Energy h Fuels 1993,7, 463-468
463
The Dynamic Nature of Coal's Macromolecular Structure: Viscoelastic Analysis of Solvent-Swollen Coals George D. Cody,*$+ Alan Davis, and Patrick G. Hatcher Department of Geosciences, The Pennsylvania State University, University Park, Pennsylvania 16802 Received December 17,1992. Revised Manuscript Received March 31, 1993
A simple molecular model of an entangled macromolecular network is used to parametrize the physical behavior of solvent-swollen coals strained under both constant and oscillatory uniaxial compressive stress. The model distinguishes between three dynamic regions covering a wide range of molecular mobility. These are a high-frequency dynamic mode associated with Brownian motion of the segments of the molecular strands, an intermediate-frequency region governed by intramolecular motion associated with contour length fluctuations between entanglement junctions or cross-links, and a low-frequency region governed by the intermolecular mobility of individual macromolecules. The two lower frequency dynamic modes have been characterized through their creep compliance behavior. Viscoelastic strain in response to a constant uniaxial compressive stress is parameterized with a cooperative diffusion coefficient, D,. Results in the present paper show that D, range from 10-6 to 10-8 cm%. Self-diffusion of individual macromolecules is greatly restricted as evident by the low degree of fluidity of solvent swollen-coals. Coefficients of viscosity, I], range from 10" to 10'3 P. The very large difference between the actual time required to reach steady-state flow and that calculated, assuming linear monodisperse chains, suggests a physical structure composed of entangled, high molecular weight, branched macromolecules with long arms. Energy dissipation in the swollen coals is significant, averaging 50 7% in the frequency range 103Hz, and results from a large contribution to the compliance of viscoelastic over elastic strain. The high degree of energy dissipation detected with such a low-frequency stress cycle is consistent with coal's rubbery behavior in the pyridine dilated state.
Introduction The physical properties of solvent-swollen coal are related to its macromolecular architecture. The dynamic nature of the macromolecular structure of coal governs the time dependence of its response during various experimental and industrial processes,e.g., the rate of mass transport during liquefaction or the velocity of steady state flow during thermal softening associated with coking. In a previous paper,' the stress-strain behavior of solventdilated coals was used to characterize the physical structure of coal. It was observed that upon application of a uniaxial compressive stress the time-dependent strain of swollen coal samples could be treated as the sum of three linearly independent strain mechanisms; an instantaneous elastic strain, a time-dependent viscoelastic strain, and a viscous strain. The presence of a purely viscous strain mechanism lead to the conclusion that the macromolecular structure of coal must be highly entangled rather than a cross-linked network. In the present paper we will investigate the dynamic characteristics of solvent-swollen coals. Molecular theories derived to describe polymer dynamics have been proposed by numerous researchers. One particularly successful approach toward deriving the molecular origins of the viscoelastic properties of polymer solutions was the Rouse model.2 Rouse's model treated individual polymer strands as a system of coupled oscillators subjected to random collisions due to Brownian t Currently at the Chemistry Division, Argonne National Laboratory, Argonne L 60439. (1) Cody, G. D.; Davis, A.; Hatcher, P. G., to be published in Energy Fuels. (2) h u e , P.E., Jr. J. Chem. Phys. 1953,21, 1272.
0887-0624193125OI-O463$04.OOf 0
motion of the surrounding solvent molecules (and/or adjacent polymer molecules); the elastic restoring forces of each oscillator are entropic in nature2 consistent with the statistical theory of rubber elasticit.9. The Rouse model has since been shown to be truly applicable only in the case of concentrated solutions of relatively small linear polymers.s In the case where molecular strands become sufficientlylong, entanglement effects manifest themselves in the viscoelastic behavior; an analysis of role of entanglement effects in polymer rheology has been thoroughly covered by Graessley.' The quantitative incorporation of entanglements into a molecular theory was formally introduced by De Gennes8 and fully established by Doi and E d w a r d ~ .Currently, ~ the entanglement concept still forms the core of theoretical studies in polymer dynamics. In the case of a swollen coal, a simple conceptual physical structuralmodel which embodies the characteristics of an entangled polymer solution or melt can be applied to reveal the relationship between molecular structure and dynamic behavior. Such a model was proposed in an earlier paper;' a simplified version is reproduced in Figure 1, the simplification being that only one molecular strand is depicted. The important features of this simple model are (1)the primitive path of the molecular strandBtracing the mean value of configurational fluctuations of the actual molec(3) Flory, P. J.; Rehner, J.J. Chem. Phys. 1944, 12, 412. (4) Wall,F. T. J. Chem. Phys. 1943,11,527. ( 5 )Treloar, L. R. G. The Physics of Rubber Elasticity; Clarendon Press: Oxford, U.K., 1975. (6) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford Science Publications: London, 1989. (7) Graessley, W. W. Ado. Polym. Sci. 1974, 16, 1. (8) De Gennes, P. G., J. Chem. Phys. 1970,55, 572.
0 1993 American Chemical Society
Cody et al.
464 Energy & Fuels, Vol. 7, No. 4, 1993
Figure 1. A simplified physical-structural model of a single molecular strand in bituminous coal. The bold line is the fundamentalpath of the molecular strand. The fine line depicts configurational fluctuations that the real chain takes about the fundamental path. The regions outlined by circles depict the location of either entanglement junctions (or cross-links). ular strand and (2) entanglement junctions distributed along the length of the molecular strand which account for the relatively high modulus in the solvent-swollenstate while ensuring liquidlike compliance at long times under a constant stress. Finally, it is assumed that solventdilated coals are still in the concentrated regime, where hydrodynamic and excluded volume effects are negligible? The model in Figure 1can be used to distinguish three distinct regions of dynamic behavior. The first involves the dynamics of the fluctuations themselves. Relatively high-frequency molecular motion in coal has been characterized using internal friction methods,lG12and nuclear magnetic resonance techniques.13-16 At the molecular level this dynamic mode involves cooperative rotations among only a limited number of segmental bonds, Le., cooperativity extends over only short contour intervals. Notwithstanding the short-range nature of this dynamic mode, it is the fluctuations in this frequency range that contribute to the relatively large "instantaneous" or time-independent strain observed in the creep-complianceanalyses described in a previous paper.1 In the present paper, the highfrequency region is not directly accessibleand we will defer from specifically addressing the dynamic characteristics of short-range molecular motion. A second dynamic mode involves longer range conformational changes among the segments between entanglement junctions, or cross-links, and in response to an applied stress. In Figure 2 the change in length of the primitive chain designated, R, results from strain of this section of the molecular strand. As will be discussed below, the dqm"cs of this behavior is quantified through a cooperative diffusion coefficient, DVl7 The importance of this dynamic mode lies in its fundamental relationship to intramolecular or segmental mobility. Moreover, D,is directly related to solvent transport in polymers17J8and, presumably, also in coal. Intermolecular mobility, the third and lowest frequency dynamic mode, involves motion of the center of gravity of (9) Doi, M.; Edwards, S.F., J. Chem. SOC.,Faraday Trans. 1978, 74, 1789. (10) Weller, M.; Wert, C. J J. Appl. Phys. 1982,53,6505. (11) Weller, M.; Wert, C. Fuel 1984,63, 2361. (12) Weller, M.; Volkl, J.; Wert, C., Int. Conf. Coal Sci., Pittsburgh, PA 1983, 283. (13) Sanada, Y; Honda, H. Fuel 1962,41, 295. (14) Jurkiewicz, A.; Marzec, A.;Pislewski, N. Fuel 1982, 61, 647. (15) Barton, W. A.; Lynch, L. J.; Webeter, D. S. Fuel 1984,63,1262. (16) Yang, X.; Garcia, A. R.; Lamen, J. W.; Silbemagel, B. G. Prepr. Pap.-Am. Chem. SOC.,Div.Fuel Chem. 1992,37,627. (17) Tanaka, T.; Hocker, L; Benedek, G. B. J. Chem. Phys. 1973,59, 5151. (18) Peters, A.; Candau, S. J. Macromolecules 1986, 19, 1952.
Figure 2. A segment of a molecular strandbetween entanglement junctions or cross-links. The time dependence of the change in length R correlates with the macroscopicviscoelastic deformation of the solvent-swollen coals.
n
W Figure 3. Path that a molecular strand may take during intermolecular motion corresponding to viscous flow. an individual molecular strand relative to the surrounding strands (Figure 3), i.e., self-diffusion. The viscous compliance or fluidity is, therefore, related to the self-diffusion characteristics of the system and may under some circumstances be parametrized through the self-diffusion coefficient, Ds;2p'39839as will be discussed below, the facility of self-diffusion is strongly related to the topology of the macromolecular strands. In this paper the time-dependent viscoelastic behavior of solvent-dilated coals will be characterized by applying the model described above as a basis for interpretation of molecular-scale phenomena in terms of various dynamic parameters. The creep compliance experiment described below cannot resolve aspects of the "instantaneous" elastic strain behavior of the solvent-swollencoals, and so analysis of the first and fastest of the dynamic modes will be neglected. The other two dynamic modes are critical to coal's mass transport behavior (diffusivity) as well as its physical behavior at elevated temperatures (softening and fluidity) and constitutes an important research area. A second experiment, however, involving cyclic stress-strain behavior probes aspects of the higher frequency compliance characteristics of the solvent dilated coals.
Experimental Section The coal samples are hand-picked vitrain 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 have been detailed in a previous paper.' In addition ' to the coal samples, experiments were also run on a piece of cross-linked rubber taken from a standard laboratory test-tube stopper. The specimen preparation protocol has also been described in detail previously.' Each specimen was exhaustively extracted in pyridine prior to analysis. Also, each selected specimen was free of cracks, as judged by microscopic examination following extraction. The specimen geometries are rectangular, with a volume on the order of 1 "3. Linear DisplacementMeasurement. The microdilatometer employed in this work has been described in detail previously.1B The important characteristics are that the instrument uses a linearly variable differential transformer (LVDT) to obtain a
Energy & Fuels, Vol. 7, No. 4, 1993 465
Macromolecular Structure of Coal Table I
R(M),~
G log7 lard (kg/cm2) (P) 13.1 8.8 0.37 78.9 20.25 12.4 8.0 79.1 22.56 0.47 0.53 17.32 12.4 8.2 79.8 0.57 81.4 10.45 12.1 8.1 11.8 8.1 81.0 5.26 0.58 16.64 12.0 7.8 0.63 82.1 10.41 12.3 8.3 0.65 81.9 12.4 8.4 0.66 81.9 10.77 11.6 7.8 0.81 85.4 5.95 11.6 7.7 0.94 87.3 7.44 a PSOC samples. b Vitrinita reflectance measured in oil. Calculated. Actual times to steady-state flow are on the order of 109 8.
sample 1316 1318 772 1270 1539 1274 746 223 1171 1336
(%)
5%
CDAF
L
-i
' A
12
A
sensitive measurement of linear displacement of a piston which rests upon the surface of the coal specimen. The limit in sensitivity is 0.5 pm, imposed not by the LVDT but rather by an inability to isolate the microdilatometer from vibrations in our laboratory. The displacements typically measured are on the order of 10s to 100s of micrometers. The compliance of the instrument is extremely low, less than the limits in detection. Mechanical Measurements of Solvent-Swollen Coals. Two types of experiments were conducted. In the first, creep compliance runs were used to obtain the time dependent compliance of the solvent-swollencoals under a uniaxial stress. During a creep compliance experiment, the compliance of a solvent-swollencoal sample is continuously monitored following the application of a uniaxial compressivestress. The magnitude of the stress is on the order of 0.5-1.0 kg/cm2. The maximum strain is on the order of 0.04. The second experiment investigated the magnitude of energy dissipation when the following experimental protocol was applied. Uniaxial compressive stress was incrementally applied to the solvent-dilated coal to a maximum of about 6 kg/cm2. Upon reaching the maximum, the stress was then progressivelyreduced. The stress rate during both loading and unloading was on the order of 3 kg/cmS/min resulting in a stress cycle with a frequency of 4.2 X 1o-SHz. To avoid introducing uncertainty in the position of the dilatometer piston upon complete loading, it was necessary to unload to only 1kg/cm2 before initiating the next stress cycle, which was begun only after the strain recovery rate reached zero.
Results and Discussion The results of viscoelastic measurements on solventswollen coals can be parametrized in terms of various dynamic coefficients. The amount of recoverable strain is defined as the steady-state compliance, J E O , where
JEo= GE-l
(1)
and GE is the steady-state elastic shear modulus. In a previous paper1 values for GE were obtained from sequential creep-compliance experiments. A troublesome aspect of this approach was the potential problem of osmotic compressibility contributing to the compliance. In the present study we will assume that the elastic compliance is entirely incompressible. As will become apparent below, this assumption is not crucial to our understanding of the dynamic aspects of viscoelasticstrain in solvent-dilated coals. In the previous work the shear modulus, GE, was corrected for solvent effects. In the present work we are interested in the experimentally derived modulus, uncorrected for solvent. These values are listed in Table I. Another important dynamic parameter is the coefficient of viscosity, q, obtained from the compliance of coal at (19) Cody, G.D.;Davia,A.; Hatcher, P. G.;Eseer,S.Prepr. Pap.-Am. Chem. SOC.,Diu. Fuel Chem. 1991,36,1307.
A
11.51*,'I 0.3 0.4 I
1
I
I
4
1
,
0.5
'
1
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I
I
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8
1
'
I , , ' ,
0.7
0.8
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,
~
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'
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~ 8
~
1
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Figure 4. Log of the coefficient of viscosity, 7, derived from the linear stress-strain region plotted against mean maximum reflectance in oil.
long times while it is subjected to a constant uniaxial compressivestress. It has been shown that solvent-swollen coals exhibit viscous strain in addition to the purely elastic and viscoelastic deformation. Solvent-swollencoals, therefore, behave like extremely viscous fluids. Linearly viscous, Newtonian, fluids exhibit the following relationship between stress and strain rate 3 = u,l~, (2) Strictly speaking, q is the shear viscosity, is is the shear rate, and a, is the shear stress. In the creep compliance experiments, however, e, is the strain rate in the z direction; e, is proportional to i, and us is proportional to UO. Therefore, using i,and uoas order of magnitude estimates for 0, and us, q, the longitudinal viscosity, is readily obtained. The calculated values are presented in Table I and plotted against rank in Figure 4. The magnitude of the viscosity is relatively high, ranging from 10" to 1013 P. Moreover, a trend toward reduced viscosity with increased rank is clearly evident. In the case of a monodisperse system of high molecular weight linear macromolecules,the steady-state compliance, JE', and the coefficient of viscosity, q , are related through a characteristic relaxation time, Td, typically referred to as the disengagement time, Le., the time it takes to achieve steady-state flow.612b22
(3) The magnitude of calculated Td is presented in Table I; the actual time required to reach steady-state flow is on the order of 103 s. The disparity between the calculated vs observed values of Td presumably results from either the assumption of monodispersity or the assumption of chain linearity or both. In the case of linear chains the effect of polydispersity is to increase JE', while the magnitude of q becomes dependent on the weight average of the constituenta in the mixture.20 The influence of chain branching on the magnitude of the dynamic parameters J E O and q is highly significant if the chain branches are (20) Ferry, J. D. The ViscoelasticProperties of Polymers; John Wiley & Sons: New York, 1980. (21) De Gennes, P. G. Macromolecules 1976,9,587. (22) Graessley, W. W. J. Polym. Sci. Polym. Phys. Ed. 1980,18, 27. (23) Bueche, F. Physical Properties of Polymers; Interscience: New York, 1962.
I
~
466 Energy & Fuels, Vol. 7,No. 4, 1993
Cody et al.
long. In the entangled concentration regime, both q and
spherical geometry is inappropriate. Therefore, consid-
JE' increase with the degree of chain b r a n ~ h i n g .A~ ~ ~ ~ering ~ only the component of stress and strain in the particularly important point, however, is that in the case of long-chain branches the fundamental mechanism of self-diffusion of individual macromolecules change from one of translational diffusion resulting from reptation along chain contours to a mechanism involving complete topological reconfiguration,9-2"26 in this latter case the relationship between J E O , Q, and 7d, given in eq 3, is no longer valid. Clearly,the assumptions of linearity and monodispersity of macromolecular strands in coal are not supported by the data. The relative magnitudes of the dynamic parameters in Table I are consistent, however, with the macromolecular structure of coal being composed of large, entangled, and branched molecules with long arms. Qualitatively, it is observed that the rate of self-diffusion increases with rank. The magnitudes of q, JE', and 7d reveal a situation where inter and intrasegmental mobility between entanglements is much greater than a t entanglement junctions. In the present work, therefore, we will consider the dynamics of viscoelastic strain separate to the dynamics of selfdiffusion; this separation is justified by the observation that in this case each mode of strain operates on significantly different time scales. In the case of theviscoelastic strain of branched polymers, it has been theorized6 that stress relaxation occurs exclusively through contour length fluctuations, where the dynamics of such fluctuations are governed by a cooperative diffusion coefficient, D,. In order to calculate the magnitude of the cooperative diffusion coefficient,D,, in the present study, the approach of Tanaka et al.17 has been used. Tanaka et al. originally considered the problem of dynamic light-scattering of swollen poly(acry1amide)gels, where the rate of decay of the autocorrelated signal intensity was given as a function of the cooperative diffusion coefficient. The essence of their approach is to parametrize the time dependence of strain by solving the equations of motion which describe the system. The following equation was proposed to describe their system:
a2U/at2= vu - f auiat (4) where u is the displacement vector,27t is time, p is density, Vu is divergence of the stress tensor, and f is a friction coefficient. Equation 4 is an expression of Newton's second law. In the case of poly(acry1amide)-water or coalpyridine, the frictional terms are far greater than the inertial terms; thus, eq 4 can be simplified to
vult = auiat (5) The standard expression for the stress tensor of an isotropic elastic solid is given by eq 6, where K is the bulk
compressibility modulus, eij is the strain gradient tensor (exx = au,/dx), G is the shear modulus, and 6ij is the Kronecker delta. In the application of Tanaka et al.," only the radial component of the stress in a spherical coordinate system was considered. In the present case, a ~
~
(24) Graessley, W. W.; Masuda, T.;Roovers, J. E. L.; Hadjichristidie, N. Macromolecules 1976, 9, 127. (25) Klein, J. Macromolecules 1986, 19, 105. (28) De Gennes, P. G. J. Phys. (Paris) 1974,36,1199. (27) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon Press: New York, 1959.
direction of applied stress (the z direction in a Cartesian coordinate system) one obtains the following equation
swat = D, d2u/az2
(7) where D,, the cooperative diffusion coefficient, is the ratio of the elastic and friction parameters,17 the exact formulation being dependent on whether the viscoelastic strain is compressible or incompressible. The solution of eq 7 requires the followinginitial and boundary equations. First, at t > 0 the stress on the surface of the sample is equal to UO, where the surface is at z = L . At the base of the sample, z = 0, u = 0 for all values oft. The initial condition is u(z,O) = z + %AIL,where A is the equilibrium or steadystate elastic strain at t = a. The solution to eq 7 then becomes
where X, = ( n
+ l)?r/2L
and for the special case where z = L
At t =
m,
u(L,t) = A, and the strain (e,) is defined as e, = AILo
(10) where LO is the unstrained length. Furthermore, by definition, the equilibrium or steady-state elastic strain is related to the applied stress uo through the equilibrium or steady-state elastic compliance, J E O
A = UOJ,~L, (11) It follows that the time-dependent strain e(t) is then described through the following expression,
where
7,
is the characteristic relaxation time given by 7,
= 4L2/(n+ 1)2Dc?r2
(13)
For large values of t , ~ ( t=) uOJEO = em. Furthermore, at long times the summation in eq 12 is dominated by the largest relaxation time, TO, at n = 0. Therefore a semilog plot of the time-dependent strain ratioed to the strain at t= w ill exhibit linear behavior at long times, where the slope of the linear region = -1170. The cooperative diffusion coefficient,D,, is then obtained from TO,as follows
D, = 4L2/?r2T0 (14) Figure 5 presents the semilog behavior of the timedependent compliance vs time. Analysis of the timedependent compliance of all of the coais in this study yields a narrow range of D,. The results are presented in Figure 6 in which log D, is plotted as a function of rank, where the rank parameter is the mean maximum vitrinite reflectance in oil. The values for D, range from 10-6 to 1W cm2/s. There does appear to be a decrease with respect to rank indicating either a reduction in the elastic modulus with rank (see Table I, such a reduction has also been
Macromolecular Structure of Coal
Energy & Fuels, Vol. 7, No. 4,1993 467
7 t
.
t
h
w
U
52
Figure 7. A linear viscoelastic model of a standard solid composed of an elastic spring with a compliance, J1,in series with an element composed of an elastic spring in parallel with a Newtonian dash-pot. This last element has a compliance Jz.
t
t
-8I 0
I
40 60 t (min) Figure 6. Log, of the strain vs time. The slope of the linear region corresponds to the longest relaxation time, TO.
-4.7
-4.9
-Ff
-5.1
20
4
I
1.
.
1
J f
.
.
Jf(w)= J , + J z / ( l + w%') (18) whereas the complex part of the compliance is given by the equation
-5.5 0.4
0.5
0.6
0.7
0.8
0.0
1.0
Figure 6. Cooperative diffusion coefficient, Do, plotted against mean maximum reflectance in oil as a rank parameter. previously indicated'), or an increase in the friction factor (eq 5 ) , perhaps indicating a decrease in intramolecular or segmental mobility with rank. Similar analysis of a cross-linked rubber, both dry and swollen in toluene, yields D,'s of 1We8and 106.z cmz/s, respectively. The increase in D, with increased polymer concentration has been predicted based upon theoretical considerations.6 Energy Dissipation during Stress Cycling. The second set of experiments was designed to quantify the magnitude of energy dissipation during the compressive deformation of solvent-swollen coals. Energy dissipation is a consequence of the viscoelastic nature of solvent swollen coals; its magnitude, however, is frequency dependent. For viscoelastic materials, under the conditions of oscillatory stress, the resultant strain e(w) is equal in frequency but shifted in phase;z01z6it is the magnitude of the phase shift angle that determines the magnitude of the energy dissipation. Consider the following stress cycle u = uo cos(&)
Jff
where J', the real part of the compliance, is given by the equation
-5.31
0.3
= 'o cos(wt - 6)
(16) To further investigate the frequency-dependent compliance of solvent-swollen coals a suitable viscoelastic model may be chosen to replicate coal's behavior. An appropriate model, typically referred to as a "standard solid", is depicted in Figure 7. This model incorporates both the elastic and viscoelastic components of strain. To a first approximation the viscous component of strain can be ignored because of its extremely low frequency behavior. Under the conditions of oscillatory stress, the compliance of the standard solid model is complex J* 3 - i (17) €
1
I
(15)
The corresponding strain will be equal in frequency but shifted in phase by an angle 6;z8 this strain is given by
Jff(w)= Jzw7/(1+ ~'7') (19) where J1 and Jz correspond to the compliancesof the elastic elements in the standard solid, while T is the retardation time of the model given by Jzq, and where q is the viscosity of the dash-pot element in Figure 7. In reality, such a simple model can not be expected to recreate the frequency-dependent compliance behavior of a complex material such as coal. The above models may be expanded, however, to incorporate a spectrum of retardation times which will exactly fit the experimental data.20 The phase shift angle, 6, is related to J' and J" through the relationshipz8 tan 6 = Jf'/J' (20) Furthermore, the magnitude of the phase shift angle is directly related to viscous losses, i.e., energy dissipation, through the following expression, AW/W=2asin6 (21) where W is the maximum work done on the sample and A W is the difference between the maximum work done and the work recovered following one stress-strain cycle. A W/ W is the fraction of energy dissipated during each cycle. Given a single cycle of stress, as in the present study, the work done in compressively straining the material is proportional to the integral of stress over the strain, i.e., W - JudX
(22)
The dissipation energy is therefore proportional to the difference in work between the compression and decom(28) Findlay, W.N.; Lai, J. S.; Onaran, K.Creep and Relaration of Nonlinear Viacoelastic Materials; Dover Publications,Inc.: New York, 1989.
468 Energy &Fuels, Vol. 7, No. 4, 1993
1
0.8 h
F Q
wC
0.6
.-E
0.4
C
0
I-
8
(II
V)
E
0.97
0.98
0.99
1 .oo
"1
0.0
1.01
1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
ULo
Figure 8. Cyclic stress-strain behavior, three cycles.
pression cycle. At the molecular level, the magnitude of the dissipation energy reflects the magnitude of barriers (temporal in nature) which oppose intramolecular mobility, e.g., temporary entanglements. The microdilatometer employed in this study does not have the capability to systematically vary the frequency of the stress while monitoring strain. For the purpose of comparison with other polymeric materials, however, it is possible to measure the fraction of work lost through viscous dissipation by conducting repeated single-cycle compression/ decompression runs over identical periods (240 8). The stress cycle is created by sequential loading and unloading. Figure 8 is an example of a typical run involving three stress cycles, with each cycle followed by a recovery period. There are several interesting features. First, the strain at the beginning of each cycle appears not to be zero, Le., X # 1.0. This is an artifact of the compression cycle data being fitted to a linear equation. For reasons that are not currently understood, the second derivative of the strain curve for the first step of the compression cycle for solventswollen coals was always negative. This behavior was observed to a greater or lesser degree for all samples, but not the cross-linked dry rubber. It is possible, therefore, that such behavior relates to the presence of solvent in the coal. The energy dissipation as evident in the shift in strain corresponding to a specific stress during a given cycle is clearly large. Figure 9 presents the fractional energy dissipation derived from a stress cycle frequency of 4.2 X H z plotted against R,9% as a rank parameter. Approximately,35-75 9% of the stored energy is lost through viscous dissipation. There is no apparent rank trend in the magnitude of the energy dissipation and a fair amount of scatter. On average, the solvent-swollen coals exhibit 50% dissipation. The cross-linked rubber swollen in toluene, for comparison, exhibited an approximately 30 9% reduction in recoverable work. In terms of the standard solid model (Figure 7)and eqs 18-20, the difference in the dissipation energy between the coals and the cross-linked rubber at this frequency lies in the relative magnitudes of the instantaneous elastic compliance as opposed to viscoelastic compliance. In the case of the cross-linked rubber swollen in toluene there is a large contribution from the instantaneous compliance, whereas for coals a propor-
Figure 9. Dissipation energy plotted against mean maximum reflectance in oil.
tionally greater amount of compliance was from the viscoelastic modes. In arelated experiment, Weller and We+ used internal friction methods to reveal the magnitude of energy dissipation as a function of temperature for dry coals. The dissipative modes detected were relatively high frequency (103-104 Hz at ambient temperatures) consistent with a glassy structure. In the present study, however, high levels of energy dissipation are observed at frequencies over 6 orders of magnitude lower; this behavior is consistent with the rubbery characteristics of solvent swollen coals.
Conclusions If one accepts that the appropriate physical structure of bituminous coals is a branched and highly entangled network, then the viscous compliance is directly related to the mobility of individual molecules within the swollen gel. Given the very large magnitudes of q in these solventswollen coals, however, it is clear that intermolecular mobility in coals is significantlyretarded. Such low degrees of translational mobility explains the following (apparently paradoxical) situation; the extractability of coal reaches an apparent asymptotic limit during Soxhlet extraction. If the cross-link density is zero then the coals should be 100% extractable, assuming equilibrium behavior. Enormous barriers to diffusion, however, exist for the larger molecules in coal, as is indicated by the very large viscous coefficients. This suggests that while complete extraction may be theoretically possible it is kinetically improbable given the constraint of time. With continuous flushing with fresh solvent and infinite time coal might very well be 100% extractable. Indeed, such a possibility has been suggested by Painter et al.% In practical terms, however, coal's extractability is limited to the removal of smaller molecules whose self-diffusion coefficients are sufficiently large. Acknowledgment. This work was partially funded through the support of the Cooperative Program in Coal Research a t Penn State. G.D.C. gratefully acknowledges support in the form of a Fellowship from the Mining and Minerals Research Institute (MMRI). (29) Painter, P.C.; Park, Y.;Coleman, M.M.Energy Fuels 1988,2,
693.
