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Characterization of the morphology of rubber grade carbon blacks by thermoporometry
tMOM223i92 %f.OO+ .OO Copyright 0 1991 Pergamon Press plc
Vol. 30. No. 1, pp. 31-40. 1992 in Great Britain.
CHARACTERIZATION OF THE MORPHOLOGY RUBBER GRADE CARBON BLACKS BY THERMOPOROMETRY
OF
FRANCOISE E~RBURGER-ROLLE Centre. de Recherches sur la Phyico-Chimie des Surfaces Solides, CNRS, 24 Avenue du President Kennedy, F-68200 M&house, France
and SHINJI MISONO
Fuji Research Laboratory, Tokai Carbon Co., Ltd., 394-l Subashiri Oyama-Cho, !&to-Gun, Shizuoka-ken 410-14, Japan (Received 8 April 1991; accepted in revised form 3 June 1991)
Abstract-The intra-aggregate void size distribution (VSD) and void volume, and the fractal dimension D of rubber grade carbon blacks dispersed in water, were measured by thermoporometry. It is shown that the value of R,,, corresponding to the maximum of the VSD curve is proportional to the DBP volume and to the size R, of the primary particles constituting the aggregates. It follows that the ratio R,,,/R, can be considered as a morphological parameter leading, with the vatue of the fractal dimension D, to the characterization of the carbon black aggregates. The intra-aggregate void volume, measured by thermoporometry, is compared with that of the occluded volume, calculated by Medalia. It appears that the experimental and calculated values agree fairly well for carbon blacks exhibiting small aggregates, but that the occluded volumes, calculated for larger aggregates, are overestimated. From the determination of the ratio f/6, between the value of the volume fraction of carbon black f and the volume fraction of aggregates 4, and from the comparison with its theoretical value (R,,,IR,)“-“, it is shown that the behavior of the systems, with increasing f, is mainly controlled by the interpenetration of the aggregates. Key Words-Carbon porometry.
black, fractal aggregates,
void size distribution,
1. I~RODU~ON
occluded volume,
thermo-
parameter is the morphology of the aggregates and their arrangement within a given medium. This explains the large number of experimental works[1,2,13] devoted to the characterization of the structure of carbon black aggregates. One can consider mainly two different approaches: either analyze one by one isolated aggregates and extrapolate to the properties of the assembly, or measure directly the voidage of the assembly of aggregates and deduce morphological parameters. The first approach consists mainly in the analysis of the shape of individual isolated aggregates on electron micrographs. This method was particularly developed by Medalia~l3,14,lj] and leads’ to morphological parameters such as anisometry, bulkiness, shape factor, or equivalent diameter. More recently, the concept of fractal dimension, pertinent for the characterization of solid particle aggregates [16,17], allowed the problem to be seen in a new light, as shown for soot[l7] and carbon black aggregates[l8,19]. The fractal dimension of the aggregates can also be determined by different scattering techniques[20], particularly Small Angle Neutrons Scattering (SANS)[21], providing the dispersion is very dilute in order to avoid inte~enetration of the aggregates. Among the second type of method, the most terminant
Carbon black is widely used as a filler to modify the mechanical, electrical, or optical properties of the medium (mainly elastomers, rubbers, or paints) in which it is dispersed[ 1,2]. The physical properties of the composite material depend on the amount of carbon black, the morphology of the carbon black aggregates, and also on the carbon black-matrix interactions. In most situations, the modifications of the behavior of the matrix, due to the presence of the filler, appear only above a critical value and vary nonlinearly with the carbon black concentrationf3,4]. It is now accepted that, at least in problems of conductivity, the behavior of the composite [5,6,7,8], as well as that of the dry powder[9], can be described by a percolation law. The percolation threshold remains generally much lower than the theoretical value ($* = 0.17) determined by Scher and Zallen[lO] in the case of percolation of spheres in a continuous medium. Moreover, for a given amount of a given carbon black, the value of the conductivity[ll] or of the mechanical moduli[ 1,2,3] vary largely with the nature of the matrix. Conversely, the conductivity of a matrix loaded with the same weight percent of different carbon blacks is very different[l2]. From these observations, it follows that the de31
32
F. EHRBURGER-DOLLEand S. MISONO
widely used is the absorption of dibutyl phtalate (DBP) to the critical end point. It consists in the measurement of the volume of DBP necessary to fill the void volume of a close packing of aggregates without breaking or deformation. It is intuitively assumed that a small or a large inter- and intra-aggregate void volume (i.e., the volume of DBP) corresponds respectively to low or high structure carbon blacks (i.e., with aggregates containing a small or a large number of primary particles). This assumption was verified by Medalia[22], who has obtained a good agreement between the void volume calculated from the shape of the aggregates on electron micrographs and the measured DBP volume, provided some empirical corrections were made. Actually, the important morphological parameter is the void volume and, particularly for reinforcement purposes, the occluded volume, which was related to the DBP volume by means of semi-empirical equations [23] obtained with some arbitrary assumptions due to the difficulty of determining precisely the void size distribution (VSD). In a recent paper[24] we have shown, for several electrical grade carbon blacks, that the VSD curves can be measured by thermoporometry and wholly describe the structure of the aggregates and their arrangement in different media. Moreover, the analysis of the shape of the curves also provides the value of the fractal dimension[25,26]. The aim of the present paper is to describe the results obtained by thermoporometry on a series of rubber grade carbon blacks. Some preliminary results were already presented at Carbone 90[27]. We have also compared our values, obtained from the VSD curves, with some morphological data calculated by Medalia.
2. EXPERIMENTAL
2.1 Carbon black samples The carbon blacks that were investigated were rubber grade furnace blacks of various surface areas and DBP volume. Their morphological characteristics are collected in Table 1. For all these blacks, the primary particles are nonporous. The carbon black surface is oleophilic rather than hydrophilic. The hygroscopic character of some
blacks can be attributed to the presence of hydroxyl and carboxyl groups onto which the water molecules are adsorbed. The adsorbed water, however, probably remains located on these sites[2], without forming a continuous adsorbed film able to screen the sticking of aggregates, unlike an alkane as shown previously[24]. It follows that, in water, the interpenetration of the aggregates is low. Therefore water was chosen as the dispersing medium. The amount of carbon black was 0.25 g per cm3 of water for all samples. This concentration corresponds to a total volume of water equal to 4 cm3 per g of carbon black, which is much larger than the DBP value for all samples, and small enough to ensure a stable, nondemixing dispersion. In order to achieve a complete dispersion of the primary (irreversible) aggregates, the carbon black was dispersed in water by ultrasonication. The ultrasonic waves were transmitted to the carbon black-water system by means of a “Bransonic 12” (lOOV, 40W) apparatus, during one minute. To avoid excessive heating of the sample, the test tube containing the carbon black-water mixture was immersed in cold water.
2.2 Thermoporometry measurements The principle of the method[28,29] and the experimental procedures have already been discussed in details[24,30]. The method is based on the lowering AT of the triple point temperature of a liquid filling a pore of radius R. AT and R are related by: R = AIAT f B where A = 66.67 and B = -0.57 for water[28,29]. The determination of the volume of liquid freezing at a given temperature To - AT below the normal freezing temperature T,, was performed by Differential Scanning Calorimetry (DSC 30 METILER) at a very low cooling rate (0.1 K/ min). The mass of the sample was always small enough to avoid internal temperature gradients or kinetic effects. Because of the low level of the DSC heat of flux under these conditions, noise was always present, leading to some uncertainties, particularly as to determination of the baseline and therefore as to onset of the peak in the low temperature side (i.e., the contribution of the smallest pores on the VSD curve). The upper limit of pore size in water is 320 nm (AT = 0.2K).
Table 1. Morphological characteristics of the carbon black samples Carbon black Seast*
N351 NH
N347 3H
N330 3
N326 300
N339 KH
N220 6
NllO 9
Nitrogen surface area (m*/g) CTAB surface area (m*/g) 1, No. (mglg) DBP absorption (cm’/g) 24M4DBP absorption (cm-‘/g) Tint (% IRB#3)
72 73 70 1.26 0.98 95
84 82 84 1.25 0.99 98
81 82 83 1.01 0.86 100
83 84 85 0.76 0.71 112
93 92 90 1.19 1.00 106
118 112 120 1.16 0.98 116
140 132 143 1.13 0.97 128
*Seast is the trade name of blacks produced by Tokai Carbon Co., Ltd.
Morphology 3. RESULTS
AND DISCUSSION
3.1 Void size ~i~tributio~ (KSD) curves Figures la and b show the VSD curves obtained for the different carbon blacks, dispersed in water (0.25 g/cm3). The total void volumes, VP, calculated by integration of the curves between R,, (onset of
’ ’
1 dV/dR
33
of carbon blacks
the VSD curve) and 320 nm, are given in Table 2. From the study of the VSD curves obtained on simmated aggregates[26], we have shown that the position of the maximum of the distribution, I-Z,,,,,, corresponds to the size of the largest internal void. When the aggregates are not isolated but surrounded by close neighbors, R,,, also represents the limit of
(4
Ccm3/g/nn)
j
-01 :
.OOl :
R (nm)
.l
dV/dR
7
(cm3/cjnn)
04
.Ol
.OOl
.OOOl 1 10 Fig. 1. Void size distribution CAR30:1-B
I 100
R (nm)
1000
(VSD) curves obtained for the different furnace blacks, ultrasonically dispersed in water (0.25 cm’ig).
34
F. EHRBURGER-DOLLE and S. MISONO Table 2. Data obtained from the analysis of the VSD S (Fig. 2). From a least square fitting, one obtains curves for the different carbon blacks (uncertainty on data the following relation: is less than about 10%) Sample
nm
Rimin Rs v* vinevcd nm nm R,,,.,IR, cm’ig cm3/g cm’/g
N326 N330 NllO N220 N339 N351 N347
33 47 30 39 50.5
17 23 16 20 20
R mm
TB#SSOO 73 206 m2/g 1.55 cm3/g TB#4500 108 57 m*/g 1.60 cm’ig XC-72 82 214 m*/g 1.76 cm3/g Acetylene 100 black 80 m’fg 2.15 cm31g
20 20 12 14 17.5
1.69 2.34 2.60 2.81 2.89 2.89 3.07
0.72 1.29 1.11 1.09 1.35 1.64 1.54
0.39 0.52 0.54 0.53 0.77 0.79 0.73
0.43 0.63 0.73 0.75 0.78 0.83 0.83
24 20 (7.9)
3.65
1.40
0.79
1.06
31 29
3.78
1.40
0.72
1.10
20 20 (7.6)
4.08
1.39
0.70
1.23
30 20
S
0.50
0.20
1.52
interpenetration of the aggregates and, furthermore, the upper limit of the fractal range. As we have shown earlier[24,25,26], the fractal dimension D can be determined from the VSD curve by using the following relation:
R max= 3824 VDBpfS
(2)
in which R,, is expressed in nm, V,,, in cm3/g, and S in m2tg. We have also indicated the value of R,, vs. V,,,/ S for the previously studied electrical grade carbon blacks[24]. The parameters of the fit are almost unchanged when the values obtained for acetylene black and TB#4500 are taken into account. On the contrary, the high surface area carbon blacks, TB#5500 and XC-72, do not fit with eqn (2). However, as indicated by their CTAB surface areas[l2], these blacks are slightly microporous. Such a microporosity leads to a high surface area, but to a micropore volume that can be neglected in comparison with the DBP volume. We may now assume that only the external surface area has to be taken into account in eqn (2) and calculate the value of S,,, from eqn (2) and the experiments R,, value. One obtains for both carbon blacks S = 81 m*/g. As will be seen further on, this value agrees well with other observations. The ratio V&S can be considered as the thickness, eosP of the solvent covering the whole surface. Using a coherent system of units, eqn (2) becomes: Rmax= 3.824 eDsp
(2a)
eDr,P= 0.26 R,,,.
@b)
or
a [(Lx -
R)I(R - Rmin)13-‘* (1)
It appears that all these values lie between 1.65 and 1.80. Further refinements are needed, however, to ascertain any signi~cant difference in the values of D for the considered carbon blacks. Therefore, in the following, a mean value of D = 1.75 will be used. These values are also comparable to the fractal dimensions obtained previously[l9,24] for electrical grade furnace blacks, and agree with a cluster-cluster brownian aggregation process[l6]. It follows that all these carbon blacks are self similar, in a first approximation. Therefore, the differences in their morphological chara~te~sti~s cannot be attributed to differences in their fractal dimension but, as will be discussed in the following, to their size.
It means that the mean distance between the aggregates is equal to 0.52 R,,. On Fig.2, we have also plotted the value of RO, defined by ~ed~ia~l3] as the radius of the solid sphere containing the same volume of carbon black as the aggregate and related to V,,, and S through the following relation[31]: R, = (2540 f 7100 VDBp)/2S.
(3)
It appears that the values of R, are slightly larger than that of R,. The value of R, calculated for TI3#5500 and XC-72 seems to fit with that obtained for the other carbon blacks. However, if R, is supposed to characterize the morphology of the aggregates in terms of their physical behavior, these values are unrealistic, as the properties of these two blacks are closer to that of TB#4500 and acetylene black than that of the rubber grade NllO carbon black.
3.2 Relation between R_, the DBP volume V,,, and the BET sueace area S From Table 1 and 2, it appears that the value of R,, obtained for carbon blacks of similar surface area (N326, N330, and N347) increases with VDBp 3.3 Characterization of the aggregates by means of and decreases with the surface area S, for carbon blacks of similar V,,, values (for example, N220, &JR For the nonporous carbon blacks, the radius of N339, N347, and N351). Therefore, we will now examine the variation of R,,, as a function of VDBp/ the primary particles can be approximated from the
Morphology
R max
140
of carbon btacks
hm’
R,= (2540*7lOOV,,,)/2SCnrnl
x
XC-T?
80
Ti3#!iSKt
-0
X
15
10
5
20
25
30
Fig, 2. Evolution of R,,, vs. (IO00 V&S) for the rubber grade (A) and the electrical grade (X ) carbon blacks. The solid line is obtained by a least square fitting of all points (except TB#5500 and XC-72) (3 = 0.33)_ The value of R, calculated by means of the relation R, = (2540 t 7100 VD,)/2S, after Medalia, are also indicated for rubber grade (@i and electrical grade (*> carbon blacks. BET
surface area
by means
of the
folk~~wing mass-
v&.ime relation : R, = 3 1o”:Slp
f4)
in which R, is expressed in nm and S in m*/g. p is the true specific gravity of the primary particles. For all blacks (except acetylene black for which p = 1.87 cm’/g), a mean value of p = 1.84 g/cm3 was used. From eqns (2) and (4), one obtains: &JR,
= I_275 VDBpfus
(51
where v, = lip is the true specific volume of the
carbon black particles (us = 0.543 cm31g). The value of I?, and R,,,IR, are indicated in Table 2. The importance of the ratio R,,,IR,, which can be considered as a characteristic length, has already been evidenced, for electrical grade carbon blacks, in the value of the volume fraction of carbon black at the percolation threshold in the dry state[9]. Our previous study of the WD curves of simulated aggregates of a given fractal dimension[26] has also shown that &JR, increaxs with the number of particles, N, in the aggregate. Unfortunately, the determination of N from rhe value of R,,.JR, lacks precision in the present range of sizes. To give an
order of magnitude, values of R,,IR, ranging between 3 and 4 correspond approximately to vahtes of N between 64 and 128 particles. From the geametrical analysis of the carbon black aggregates on electron microscope photos, Medaha obtained the following relation: P&d = 1..432 (1 -t- 2.139l~‘,,,)~.~~,
(61
which is generally used in a simplified form: D,ld = 1.321 + 3.7 v,,,.
@a)
In this refation, D, ( = 2R,) is the diameter of the solid sphere introduced above and d is the measured diameter of the primary particles. Medaha deduced eqn (3) from eqn (6a) by assuming that the number average diameter d, (in relation (6) and (6a)) is related to the volume average diameter d, ( = 2R,) by: n’, = dJl.68
= 1920/S.
(7)
This explains why the values of DJd calculated for a given V,,, are much larger than &JR,. The factor I.58 was introduced by P&d&a in order to include the confr~hu~jon from the effect of particte fusion and the particte size distribution.
36
F. EHRBURGER-DOLLE and S. MISONO
The above comments raise the question of the real size of the primary particle constituting the aggregate, which will now be discussed and justified by the size of the smallest void R,, obtained experimentally. In an assembly of spheres of radius R*, the radius of the cavity depends on the volume fraction of spheres (i.e., the coordination number). Particularly, it remains smaller than R* when the coordination number is larger than 4 (tetragonal arrangement). From the low value of the fractal dimension and from the examination of pictures obtained by electron microscopy, it is clear that the mean coordination number of the particles within the aggregate is smaller or at least equal to 4. It follows that the size of the particle R, is expected to be close to R,,. Table 2 indicates that R, is generally slightly smaller than R,,,. In fact, R,,, is determined in the DSC measurements, from the onset of the deviation of the baseline. Therefore, it is subjected to an uncertainty of about 2 nm on this range of R values. Moreover, the contribution of the smallest pores (of size R,) may be too low to be accurately measured, particularly with decreasing R, as observed for N220 and NllO. Thus, we will now assume that the size of the particles constituting the aggregates (i.e., that of the particles really involved in the aggregation process leading to the characteristic morphology of the aggregates[ 161) is correctly evaluated by the value of R, calculated from the BET surface area, except in the case of microporous carbon blacks such as TB#5500 or XC-72. However, the value of R, ( = 20 nm) calculated by using S = 81 m2/g obtained above by means of eqn (2) agrees with that of R,,.
3.4 Determination of the total and internal void volume
By integration of the VSD curve between R,, and R = 320 nm, one obtains the total void volume VP (Table 2). For three blacks (NllO, N220, and N330) VPis close to the DBP volume. For these samplesthe VSD curves show a second family of pores at higher R values, which can be attributed to interaggregate voids. The measured value of VP includes this contribution. For the electrical grade carbon blacks, the value of VPremains close to 1.40 cm3/g and is smaller thanVDsP whereas for all other rubber grade carbon blacks the total void volume is systematically larger than V,,,. This observation will be discussed in a later paragraph. From the assumption that R,, is the size of the largest internal voids, it follows that the internal void volume V,,, is determined by integration of the VSD curve between R,,, and R,,, (Table 2). It follows also that Vi,, could be considered as the occluded volume, derived from the DBP volume by Medalia[23] by means of the following relation: V,,,,,lu, = (VD,, - 0.215)/0.6825.
(8)
The values of V,,, calculated by means of eqn (8) are indicated in Table 2. For most of the rubber grade carbon blacks, the values of Vi,, agree fairly well with the calculated value of VO,,,. For N220 and NllO, however, these values differ largely and the discrepancies continue to increase with increasing V DBP,for the electrical grade carbon blacks. On Fig. 3 we have plotted the dimensionless ratio V,,,lu, vs. R,,,IR,. The solid line corresponds to eqn
Fig. 3. Evolution of the internal void volume V,.,lus (A, x) measured between R,,s” and R,,. and the total void volume VP/us (m,*)measured up to 320 nm for the different samples, versus R,,,IRs and V,,,,/u, = (0.78R,,,/Rs - 0.215)10.6825 (after comparison with: -.-. VDBp/vs = 0.78 RmaxlRs calculated for D = 1.75 and D = 2. Medalia, eqn (8) and eqn (5)) -e.-..(R,,,IRs)D~3
37 Morphology of carbon blacks 320 nm, are taken into account for the calculation (8) in which V,,, was substituted by its value in terms of the void volume, then eqn (10) and (11) become: of R,,,IR, calculated with eqn (5). This plot suggests a stepwise increase of VJv, when R,,IR, increases. For R,,,IR, ranging between about 2 and 2.84 (i.e., & = (I + V,/~,)/(V,&, + I) (12) for NllO, N220, and N330), V,,,/u, remains almost f/4, = l/(V,/u, + 1). (13) constant, close to 1. Above R,,,IR, = 2.84 and up to about 3.7, V,,,lu, is close to 1.4. When R-/R, In the present experiments, the amount of carbon continues to increase, VJu, starts to decrease, black per cm3 of water is 0.25 g, thus f = 0.12. In reaching a very low value for acetylene black. Howthe case of the electrical grade carbon blacks, the ever, the fractal dimension of this black (D = 2[19]) amount of carbon black was 0.15 g/cm3, therefore is different from that of the other blacks (D = 1.75). {~~l.p. The experimental values are indicated in From the above observations, one may conclude that V,,, is the occluded volume of the aggregate and that the calculation of VoClby eqn (8) leads to overestimated values for large aggregates. 3.5 Determination of the volume fraction qbof aggregates
Up to now, the analysis of the data was mainly focused on the morphology of the aggregates considered as isolated nearly up to length R,,IR,. Let us now examine the arrangement of the assembly of aggregates within the system, by considering the volume fraction of carbon black particles, f, and the volume fraction of aggregates, 4. By definition:
= l/(VJu,
When the contribution
(14)
- (R,,,/RY
and the void volume per particle equals V/n: V/n = (R,,,/R,)3-D
-1
= VJu,.
+ 1)
+ 1).
f/&t
(10)
(11)
of the larger voids. up to
= (Rmax/R,Yi.
(16)
From Fig. 4, it appears clearly that the void volume Vi”,,,> measured by thermoporometry (dotted surface), is smaller than the void volume of the boxes covering the aggregates (dashed surface), as the external void space cannot be measured. This explains why eqn (15) is not verified (Fig. 3). Moreover, in-
Table 3. Volume fractions of aggregates and comparison with (R,,,/R,)D~3, calculated with D = 1.75 for all carbon blacks but acetylene black (for which D = 2). From the possible fluctuations of D (between 1.65 and 1.85) observed experimentally for the different furnace blacks, the range of uncertainty on (R,,.IR,)D~3 is & 0.03 Sample 7:
0.12 - 4 cm’ig
N326 N330 NllO N220 N339 N351 N347 p:
1.69 2.34 2.60 2.81 2.89 2.89 3.07
0.21 0.23 0.24 0.24 0.29 0.29 0.28
0.28 0.40 0.36 0.36 0.42 0.48 0.46
0.58 0.51 0.50 0.51 0.42 0.41 0.43
0.43 0.30 0.33 0.33 0.29 0.25 0.26
0.52 0.35 0.30 0.27 0.27 0.27 0.25
0.42 0.35 0.32 0.32 0.31 0.30 0.30
3.65 3.78 4.08 5.00
0.18 0.18 0.17 0.10
0.27 0.27 0.27 0.13
0.40 0.43 0.44 0.73
0.28 0.28 0.28 0.58
0.20 0.19 0.17 0.20
0.26 0.25 0.24 0.20
0.075 - 6.67 cm3/g
TB#5500 TB#4500 XC-72 Acetlylene black
(15)
(9)
It follows: f/&l
V = (R,,,/RJ3
Thus eqn (11) becomes:
f = l/(vrrZOluS + 1) +,., = (1 + Vi.,/W(VW,/~~
One of the methods to measure fractal objects is to cover it with boxes of increasing size, R/R, (Fig. 4). The number of particles, n, within one box scales as (R/R#’ and the volume of the box equals (RIR,)3. It follows that the void volume in a box of size (R,,,l R,) is:
38
F. EHRBURGER-DOLLEand S. MISONO indicates that, for a given carbon black, the ratio between the volume fractionfof carbon black within the system (which is experimentally imposed by the weight of carbon black within the system) and the volume fraction + of aggregates, which is measured by &,,( or, more realistically, by & is a constant determined by the size and the fractal dimension of the aggregates. The validity of eqn (16) implies that the boxes of size R,,,IR, do not interpenetrate each other (i.e., they behave as hard core boxes) when f increases. It follows that if some boxes covering the external parts of the aggregate interpenetrate (behaving as soft core boxes), the ratio f/+ should become larger than (R,,IR,)D-3. The validity of eqn (16) over the whole range of f values in which the system behaves as a continuum, will now be discussed. Near the percolation
Fig. 4. Example of an aggregate of N347 (magnification 225,000) covered by boxes of size 2R,,, = 120 nm (equivalent to the dashed surface). The dotted region corresponds to the void volume actually measured by thermoporometry up to R = R,,,. u corresponds to a void of diameter corresponds to the convex hull, inequal to 2R,,,. ---cluding voids of diameters larger than 2R,,,.
creasing the size of fractal aggregates is accompanied by a larger development of “arms” (i.e., by nonmeasurable void volume). Therefore, the discrepancy is expected to increase with the increase of R,,,l R,, as observed on Fig. 3. Taking into account the void volume, V, of the larger voids, partly balances the difference. In the case of small aggregates, however, part of the larger voids are inter-aggregate voids (as evidenced by the presence of a secondary peak in the VSD curve (Fig. l), and the real volume will be located somewhere between Vi”, and VP. 3.6 Correlations between f/4, fDBp and (R,,IR,)D-3 A disordered assembly of individual objects behaves as a continuous medium between two limits of volume fraction:
threshold
fc in
the dry state.
It may be assumed that, at the mechanical percolation threshold, the volume fraction $I of aggregates (i.e., the volume fraction of boxes of size R,,,IR,), is equal to the critical value & = 0.17[ lo] calculated for spheres. It follows from eqn (16) that the critical volume fraction of carbon black, fc,should be equal to 0.17 (R,,IR,)D-3. Such a relation was actually verified for electrical grade carbon blacks[9] in the dry state by uniaxial compaction at very low pressure. At intermediate volume fractions in water. It was shown [9,24] that, for the electrical grade carbon blacks presented here, the threshold f” above which a stable dispersion in water is achieved is slightly larger than fc.This was attributed to an interpenetration of the external boxes covering the aggregates, as also suggested by the observation that f/+ > (R,,,l RJDm3, in such a way that the volume fraction of aggregates becomes close to 0.27, as seen in Table 3, for systems measured with f = 0.075, just above the threshold. This is in agreement with the threshold (& = 0.29) expected for soft core percolating spheres[32]. It is also interesting to note that this value corresponds to an internal volume fraction (i.e.,to occluded liquid) equal to 0.17. Obviously, the very high value of fl+ obtained for acetylene black results from the small values of V, or VP measured for this black. This is probably due to the shape (characterized by D = 2) of its aggregates which is, however, very different from that of chemical aggregates[l6] having almost the same fractal dimension. From the comparison of fl$ and (R,,,IR,)D-3, for the rubber grade carbon blacks, the following information can be deduced:
l a lower limit, fc, which may be a percolation threshold; l an upper limit which is the tightest arrangement of the objects without breaking. Such an arrangement is actually measured by the DBP end point l For small aggregates (N326, N330), the value method and corresponds to a volume fraction fDBP of (R,,,/R,)D-3, calculated with D = 1.75 for all of carbon black, related to the measured VDBPby eqn blacks, is between f/9., and f/c&. Because in this case V, also includes a contribution from the inter(9): &BP = l/(VLW&, + 1). aggregate void volume, f/4+, could be underestimated. The real value of f/c+is therefore probably It is observed that~fc and fDBplargely change from close to that of (R,,,IRS)D-3, suggesting no interpeone carbon black to the other and are obviously netration. Such a result would be consistent with the related to the structure of the aggregates. Eqn (16)
39
Morphology of carbon blacks fact that the size of the covering boxes, R,,,/R,,
is only slightly larger than that of the particles, leaving therefore ony a small external void volume. l For the other rubber grade carbon blacks, f/d, is generally slightly larger than (R,,,/RJDe3, depending mainly on the value of D, ranging between 1.65 and 1.85, used for the calculation (Table 3). Near the DBP end point. By de~nition (eqn (9)), fDBp= l/(1 + V&u,) (i.e., by using eqn (5)): fDBB= l/(1 With the assumption verified for fDBp:
+ 0.78R,,,iR,).
(17)
of hard core boxes, eqn (lo) is
~I,BP/+DBP= (RmaxlRsY3
(18)
where &,ep is the maximum volume fraction of aggregates. As a first approximation, this value will be taken as unity: QtDBp= 1. The graphic resolution of the system of equations (17) and (18) leads to the following solutions: e for D = 1.75 l
forD
= 2
R,,,IR,
=2.25
fDBp = 0.363
R,,,IR,
= 4.54
fDBp= 0.220.
From Table 3, it appears that this is experimentally verified for N330. The result obtained for acetylene black is also consistent as the difference between the values of fDBp calculated with R,,,IR, = 5 by eqn (17) (fnsp = 0.202) and by eqn (18) (fosP = 0.200) is negligible in this range of R,,,IR, values. It may therefore be concluded that there is a limiting size of aggregates, which depends only on the fractal dimension D, below which the aggregates behave as hard spheres and above which they behave as soft, interpenetrable spheres, over a limited thickness. The question that now arises is why eqn (17) describes this interpenetration. We think that the answer could be obtained from the experimental study of dispersions of carbon blacks in undecane, for which the interpenetration is evidenced by a displacement of the VSD curve toward smaller void sizes, resulting from the increase of the number of smaller interaggregate voids in the overlapping layers.
the fractal dimension D; the size of the largest internal voids R,,,IR, (depending also on the BET surface area). l
l
Furthermore, these experimental parameters allow characterization of the organization of the aggregates in an assembly over the whole domain over which it is a continuum, from the percolation threshold up to the DBP end point. It is also interesting t o note that the DBP volume is actually a classification parameter for the carbon blacks. However. we have evidence that, particularly at low volume fractions, near the percolation threshold, the properties of the system are completely described only if the fractal dimension of the aggregates is taken into account. Acknowledgements-The
authors would like to thank Jacques Lahaye for fruitful discussions. REFERENCES
1. J. B. Donnet and A. Voet, Carbon Black: Physics, Chemistry and Elastomer Reinforcement, Marcel Dek-
ker, New York (1976). A. I. Medalia and D. Rivin, In Characterization of Powder Surfaces (Edited by G. D. Parfitt and K. S. W. Sing), pp. 279-351. Academic Press, London (1976). 3. L. P. Fox, In Carbon Black-Polymer Composites (Edited by E. K. Sichel), pp. 163-187. Marcel Dekker, New York(1982). 4. F. Carmona, Ann.Chim.Fr. 13, 395 (1988). 5. I. Balberg and S. Bozowski, Solid State Comm. 44, 551 (1982). 6. F. Carmona, F. Barreau, and R. Canet, J. PhysLetters 2.
41, L534 (1980). 7. 8. 9.
10. 11. 12.
13. 14.
L. Benguigui, J. Yacubowicz, and M. Narkis,.!. Polym. Sci. Polym. Phys. 25, 127 (1987). I. Balberg, P&s. Rev. Letters 59, 1305 (1987). F. Ehrburaer-Dolle. S. Misono. and J. Lahave. SEEClub 11 “Etudes Gentrales” Paris, 10 Octobre 1990, pp. 197-204. H. Scher and R. Zallen, J. Chem. Phys. 53, 3759 (1970). K. Miyasaka, K. Watanabe, E. Jojima, H. Aida, M. Sumita, and K. Ishikawa, J. Mater.&?. 17,161O (1982). W. F. Verhelst, K. G. Volthuis, A. Voet, P. Ehrburger, and J. B. Donnet, Rubber Chem. Technot. SO. 735 (1977). A. I. Medalia, In Carbon Black-Polymer Composites (Edited bv E. K. Sichelf, DR. l-49. Marcel Dekker. ” L* New York (1982). A. I. Medalia and F. A. Heckman, Carbon 7, 567 (1969).
15. 16.
4. CONCLUSION
The experimental results described in the present paper clearly evidence the interest of the measurement of the actual void size distribution and occluded volume in carbon black aggregates. Our approach, by using the principles of fractal geometry, leads to a more pertinent analysis of the morphology of the aggregates, which is now completely characterized by two parameters:
17.
A. I. Medalia, Powder Technot. 4, 117 (1970). R. Jullien and R. Botet, Aggregation and Fractal Aggregates, World Scientific, Singapore (1987). R. J. Samson, G. W. Mulholland, and J. W. Gentry, Langmuir 3, 272 (1987).
18. 19.
X. Bourrat, A. Oberlin, H. Van Damme, C. Gateau, and R. Bachelar, Carbon 26. 100 11988). F. Ehrburger-Dolle and M.’ Ten&, Carbon 28, 448 (1990).
20.
21. 22.
P. W. Schmidt, In The Fractal Approach to Heterogeneous Chemistry (Edited by D. Avnir), pp. 67-79. Wiley, Chichester (1989). L. Salomt and F. Carmona, Carbone 90, Paris 16-20 3rd~ 1990, Extended Abstracts pp. 314-315. A. I. Medalia, J. Colbid Interface Sci. 32, 115 (1970).
40
F. EHRBURGER-DOLLEand S. MISONO
23. A. I. Medalia, Rubber C/rem. Technol. 45,1171(1972). 24. E Ehrburger-Dolle, S. Misono, and J. Lahaye, J. Colloid Interface Sci. l35, 468 (1990). 25. E Ehrburger and R. Jullien, In Characterization of Porous Solids (Edited by K. K. Unger et al.) pp. 441-
449. Elsevier, Amsterdam (1988). 26. F. Ehrburger-Dolle, P. Mors, and R. Jullien, J. Colloid Interface Sci. (in press). 27. S. Misono and F. Ehrburger-Dolle, Carbone 90, Paris 16-20 July 1990, Extended Abstracts, pp. 112-l 13.
28. M. Brun, A. Lallemand, J. F. Quinson, and C. Eyraud, 7’hermochimica Acta 21, 59 (1977): 29. J. F. Quinson and M. Brun, In Characterization of Porous Solids (Edited by K. K. Unger et al.) Elsevier, Amsterdam (1988), pp. 307-315. 30. F. Ehrburger and J. Lahaye, Colloids Surf. 23, 105 (1987).
31. A. I. Medalia and L. W. Richards, J. Colloid Interface Sci. 40, 233 (1972). 32. F. Carmona, Physica A 157, 461 (1989).
Vol. 30. No. 1, pp. 31-40. 1992 in Great Britain.
CHARACTERIZATION OF THE MORPHOLOGY RUBBER GRADE CARBON BLACKS BY THERMOPOROMETRY
OF
FRANCOISE E~RBURGER-ROLLE Centre. de Recherches sur la Phyico-Chimie des Surfaces Solides, CNRS, 24 Avenue du President Kennedy, F-68200 M&house, France
and SHINJI MISONO
Fuji Research Laboratory, Tokai Carbon Co., Ltd., 394-l Subashiri Oyama-Cho, !&to-Gun, Shizuoka-ken 410-14, Japan (Received 8 April 1991; accepted in revised form 3 June 1991)
Abstract-The intra-aggregate void size distribution (VSD) and void volume, and the fractal dimension D of rubber grade carbon blacks dispersed in water, were measured by thermoporometry. It is shown that the value of R,,, corresponding to the maximum of the VSD curve is proportional to the DBP volume and to the size R, of the primary particles constituting the aggregates. It follows that the ratio R,,,/R, can be considered as a morphological parameter leading, with the vatue of the fractal dimension D, to the characterization of the carbon black aggregates. The intra-aggregate void volume, measured by thermoporometry, is compared with that of the occluded volume, calculated by Medalia. It appears that the experimental and calculated values agree fairly well for carbon blacks exhibiting small aggregates, but that the occluded volumes, calculated for larger aggregates, are overestimated. From the determination of the ratio f/6, between the value of the volume fraction of carbon black f and the volume fraction of aggregates 4, and from the comparison with its theoretical value (R,,,IR,)“-“, it is shown that the behavior of the systems, with increasing f, is mainly controlled by the interpenetration of the aggregates. Key Words-Carbon porometry.
black, fractal aggregates,
void size distribution,
1. I~RODU~ON
occluded volume,
thermo-
parameter is the morphology of the aggregates and their arrangement within a given medium. This explains the large number of experimental works[1,2,13] devoted to the characterization of the structure of carbon black aggregates. One can consider mainly two different approaches: either analyze one by one isolated aggregates and extrapolate to the properties of the assembly, or measure directly the voidage of the assembly of aggregates and deduce morphological parameters. The first approach consists mainly in the analysis of the shape of individual isolated aggregates on electron micrographs. This method was particularly developed by Medalia~l3,14,lj] and leads’ to morphological parameters such as anisometry, bulkiness, shape factor, or equivalent diameter. More recently, the concept of fractal dimension, pertinent for the characterization of solid particle aggregates [16,17], allowed the problem to be seen in a new light, as shown for soot[l7] and carbon black aggregates[l8,19]. The fractal dimension of the aggregates can also be determined by different scattering techniques[20], particularly Small Angle Neutrons Scattering (SANS)[21], providing the dispersion is very dilute in order to avoid inte~enetration of the aggregates. Among the second type of method, the most terminant
Carbon black is widely used as a filler to modify the mechanical, electrical, or optical properties of the medium (mainly elastomers, rubbers, or paints) in which it is dispersed[ 1,2]. The physical properties of the composite material depend on the amount of carbon black, the morphology of the carbon black aggregates, and also on the carbon black-matrix interactions. In most situations, the modifications of the behavior of the matrix, due to the presence of the filler, appear only above a critical value and vary nonlinearly with the carbon black concentrationf3,4]. It is now accepted that, at least in problems of conductivity, the behavior of the composite [5,6,7,8], as well as that of the dry powder[9], can be described by a percolation law. The percolation threshold remains generally much lower than the theoretical value ($* = 0.17) determined by Scher and Zallen[lO] in the case of percolation of spheres in a continuous medium. Moreover, for a given amount of a given carbon black, the value of the conductivity[ll] or of the mechanical moduli[ 1,2,3] vary largely with the nature of the matrix. Conversely, the conductivity of a matrix loaded with the same weight percent of different carbon blacks is very different[l2]. From these observations, it follows that the de31
32
F. EHRBURGER-DOLLEand S. MISONO
widely used is the absorption of dibutyl phtalate (DBP) to the critical end point. It consists in the measurement of the volume of DBP necessary to fill the void volume of a close packing of aggregates without breaking or deformation. It is intuitively assumed that a small or a large inter- and intra-aggregate void volume (i.e., the volume of DBP) corresponds respectively to low or high structure carbon blacks (i.e., with aggregates containing a small or a large number of primary particles). This assumption was verified by Medalia[22], who has obtained a good agreement between the void volume calculated from the shape of the aggregates on electron micrographs and the measured DBP volume, provided some empirical corrections were made. Actually, the important morphological parameter is the void volume and, particularly for reinforcement purposes, the occluded volume, which was related to the DBP volume by means of semi-empirical equations [23] obtained with some arbitrary assumptions due to the difficulty of determining precisely the void size distribution (VSD). In a recent paper[24] we have shown, for several electrical grade carbon blacks, that the VSD curves can be measured by thermoporometry and wholly describe the structure of the aggregates and their arrangement in different media. Moreover, the analysis of the shape of the curves also provides the value of the fractal dimension[25,26]. The aim of the present paper is to describe the results obtained by thermoporometry on a series of rubber grade carbon blacks. Some preliminary results were already presented at Carbone 90[27]. We have also compared our values, obtained from the VSD curves, with some morphological data calculated by Medalia.
2. EXPERIMENTAL
2.1 Carbon black samples The carbon blacks that were investigated were rubber grade furnace blacks of various surface areas and DBP volume. Their morphological characteristics are collected in Table 1. For all these blacks, the primary particles are nonporous. The carbon black surface is oleophilic rather than hydrophilic. The hygroscopic character of some
blacks can be attributed to the presence of hydroxyl and carboxyl groups onto which the water molecules are adsorbed. The adsorbed water, however, probably remains located on these sites[2], without forming a continuous adsorbed film able to screen the sticking of aggregates, unlike an alkane as shown previously[24]. It follows that, in water, the interpenetration of the aggregates is low. Therefore water was chosen as the dispersing medium. The amount of carbon black was 0.25 g per cm3 of water for all samples. This concentration corresponds to a total volume of water equal to 4 cm3 per g of carbon black, which is much larger than the DBP value for all samples, and small enough to ensure a stable, nondemixing dispersion. In order to achieve a complete dispersion of the primary (irreversible) aggregates, the carbon black was dispersed in water by ultrasonication. The ultrasonic waves were transmitted to the carbon black-water system by means of a “Bransonic 12” (lOOV, 40W) apparatus, during one minute. To avoid excessive heating of the sample, the test tube containing the carbon black-water mixture was immersed in cold water.
2.2 Thermoporometry measurements The principle of the method[28,29] and the experimental procedures have already been discussed in details[24,30]. The method is based on the lowering AT of the triple point temperature of a liquid filling a pore of radius R. AT and R are related by: R = AIAT f B where A = 66.67 and B = -0.57 for water[28,29]. The determination of the volume of liquid freezing at a given temperature To - AT below the normal freezing temperature T,, was performed by Differential Scanning Calorimetry (DSC 30 METILER) at a very low cooling rate (0.1 K/ min). The mass of the sample was always small enough to avoid internal temperature gradients or kinetic effects. Because of the low level of the DSC heat of flux under these conditions, noise was always present, leading to some uncertainties, particularly as to determination of the baseline and therefore as to onset of the peak in the low temperature side (i.e., the contribution of the smallest pores on the VSD curve). The upper limit of pore size in water is 320 nm (AT = 0.2K).
Table 1. Morphological characteristics of the carbon black samples Carbon black Seast*
N351 NH
N347 3H
N330 3
N326 300
N339 KH
N220 6
NllO 9
Nitrogen surface area (m*/g) CTAB surface area (m*/g) 1, No. (mglg) DBP absorption (cm’/g) 24M4DBP absorption (cm-‘/g) Tint (% IRB#3)
72 73 70 1.26 0.98 95
84 82 84 1.25 0.99 98
81 82 83 1.01 0.86 100
83 84 85 0.76 0.71 112
93 92 90 1.19 1.00 106
118 112 120 1.16 0.98 116
140 132 143 1.13 0.97 128
*Seast is the trade name of blacks produced by Tokai Carbon Co., Ltd.
Morphology 3. RESULTS
AND DISCUSSION
3.1 Void size ~i~tributio~ (KSD) curves Figures la and b show the VSD curves obtained for the different carbon blacks, dispersed in water (0.25 g/cm3). The total void volumes, VP, calculated by integration of the curves between R,, (onset of
’ ’
1 dV/dR
33
of carbon blacks
the VSD curve) and 320 nm, are given in Table 2. From the study of the VSD curves obtained on simmated aggregates[26], we have shown that the position of the maximum of the distribution, I-Z,,,,,, corresponds to the size of the largest internal void. When the aggregates are not isolated but surrounded by close neighbors, R,,, also represents the limit of
(4
Ccm3/g/nn)
j
-01 :
.OOl :
R (nm)
.l
dV/dR
7
(cm3/cjnn)
04
.Ol
.OOl
.OOOl 1 10 Fig. 1. Void size distribution CAR30:1-B
I 100
R (nm)
1000
(VSD) curves obtained for the different furnace blacks, ultrasonically dispersed in water (0.25 cm’ig).
34
F. EHRBURGER-DOLLE and S. MISONO Table 2. Data obtained from the analysis of the VSD S (Fig. 2). From a least square fitting, one obtains curves for the different carbon blacks (uncertainty on data the following relation: is less than about 10%) Sample
nm
Rimin Rs v* vinevcd nm nm R,,,.,IR, cm’ig cm3/g cm’/g
N326 N330 NllO N220 N339 N351 N347
33 47 30 39 50.5
17 23 16 20 20
R mm
TB#SSOO 73 206 m2/g 1.55 cm3/g TB#4500 108 57 m*/g 1.60 cm’ig XC-72 82 214 m*/g 1.76 cm3/g Acetylene 100 black 80 m’fg 2.15 cm31g
20 20 12 14 17.5
1.69 2.34 2.60 2.81 2.89 2.89 3.07
0.72 1.29 1.11 1.09 1.35 1.64 1.54
0.39 0.52 0.54 0.53 0.77 0.79 0.73
0.43 0.63 0.73 0.75 0.78 0.83 0.83
24 20 (7.9)
3.65
1.40
0.79
1.06
31 29
3.78
1.40
0.72
1.10
20 20 (7.6)
4.08
1.39
0.70
1.23
30 20
S
0.50
0.20
1.52
interpenetration of the aggregates and, furthermore, the upper limit of the fractal range. As we have shown earlier[24,25,26], the fractal dimension D can be determined from the VSD curve by using the following relation:
R max= 3824 VDBpfS
(2)
in which R,, is expressed in nm, V,,, in cm3/g, and S in m2tg. We have also indicated the value of R,, vs. V,,,/ S for the previously studied electrical grade carbon blacks[24]. The parameters of the fit are almost unchanged when the values obtained for acetylene black and TB#4500 are taken into account. On the contrary, the high surface area carbon blacks, TB#5500 and XC-72, do not fit with eqn (2). However, as indicated by their CTAB surface areas[l2], these blacks are slightly microporous. Such a microporosity leads to a high surface area, but to a micropore volume that can be neglected in comparison with the DBP volume. We may now assume that only the external surface area has to be taken into account in eqn (2) and calculate the value of S,,, from eqn (2) and the experiments R,, value. One obtains for both carbon blacks S = 81 m*/g. As will be seen further on, this value agrees well with other observations. The ratio V&S can be considered as the thickness, eosP of the solvent covering the whole surface. Using a coherent system of units, eqn (2) becomes: Rmax= 3.824 eDsp
(2a)
eDr,P= 0.26 R,,,.
@b)
or
a [(Lx -
R)I(R - Rmin)13-‘* (1)
It appears that all these values lie between 1.65 and 1.80. Further refinements are needed, however, to ascertain any signi~cant difference in the values of D for the considered carbon blacks. Therefore, in the following, a mean value of D = 1.75 will be used. These values are also comparable to the fractal dimensions obtained previously[l9,24] for electrical grade furnace blacks, and agree with a cluster-cluster brownian aggregation process[l6]. It follows that all these carbon blacks are self similar, in a first approximation. Therefore, the differences in their morphological chara~te~sti~s cannot be attributed to differences in their fractal dimension but, as will be discussed in the following, to their size.
It means that the mean distance between the aggregates is equal to 0.52 R,,. On Fig.2, we have also plotted the value of RO, defined by ~ed~ia~l3] as the radius of the solid sphere containing the same volume of carbon black as the aggregate and related to V,,, and S through the following relation[31]: R, = (2540 f 7100 VDBp)/2S.
(3)
It appears that the values of R, are slightly larger than that of R,. The value of R, calculated for TI3#5500 and XC-72 seems to fit with that obtained for the other carbon blacks. However, if R, is supposed to characterize the morphology of the aggregates in terms of their physical behavior, these values are unrealistic, as the properties of these two blacks are closer to that of TB#4500 and acetylene black than that of the rubber grade NllO carbon black.
3.2 Relation between R_, the DBP volume V,,, and the BET sueace area S From Table 1 and 2, it appears that the value of R,, obtained for carbon blacks of similar surface area (N326, N330, and N347) increases with VDBp 3.3 Characterization of the aggregates by means of and decreases with the surface area S, for carbon blacks of similar V,,, values (for example, N220, &JR For the nonporous carbon blacks, the radius of N339, N347, and N351). Therefore, we will now examine the variation of R,,, as a function of VDBp/ the primary particles can be approximated from the
Morphology
R max
140
of carbon btacks
hm’
R,= (2540*7lOOV,,,)/2SCnrnl
x
XC-T?
80
Ti3#!iSKt
-0
X
15
10
5
20
25
30
Fig, 2. Evolution of R,,, vs. (IO00 V&S) for the rubber grade (A) and the electrical grade (X ) carbon blacks. The solid line is obtained by a least square fitting of all points (except TB#5500 and XC-72) (3 = 0.33)_ The value of R, calculated by means of the relation R, = (2540 t 7100 VD,)/2S, after Medalia, are also indicated for rubber grade (@i and electrical grade (*> carbon blacks. BET
surface area
by means
of the
folk~~wing mass-
v&.ime relation : R, = 3 1o”:Slp
f4)
in which R, is expressed in nm and S in m*/g. p is the true specific gravity of the primary particles. For all blacks (except acetylene black for which p = 1.87 cm’/g), a mean value of p = 1.84 g/cm3 was used. From eqns (2) and (4), one obtains: &JR,
= I_275 VDBpfus
(51
where v, = lip is the true specific volume of the
carbon black particles (us = 0.543 cm31g). The value of I?, and R,,,IR, are indicated in Table 2. The importance of the ratio R,,,IR,, which can be considered as a characteristic length, has already been evidenced, for electrical grade carbon blacks, in the value of the volume fraction of carbon black at the percolation threshold in the dry state[9]. Our previous study of the WD curves of simulated aggregates of a given fractal dimension[26] has also shown that &JR, increaxs with the number of particles, N, in the aggregate. Unfortunately, the determination of N from rhe value of R,,.JR, lacks precision in the present range of sizes. To give an
order of magnitude, values of R,,IR, ranging between 3 and 4 correspond approximately to vahtes of N between 64 and 128 particles. From the geametrical analysis of the carbon black aggregates on electron microscope photos, Medaha obtained the following relation: P&d = 1..432 (1 -t- 2.139l~‘,,,)~.~~,
(61
which is generally used in a simplified form: D,ld = 1.321 + 3.7 v,,,.
@a)
In this refation, D, ( = 2R,) is the diameter of the solid sphere introduced above and d is the measured diameter of the primary particles. Medaha deduced eqn (3) from eqn (6a) by assuming that the number average diameter d, (in relation (6) and (6a)) is related to the volume average diameter d, ( = 2R,) by: n’, = dJl.68
= 1920/S.
(7)
This explains why the values of DJd calculated for a given V,,, are much larger than &JR,. The factor I.58 was introduced by P&d&a in order to include the confr~hu~jon from the effect of particte fusion and the particte size distribution.
36
F. EHRBURGER-DOLLE and S. MISONO
The above comments raise the question of the real size of the primary particle constituting the aggregate, which will now be discussed and justified by the size of the smallest void R,, obtained experimentally. In an assembly of spheres of radius R*, the radius of the cavity depends on the volume fraction of spheres (i.e., the coordination number). Particularly, it remains smaller than R* when the coordination number is larger than 4 (tetragonal arrangement). From the low value of the fractal dimension and from the examination of pictures obtained by electron microscopy, it is clear that the mean coordination number of the particles within the aggregate is smaller or at least equal to 4. It follows that the size of the particle R, is expected to be close to R,,. Table 2 indicates that R, is generally slightly smaller than R,,,. In fact, R,,, is determined in the DSC measurements, from the onset of the deviation of the baseline. Therefore, it is subjected to an uncertainty of about 2 nm on this range of R values. Moreover, the contribution of the smallest pores (of size R,) may be too low to be accurately measured, particularly with decreasing R, as observed for N220 and NllO. Thus, we will now assume that the size of the particles constituting the aggregates (i.e., that of the particles really involved in the aggregation process leading to the characteristic morphology of the aggregates[ 161) is correctly evaluated by the value of R, calculated from the BET surface area, except in the case of microporous carbon blacks such as TB#5500 or XC-72. However, the value of R, ( = 20 nm) calculated by using S = 81 m2/g obtained above by means of eqn (2) agrees with that of R,,.
3.4 Determination of the total and internal void volume
By integration of the VSD curve between R,, and R = 320 nm, one obtains the total void volume VP (Table 2). For three blacks (NllO, N220, and N330) VPis close to the DBP volume. For these samplesthe VSD curves show a second family of pores at higher R values, which can be attributed to interaggregate voids. The measured value of VP includes this contribution. For the electrical grade carbon blacks, the value of VPremains close to 1.40 cm3/g and is smaller thanVDsP whereas for all other rubber grade carbon blacks the total void volume is systematically larger than V,,,. This observation will be discussed in a later paragraph. From the assumption that R,, is the size of the largest internal voids, it follows that the internal void volume V,,, is determined by integration of the VSD curve between R,,, and R,,, (Table 2). It follows also that Vi,, could be considered as the occluded volume, derived from the DBP volume by Medalia[23] by means of the following relation: V,,,,,lu, = (VD,, - 0.215)/0.6825.
(8)
The values of V,,, calculated by means of eqn (8) are indicated in Table 2. For most of the rubber grade carbon blacks, the values of Vi,, agree fairly well with the calculated value of VO,,,. For N220 and NllO, however, these values differ largely and the discrepancies continue to increase with increasing V DBP,for the electrical grade carbon blacks. On Fig. 3 we have plotted the dimensionless ratio V,,,lu, vs. R,,,IR,. The solid line corresponds to eqn
Fig. 3. Evolution of the internal void volume V,.,lus (A, x) measured between R,,s” and R,,. and the total void volume VP/us (m,*)measured up to 320 nm for the different samples, versus R,,,IRs and V,,,,/u, = (0.78R,,,/Rs - 0.215)10.6825 (after comparison with: -.-. VDBp/vs = 0.78 RmaxlRs calculated for D = 1.75 and D = 2. Medalia, eqn (8) and eqn (5)) -e.-..(R,,,IRs)D~3
37 Morphology of carbon blacks 320 nm, are taken into account for the calculation (8) in which V,,, was substituted by its value in terms of the void volume, then eqn (10) and (11) become: of R,,,IR, calculated with eqn (5). This plot suggests a stepwise increase of VJv, when R,,IR, increases. For R,,,IR, ranging between about 2 and 2.84 (i.e., & = (I + V,/~,)/(V,&, + I) (12) for NllO, N220, and N330), V,,,/u, remains almost f/4, = l/(V,/u, + 1). (13) constant, close to 1. Above R,,,IR, = 2.84 and up to about 3.7, V,,,lu, is close to 1.4. When R-/R, In the present experiments, the amount of carbon continues to increase, VJu, starts to decrease, black per cm3 of water is 0.25 g, thus f = 0.12. In reaching a very low value for acetylene black. Howthe case of the electrical grade carbon blacks, the ever, the fractal dimension of this black (D = 2[19]) amount of carbon black was 0.15 g/cm3, therefore is different from that of the other blacks (D = 1.75). {~~l.p. The experimental values are indicated in From the above observations, one may conclude that V,,, is the occluded volume of the aggregate and that the calculation of VoClby eqn (8) leads to overestimated values for large aggregates. 3.5 Determination of the volume fraction qbof aggregates
Up to now, the analysis of the data was mainly focused on the morphology of the aggregates considered as isolated nearly up to length R,,IR,. Let us now examine the arrangement of the assembly of aggregates within the system, by considering the volume fraction of carbon black particles, f, and the volume fraction of aggregates, 4. By definition:
= l/(VJu,
When the contribution
(14)
- (R,,,/RY
and the void volume per particle equals V/n: V/n = (R,,,/R,)3-D
-1
= VJu,.
+ 1)
+ 1).
f/&t
(10)
(11)
of the larger voids. up to
= (Rmax/R,Yi.
(16)
From Fig. 4, it appears clearly that the void volume Vi”,,,> measured by thermoporometry (dotted surface), is smaller than the void volume of the boxes covering the aggregates (dashed surface), as the external void space cannot be measured. This explains why eqn (15) is not verified (Fig. 3). Moreover, in-
Table 3. Volume fractions of aggregates and comparison with (R,,,/R,)D~3, calculated with D = 1.75 for all carbon blacks but acetylene black (for which D = 2). From the possible fluctuations of D (between 1.65 and 1.85) observed experimentally for the different furnace blacks, the range of uncertainty on (R,,.IR,)D~3 is & 0.03 Sample 7:
0.12 - 4 cm’ig
N326 N330 NllO N220 N339 N351 N347 p:
1.69 2.34 2.60 2.81 2.89 2.89 3.07
0.21 0.23 0.24 0.24 0.29 0.29 0.28
0.28 0.40 0.36 0.36 0.42 0.48 0.46
0.58 0.51 0.50 0.51 0.42 0.41 0.43
0.43 0.30 0.33 0.33 0.29 0.25 0.26
0.52 0.35 0.30 0.27 0.27 0.27 0.25
0.42 0.35 0.32 0.32 0.31 0.30 0.30
3.65 3.78 4.08 5.00
0.18 0.18 0.17 0.10
0.27 0.27 0.27 0.13
0.40 0.43 0.44 0.73
0.28 0.28 0.28 0.58
0.20 0.19 0.17 0.20
0.26 0.25 0.24 0.20
0.075 - 6.67 cm3/g
TB#5500 TB#4500 XC-72 Acetlylene black
(15)
(9)
It follows: f/&l
V = (R,,,/RJ3
Thus eqn (11) becomes:
f = l/(vrrZOluS + 1) +,., = (1 + Vi.,/W(VW,/~~
One of the methods to measure fractal objects is to cover it with boxes of increasing size, R/R, (Fig. 4). The number of particles, n, within one box scales as (R/R#’ and the volume of the box equals (RIR,)3. It follows that the void volume in a box of size (R,,,l R,) is:
38
F. EHRBURGER-DOLLEand S. MISONO indicates that, for a given carbon black, the ratio between the volume fractionfof carbon black within the system (which is experimentally imposed by the weight of carbon black within the system) and the volume fraction + of aggregates, which is measured by &,,( or, more realistically, by & is a constant determined by the size and the fractal dimension of the aggregates. The validity of eqn (16) implies that the boxes of size R,,,IR, do not interpenetrate each other (i.e., they behave as hard core boxes) when f increases. It follows that if some boxes covering the external parts of the aggregate interpenetrate (behaving as soft core boxes), the ratio f/+ should become larger than (R,,IR,)D-3. The validity of eqn (16) over the whole range of f values in which the system behaves as a continuum, will now be discussed. Near the percolation
Fig. 4. Example of an aggregate of N347 (magnification 225,000) covered by boxes of size 2R,,, = 120 nm (equivalent to the dashed surface). The dotted region corresponds to the void volume actually measured by thermoporometry up to R = R,,,. u corresponds to a void of diameter corresponds to the convex hull, inequal to 2R,,,. ---cluding voids of diameters larger than 2R,,,.
creasing the size of fractal aggregates is accompanied by a larger development of “arms” (i.e., by nonmeasurable void volume). Therefore, the discrepancy is expected to increase with the increase of R,,,l R,, as observed on Fig. 3. Taking into account the void volume, V, of the larger voids, partly balances the difference. In the case of small aggregates, however, part of the larger voids are inter-aggregate voids (as evidenced by the presence of a secondary peak in the VSD curve (Fig. l), and the real volume will be located somewhere between Vi”, and VP. 3.6 Correlations between f/4, fDBp and (R,,IR,)D-3 A disordered assembly of individual objects behaves as a continuous medium between two limits of volume fraction:
threshold
fc in
the dry state.
It may be assumed that, at the mechanical percolation threshold, the volume fraction $I of aggregates (i.e., the volume fraction of boxes of size R,,,IR,), is equal to the critical value & = 0.17[ lo] calculated for spheres. It follows from eqn (16) that the critical volume fraction of carbon black, fc,should be equal to 0.17 (R,,IR,)D-3. Such a relation was actually verified for electrical grade carbon blacks[9] in the dry state by uniaxial compaction at very low pressure. At intermediate volume fractions in water. It was shown [9,24] that, for the electrical grade carbon blacks presented here, the threshold f” above which a stable dispersion in water is achieved is slightly larger than fc.This was attributed to an interpenetration of the external boxes covering the aggregates, as also suggested by the observation that f/+ > (R,,,l RJDm3, in such a way that the volume fraction of aggregates becomes close to 0.27, as seen in Table 3, for systems measured with f = 0.075, just above the threshold. This is in agreement with the threshold (& = 0.29) expected for soft core percolating spheres[32]. It is also interesting to note that this value corresponds to an internal volume fraction (i.e.,to occluded liquid) equal to 0.17. Obviously, the very high value of fl+ obtained for acetylene black results from the small values of V, or VP measured for this black. This is probably due to the shape (characterized by D = 2) of its aggregates which is, however, very different from that of chemical aggregates[l6] having almost the same fractal dimension. From the comparison of fl$ and (R,,,IR,)D-3, for the rubber grade carbon blacks, the following information can be deduced:
l a lower limit, fc, which may be a percolation threshold; l an upper limit which is the tightest arrangement of the objects without breaking. Such an arrangement is actually measured by the DBP end point l For small aggregates (N326, N330), the value method and corresponds to a volume fraction fDBP of (R,,,/R,)D-3, calculated with D = 1.75 for all of carbon black, related to the measured VDBPby eqn blacks, is between f/9., and f/c&. Because in this case V, also includes a contribution from the inter(9): &BP = l/(VLW&, + 1). aggregate void volume, f/4+, could be underestimated. The real value of f/c+is therefore probably It is observed that~fc and fDBplargely change from close to that of (R,,,IRS)D-3, suggesting no interpeone carbon black to the other and are obviously netration. Such a result would be consistent with the related to the structure of the aggregates. Eqn (16)
39
Morphology of carbon blacks fact that the size of the covering boxes, R,,,/R,,
is only slightly larger than that of the particles, leaving therefore ony a small external void volume. l For the other rubber grade carbon blacks, f/d, is generally slightly larger than (R,,,/RJDe3, depending mainly on the value of D, ranging between 1.65 and 1.85, used for the calculation (Table 3). Near the DBP end point. By de~nition (eqn (9)), fDBp= l/(1 + V&u,) (i.e., by using eqn (5)): fDBB= l/(1 With the assumption verified for fDBp:
+ 0.78R,,,iR,).
(17)
of hard core boxes, eqn (lo) is
~I,BP/+DBP= (RmaxlRsY3
(18)
where &,ep is the maximum volume fraction of aggregates. As a first approximation, this value will be taken as unity: QtDBp= 1. The graphic resolution of the system of equations (17) and (18) leads to the following solutions: e for D = 1.75 l
forD
= 2
R,,,IR,
=2.25
fDBp = 0.363
R,,,IR,
= 4.54
fDBp= 0.220.
From Table 3, it appears that this is experimentally verified for N330. The result obtained for acetylene black is also consistent as the difference between the values of fDBp calculated with R,,,IR, = 5 by eqn (17) (fnsp = 0.202) and by eqn (18) (fosP = 0.200) is negligible in this range of R,,,IR, values. It may therefore be concluded that there is a limiting size of aggregates, which depends only on the fractal dimension D, below which the aggregates behave as hard spheres and above which they behave as soft, interpenetrable spheres, over a limited thickness. The question that now arises is why eqn (17) describes this interpenetration. We think that the answer could be obtained from the experimental study of dispersions of carbon blacks in undecane, for which the interpenetration is evidenced by a displacement of the VSD curve toward smaller void sizes, resulting from the increase of the number of smaller interaggregate voids in the overlapping layers.
the fractal dimension D; the size of the largest internal voids R,,,IR, (depending also on the BET surface area). l
l
Furthermore, these experimental parameters allow characterization of the organization of the aggregates in an assembly over the whole domain over which it is a continuum, from the percolation threshold up to the DBP end point. It is also interesting t o note that the DBP volume is actually a classification parameter for the carbon blacks. However. we have evidence that, particularly at low volume fractions, near the percolation threshold, the properties of the system are completely described only if the fractal dimension of the aggregates is taken into account. Acknowledgements-The
authors would like to thank Jacques Lahaye for fruitful discussions. REFERENCES
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4. CONCLUSION
The experimental results described in the present paper clearly evidence the interest of the measurement of the actual void size distribution and occluded volume in carbon black aggregates. Our approach, by using the principles of fractal geometry, leads to a more pertinent analysis of the morphology of the aggregates, which is now completely characterized by two parameters:
17.
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F. EHRBURGER-DOLLEand S. MISONO
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