approximation for the partition function. For values of N, the number of “statistical segments” (freely hinged and rotating) between cross-link points, e2, the Gaussian results cannot apply. For small deformations of networks with N g 3, the Gaussian approximation is shown to be valid. This latter result will be employed in studies of the deformation of swollen coal gels in future work.

Introduction For many years the “standard model” of coal structure has been based on the premise that coal is a covalently cross-linked network with a cross-link density and “sol” fraction that varies systematically with rank (this model excludes anthracites and other high-rank coals that are more graphitic in their structure). Although the nature of cross-linking in coal is still the subject of some controversy, by and large this has been a fruitful approach, allowing the application of theories developed to describe the structure and properties of synthetic polymer networks. Recently, this view has been challenged, and an alternative model that considers coal to be a largely associated structure has been proposed.1-5 Ironically, this historic progression of models is in reverse order to the development of ideas concerning the nature of natural and synthetic polymers, where models based on associated or colloidal structures were initially challenged and finally replaced by the macromolecular hypothesis.6 It is our view that association models for coal structure are equally flawed and based on an incorrect analysis of the evidence. Our arguments are presented elsewhere,7 however, and in this paper our concern is the continued development of the “standard model” approach. Nevertheless, the theory described here and applied to data on the compression of swollen coal particles in a following paper will allow a further test of the validity of the association model. If coal is an aggregate of smaller molecules held together by secondary forces, its response to a load will be largely energetic. Conversely, if coal consists of a cross-linked X Abstract published in Advance ACS Abstracts, October 1, 1997. (1) Nishioka, M.; Gebhard, L. A.; Silbernagel, B. G. Fuel 1991, 70, 341. (2) Nishioka, M. Fuel 1991, 70, 1413. (3) Nishioka, M. Energy Fuels 1991, 5, 487. (4) Nishioka, M. Fuel 1992, 71, 941. (5) Takanohashi, T.; Iino, M.; Nishioka, M. Energy Fuels 1995, 9, 788. (6) Morawetz, H. Polymers: The Origins and Growth of a Science; John Wiley & Sons: New York, 1985. (7) Painter, P. C.; Sobkowiak, M.; Coleman, M. M. Prepr. Pap.sAm. Chem. Soc., Div. Fuel Chem. 1997.

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network, at least part of its response would be entropic. These alternatives can be probed by an examination of the temperature dependence of the compression modulus. Initial treatments of coal as a cross-linked macromolecular network applied the Flory-Rehner theory for the swelling of polymer networks. This theory assumes that the deformation of the elementary chains of the network is in some fashion affine with the macroscopic deformation of the sample. For the swelling of lightly cross-linked synthetic networks it is now clear that this assumption is unrealistic (but it may still be appropriate when applied to small deformations in stress/strain experiments). Neutron scattering results8 have demonstrated that the chains in swollen networks have approximately the same dimensions as equivalent noncross-linked chains in solutions of the same concentration, so that the measured chain expansion is far less than would be predicted on the basis of affine deformation of the network. An alternative approach, based on a topological rearrangement of cross-link junctions and the c* blob models of de Gennes, provides a much better agreement with the experimental observations. In previous work in this laboratory we have modified the classical Flory-Rehner model for swelling by replacing the affine assumption with the packing condition of the c* model in order to obtain a relationship between the degree of swelling and chain extension. This model fits experimental data obtained on lightly cross-linked synthetic networks very well,9,10 but when applied to coal, it results in a major internal inconsistency. Essentially, the model predicts, reasonably enough, that there are very few statistical segments between cross-link junctions, but this naturally violates the central assumption of the theory that the chains obey Gaussian statistics. (8) Bastide, J.; Picot, C.; Candau, S. J. Macromol. Sci., Phys. 1981, B19, 13. (9) Painter, P. C.; Shenoy, S. L. J. Chem. Phys. 1993, 99, 14091418. (10) Painter, P. C.; Shenoy, S. L. Energy Fuels 1995, 9, 364-371.

© 1997 American Chemical Society

Statistical Mechanics of Coal

Although various models have been developed to describe modified Gaussian or non-Gaussian behavior,11,12 these usually rely on the assumption that N, the number segments of the elementary chains of the networks (i.e., the average length of a path between cross-links) is large. Treloar (see ref 13, Chapter 6) obtained an exact expression for the partition function of a freely jointed chain for arbitrary N. However, he also was primarily interested in the loosely cross-linked (large N) limit. Despite its long history which is now more than a century old (see, e.g. ref 14 ), the issue of Gaussian or non-Gaussian behavior of dense networks seems to be not universally agreed upon. Therefore, there is a certain need in the rigorous treatment of dense networks. We think that this treatment should be as free of assumptions as possible and start from the first principles of statistical physics. Here we will present such a treatment of densely connected networks. Strangely enough, this approach resurrects the affine assumption and for small deformations of networks, where the number N of freely hinged segments between cross-links is greater or equal to 3, the simple Gaussian result is obtained. Free Energy of a Strained Network We will start by making the usual assumption that one can define a statistical segment, which contains a sufficient number of “chemical repeat units” or, in coal, aromatic “clusters” linked by methylene and ether bridges, and this segment behaves like a freely joined bond of length b. This assumption is not necessary for the first part of the theoretical development but will be used later. We also assume that the elastic component of the free energy is independent of any liquidlike contributions (responsible for van der Waals interactions, excluded volume effects, etc.). If there are M cross links, each of functionality f, then the number of chains in the network is ν ) fM/2, while the total number of segments is gMN h /2, where N h is the average number of segments between cross-link points. Our goal is to establish an equation of state for such system. A major advantage of Gaussian statistics is that the method used to average the free energy is often unimportant, because most averages, obtained by different ways, coincide. When dealing with non-Gaussian systems we will have to be far more careful. So we will first derive the usual formulas of polymer network theory in a fashion that will be more useful when we consider the densely cross-linked limit. Let ri, i ) 1, 2, ..., ν, be the vector connecting the ends of the ith chain. The set of vectors ri are not independent: if we are going along any cycle of the network, we ultimately return to the starting point. Accordingly, the sum of the corresponding vectors ri taken along any cycle is zero. If ξ is the cycle rank of the network,15 then we have exactly ξ linear conditions imposed on ν vectors ri. (11) Kovac, J. Macromolecules 1978, 11, 362. (12) Barr-Howell, B. D.; Peppas, N. A. Polym. Bull. 1985, 13, 91. (13) Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd ed.; Monographs on the Physics and Chemistry of Materials; Clarendon Press: Oxford, U.K., 1975. (14) Chandrasekhar, S. Stochastic, Statistical, and Hydromagnetic Problems in Physics and Astronomy; Chandrasekhar, S. Selections; University of Chicago Press: Chicago, 1989; Vol. 3.

Energy & Fuels, Vol. 11, No. 6, 1997 1251

Figure 1. A stretched chain.

In the usual approach (see, e.g. ref 16 ) it is customary to introduce a partition function for polymer chain Z(r) that depends on the vector r connecting the ends of the chain (see Figure 1). In our situation the partition function of the network should be a product of ν partition functions Zi(ri), corresponding to each of the ν chains in the networks. However, since we have ξ constraints, we must reject all configurations that correspond to chains that are not connected to one another at cross-link points. The simplest way to do this is to multiply the partition function by the product of δ-functions, each of which corresponds to a linear constraint on the vectors ri. The partition function for the unstrained network is therefore ν

ν

i)1

i)1

∫∏dri∆(r1, ..., rν)∏Zi(ri)

Ξ0 ) V

(1)

where V is the volume of the system, ∆(r1, ..., rν) is the product of ξ δ-functions, corresponding to ξ linear conditions on the vectors ri, and Zi(ri) is the partition function of a free ith chain. The probability that the set of ri vectors has certain values r1, r2, ..., rν is then

P0(r1, ..., rν) )

1

ν

Zi(ri) ∏ i)1

∆(r1, ..., rν)

Ξ0

(2)

Let us now strain the network. Following Flory15 we will do this in two stages: stage (i): an affine deformation of the vectors ri; this stage can be described by the formula

ri f Mri

(3)

where M is a tensor describing the deformation stage (ii): relaxation of the junctions to their equilibrium positions. In stage (i) we imagine moving the positions of the chain “end points” or junction zones and then allowing the chains to “relax” by bond rotations to accommodate this new configuration; the junction points themselves are not allowed to relax. For simplicity, consider a simple chain in this network and the one-dimensional problem of deformation parallel to some axis in a Cartesian coordinate system (Figure 2). Before defor(15) Flory, P. J. Proc. Roy. Soc. London A 1976, 351, 351-380. (16) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, U.K., 1986.

1252 Energy & Fuels, Vol. 11, No. 6, 1997

Veytsman and Painter

Figure 2. A one-dimensional stretching problem.

mation the distance between the ends lies in the interval [r, r + dr] and the partition function of this system is Z(r) dr. For a simple one-dimensional affine stretching of the chain, the new partition function is then simply obtained by multiplying by the constant factor M and can be written as Z(Mr)M dr. The distance between the chain ends now lies in the interval [Mr, Mr + M dr]. In the three-dimensional case the partition function of an individual chain becomes det M Z(Mr) where the multiplication by the factor det M follows in the same way as multiplication by the factor M in the onedimensional case. To obtain the partition function of the whole network we must multiply the contributions from the individual chains and account for the constraints. The partition function of the strained network is therefore

Ξ(r1, ..., rν; M) ) ν

ν+1

(det M)

Zi(Mri) ∏ i)1

∆(Mr1, ..., Mrν)

environment. A priori one can imagine two limiting cases for this process: either a complete relaxation (“phantom networks”) or a complete lack of relaxation (“affine networks”). The constrained junctions model of Flory17-19 treats the intermediate case, but it uses the fact that the convolution of Gaussian distributions (describing the average chain configuration and fluctuations, respectively) is a Gaussian distribution. This allows the calculation of the contribution of relaxed junctions to the free energy in a simple and elegant way,15 but we cannot use this result for non-Gaussian distribution functions. Fortunately, in densely crosslinked networks the fluctuations of the junctions are presumably severely limited, so that the assumption of affine deformations would seem to be reasonable. Moreover, the deformation experiments that will be of immediate interest to us involve relatively small deformations (for elastic networks) in compression, where even loosely cross-linked network exhibit behavior close to the affine limit.13 The affine assumption allows us to neglect stage (ii) of the deformation process. The averaging described in eq 8 is now easily performed, if we recall that the network was formed from the relaxed chains. Therefore, if A(ri) is a function of the vector ri only, then averaging with the probability distribution (2) coincides with averaging with the single-chain probability distribution Zi(ri):

(4) 〈A(ri)〉0 )

Since ∆ is a product of ξ δ-functions, we can write

∆(Mr1, ..., Mrν) ) (det M)-ξ∆(r1, ..., rν)

(5)

Accordingly, the free energy change following stage (i) of the deformation is ν

[(Zi(Mri)) - ln(Zi(ri))] ∑ i)1

G(r1, ..., rν; M) ) -kT

kT(ν - ξ + 1) ln(det M) (6) Equation 6 describes the free energy of a system that was in a given initial state r1, ..., rν. To obtain the average free energy we must average this equation using the probability distribution given in eq 2. Denoting the resulting average 〈...〉0;

〈A(r1, ..., rν)〉0 ) 1 Ξ0

ν

∫

Zi(ri) dri ∏ i)1

∆(r1, ..., rν)A(r1, ..., rν)

(7)

we obtain

G ) 〈G(r1, ..., rν;M)〉0

∫

1 A(ri)Zi(ri) dri Ξi

(9)

where

Ξi )

∫Zi(ri) dri

(10)

For symmetry reasons Zi(ri) will depend only on the length of the vector ri, but not on its direction. Instead of Zi(ri) and Zi(Mri) we can therefore use the expressions Zi(r2) and Zi(|Mri|2).

Limit of Small Deformations For the limiting case of small deformations we can expand the expression in the angular brackets in eq 8 in series in the small parameter M - E, where E is the identity transformation. We will see below that this expansion does not converge for all cases, but for now we will assume that it does. Let Mab be the components of the tensor M, and u be the square of the vector r after the transformation, so

u ) |Mr|2 ) MabMacrbrc, u0 ) |r|2 ) rara (11)

ν

〈ln(Zi(Mri)) - ln(Zi(ri))〉0 ∑ i)1 kT(ν - ξ + 1) ln(det M) (8)

where a summation over repeated subscripts is assumed. Using eq 9 and the fact that Zi depends only on u and u0, we obtain in the lowest order in Mab - δab,

The second stage of the deformation presents some difficulties for this theory. The degree of relaxation of the ξ junctions depends on the details of their local

(17) Flory, P. J. J. Chem. Phys. 1977, 66, 570-5729. (18) Erman, B.; Flory, P. J. J. Chem. Phys. 1978, 68, 5363-55369. (19) Flory, P. J.; Erman, B. Macromolecules 1982, 15, 800-806.

) -kT



where δab is the Kronecker delta:

〈ln(Zi(Mr)) - ln(Zi(r))〉0 ) ∂ ln Zi(u) 1 (u - u0) dr Zi(r) Ξi ∂u

∫

)

1 ∂ Zi(r) 1 r r dr (MabMac - δbc) Ξi 2r ∂r b c

∫

(12)

where we used the fact that ∂/(∂u) ) (1/2r)∂/(∂r). From symmetry considerations it follows that we can substitute rbrc in this equation by r2δbc/3. Then we have

〈ln(Zi(Mr)) - ln(Zi(r))〉0 )

)-

(

)∫ Z (r)r

6π 1 2 M -1 Ξi 3 ab

(

∞∂Zi(r) 3 r 0 ∂r

)∫

2π 1 2 M -1 Ξi 3 ab ∞

0

2

i

1 2 ) - (Mab - 3) 2

dr

dr (13)

From eq 8 we obtain

G ) 1/2kTν(Mab2 - 3) - kT(ν - ξ + 1) ln(det M) (14) The most common deformation experiments involve unidirectional stretching at constant volume and symmetric swelling. For tensile deformation by a relative amount R ) l/l0 the tensor M is

(

R 0 0 -1/2 M) 0 R 0 0 R-1/2 and eq 14 gives

)

(

2

( )

The statistical mechanics of networks consisting of freely jointed polymer chains was discussed in the seminal work of Flory,20 but in the Gaussian limit of large N. In this section we will therefore obtain an equation of state for a single chain that is appropriate for all N. Consider a freely jointed chain with N segments, each of length b. Following Flory20 we introduce a partition function Zr for a polymer chain with a given vector r connecting its ends, and a partition function Zf describing a chain stretched by a given force f (see Figure 1). Let us introduce thermodynamic potentials

Gr ) -kT ln Zr, Gf ) -kT ln Zf

(20)

The partition function Zf is rather well-known:20

)

Zf )

(16)

If we assume that the chains also deform a relative amount R in symmetric swelling,

R 0 0 M) 0 R 0 0 R

Arbitrary Deformations of Freely Jointed Polymer Chains

(15)

2 G ) /2kTν a + - 3 R 1

this derivation is that we have shown that neither an assumption of Gaussian statistics nor the Langevin function approximation20 is necessary in order to obtain this result. The final expression for the free energy does not depend on the details of chain flexibility or the topology of the network (as long as it is dense enough for the deformation to be affine). The only prerequisite for the validity of eqs 16 and 18 is the convergence of the series expansion in eq 8. Unfortunately this assumption is not always satisfied. To show this we will calculate the right-hand side of eq 8 exactly for the freely jointed model. To do this we will explore the statistics of small polymer chains.

(17)

and therefore

G ) 3/2kTν(R2 - 1) - 3kT(ν - ξ + 1) ln R (18) If R is close to 1 (the case of small swelling), the first term in this equation can be written as 3

/2kTν(R2 - 1) ) 3/2kTν[(1 + R - 1)2 - 1] ≈ 3(R - 1) (19)

At this point we would like to note that there exists another hypothesis of swelling of dense networks. According to this hypothesis,8 the swelling of the sample is largely due to cross-link “disinterspersion”, and in certain circumstances (i.e., low degrees of swelling) the chains may not change their conformational distribution, so that R ) 1. We will discuss this in more detail below, but the crucial point we wish to make is that eqs 16 and 18 are the usual results obtained in theories of affine polymer network deformation. What is new in

fb sinh (4πkT fb kT)

N

(21)

and there exists an approximate method20 to obtain Zr from Zf (using the inverse Langevin function). Unfortunately, this method works only for large N J 10. Since we are mostly interested in small N, this method is not appropriate in our case. We start by letting Γ be some set of angles between segments that fully characterizes the state of the chain, and rˆ be the corresponding vector between the first and last links. Then by definition

Zr )

∫ exp(-U(Γ)/kT)δ(rˆ - r) dΓ

(22)

where U(Γ) is the energy of the chain in the state Γ (for the freely jointed model U ≡ 0, but we will preserve the factor exp(-U(Γ)/kT) for generality), and δ is the Dirac δ-function. Using the well-known integral representation21

δ(rˆ - r) )

du ∫ exp(iu(rˆ - r))(2π) 3

(23)

(20) Flory, P. J.; Hoeve, C. A. J.; Ciferri, A. J. Polym. Sci. 1959, 34, 337-347. (21) Barton, G. Elements of Green Functions and Propagation: Potentials, Diffusion and Waves; Clarendon Press: Oxford, U.K., 1989.



we obtain

Zr )

du ∫ exp(-iur) [∫ exp(iurˆ - U(Γ)/kT) dΓ](2π) 3 (24)

The integral in the square brackets in this equation is nothing else than Zf, with the properly defined f, and therefore we have

Zr )

du ∫ exp(-iur)Zf(ikTu)(2π) 3

(25)

It is interesting to follow how eq 25 gives Gaussian statistics if N f ∞. In this limit, eq 21 gives N

2 2

Figure 3. Partition function for freely jointed chains, N ) 2: (1) exact formula, (2) Gaussian approximation.

3. N ) 3. For this case

{ {

N

Zf(ikTu) ≈ (4π) (1 - u b /6) ≈ N

reb 1/(8πb3), Zr ) (3b - r)/(16πb3r), b < r e 3b 0, r > 3b

2 2

(4π) exp(-Nu b /6) (26) Substituting into eq 25 and integrating, we obtain

(34)


3kTr2 +C Gr ) 2Nb2

(27)

where C does not depend on r. This is the free energy of a Gaussian coil. Let us return to the general case of N not necessarily large. If θ is the angle between the vectors u and r, then ur ) ur cos θ. Since Zf does not depend on the direction of the vector f, we can integrate eq 25 over the angular coordinates and obtain

Zr )

∫0∞Zf(ikTu) sin(ur)u du

1 2π2r

(28)

For a freely jointed network, eq 21 gives

Zf(ikTu) )

4π sinh(iub)) (iub

N

(

)

sin(ub) ub

(29)

N

(30)

This gives

Zr )

∫0∞(

)

sin(ub) N sin(ur)u du ub

1 2π2r

(31)

A result obtained by Flory22 by a different methodology. Let us now discuss several special cases. 1. N ) 1. For this case

1 δ(r - b) Zr ) 2πbr

(32)

This equation reflects the fact that a single segment in a freely jointed chain cannot be stretched. 2. N ) 2. For this case

{

1/(8πb2r), r e 2b Zr ) 0, r > 2b

(35)


{

Zr ) (5b2 - r2)/(64πb5), (2r3 - 15br2 + 30b2r - 5b3)/(192πb5r), (5b - r)3/(384πb5r), 0,

r e 2b 2b < r e 3b 3b < r e 5b r > 5b (36)


or, dropping the factor 4π that does not depend on u

Zf(ikTu) )

(8b - 3r)/(64πb4), r e 2b Zr ) (4b - r)/(64πb4r), 2b < r e 4b 0, r > 4b

{

Zr ) (5r3 - 24br2 + 96b3)/(1536πb4), (-5r4 + 72br3 - 360b2r2 + 672b3r 240b4)/(3072πb6r), 4 6 (6b - r) /(3072πb r), 0,

r e 2b 2b < r e 4b 4b < r e 6b r > 6b (37)

Other cases can be easily treated in the same way, but the resulting expressions become progressively longer. In Figures 3-7 we plot the corresponding functions Zr and compare with the corresponding Gaussian approximation (see eq 27)

Zr )

(

) (

27 8πNb2

3/2

exp -

3r2 2Nb2

)

(38)

It can be seen that as N increases, the exact solution quickly approaches the Gaussian one. Stretching of “Regular” Networks

(33)

(22) Flory, P. J. Statistical Mechanics of Chain Molecules; John Wiley & Sons: New York, 1969.

We will call networks that have exactly the same number of segments between each pair of connected junctions regular networks. Of course, a regular network is an idealization: real chemical networks, par-



Figure 8. A couple of regular networks with n ) 1.

Figure 9. An almost extended chain with two segments.




We will start from the case N ) 1. Since the partition function (32) includes δ-functions, eq 8 gives an infinite free energy change upon stretching. To see why this is so it is enough to consider the regular networks shown in Figure 8. The first network (a “honeycomb”) cannot be stretched at all. The second allows only a skew deformation. While expression 14 gives some value even for N ) 1, it is obvious that this answer is wrong. This is because the expansion of eq 8 in series is illegal. In other words, expression 14 implies that the actual deformation of the network is small compared to the maximal deformation that the network can withstand without breaking. Since at N ) 1 the network cannot be stretched at all, the maximal deformation is zero, and no deformation is small enough. Let us now consider the case where N ) 2. There is still a problem with eq 8 because it requires integrating ln Zr, and Zr can be zero (see eq 33). The reason for this difficulty is evident from Figure 9. There is a divergence as the chain approaches the fully extended state (r f 2b). This divergence is an artifact of using the freely jointed model in conjunction with the affine deformation hypothesis. The simplest cure is to change the affine deformation hypothesis to accommodate extended chains. Accordingly, we will assume that the partition function of the stretched chains is equal to Zr(2b) if |Mr| > 2b. For simplicity we will discuss only the case of swelling with the tensor M given by eq 17. Substituting r′i instead of Mr in eq 8, we obtain

G ) kTν

R2 - 1 - 3kT(ν - ξ) ln R R2

(39)

For R close to 1 the first term in this formula gives 2kTν(R - 1). This is 1.5 times lower than the expression 3kTν(R - 1), predicted by eq 18! This discrepancy can be explained by a close examination of expression 33. There is a discontinuity at r ) 2b. This discontinuity makes the integration by parts in eq 14 illegal. Let us now discuss the case N ) 3. Once again we have a problem with almost fully extended chains. We correct it by adjusting the affine hypothesis in the same way as above. Then the integration of eq 8 gives Figure 7. Partition function for freely jointed chains, N ) 6: (1) exact formula, (2) Gaussian approximation.

G ) kTνF - 3kT(ν - ξ) ln R

(40)

where F is a rather long expression: ticularly heterogeneous materials like coal, have statistical distributions of chain lengths between connected junctions. Nevertheless, it is a useful approximation and it allows us to demonstrate some interesting peculiarities of the freely jointed model.

(

)

17 21 27 9 27 5 + + × F ) ln(2R) 6 8R 4R2 8R3 8R2 4R3 3(R - 1) 7 9 + ln(3 - R) - ln(3(R - 1)) (41) ln 3-R 24 8



As R f 1 we have

F ∼ 3(R - 1)

(42)

which coincides with the Gaussian prediction for small deformations (19). Since at N g 3 the function Zr(r) is continuous, we recover the Gaussian prediction at N g 3. This resultsthe failure of the Gaussian prediction for N ) 2 and its success for N g 3scan be easily understood just from a glance on Figures 3-7. Indeed, for N g 3 Gaussian curves seem to be a good approximation of the exact result, while for N ) 2 the discrepancy is glaring. Stretching of Irregular Networks We now turn our attention to irregular networks. If the average number of chain segments between the

junctions is long enough (say, 10 or more), we can assume that chains with one or two segments are rare and use formula 14. This means that the classic theory developed for Gaussian chains is also valid for nonGaussian ones. For small chains the deviations from eq 14 have different signs for N ) 1 and N ) 2, so that these effects will compensate each other. For N g 3 we recover the Gaussian prediction at small enough deformations. We will use this latter result to interpret the data on small deformations of swollen coal gels. Acknowledgment. The authors gratefully acknowledge the support of the Office of Chemical Sciences, U.S. Department of Energy, under grant No. DE-F602-86ER1353. EF970066O

Statistical Mechanics of Coal Treated as a Densely Cross-Linked

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