Energy & Fuels 1993, 7, 455-462
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Physical Structural Characterization of Bituminous Coals: Stress-Strain Analysis in the Pyridine-Dilated State George D. Cody,'v+ Alan Davis, and Patrick G. Hatcher Department of Geosciences, The Pennsylvania State University, University Park, Pennsylvania 16802 Received November 23,1992. Revised Manuscript Received March 11, 1993
Measurement of the visocelastic properties of bituminous coals in the pyridine-dilated state reveals that the characteristic behavior of the coals is similar to that of rubbery materials. This substantiates similar observations and conclusions made by Brenner. Quantification of the viscoelastic parameters indicate that the modulus of coal and its solidlike behavior results from either a cross-linked network or extensive entanglement coupling of noncovalently bound macromolecular strands. Viscous compliance observed after long experimental times is consistent with the physical structure of coal being an entangled network; however, a cross-linked network capable of steady-state bond rupture and re-formation would also explain the presence of viscous compliance a t long times. In the case of an entangled physical structure the cross-link density in the coals studied would be very low and could conceivably even be zero. The number-averaged molecular weights between entanglement junctions or cross-links are calculated using the statistical theory of rubber elasticity and considering two extreme cases of viscoelastic behavior (one case including osmotic compressibility and the other assumes zero compressibility). The values span the range of previously reported values obtained through solvent-swelling studies assuming cross-linked networks.
Introduction Although it is generally accepted that coals of intermediate rank are macromolecular solids, details of their network architecture are still lacking. Measurement of coal's viscoelastic properties as a means of elucidating its macromolecular structure has been investigated by several researchers. Larsen and Kovacl were the first to suggest that coal's elastic modulus could be related to fundamental polymeric characteristics, such as the number-averaged molecular weight between cross-links. In their survey of past work, however, they concluded that results obtained through stress-strain measurements and solvent swelling were not consistent, although they clearly should be. They noted that the values for the number-averaged molecular weight between cross-links calculated from the magnitude of the shear modulus were often much lower than those derived from solvent-swelling measurements. It was soon recognized that one of the inherent difficulties with the stress-strain approach is that coal exists in a glassy state at room temperaure.2 The mechanical behavior of coal, undiluted with solvent, has been measured by several researchema For the most part, such measurements detect only short-range, elastic, deformational mechanisms, consistent with coal's glassy characteristics. Although these measurements reveal aspects of the local t Currently at Chemistry Division, Argonne National Laboratory, Argonne, IL 60439. (1) Larsen, J. W.; Kovac, J. in Organic Chemistry of Coal; Larsen, J. W., Ed.;Am. Chem. Soc. Symp. Ser. 71; American Chemical Society: Washington DC, 1978. (2) Brenner,D.Proe.Int. Conf.Coal Sci.,1981 (Gluckauf,Essen) 1981, 163. (3) Bangham, D. H.; Maggs, F. A. P. in Proceedings of the Conference on the Ultrafine Structure of Coals and Cokes; British Coal Utilization and Research Association; Lewis, H. K., and Co, Ltd.: London, 1944. (4) Morgans, W. T. A.; Terry, N. B. Fuel 1958,37, 201. (5) van Krevelan, D. W.; Cherminn, H. A. G.; Schuyer, J. Fuel 1959, 38,438. (6) Weller, M.; Wert, C. J. Appl. Phys. 1982,53,6505.
cooperative rotational characteristics involving only several bonds, they fail to yield information on the larger scale tertiary structure of the coal macromolecule. Detection of long-range and lower frequency viscoelastic modes requires that coal exists in a rubbery state. A glass to rubber transition in coal has been postulated to exist at elevated tempe~atures.~-l*Investigations of coal's mechanical behavior a t temperatures above the range of the proposed glass transition should measure viscoelastic deformation mechanisms in the rubbery state. Such investigations have been carried out.11-12 At temperatures greater than 300 O C , coal exhibits significant irreversible compliance under a constant stress which presumably results from thermolytic degradation occurring even a t these temperatures.lz The existence of parallel, timedependent processes such as pyrolysis greatly interferes with the quantification of such viscoelastic phenomena. Under conditions of thermal degradation, viscoelastic parameters such as elastic moduli and coefficients of viscosity are no longer constants; rather they become a time- and temperature-dependent function of the state of the macromolecular network. The optimum experiment would be to characterize coal in a rubbery state a t temperatures below the thermal degradation threshold. Brenner13demonstrated that coals immersed in appropriate solvents, such as pyridine, exhibit rubberlike characteristics. He subsequently quantified this observation by estimating coal's dynamic Young's (7) Sanada, Y.; Honda, H. Fuel 1963,42, 479. (8)Sanada, Y.; Honda, H. Fuel 1966,45, 295. (9) Lucht, L. M.; Larsen, J. M.; Peppas, N. A. Energy Fuels 1987,1, 56. (10) Yun, Y.; Suuberg, E. Prepr. Pap.-Am. Chem. SOC.,Diu. Fuel Chem. 1992, 37, 856. (11) Sanada, Y.; Honda, H. Fuel 1962,41,437. (12) Howell, J. M.; Peppas, N. A. Fuel 1987,66,810. (13) Brenner, D. Fuel 1985, 64, 167.
0887-0624/93/2507-0455$04.00/00 1993 American Chemical Society
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456 Energy & Fuels, Vol. 7, No. 4, 1993
sample 1 2 3
4 5 6 7 8 9 10
PSOC no. 1316 1318 772 1270 1539 1274 746 223 1171 1336
Table I. Compositional and Bulk Characteristics of Samples seam name State %ca %H %O No. 6 IL 78.9 5.5 11.1 No. 6 IL 79.1 5.4 12.2 L. Kittanning OH 79.8 6.0 11.6 OH 81.4 L. Kittanning 5.6 9.4 No. 6 5.5 IL 81.0 9.4 L. Kittanning OH 82.1 8.0 5.9 No. 6 No. 6 L. Kittanning
KY KY PA
81.9 81.9 85.4 87.3
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Q 0.37 0.47 0.53 0.57 0.58 0.63 0.65 0.66 0.81 0.94
2.0 1.8 2.0 2.1 2.3 1.9 2.1 1.9 2.2 2.1
L. Kittanning PA Mean maximum reflectance of vitrinite in oil. e Swollen volume to dry volume ratio in pyridine. 4 Elemental composition on a daf basis. modulus while the coal was in the swollen state;I4 the magnitude of this modulus was shown to lie in the range of stiff rubber. Thus, in the solvent-swollen state the potential exists for coal‘s low-frequency viscoelastic behavior to be measured. In the present paper, the approach of Brenner is extended to a larger set of samples and experiments. The samples were selected to provide a window into the evolutionary processeswhich modify coal’s macromolecule structure as it proceeds through the early stages of diagenesis. The experiments are designed to elucidate specific aspects of each coal’s viscoelastic behavior; with the ultimate goal of deriving from this overall behavior an increased understanding of the architecture of coal’s molecular structure.
Experimental Section Samples. The coal samples were selected from single blocks of coal obtained from the Penn State Coal Sample Bank. Reflectance, ultimate analysis, and other pertinent data on the whole coals are presented in Table I. In addition to the vitrain samples, parallel experiments were run on a piece of cross-linked rubber obtained from a standard laboratory test-tube stopper. Sample Preparation Protocol. For the purposes of this study a given sample had to meet two conditions in order to be considered acceptable for analysis. First, the sample had to be composed exclusivelyof vitrinite. To achievethis, homogeneous vitrain bands were selected from within each coal block. These bands were sectioned off the whole block using a diamond wafer saw. The faces of the resulting slabs were ground parallel using n Sic rotating disk. Smallsamples,typically 1mms, were obtained from the slab using a razor knife. The second essential requirement for the samples was that they be exhaustively extracted in pyridine. This was to ensure that over the time of the experiment the solvent’schemical potential remained constant; i.e., dilution of the solvent with extract does not occur. It is well-known, however, that extraction, hence swelling, with pyridine can lead to the formation of cracks. To avoid this, each sample was first equilibrated with pyridine vapor for a period of 2-3 days. Each sample was then extracted in an excess of pyridine liquid for period of over 1week. Deswelling was performed by sequential dilution of the liquid pyridine by the addition of toluene; each dilution was followed by an reequilibration time delay of several hours. The samples were removed from the mostly toluene solution and equilibratedwith toluene vapor. Finally,the samples were dried to constant weight in a vacuum oven and stored under nitrogen. This protocol generallyresulted in samples free of microcracks. Prior to reswelling, each sample was inspected with a lowmagnification microscope for the presence of microcracks. If microcracks were observed, it was usually possible to carve out crack-freeregions from the originalsamples using the microscope, microtweezers, and a razor knife. The samples were stored in sealed jars under argon until use. (14) Brenner, D. Prepr. Pap.-Am. Chem. SOC.Diu.Fuel Chem. 1986, 31, 17.
Linear Displacement Measurement. The microditometer employedin this study was originallyused to study the expansion of particulate coals under gasification conditions.16 It has since been modified to measure linear expansion of single particles of coal during solvent swelling.I6 Changes in sample dimension are measured by monitoringthe verticalpositionof a soft iron cylinder mounted on a glass rod which sits on the sample surface. The iron cylinder is the core of a linearly variable differential transformer (LVDT). When the rod and iron piston assembly are counterbalanced, the LVDT apparatus is highly sensitive to vibrationswhichtypicallyoccur in a laboratorysetting. Evidently the viscosity of the pyridine solvent is not sufficient to dampen these vibrations. Consequently, there is a practical lower limit of resolution of about 0.5 pm displacement under the hydrostatic conditions of solvent swelling. Under applied uniaxial stresses the piston is sufficiently coupled to the swollen coal sample that these vibrations are totally dampened. In the experiments described, the measured displacements are on the order of 10s to 100s of micrometers; therefore, all the compliance measurementa reported in this study are far above the limit of detection. The compliance of the instrument, determined from displacement at maximum load (approximately 6 kg/cmZ), is below the limit of detection. Mechanical Measurements of Solvent-Swollen Coals. The experimental protocol employed in this study is a straightforward creep complianceexperiment which involved monitoring the time-dependent strain while the coal was subjected to a constant uniaxial stress. By monitoring the time dependence of the strain, it is possible to elucidate several different strain mechanisms, as described below. After the strain rate reached a constant value, the stress was incrementally increased to a new constant valuewhile continuouslymonitoringthe resultant strain. This protocol was repeated four times; the results alloy for separation of viscous from viscoelastic deformation and yields, ultimately, the equilibrium or steady-state elastic compliance.
Results and Discussion Creep Compliance following Uniaxial Stress. Following an incrementally applied compressive stress,three strain elements were observed in all the coal samples used in this study. First there was an instantaneous,essentially elastic,strain. This was followed by time-dependent strain (viscoelastic strain) which decayed exponentially to a constant strain rate. The final compliance was linear with time and irreversible; hence i t is a purely viscous strain element. The overall behavior is generalized in Figure 1. Generally, the creep compliance runs were recorded only for 2-3 h, because the viscoelastic mode typically reached a steady-state value within 15-60min. However, the strain has been monitored for over 72 h in order to detect any plateau in the total compliance, Le., a region where the strain rate becomes zero; no plateau was observed. This is not the case with cross-linked polymers. The creep (15)Jenkins, R.G., personal communication. (16)Cody,G. D.; Davis, A.; Eeer, S.; Hatcher, P. G. Prepr. Pap.-Am. Chem. SOC.,Diu.Fuel Chem. 1991,36, 1307.
Energy & Fuels, Vol. 7,No. 4, 1993 467
Characterization of Bituminous Coals
d-l
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Figure 1. Generalized response of pyridine-dilated coal to an uniaxial compressive stress. An initial, instantaneous elastic strain, CE,is followed by a viscoelastic strain, CEV. The strain rate decays until it reaches a constant rate which results from viscous, irreversible compliance, ev.
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lime (mln) Figure 2. The creep compliance behavior of cross-linkedrubber; dry (H) and swollen in toluene (+), compressed under uniaxial stress. compliance response of cross-linked rubber, both dry and swollen in toluene, reaches a plateau compliance after a short period of time (Figure 2). Cross-linked polymer networks yield a plateau compliance which is the elastic limit of strain associated with a given stress. Entangled polymer solutions or melts, however, yield a steady-state compliance following elastic strain. This steady-state strain results from the fully developed flow of a viscous fluid. Solvent-swollen coals, therefore, behave like entangled systems and not like a cross-linked macromolecular network. Compliance a t long times does not necessarily have to involve the intermolecular motion of macromolecular strands. Other materials such as metals and crystalline solids also exhibit regions of steady-state compliance which result from diffusion and migration of atomic-scale defects as well as other atomic-scale phenomena such as slip and twinning." These mechanisms are high-energy phenomena, however, and given the relatively low temperatures and low magnitudes of the applied stresses in the present experiments it is not likely that such mechanisms are involved in the constant strain rate behavior associated with pyridine-swollen coals. Another potential mechanism which yields results in compliance a t long times is strain associated with crack propagation in a microfractured material. The physical behavior indicative of this type of (17) Hertzberg, R. W. Deformation and Fracture of Engineering Materials; John Wiley and Sons: New York, 1988.
Figure 3. Schematic of a degenerate,four-element Kelvin-Voigt model of linear viscoelastic strain. The elements ev, em, and correspond to viscous, viscoelastic, and elastic strain; depicted by a Newtonian dashpot, a Kelvin-Voigt element, and a Hookean spring, respectively. compliance mechanism, however, is an accelerating strain rate, rather than a constant strain rate. In the samples studied there was no evidence of cracks visible under the optical microscope, nor was there any indication of strain rate acceleration. The viscous deformation observed in the present study, therefore, may result from intermolecular mobility in a physically entangled macromolecular system such as described in the theories of Rouse,18DeGennes,lg Klein,20 and Doi and Edwards.21 Another theory exists, however, which considers mechanism of steady-state compliance in cross-linked systems. In the transient network theory proposed by Tobolsky and steady-state compliance results from a structure capable of steady-state bond rupture and reformation under a constant stress. In the present chapter the relative merits of either of the above theories to explain viscous flow in coals will not be considered; rather, the focus of the present chapter is in elucidating the structural basis for the relatively large amount of elastic energy stored prior to steady state strain. The structural consequences of each molecular theory of viscous flow, however, will be considered in a later paper. The ensemble of independent strain mechanisms deduced from the behavior generalized in Figure l can be modeled using a degenerate four-element Kelvin-Voigt modelz8which describes linear viscoelastic strain (Figure 3); this model is typically referred to as a Burger's fluid. Although this model oversimplifies the viscoelastic properties of polymer fluids it serves to characterize, phenomenologically,the time-dependent compliance through a linear combination of the constitutive equations for elastic solids and Newtonian liquids. It is a simple task to deconvolute the stressatrain data and obtain the viscous and elastic parameters of each strain element. The time-dependent strain c as a function of a constant axial stress uo is given by the expression, where J ( t ) = t / q + J~ + ~ ~ - e-''') ( 1 (2) In eq 2 the first term on the right describes the viscous strain, where 9 is the coefficient of viscosity (this term is zero for cross-linked macromolecular networks) and t is time. (18) Rouse, P. E.J. Chem. Phys. 1963,21, 1272. (19) DeGennes, P.G. J. Chem. Phys. 1970,55, 572. (20) Klein, J. Macromolecules 1978, 11, 852. Edwards, S. F. J . Chem. SOC.,Faraday Trans. 1978, 74, (21) Doi, M.; 1789. (22) Tobolsky, A. V.;Andrews, R. D. J. Chem. Phys. 1945,13, 3. (23) Green, M.S.;Tobolsky, A. V. J. Chem. Phys. 1946,14,80. (24) Stern, M.D.;Tobolsky, A. V. J . Chem. Phys. 1946, 14, 93. (25) Scott, K. W.;Stein, R. S. J. Chem. Phys. 1963,21, 1281. (26) Lodge, A. S. Trans. Faraday SOC.1956,6,120. (27) Flory, P.J. Trans. Faraday SOC.1960, 56, 722. (28) Calcote, L. R. Introduction to Continuum Mechanics; D. Van Nostrand Co., Inc.: Princeton, NJ, 1968.
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458 Energy & Fuels, Vol. 7,No. 4,1993
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Figure4. Typical creep compliance behavior of pyridine-swollen coal compressed under a constant uniaxial stress.
J1 is the instantaneous elastic compliance, JZ is the maximum value of the viscoelastic compliance, and T is the retardation time constant for the viscoelastic strain. The time-dependent compliance may, therefore, be treated as the sum of contributions to the compliance from each strain element. In coal and polymers, the elastic compliance (JE)has important structural significance with regard to the architecture of the macromolecular network. In cross-linked polymer networks, the elastic compliance may be related to the number-averaged molecular weight between cross-linksthrough the statistical theory of rubber elasticity.8 In noncross-linked polymer liquids, JE is related to the molecular weight of the polymer chains and indicates the magnitude of energy stored during steadystate flow a t a given velocity.30 To obtain a value for JE from creep compliance runs it is necessary to remove the contribution of viscous strain from the total strain. At long times,
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Therefore, the data can be corrected for the viscous strain to obtain a value of JE;obviously no such corrections are necessary for cross-linked macromolecular networks. Figure 4 presents an example of creep compliance data; clearly evident a t long times ( t > 40 min) is the deformation a t constant strain rate, Le., the viscous deformation. Each coal in this series exhibited similar viscous strain. Correcting the total strain for this viscous deformation results in data such as those presented in Figure 5, where the equilibrium compliance is seen to be nearly one-half the total compliance observed a t long times in Figure 4. All of the coals in this study were subjected to this correction protocol for the purpose of obtaining the relationship between u and JE. Elastic Shear Modulus Derived from Stress-Strain Behavior. The corrected strain behavior after each sequential loading of an axial stress reveals details of the (29) Treloar, L. R. G. The Physics of Rubber Elasticity; Clarendon Press, Oxford , U.K., 1975. (30) Ferry, J.D.The Viscoelastic Properties of Polymers; John Wiley & Sone: New York, 1980.
corrected for viscous strain.
fundamental nature of coal's elastic free energy, Fel. In the case of typical elastic solids, the change in free energy with strain corresponds to an increase in internal energy;31 the relationship between stress and free energy for elastic isotropic solids is given by
where uijis the stress tensor, qj is the strain gradient tensor, G is the shear modulus, K is the compressibility modulus, and bij is the Kronecker delta.31 Stress-strain relationships in noncrystalline polymeric materials are nonlinear. Given small strains, the deformational behavior of rubbery materials can often be described through a simple Gaussian elastic model which relates the strain to the displacement of the end-to-end distance of molecular chain lengths from their most probably configuration in response to an applied stress. The change in free energy due to strain then results from a reduction in e n t r ~ p y For . ~ ~cases ~ ~ of ~ uniaxial strain, assuming incompressibility (K = m), the stress-strain relationship is given by u = dFEJaX = G(X -
(5)
where X is the ratio of the strained to unstrained length, measured parallel to the applied stress. Equation 5 is derived from the statistical theory of rubber elasticity, which hereafter will be referred to as the Gaussian model. In addition to the Gaussian model there exists a variety of theories of rubber elasticity which incorporate more realistic assumptions about molecular scale behavior. All of the "modified" elasticity theories were derived to explain elastic behavior a t large strains. Given small strains, all of the theories exhibit the same stress-strain behavior as described by eq 5.29+34 In the case of solvent-swollen coal or polymers, it cannot be assumed that the strain is incompressible, K = 03. For (31) Landau, L. D.;Lifshitz, E. M. Theory of Elasticity; Pergamon Press: New York, 1959. (32) Flory, P. J. The Principles of Polymer Chemistry; Cornell University Prees: Itheca, NY, 1953. (33) Wall, F. T. J. Chem. Phys. 1943,11, 527. (34) Kovac, J. Macromolecules 1978, 1 1 , 362.
Characterization of Bituminous Coals
Energy & Fuels, Vol. 7, No. 4,1993 469
solvent-swollen systems, K is the osmotic bulk compressibility modulus36 and is defined by
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uz(aT/au2) (6) where u2 is the volume fraction of the polymer in the gel and T is the osmotic pressure. The compressibility results from stress-induced changes in the solvent concentration in the swollen gel. The stress-strain relationship in this case has an osmotic character and is given by where Vm is the molar volume of the solvent, nl is the number of solvent molecules, and F d is the free energy of dilution associated with swelling. If K , >> G, then the stress-strain relationship can be described using eq 5. This is the case, for example, of cross-linked poly(acry1amide) swollen in waterqSeOn the other hand, for some polymersolvent systems, K is only slightly larger than G.36~~’ The magnitude of K relative to G for solvent swollen coals has not been determined; however, the theoretical range in K is from K = 2/3G to K = 0 0 . ~ Using ~ this fact and careful analysis of the creep compliance data for the coals and the cross-linked rubbers, the shear modulus may still be constrained to a sufficiently narrow range for the purpose of ultimately characterizing coal‘s macromolecular structure. Consider the instantaneous elastic compliance evident in Figures 1, 2, 4, and 5. Any osmotic deswelling which results in compressibility necessarily requires the solvent to diffuse out of the network. It is physically impossible, therefore, for the instantaneous elastic strain to be the result of osmotic compressibility. To confirm this, the instantaneous lateral strain accompanying the instantaneous longitudinal strain was measured using a microscope with a calibrated reticle (at 50X magnification) for a pyridine-swollen coal, a toluene-swollen cross-linked rubber, and a dry cross-linked rubber. In the former case, an instantaneous longitudinal strain of -0.05 was accompanied by an instantaneous lateral strain of +0.03 f 0.01. Both rubbers exhibited an instantaneous longitudinal strain of -0.04 and yielded associated lateral strains of +0.02 f 0.01. The instantaneous elastic strain is, essentially, incompressible. The time-dependent elastic strain, CEV,however, may or may not be incompressible, depending upon the magnitude of K , relative to G. In the case of both the swollen rubber in toluene- and the pyridine-swollen coals, the lateral strain continued to increase with time. In the former case, this clearly demonstrates a low degree of compressibility because there was no viscous deformation. In the case of coal, however, the presence of a viscous strain element complicates matters because the viscous deformation is presumably incompressible and is superposed on the viscoelastic deformation. Therefore, it is not possible to assert unequivocally that solvent-swollen coals have a low degree of compressibility. For the purpose of constraining the shear moduli to a range, two extreme cases are considered. A t the one extreme, case 1, all of the elastic strain following the instantaneous elastic strain is treated as being purely (35) Candau, 5.;Bastide, J.; Delsanti, M. Adu. Polym. Sci. 1982, 44, 30. (36) Tanaka, T.;Hocker, L. 0.;Benedek, G. B. J. Chem. Phys. 1973, 69,5151. (37) Bastide, J.;Dupleseix, R.; Picot, C.; Candau, S. Macromolecules 1984,17,83.
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Figure 6. Schematic cross section through a sample during the two stages of case 1type elastic deformation under a uniaxial compressive stress a,. The first, instantaneous elastic stage is incompressible and depicted with the solid lined rectangle. The second, viscoelastic stage is purely compressible, designated by the dashed line.
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460 Energy & Fuels, Vol. 7, No. 4, 1993 301
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Figure 10. Shear moduli (corrected for solvent), G, vs mean maximum reflectance in oil, R(M), % ; calculated assuming case 2 behavior. 1and 2. Beyond this range, however, it is not possible to determine the precise behavior of solvent-swollen coals without an independent means to determine the magnitude of the osmotic bulk modulus. Estimate of Number Averaged Molecular Weight between Cross-Links (Entanglements). Embedded within all of the statistical theories of rubber elasticity are molecular-scale parameters such as the number averaged molecular weights of polymer chains and/or the number of chemical repeat units. Therefore, it is reasonable to use the creep compliance data reported above to derive an estimate for such molecular-scale parameters for solventswollen coals. It is important to recall that the equilibrium compliance reported in this study is obtained by correcting for linear viscous strain a t long times. The equilibrium compliance, therefore, actually constitutes a steady-state condition, which, in a strict sense, is not an equilibrium state. It may be debated that the steady state nature of the present data rather than equilibrium elastic data invalidates the use of the statistical theory of rubber elasticity for the purpose of characterizing solvent-swollen coals. Moreover, there is the question as to the molecular-scale origin of the viscous strain; i.e., is it due to entanglement friction or bond rupture? As it stands, neither mechanism invalidates the use of the statistical theory of rubber elasticity. In the case of the entangled network, consider the following simple model for coal, proposed to explain the elastic qualities of pyridine-swollen coals. In Figure 11,long macromolecular strands interact with each other a t numerous entanglement points distributed along their lengths. Note that for the sake of simplicity the probable presence of chain branches has been neglected. The macroscale elasticity of such a material results from stress-induced changes in chain configuration between entanglement points in the same manner that the elasticity of a cross-linked network relates to stress-induced changes in chain configuration between cross-links. The problem of rubber elasticity is, therefore, treated in the same manner as for cross-linked networks. S. F. Edwards39 has considered this problem specifically with regard to the application of the statistical theory of rubber elasticity to unvulcanized or %repen rubber. His conclusion was that the elastic energy, Fellis related to the (39)Edwards, S. F. R o c . Phys. SOC.1967, 92, 9.
Characterization of Bituminous Coals A
Energy & Fuels, Vol. 7, No. 4, 1993 461 Table 11. Shear Modulus and Number-Averaged Molecular Weight between Entanglements
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61 59 72 37 20 62 30 40 39 47
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Figure 11. A physical-structural model illustrating four molecular chains in coal. The heavy lines delineate the fundamental path of each molecular strand. The fine lines depict the configuration of the actual chain about the fundamental path. Each chain interacts with the others at entanglement points. The elasticity of the overall system is related to stress-induced changes in the configuration of the chains between entanglement points.
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i concentration of entanglements. The behavior of entangled systems, therefore, is equivalent to that of cross-linked O ~ L , , networks where the elastic free energy, Pel,is related to 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 the average molecular weight between cross-links. In R,% entangled systems, however, Fel is related to the average Figure 12. Number-averaged molecular weight between enmolecular weight between entanglement points. tanglements or cross-links,(Mc(E)), vs mean maximum reflectance If the viscous effects are associated with bond rupture in oil, R ( M )calculated using the Gaussian theory of rubber as was proposed by Tobolsky and Andrews22there still is elasticity assuming case 1 behavior. no problem with the application of the statistical theory of rubber elasticity. Provided that the viscous strain rate presented, it is worthwhile to see whether they are is considerably less than the viscoelastic strain rate then comparable with values that have been previously reported the steady-state compliance again relates to an numberfor coal using methods other than stress-strain analysis. averaged molecular weight between cros~-links.~3 In the Gaussian theory of rubber elasticity, the shear A variety of elastic equations of state may be used to modulus is related to the number averaged molecular calculate a number-averaged molecular weight between weight between cross-links (entanglements) through the cross-links (entanglements). The simplest approach, the relation Gaussian theory of rubbery e l a ~ t i c i t y applies , ~ ~ ~ ~the ~~~~ statistics of random walk and the assumption of affine = pRT/ (MC(E)) (9) deformation to a freely jointed chain. This theory does where p is the dry density of the polymer, R is the gas not consider specific characteristics of the macromolecular constant, Tis the temperature, and (&E)) is thenumberchain, such as fixed bond angles, the nature of the chemical averaged molecular weight between cross-links (entanrepeat units, rotational potentials, van der Waals repulsion, glements). In using this theory one is still confronted with etc. Other statistical theories of rubber elasticity have the question of which set of Go's (case 1or case 2) to use. been formulated which account for finite extensibility, Using the case 1-derived values, and a value of 1.33 g/cm3 while still retaining the statistics of the freely jointed as reasonable for the density of bituminous vitrain,42 a ~hain3~140-41 for mathematical simplicity. In the modified from -500 up to 1600 g/mol is obtained range in (Mc(E)) Gaussian theories, an assumption about the size of the 11). These values are presented in Figure 12 as a (Table chemical repeat unit is required. All of the theories, function of rank. Case 2-derived shear moduli yield from however, predict similar behavior in the region of small -1200 to 4800 g/mol (Figure 13,Table 11). In this latter strains; it is only a t significant deformations that deviations case there appears to be a significant increase in (Mc(E)) from the simple Gaussian approach are observed. For the with rank. It should be reiterated that coal's actual purposes of this argument, therefore, the Gaussian theory behavior is expected to lie between these two cases. will be applied to estimate the number-averaged molecular The range in values between the two cases overlap those weight between entanglements, acknowledging all of the obtained by other research groups but are still within the limitations of the Gaussian theory. Whereas no great same order of magnitude. For example, Larsen and Kovac' confidence is placed in the absolute magnitude of the values report values in the range of 1500-1800based on solventswelling studies; Kirov et al.42reported values of 900(40)Lucht, L. M.; Peppas, N. A. In New Approaches in Coal Chemistry; ,
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Am. Chem. Soc. Symp. Series No. 169; American Chemical Society: Washington, DC, 1981; p 44. (41) Lucht, L. M.; Peppas, N. A. Fuel 1987,66,803.
(42) Krevelen, van, D. W. Coal; Elsevier: Amsterdam, 1960. (43) Kirov, N. Y.; OShea, J. M.; Sergeant, G. P. Fuel 1967, 46,415.
462 Energy &Fuels, Vol. 7, No.4, 1993 so00
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Summary Measurement of coal's viscoelastic properties in the pyridine-swollen state reveals that coal's characteristics are similar to those of rubbery materials. This substantiates similar observations and conclusions made by Brenner.13 Quantification of the viscoelastic parameters indicate that coal's modulus and solidlike behavior may result either from a cross-linked network or from extensive entanglement coupling. The viscous compliance,however, results either from coal's molecular structure being a highly entangled network or through a bond rupture and reformation mechanism. In the case of an entangled system, the actual cross-link density in these bituminous coals would be very low and could conceivably even be zero. The number-averaged molecular weights between entanglements or cross-links, calculated considering two extreme cases of viscoelastic behavior (one case including osmotic compressibility, the other assumes zero compressibility) span the range of previously reported values obtained through solvent swelling studies assuming a crosslinked network. Finally, if coal's molecular structure is an entangled network, then, consequently, the numberaveraged molecular weight of individual molecular chains in coal must be relatively large.
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