The molecular scaling of raw pitches by oscillatory rheometry

Carbon 37 (1999) 1189–1197

The molecular scaling of raw pitches by oscillatory rheometry E. Daguerre, F. Nauguier, X. Py* ´ ´ ´ ´ , 52 Avenue de Villeneuve, 66860 Perpignan cedex, France des Materiaux et Procedes C.N.R.S.-I.M.P., Institut de Science et Genie Received 24 June 1998; accepted 25 November 1998

Abstract Alpha relaxation and viscosity measurements performed above the glass transition region have been used for raw pitch molecular scale characterization. The characteristic time t a of the alpha relaxation is known to be similar to the molecular mobility characteristic time t m . Viscosity–temperature and t m –temperature data allow molecular equivalent hydrodynamical diameter estimation through well known Stokes–Einstein relations. This procedure applied to two petroleum and two coal tar ˚ hydrodynamical diameter range. Those results are in good agreement with basic structural units pitches led to a 9.5–20 A (BSU) theoretical molecular models or other experimental studies available in the literature. Pitch molecular scale seems similar to the coronene and ovalene monomers, dimers and trimer sizes. Moreover, an extrusion effect has been observed on coal tar pitch resulting in a possible edge-to-edge orientation. A correlation between a, b and g resins and BSU diameter range is proposed.  1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Pitch; C. Rheometry

1. Introduction An experimental method leading to molecular size would be very important and useful for the elucidation of the viscoelastic properties and chemical reactivity of pitches. For years, authors have proposed various molecular models according to their experimental approach and tools. Oberlin et al. identified the presence of basic structural units (BSU) in kerogens [1,2] and pitches [3–5]; ˚ these consisted of small stacks of molecules less than 10 A in diameter. Light hydrogenated molecules act as a suspensive medium for the BSU considered as colloidal micelles [6] (colloidal dispersions are suspensions of solid-like particles or molecules in a continuum generally made of smaller molecules). BSU were visualized by HRTEM by Oberlin et al. in experimental cokes [3] and in heavy petroleum products [7]. BSU are planar aromatic structures (4–12 rings) single or piled-up approximately in parallel to form stacks of 2–3 layers in thickness. As the pitch is heat-treated, the stacks become much longer and exhibit a characteristic misorientation. The usual characterization of pitches by fractionation in toluene and quinoline leading to a, b, g resins can be interpreted as follows [8]: BSU in g resins are distributed randomly in the light medium forming an isotropic toluene-soluble sol. b resins are *Corresponding author. Fax: 133-4-6866-2141.

known to present edge-to-edge BSU associations. Such an increase of organization is responsible for the insolubility in toluene while the weakness of these associations explains quinoline solubility. a resins are made of wellorganized face-to-face associated BSU forming anisotropic spheres as visualized by Brooks and Taylor [9]. This pitch colloidal model was used by Lafdi to explain the properties of anisotropic pitches produced either by solvent extraction [10] or sparging [11]. However, the BSU gel behaviour of such materials was denied by Fitzgerald et al. [12]. Coal tar pitch BSU always appear larger than petroleum ones [4] and aromatic ring structures fitting with TEM images lead to small molecules such as coronene. Recent molecular modelling computational techniques have been used to investigate the preferred stacks configurations [13]. Such molecular modelling has been applied to molecules typically found in petroleum and coal tar pitches. Homologous aromatic hydrocarbons associate strongly face-toface, in a parallel shifted stack arrangement with a ˚ Heteromerous aromatic displacement of about 4.7 A. hydrocarbons also prefer a shifted stack configuration of two or three molecules high. A third or fourth molecule added to a stack will prefer to orient perpendicular to the stack such that its face is against the edge of the stack. These theoretical results support the colloidal model for the behaviour of pitches. To our knowledge, only two experimental attempts have been done for molecular size

0008-6223 / 99 / $ – see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S0008-6223( 98 )00311-X

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Table 1 Physicochemical properties of the investigated pitches Parameter

a-resins %wt.

b-resins %wt.

g-resins %wt.

T S (8C)

T Sm (8C)

Petroleum A240 Petroleum HGDP Coal tar HGD1 Coal tar HGD2

,0.1 ,0.1 12 1.9

4 ,0.1 16 13.5

96 100 72 84.6

119 – 110 88.8

120 119.4 112 92

and shape determinations in pitches by diffusivity measurements [14–16] and by TEM observations [4,5].

2. Apparatus and experimental procedure The rheological experimental data have been obtained using the cone / plate Physica-Rheolab MC100 rheometer described previously [17,18]. Measurements analyzed in the present paper were made under controlled stress at temperatures in the range of 65 to 1108C and at frequencies in the range of 1 to 101 rad s 21 . No treatment has been applied on the pitches before characterization, 130 to 170 mg of the raw sample were directly loaded on the plate and heated to the working temperature.

3. Materials studied Four different raw pitches have been investigated: two petroleum (A240, HGDP) and two coal tar pitches (HGD1, HGD2). The A240 petroleum pitch and the extruded HGD1 coal tar pitch have been already presented [17,18], the HGD2 coal tar pitch has been manufactured by HGD1 centrifugation in order to withdraw the corresponding QI content and has been extruded too. The main characteristics of those four materials are gathered in Table 1.

Fig. 1. Temperature dependence of petroleum pitches complex viscosities.

4. Pitch viscosity temperature dependence Within a pitch, each molecule is surrounded by a suspensive medium of intrinsic viscosity h i which is a function of temperature and composition. When heating the pitch, which is a glassy material, the viscosity of the suspensive medium decreases and consequently the molecular mobility characteristic time t m of each molecule decreases too. Therefore, molecular mobility of different pitches could not be compared without any viscosity consideration. As raw pitches are very diluted with respect to BSU, we will consider the apparent viscosity of the whole pitch as equivalent to the intrinsic viscosity of the suspensive medium. As previously mentioned [17,18], pitches steady flow and complex viscosities are equivalent. In Fig. 1 and Fig. 2 are gathered the petroleum and coal tar pitches viscosities against temperature for comparison.

Fig. 2. Temperature dependence of coal tar pitches complex viscosities.

E. Daguerre et al. / Carbon 37 (1999) 1189 – 1197

Those results illustrate the difference between coal tar and petroleum pitches viscosities: at a given temperature, the coal tar pitch presents the lower one. Moreover, the removal of the heavier fraction of HGD1 by centrifugation led to the less viscous pitch HGD2. Although A240 and HGDP are different in origins, their viscosities are very similar in behaviour and in order of magnitude. From those results, the usual Ring and Ball softening point characteristic temperature T S can be estimated as being the isoviscous temperature T Sm at 10 3 Pa s [19]. In Table 1 isoviscous temperature T Sm at 10 3 Pa s and available T S from the manufacturer are gathered for comparison. Perfect agreement (2% of mean deviation) is obtained between them. Moreover, extrapolated viscosities at the glass transition temperature T g led to values close to 10 10 Pa s which is characteristic of fragile-pattern liquids as discussed by Angell [20]. Consequently, and considering the Stokes– Einstein relation, the diffusion coefficient D is given by: D 5 l 2 /t

(1)

in which the time t can be replaced by the relaxation time t a , the mean distance l moved in this time t a is of the order of magnitude as the molecular diameter [20].

l ¯ dh

(2)

This relation between relaxation time measured by oscillatory rheometry and molecular diameter is the basis of the characterization technique developed in the present paper.

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[21]. Therefore, direct t a experimental values are limited to low temperature isotherms close to the glass transition. At higher temperatures, the G0 isotherms do not present any maximum within the available experimented frequency range. Nevertheless, those maxima could be estimated by using the time–temperature superposition procedure and specially the corresponding shift factors calculated with the reference T r . G0 measured at frequency v T and temperature T is equivalent to G 99 r measured at frequency v T .a T and temperature T r . Therefore,

vT 5 vr /a T

(3)

As previously mentioned [17,18], the use of this time– temperature superposition procedure should be restricted to viscoelastic materials which do not present such relaxation or to the corresponding time–temperature domain. Therefore, all the analyzed data in the present paper are direct measured data. Consequently, at each experimented temperature, a characteristic time of motion can be associated to the corresponding relaxed molecule and the molecular length scale distribution can be obtained by temperature scan. In Fig. 3 and Fig. 4, alpha relaxation characteristic time of the investigated pitches (petroleum and coal tar pitches, respectively) are presented versus temperature. Those results illustrate the order of magnitude of the time required for the concerned structural unit to move from a distance equivalent to its own diameter, namely the molecular time of motion. Nevertheless, the ability of a molecule to move within its suspensive medium is not only a function of its diameter but also and mainly of the

5. Molecular time of motion characterization

5.1. Alpha relaxation identification The alpha relaxation, or primary relaxation, has been already presented for the isotropic A240 petroleum pitch [17] and the HGD1 coal tar pitch [18]. They were illustrated by typical G9 isochrone three decades steps and G0 or tan(d) isochrone peaks. The a-relaxation identification was based on two criteria: the non-Debye and nonArrhenius behaviours both fulfilled by the four investigated raw pitches. With respect to the corresponding molecular diameter, the considered molecule presents a characteristic time of motion equivalent to the experimental characteristic time scale for a definite temperature or frequency associated to the suspensive medium viscosity.

5.2. Alpha relaxation characteristic time The a-relaxation characteristic time is determined experimentally from the G0 isotherms maxima at which vt a 51 i.e. the experimental characteristic time (v 21 ) and the molecular mobility characteristic time are equivalent

Fig. 3. a-relaxation characteristic time t a of petroleum pitches versus temperature.

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Fig. 4. a-relaxation characteristic time t a of coal tar pitches versus temperature.

Fig. 6. a-relaxation characteristic time t a of coal tar pitches versus viscosity.

suspensive medium viscosity. Therefore, the obtained curves for the four different materials have not to be compared without viscosity–temperature considerations: the coal tar pitch and the petroleum pitch present very different viscosities at the same temperature. In Fig. 5 and Fig. 6, the alpha relaxation characteristic times are presented with respect to viscosities for the two petroleum and the two coal tar pitches, respectively. Those last four figures illustrate how different the relaxation times of two pitches could appear to be when

plotted versus temperature and how well they superpose when plotted versus viscosity. Therefore, each type of pitch presents similar relaxation time behaviour with respect to viscosity but petroleum and coal tar origins lead to differences in homogeneity and curvature.

6. Molecular scale characterization

6.1. Theoretical considerations Up to now, there is a lack of rigorous theory for diffusion in liquids. Nevertheless, two rough theories, namely the hydrodynamical and the Eyring ones, are usually used and described in the literature [22]. Considering that the former one is acknowledged to be fairly good for describing the diffusion of large spherical particles or molecules within a solvent described as a continuum, it has been used for data analysis of our study. Moreover, Sakai et al. [14] have shown, from intrinsic viscosity study, that the molecular shape of pitch molecules is closer to a spherical rather than a planar one. For a rigid sphere moving in creeping flow (i.e. at low Reynolds number Re<1), the needed force F to apply to the molecule with hydrodynamical diameter d h to move it within a suspensive medium of viscosity h at a given velocity u is: F 5 zu

(4)

with Fig. 5. a-relaxation characteristic time t a of petroleum pitches versus viscosity.

z 5 3p d hh(4h 1 d h f )(6h 1 d h f )21

(5)

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where z and f are the friction and the sliding friction coefficients, respectively. Two limiting cases have to be considered with respect to the last coefficient f:

1. there is no tendency for the fluid to slip at the surface of the particle, in this case, f5`, and we get the well known Stokes law, F 5 3p d hhu and z 5 3p d hh 2. there is no tendency for the fluid to stick at the surface, in this case, f50 and we get, F 5 2p d hhu and z 5 2p d hh (7) Moreover, Einstein has related the diffusion coefficient D to the Stokes force F and velocity u, D 5 kTuF 21

(8)

where k is the Boltzmann constant (1.3806310 223 J K 21 ). According to Eq. (1) in which l and t are replaced by d h and t a , the corresponding diffusional coefficient can be expressed as follows [21], D 5 d 2ht a21

(9)

Therefore, combining those expressions, d h 5 [kTta /(3ph )] 1 / 3 f 5 `

(10)

d h 5 [kTta /(2ph )] 1 / 3 f 5 0

(11)

Eq. (10) and Eq. (11) allow to estimate the hydrodynamical molecular scale from t a and h experimental values. In his study about relaxation in liquids, polymers and plastic crystals, Angell [20] has used Eq. (10). Moreover, a resins in pitches are known to be surrounded by heavy b resins belonging to the suspensive medium. Therefore, the suspensive fluid could not have any tendency to slip at the surface of the concerned molecules and the use of Eq. (10) is preferred. Anyhow, the deviation between diameters calculated from the two expressions has a mean value equal to 3% and does not exceed 6%.

Fig. 7. Hydrodynamically equivalent sphere diameter versus T for petroleum pitches.

The petroleum pitches mainly composed of g resins present a narrow and low BSU diameter range from 10.6 to ˚ The extrudated HGD1 coal tar pitch (containing 13.3 A. significative b and a resins) presents a wide and continu˚ As the HGD1 a ous BSU diameter range from 9.5 to 20 A. resins are mostly primary QI, only corresponding associated heavy b resins can lead to experimental high value diameter measurements. Those whole results illustrate the well known fact that raw coal tar pitches contain larger molecules than petroleum pitches.

6.2. Experimental results The diameter d h of the hydrodynamically equivalent sphere of the relaxed units has been calculated using Eq. (10) for the four experimented pitches. In Fig. 7 and Fig. 8, the corresponding results are presented as a function of temperature: increasing the temperature, the viscosity of the suspensive media decreases allowing larger molecules to relax. Moreover, it has to be considered that those species do not relax at the same frequency. Anyhow, time–temperature superposition allows to use T or v indifferently.

Fig. 8. Hydrodynamically equivalent sphere diameter versus T for coal tar pitches.

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The HGD2 coal tar pitch, obtained by HGD1 centrifugation and extrusion, does not present such a wide range of ˚ values. equivalent diameters but only 10.6 to 12.9 A Compared to the HGD1 data, the HGD2 equivalent diameters are shifted to the left because of the difference in viscosity. Therefore, the total removal of heavy b resins associated to primary QI from the HGD1 coal tar pitch can be correlated to the disappearance of all diameters larger ˚ This experimental observation leads to the than 12.9 A. identification of heavy b resins associated to primary QI as ˚ hydrodynamical diameter range species. 13 to 20 A The two petroleum pitches and the HGD2, all mainly ˚ composed of g resins and light b resins, present an 11.8 A diameter as a mean and rather uniform value. At constant hydrodynamical diameter and using Eq. (10), data on such materials should confirm the Stokes– Einstein relation written as follows:

ta 5 3p d 3h k 21 (hT 21 )

(12)

This expression is illustrated in Fig. 9 by the plot t vs. (h /T ). The good linearity of this graph for both petroleum pitches (regression coefficient 0.98) confirms simultaneously the applicability of Eq. (10) and the narrowness of the petroleum pitches and HGD2 BSU molecular scale. Concerning the HGD2, the two experimental points associated to the highest characteristic time (and therefore to the larger molecular diameters) are probably linked to the beginning of deviation due to a variation of d h . Nevertheless, the centrifugated coal tar pitch presents a Stokes– Einstein linear relationship superposed to those observed for petroleum pitches. This confirms the identification of

˚ range as heavy fraction of b resins the 13 to 20 A associated to primary QI diameters and consequently the ˚ range as g resin diameters. 10 to 13 A

7. Comparison with literature The above results can be compared to the already published experimental works and theoretical molecular modelling.

7.1. Experimental studies Experimental studies on this subject are scarce and usually not focused directly on molecular scaling. Historically, the papers of Sakai et al. [14–16] are the first and most concerned by this aspect and based on intrinsic viscosity and diffusivity measurements on extracted fractions of petroleum pitches. The shape of the pitch molecules has been first studied [14] through the StaudingerSakurada-Houwink’s equation exponent p:

hi 5 KM p

this equation relates the intrinsic viscosity h i to the average molecular weight M. The p exponent has been theoretically related to the shape of the molecules: 2.0 for rod-like molecules, 0.5 for planar-like molecules and 0.0 for impermeable rigid spheres. The authors found a value mostly close to the 0.0 limit. Therefore they concluded that the molecular shape of their pitch molecules is closer to a spherical rather than a planar one. Moreover, they have estimated the corresponding hydrodynamically equivalent sphere diameter by using the Einstein viscosity equation for impermeable sphere, d h 5 2((3Mhi ) /(10p N))1 / 3

Fig. 9. Stoke–Einstein plot for petroleum pitches and HGD2 centrifugated coal tar pitch.

(13)

(14)

where M is the mean molecular weight of the concerned fraction and N is Avogadro’s number. The obtained values for the investigated fractionated pitches ranged from 10.8 ˚ They proposed that the heavier fraction was to 14.4 A. composed of stacking structures of about fourplanes of condensed rings, each plane having about four benzene rings, and being combined by short side chains. The authors have completed their work by the characterization of the axial ratio of pitch molecules under the oblate ellipsoid approximation through simultaneous measurements of the intrinsic viscosity and diffusion coefficient of pitch molecules in dilute chloroform solutions [16]. Those investigations led to the fact that the shape of their pitch molecules becomes more oblate ellipsoidal with increasing ˚ equivalent molecular weight and confirm the 10–15 A diameter range. Other authors propose molecular models in agreement

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with other experimental facts. TEM studies published by Oberlin [5] led to the conclusion that coronene and stacked molecules piled up by two or three at most, present the best agreement with the histogram of sizes of the individual 002 scattering domains of coal tar and petroleum pitches. Recent TEM studies [23] focused on mesophase spheres clearly illustrate 5 to 8 BSU dicoronene-like distorted columns. Lafdi et al. [8,24] have proposed from TEM observations using a thin sectioning based method, edge-to-edge associations of aromatic structures due to stresses occurring during either filtration or flow in coal tar b resins.

7.2. Theoretical studies The most extensive theoretical study available in the literature concerning the molecular structure of the pitches has been done by Vorpagel et al. [13]. In general, the preferred interaction between homologous polynuclear aromatic hydrocarbons is that of parallel shifted stack analogous to b-graphite-like crystal packing. This preferred geometrical arrangement is also seen in coronene and ovalene crystal structures [25,26] and well reproduced with the molecular mechanics force field analysis. According to this theoretical analysis, dimers between analogous or different molecules show the same preference for a shifted stack configuration mostly imposed by the electrostatic force induced by C–H bond dipole moment interaction. Such an arrangement is illustrated in Fig. 10 for coronene molecules. Homologous trimers present the shifted stack configuration as most stable too. Heteromerous molecules present different preferred geometric arrangements according to the monomer nature: one ovalene and two coronene molecules arrange themselves in order to have the ovalene in between the coronene molecules in a shifted stack configuration while the preferred geometric arrangement between one coronene and two ovalene molecules has the coronene arranged perpendicular to the ovalene shifted stack. The molecules which make up pitches are far less regular and have a much greater size distribution than the model compounds considered here. However, their behaviour in some respects can be understood by the model compounds. These results support the colloidal model for the behavior of pitches [8,24]. For full comparison, isolated BSU models should be completed by colloidal micelle structure. Therefore, the colloidal BSU hydrodynamical diameter should be larger than the isolated structure. Moreover, experimental results can be compared to both equatorial or rotational radii or to calculated hydrodynamically equivalent sphere radius. The shape of the diffusing species may well be important since the friction factor increases by a factor 2 as the length to width ratio of a body goes from 1 to 10. Therefore, similar calculations have to be done assuming three dimensional structure as oblate ellipsoid with axial

Fig. 10. Vorpagel most stable structure of coronene monomer, dimer and trimer.

ratio of g5L /w where L is the equatorial diameter and w the diameter along the rotation axis. In this case, the friction coefficient is written as follows:

z 5 3phw( g 2 2 1)0.5 [tan 21 ( g 2 2 1)0.5 ] 21

(15)

the diameter d D of the hydrodynamically equivalent sphere estimated from D is related to the axial ratio and the diameter along the rotation axis by, 2

0.5

d D 5 w( g 2 1) [tan

21

2

0.5 21

( g 2 1) ]

(16)

and the diameter d m of the hydrodynamically equivalent sphere estimated from h is related to the axial ratio and the diameter along the rotation axis by [16], d m 5 wg 2 / 3

(17)

This last expression will be used to compare the experimental results to calculated hydrodynamical diameters of the equivalent sphere from theoretical structural models available in the literature. According to Sakai et al. [15,16] the axial ratio ranges

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Table 2 Molecular scale parameters for monomers, dimers and trimers of ovalene (o) and coronene (c) (estimated from Vorpagel [13] and Sakai et al. [14–16]) Species

c

o

c/c

o/o

c/o

c/o/c

c/c/c

o/o/o

c/o/o

˚ w (A) ˚ L (A) g ˚ d D (A) ˚ d m (A)

9.5 9.5 1 – 9.5

9.2 11.6 1.26 10.79 10.74

9.5 14.2 1.49 12.6 12.42

9.2 16.3 1.77 13.86 13.47

9.5 15.25 1.61 13.28 13.02

9.5 18.9 1.99 15.65 15.03

9.5 18.9 1.99 15.65 15.03

9.2 21 2.28 16.9 15.95

9.5 19.95 2.1 16.33 15.58

from 1 to 3. As presented in Table 2, the calculated values of g from the monomers to the trimer structures proposed by Vorpagel [13] range from 1 to 2.28. As shown in Table ˚ for 2, rotational diameter w varies from 9.2 to 9.5 A monomers, dimers and trimers while equatorial diameter L ˚ The calculated hydrodynamical varies from 9.5 to 21 A. ˚ diameter of the equivalent sphere varies from 10.8 to 17 A.

7.3. Discussion Considering that the benzene insoluble fraction of the pitch was not investigated in the experimental study of Sakai et al., their results concern the lightest pitch fraction and therefore are in perfect agreement with the petroleum pitches and HGD2 hydrodynamically equivalent diameter ˚ of the present work. range (10.6–13.3 A) The calculated hydrodynamical diameters of the equivalent sphere with respect to D or h from the Vorpagel et al. structural study are gathered in Table 2 and have to be compared to experimental results gathered in Table 3. The isolated BSU contained within g resins of experimental ˚ (A240, HGDP) diameter range from 10.6 to 13.3 A correspond to coronene and ovalene monomers (9.5–10.74 ˚ and homologous and heteromerous dimers (12.4–13.5 A) ˚A). The b and a resins of experimental diameter range of ˚ are identified as homologous and heteromer13.3 to 20 A ˚ and edge-to-edge dimers (19–21.6 ous trimers (15–17 A) ˚ As a matter of fact, the HGD1 coal tar pitch investiA). gated in the present study has been extruded and its b resins content is 16% high (see Table 1). The centrifugated and extruded HGD2, which contains only g and light b Table 3 Experimental molecular diameter ranges obtained by oscillatory rheometry or published by Sakai et al. [14–16] Raw pitches

˚ d h min. (A)

˚ d h max. (A)

˚ d h mean (A)

HGD1 HGD2 A240 HGDP Petroleum from Sakai [14–16]

9.5 10.6 10.6 10.9 10.8

20.1 12.9 12.8 13.3 14.4

14.8 11.8 11.7 12.0 12.6

resins, does not present such large diameter range any more but only monomer and dimer scales. Therefore, the HGD1 coal tar estimated hydraulic diameters in the 17–22 ˚ range could correspond to the edge-to-edge structures on A associated heavy b resins to primary QI proposed by Lafdi et al. [8,24]. All those correlations between experimental hydrodynamical diameters and molecular structures are also in agreement with TEM studies. Nevertheless, the correlation ˚ and extrusion between largest diameters of 17 to 22 A treatment needs extensive work and will be the aim of further studies.

8. Conclusion Alpha relaxation characteristic time and viscosity measurements have been used to estimate molecular scale of two petroleum and two coal tar pitches. Obtained equivalent diameters are of the same order of magnitude as theoretical molecular models available in the literature and in perfect agreement with other published experimental approaches. The isotropic A240 and HGD petroleum pitches BSU scale are similar to the hydrodynamical diameters of coronene and ovalene monomers. Moreover, raw HGD1 coal tar pitch presents BSU equivalent diameters ranging from the same monomers to the corresponding trimers through dimers. Moreover, large hydrodynamical ˚ confirm edge-to-edge associadiameters of some 17–22 A tions in orientated heavy b resins previously observed by TEM. Centrifugated and extruded HGD2 coal tar pitch which contains only g resins and light b resins, presents the same BSU diameter range of petroleum pitches. Consequently, g resins and light b resins can be identified as lightest suspensive molecules and isolated coronene and ovalene-like monomers, heavy b resins associated to primary QI as dimers and trimers of those monomers. Therefore, pitch characterisation above the glass transition by oscillatory rheometry has been shown to be a useful technique for BSU diameter estimation. Further studies will be focused on extruded or heat-treated pitches, extracted resins and on the characterisation of interaction between BSU.

E. Daguerre et al. / Carbon 37 (1999) 1189 – 1197

z v

9. Notation aT dD

dh dm

D F g G0 k K L M N p Re t T Tg TS T Sm u w f l h hi t

shift factor (–) hydrodynamically equivalent sphere diameter estimated from D (m) hydrodynamical diameter (m) hydrodynamically equivalent sphere diameter estimated from h (m) diffusion coefficient (m 2 s 21 ) Stokes force (N) oblate ellipsoid axial ratio (–) loss modulus (Pa) Boltzmann constant (J K 21 ) Sakurada–Houwink’s constant (–) molecular length or equatorial diameter (m) average molecular weight (g mol 21 ) Avogadro’s number (–) Sakurada–Houwink’s exponent (–) Reynolds number (–) time (s) temperature (8C) glass transition temperature (8C) Ring and Ball softening point (8C) temperature at which m5 10 3 Pa s (8C) molecular velocity (m s 21 ) molecular width or rotation axis diameter (m) sliding friction coefficient (–) mean distance moved in the time t by diffusion (m) viscosity (Pa s) intrinsic viscosity (Pa s) characteristic time (s)

1197

friction coefficient (–) frequency (rad s 21 )

Acknowledgements This study was part of a CNRS-ECOTECH contract [ 94N80 / 0086 with the support of ADEME.

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