Jan 2, 2018 - Pairwise van der Waals (vdW) corrections have been routinely added to density functional theory (DFT) adsorption studies of inorganic or organic molecules on solid surfaces, however, comparative studies of the available pairwise correct
Jan 2, 2018 - A second extension of the DFT-D3 was made within the BeckeâJohnson (BJ) model framework as an alternative for the damping function, DFT-D3(BJ). This implementation is justified by the analytical Koide's(61) study about the dispersion
Jan 2, 2018 - Comparison of the Performance of van der Waals Dispersion Functionals in the Description of Water and Ethanol on Transition Metal Surfaces. Rafael L. H. ...... Dion , M.; Rydberg , H.; SchrÃ¶der , E.; Langreth , D. C.; Lundqvist , B. I.
Jan 2, 2018 - Comparison of Performance of van der Waals Dispersion Functionals in Description of Water and Ethanol on Transition Metal Surfaces. Rafael Luiz Heleno Freire, Diego ... Impact of Bi Heterovalent Doping in OrganicâInorganic Metal Halid
weakly bonded systems such as adsorption phenomena on surfaces,14,15 ..... systems, since the convergence of the stress tensor components as a function.
Jan 2, 2018 - Comparison of Performance of van der Waals Dispersion Functionals in Description of Water and Ethanol on Transition Metal Surfaces. Rafael Luiz Heleno Freire, Diego Guedes-Sobrinho, Adam Kiejna, and Juarez L. F. Da Silva. J. Phys. Chem.
Aug 19, 2013 - ... force preventing the escape of graphene sheets from the deep well. ..... Graphene Sheets Using Ionic Liquid Polymers ACS Nano 2010, 4, ...
Langmuir 1985,1,651-657 charge. From these results, an average value of 0 = 44.4 f 0.6) X lo4 m3 C-' (see Table 11) is obtained for a foam prepared with a solution of conductivity K, = 0.037 S m-l; the product PK, obtained in our case is thus equal to -1.6 X lo-' m2 S C-l, which agrees with the result obtained by the Russian authors,1° for the same system. This value should allow an estimate of the i potential at the filmliquid interface, by using the relationship12
PK, = 9
where E = permittivity, f = electrokinetic potential, and 9 = solution viscosity. Values for t obtained both by using our and Sharovarnikov's data are around 0.5 V, which is too high. Since eq 1has been used quite often in nonfoam systems, with reasonable success, we tend to believe that our 0 may be overestimated. It may happen that the measured dc field induced flow is due to another factor, adding up to electroosmosis: the curvature of the Plateau borders may be dependent on the applied field, since this should decrease surfactant surface concentration in negative voltage regions and increase it in positive regions. It is thus possible that dc voltages operate not only by inducing an electroosmotic current but also by changing pressure gradients throughout the foam. This hypothesis will require further experimentation since there is no data in the literature allowing us to leave it aside. From the measurements performed by using the horizontal electrodes setup we can also estimate electroosmotic volumetric transfer. The curves given in Figure 6 show that foam drainage is clearly affected by using dc fields. The "humps" noticed in the presence of dc fields can be directly related to excess liquid flow, since the liquid volume fraction in the foam D can be evaluated from13
D=BK (2) where 1.5 IB I3.0 is a factor depending on the relative amounts of liquid within films and channels, respectively, and K = Kf/K,, i.e., the ratio between conductivities of foam and liquid.
Consequently, drained liquid volume may be evaluated from conductance changes with time:
where Vf is the foam volume in between the electrodes. Taking the data from Figure 8 for 200 V dc between 5 and 8 min and 10 and 15 min and the extreme values for B (1.5 and 3.0),we obtain aVL/at in the range (2.2-4.4) x m-3 s-l , since Kc = 0.037 S m-l. These values are again in the same order of magnitude of these given in the literature.'O Electric field is only effective at early drainage stages. Slopes of the curves in Figure 3 are linearly dependent on voltage, at short times. This indicates that liquid holdup in the foam decreases linearly with voltage, at short times. After about 7 min, electrical current changes with time are rather small. We suggest that at this point bulk foam conductivity has an important contribution from film surface conductivity. The results given in this paper show that foam drainage pattern may be modified by using electroosmosis. Further developmentsmay lead to the use of dc fields to destabilize (or, eventually, to stabilize) foams. Given the growing rate of practical use of foam and froth chemical separation, it is possible that dc fields may be used to drain foams and to fractionate their chemical constituents, at once. In this regard, we should remember that 1 F generates only 1 equiv of product in an electrochemical process, but it can pump 0.4 m3 of liquid, in a foam. A final point is concerning the existence of a drainage electric potential. Since a dc voltage induces liquid transfer, gravitational spontaneous drainage should lead to the appearance of dc voltages. Moreover, these drainage-induced voltages might be responsible for drainage slowdown. These points deserve further examination.
Acknowledgment. M.E.D.Z. is a FAPESP predoctoral fellow. Registry No. SDS, 151-21-3.
Dispersion of Powders in Liquids. 1. The Contribution of the van der Waals Force to the Cohesiveness of Carbon Black Powders P. A. Hartley* and G. D. Parfittt Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 Received January 18, 1985. In Final Form: June 13, 1985 In the dispersion of powders in liquid media agglomerates are broken down into primary particles. The strength of these agglomerates is dependent on both the interparticle force and the packing structure. A model is developed which relates the tensile strength and packing density of powder compacts to the interparticle force and has been tested with carbon blacks assuming the only force of significance is the van der Waals attraction. Of the 11powders examined, 10 showed good agreement between the force derived from the experimental tensile strength data and that predicted theoretically, from which it may be concluded that the van der Waals force is responsible for the agglomeration of carbon black.
Introduction Powders that contain primary particles of submicron dimensions are normally cohesive and include both agDeceased July 7 , 1985.
glomerates and aggregates. Aggregates are defined as groups of Primary Particles joined together by a solid bridge such that the surface area is reduced significantly from the sum of the areas of the individual particles. Agglomerates are groups of primary particles and/or aggregates held together by bonding mechanisms other than 0 1985 American Chemical Society
Hartley and Parfitt
652 Langmuir, Vol. 1, No. 6, 1985 solid bridging such that the surface area is not reduced from that of its constituents. The strength of these agglomerates is dependent on both the interparticle force and the particle packing structure. The process of dispersion of powders containing submicron particles in liquid media consists of a number of stages:' (a) incorporation of powder into the liquid medium, (b) wetting of the powder, (c) breaking up of agglomerates and aggregates into primary particles, and (d) stabilization of the resulting dispersion. The efficiency of the first three stages is often referred to as the dispersibility of a particular system and may be defined in terms of the amount of energy per unit mass of powder required to achieve dispersion. There are two sources of this energy. One is chemical energy, related to the chemistry of the system and the interfacial effects. The other comes from mechanical milling. By altering the chemistry of the componentsthrough, for example, surface treatment of the powder and the use of surface-active agents to improve wetting, the chemical energy input can be optimized thereby reducing the mechanical energy demand. For efficient use of these input energies, each stage of the dispersion process must be understood. Of particular interest in this work is the disagglomeration stage. The nature and magnitude of the interparticle forces responsible for agglomeration must be assessed before the factors involved in dispersibility can be optimized. A number of interparticle forces have been identified2" and can be divided into two group^:^ (a) those that act independent of material bridges and (b) those that are a result of such bridges. Category (a) includes the van der Waals force, which is particularly important when the particles have submicron dimensions, an electrostatic force, and a magnetic force. Category (b) includes a liquid bonding force, a force due to the presence of viscous binders, and a solid bridging force which usually results in the formation of aggregates. In a particular situation any of these forces or combination thereof may dominate the cumulative interparticle force, but only the van der Waals force is always present. A simple and well-documented method of assessing the cohesiveness of powders is the measurement of the tensile strength of powder ~ompacts."~Previous models relating tensile strength to packing density are limited as they have combined all the interparticle forces into a single term. We have developed a theory for powders containing submicron particles which incorporates a specific interparticle force, and for the initial evaluation of the model we take this to be the van der Waals force. We have measured the tensile strength of several carbon black powders using a split-cell type apparatus. Carbon blacks are nonmagnetic and not susceptible to electrostatic charging and the particles have essentially hydrophobic surfaces. Thus it is reasonable to assume for these powders the only interparticle force of significance is the van der Waals force and that this force is responsible for the agglomeration of carbon blacks. (1) Parfitt, G. D. In "Dispersion of Powders in Liquids", 3rd ed.; Parfitt, G. D., Ed.; Applied Science Publishers: London, 1981; p 2. (2) Rumpf, H. In "Agglomeration"; Knepper, W. A., Ed.; Interscience: New York, 1962; p 379. (3) Hartley, P. A.; Parfitt, G. D.; Pollack, L. B. Powder Technol. 1985, 41, 35. (4) Pietsch, W. CZ-Chem.-Tech. 1972, I , 116. (5) Ashton, M. D.; Farley, R.; Valentin, F. H. H. J.Sci. Instrum. 1964, 41, 763. (6) Parfitt, G. D. "Pigment Dispersion-in Principle and Practice", 14th Fatipec Congressbook, Hungary, 1978; p 107. (7) Yokoyama, T.; Fujii, K.; Yokoyama, T. Powder Technol. 1982,32, 55.
Theory Tensile strength is defined as the maximum tensile stress required to fracture a specimen in simple tension when no shear force is applied. Models have been proposed relating the tensile strength and packing density of powder compacta to the interparticle force. One of the first was developed by Rumpf2 who derived an expression (eq 1) for equal sized spheres, assuming that (a) there are a large number of bonds broken upon tensile failure, (b) these bonds are randomly distributed, (c) the particles in the compact are randomly distributed, and (d) all interparticle forces can be considered in terms of a single mean effective force.
T is the tensile strength, t the void volume, d the particle diameter, c the coordination number, and f is the mean interparticle force. ChengSlo proposed an alternative model relating the tensile strength of a powder compact to its packing density, particle size distribution, and interparticle force (eq 2). He considered the case of particles touching at asperities of surface protrusions (surface roughness) in order to introduce the concept of an effective surface separation distance.
T=3 to sd ~ 1-t 4 v
to is the effective surface separation distance at zero tensile strength, h is the interparticle force per unit fracture area, and s, d, and u are the mean particle surface area, diameter, and volume, respectively. Cheng also related the overall interparticle force to a single energy and length parameter which allowed him to rank various powders according to their cohesiveness. This is not altogether satisfactory since the energy parameter for a given powder must be stated relative to an arbitrarily chosen reference powder. We have developed a model that relates the tensile strength and packing density of compacts of fine particle powders to the van der Waals force. It is assumed that a powder compact is composed of individual, submicron, spherical particles of uniform size, packed in a random fashion. The packing structure determines the packing density (ratio of the density of the powder compact p to the density of the solid material p , ) and the coordination number. The relationship between c and PIP, may be established using models based on either computer simulation or empirical data. Several investigated random packing of spheres using computer models and concluded that c N 6 and p l p , N 0.6 for packing structures in which each sphere must be in a stable position with respect to gravity. These models have restricted value in that they do not account for interparticle forces, which may affect the packing structure. Empirical relationships have been proposed14J5but were developed by using data obtained from packing assemblies of spheres which are large enough so that contact and near contact (8) Cheng, D. C.-H. Chem. Eng. Sci. 1968,23, 1405. (9) Cheng, D. C.-H.; Farley, R.; Valentin, F. H. H. Inst. Chem. Eng. Symp. Ser. 1968, No. 29, 14. (10)Cheng, D. C.-H. Proc. SOC.Anal. Chem. 1973, I O , 17. (11) Powell, M. J. Powder Technol. 1980, 25, 45. (12) Jodrey, W. S.; Tory, E. M. Simulation 1969, 13, 1. (13) Matheson, A. J. J. Phys. C 1974, 7, 2569. (14) Ridgway, K.; Tarbuck, K. J. Br. Chem. Eng. 1967, 12, 384. (15) Smith, W. 0.; Foote, P. 0.; Busang, P. F. Phys. Reo. 1929, 34, 1271.
Langmuir, Vol. 1, No. 6, 1985 653
Dispersion of Powders in Liquids
points can be marked and counted. The packing structure of spheres of this size is probably not influenced by interparticle forces. consequently the range of packing density that can be studied is small. Expressions currently available must therefore be extrapolated to low packing densities characteristic of compacta of submicron particles, which is questionable. Gray16stated that there is no simple relationship between c and p / p s in randomly packed beds, even with the exclusion of interparticle forces. In the absence of a model which adequately accounts for interparticle forces we shall use the simple expression attributed to Smith et al.,15 which is based on data for large lead spheres and has seemingly worked well in the models proposed by both Cheng'O and Rumpf,2 namely,
When the compact is fractured under a tensile load applied perpendicular to the fracture plane, particles within this plane are separated. The number of particle pairs per unit area of the fracture plane Nppla can be defined in terms of the number of particle pairs per unit fracture volume N, and the effective particle diameter defP According to &eng: the need to define an effective diameter stems from consideration of surface roughness. Asperities protrude from the surface and are the points where the particles make physical contact. Thus the diameter d of the particle with a smooth surface differs from the effective diameter deffby a distance t , the apparent separation distance between particle surfaces. This definition o f t in terms of surface roughness alone is inadequate for submicron particles, and in fact does not appear to be altogether reasonable for particles of somewhat larger size. Cheng'sg data for calcium carbonate (particle diameter 2.36 pm) yield an apparent surface separation of 1.15 pm, which indicates severe surface roughness. Similar results were found for other powders. We propose an alternative physical model which does not lead to these contradictions and although we consider only submicron particles the model should hold equally well for particles of larger size. It is found in practice17 that powders made up of small particles pack less closely ( p / p , < 0.4) than do large particles ( p / p , = 0.6 for randomly packed spheres). The large void space found in packings of submicron particles may in part be due to surface roughness but will largely be due to an open random packing structure.l8 In a random packing of large particles the gravity force is greater than the interparticle force, and the openness of the structure tends to be compressed. For submicron particles the gravity force is secondary to the interparticle force and the openness of the structure tends to be supported. Therefore with submicron particles the distance between adjacent particles will vary from a state where asperities are in contact, or slightly closer if deformation or interlocking of these asperities occurs, to a state of openness where not all particles make actual physical contact even though they may be within a region where interparticle forces act. Thus the apparent separation distance between particle surfaces t is physically defined as an average separation distance which does not imply actual physical contact. For submicron particles at low packing densities d = t. (16) Gray, W.A. 'The Packing of Solid Particles";Chapman & Hall: London, 1968; p 17. (17) Fowkes, R. S.;Fritz, J. F. Information Circular 8623; Pittsburgh Mining & Safety Research Center: Pittsburgh, PA, 1974. (18) Haughey, D. P.; Beveridge, G. S. G. Can. J.Chem. Eng. 1969,47, 130.
Figure 1. Sketch showing the area over which the interparticle force acts per particle pair.
The relationship between NppIa, Npplv, and deffis thus given by eq 4. Npp/a
= Npp/vdeff = Npp/v(d + t )
NpPlvis related to the number of particles per unit fracture volume Nplvby where c accounts for the packing arrangement of the particles and the '/z is to avoid counting each particle twice. The number of particles per unit fracture volume is related to its density and the particle volume by &/VU
Equation 4 may now be written as (7)
In addition to the packing characteristics of the compact we must also consider the force F which acts between the particles in particle pairs. Since the particles are assumed to be randomly packed, this force may not act parallel to the applied tensile load. The force can, however, be resolved to its component form in the direction of the tensile load F cos 8, where 8 is the angle that fixes the inclination of the force vector. Further, 8 may assume any value between f?r/2. Assuming that all positions are equally probable we may average F cos 8 and obtain a mean value for the force F which acts parallel to the tensile load.
F may be written in terms of an overall interparticle force per unit area Fppo and an area over which the interparticle force acts per particle pair A,,. F = FppaAPp
The tensile strength T of the compact is the sum of all the forces acting between particles in particle pairs in the overall area of fracture. Implicit in this statement is the assumption that all particle bonds are ruptured simultaneously.
T = CF = ~2XF,,aA,p Since A,, is a single particle pair quantity, the summation in eq 10 may be carried out by multiplying by the number of particle pairs in the fracture area NpPla. To define A,, we introduce the parameter t owhich is the apparent separation distance between particles when the tensile strength of the powder compact is zero. Thus t o is effectively a measure of the range over which interparticle forces act, as shown in Figure 1. We then are able to use simple geometric relationships to derive the following expression for A,,.
654 Langmuir, Vol. 1, No. 6,1985
Hartley and Parfitt
Since t and toare defined as average quantities, we may assume for the purpose of evaluating separation distances that the powder compact is a regular array of particles having a unique packing structure. Further, the geometric arrangement of the particles is assumed to remain the same with changes in packing density. We assume that the powder compact has a regular rhombohedral packing (c = 12, maximum packing density = 0.74) of uniform size spheres, as has been done in heat transfer studies in packed beds,lgand the following expression20is used to evaluate the separation distance. (d
0.74d3 + t)3= P/Ps
Solving for t gives eq 14.
If we evaluate this expression at T = 0, then p = po and t = towhich leads to eq 15.
replaced by the characteristic energy parameter A and eq 16 becomes po/ps is the packing density at zero tensile strength and
is found by extrapolating a plot of tensile strength vs. packing density. We have thus far related the tensile strength of a powder compact to the overall interparticle force per unit contact area by eq 11 using packing and force relationships. All quantities in this expression are known except for Fppo which can be evaluated using a law of corresponding states.21 This law states that any two-body intermolecular potential function is characterized by one energy parameter eo and one length parameter to. We assume the same method may be used to characterize the force between two particles. Since FppO has units of force per area, Q has units of force times length, and t o has units of length, the dimensionless form of Fppois ( t o 3 F p p 0 ) / ~Since o . Fppois a function o f t , the dimensionless force is a function of the dimensionless distance.
CP denotes some functional form of the dimensionless separation distance. Assuming that the interparticle force is due to van der Waals attraction the Hamaker expressionB derived for two equal sized spheres may be used. Here a = t 2+ 2dt, b Fvdw
= (A/6) d2(d
+ t ) ( ( l / a 2 ) + (1/b2) - (2/ab))
= a + d2,and A is the Hamaker constant. The distancedependent part of the Hamaker expression, which is in terms of t , is rendered dimensionless by dividing through by the identical expression evaluated at t = t,. eo may be
(l/ao2) + (1/b,2) - (2/aobo)
(19) Hill, F. B.; Wilhelm, R. H. AZChE J. 1959, 5, 486. (20) Mysels, K. J. "Introduction to Colloid Chemistry";Interscience: New York, 1959 p 120. (21) Hakala, R. W. J. Phys. Chem. 1967, 71, 1880. (22) Hamaker, H. C. Physica (Amsterdam) 1937, 4, 1058.
4a( t) t0
Substitution of this result and eq 7 and 1 2 into eq 11and rearranging gives eq 20.
(P/PS)/(l - (P/Ps))
+ t ) (to- t ) d2
(20) Here R is termed the reduced interparticle force. We may now use the left-hand side (lhs) of eq 20 to evaluate R from tensile strength and packing density data and compare it to the right-hand side (rhs) which contains the theoretical expression for the van der Waals force. The proposed model offers several signficant advances in this type of study. It provides a means of incorporating a specific interparticle force, allows for the inclusion of submicron particles by defining an average separation distance in terms of packing structure, and enables powders to be ranked on the basis of cohesiveness on an absolute scale rather than relative to an arbitrary reference powder.
Experimental Section Apparatus and Procedure. A split cell apparatus, based on previous designs of Ashton et al.? Parfitt; and Yokoyama et al.,7 was designed and built for the purpose of measuring the tensile strength of powder compacts and is described in full elsewhere.23 The apparatus (Figure 2) has a split cell (A),inside diameter 25.40 mm and 6.35 mm deep, with a grooved bottom which prevents the powder from slipping. The two halves of the cell are clamped together (B) and attached to the base unit, which consists of a fixed plate (D) and a movable plate (E) which rides upon a set of ball bearings (F), and is translated by a variable-speed Inchworm translator ( G ) . Positioned between the translator and the movable plate is a force transducer (H) which rests upon a second set of bearings (I). The transducer provides an output voltage proportional to the applied load. The procedure used is as follows. The contacting surfaces of the split cell are cleaned with acetone and clamped together. A preweighed amount of powder is poured into the bed through a guide sleeve, leveled, and compressed to a given volume. The (23) Hartley, P. A.; Parfitt, G. D. J. Phys. E 1984, 17,347.
Langmuir, Vol. 1. No. 6, 1985 655
Dispersion of Powders in Liquids Table 1. Pbynicnl Proparties of Carbon Blacks mean
Cabot Corp. 106'd Cabot Corp. pelletized 105'.' N299 Huber Corp. pelletized 111s 0.075 86'.' N650 Huber Corp. pelletized 389 0.071 73'.b N660 Huber Corp. pelletized 349 0.100 Huber Corp. pelletized 24' 55'b ARWO 4Vb N990 Huber Corp. pelletized 80 0.320 'Typical for that grade. 'For depelletized material. 'As received from manufacturer. dLhterminedby Cabt for samples used in this work.
H 0.1 pm
Figure 3. Electron micrograph showing a typical carbon black aggregate structure.
packed cell is then fixed to the base unit and the cell clamp is removed. The movable plate is then translated and the tensile stress as a function of time is recorded. The tensile stress deereases sharply when the compact fractures, except at very low packing density, thus validating the hypothesis that all bonds along the fracture plane are ruptured simultaneously. The maximum tensile stress is taken as the tensile strength. All experiments were conductedat constant hed height (6.35nun) and translation speed (5.0X mm 8.'). Materials. Several grades of carbon black (Cabot Corp. and Huber Corp.) were used in this study. These materials contain suhmicron. nonporous particles ( d e n s i p 1.86 g cn") that have essentially hydrophobic surfaces. During the manufacturing process of these blacks, primary particles flocculate while still in a viscous state and fuse together to form structured aggregates.% An electron micrograph of a typical aggregateis shown in Figure 3. These aggregateshave been described by Medalia as branched clusters with tentacle-likeprotrusions and their structure can be related to their size and shape and the number of particles per aggregate.sm A quantitative measure of the degree of structure can be obtained from oil absorption or dibutyl phthalate absorptionn (DBPA). The higher the DBPA the more structured the aggregate.=." The characteristics of the aggregate structure affect the packing density of the bulk powder" and hence the powder cohesiveness and dispersibility. Some of the blacks were received in a fluffy form while others were pelletized. The latter were depelletized in a high shear (24)Medelia. A. I.; Rivin. D. In "Charaetlrimtiona l P d e r Surfam"; W ,Fds :Academic Press: New York. 1918: p
Pditt. C D ,Sing. K S 777
(25)Medalia. A. 1. J. Colloid Interface Sei. 1970,32, 115. (26)Medalia. A. 1.: Richards. L.W.J . Colloid Interface Sei. IW2.40. 233. (27)Dibutyl PhthalateA h r p t i o n Number of Carbon Black ASTM Procedure D2414-25T. (28)Dannenberg. E.M.In 'Kirk-Othmer: heyclopdia of Chemical Technology", 3rd ed.; Wiley-Intemienee: New York. 1978: p 631.
PACKINQ DENSITY. P/P,
Figure 4. Tensile strength vs. packing density: (0)Regal 300 R and (m) AR060.
grinder. The carbon blacks contain inorganic impurities (ash) and have chemisorbedoxygen on the surface (volatile content). DBPA values, volatile and ash contents. mean particle diameter8 (determined by assessing a minimum of 100 particles from transmission electron micrographs),and BET surface areas are given in Table I.
Results and Discussion Typical plota of tensile strength vs. packing density are shown in Figure 4 for two of the carbon blacks, AR060 and Regal 300 R. A nonlinear regression analysis was used to determine the best fit lines through these data. The lines are extrapolated to give the packing density p o / p . at zero tensile strength which is used to calculate values oft, The two curves are parallel, as are the curves for most of the other blacks, which suggests that the nature of the force is the same for each powder. The curve for AR060 is shifted to the right of that for Regal 300 R indicating a difference in the packing structure. Both the increased particle size and the decreased aggregate structure of the AR060 are in part responsible for this shift. This is true in general since particles with smaller dimensions pack more openly than do larger particles." Also, it has been found25.26that the larger the particle aggregate, the more branched is the aggregate structure and hence the more
Hartley and Parfitt
656 kngmuir, Vol. I , No, 6, 1985
SEPARATION DISTANCE, 1. p m
Figure 5. Reduced interparticle force vs. separation distance, eq 20. Regal 300 R (0)Lhs (experimental),(-) rhs (theoretical), A = 1.1 X J. AR060 (m) Ihs (experimental),(-) rhs theoretical, A = 1.7 X J. Table 11. Values of Packing Density at Zero Tensile Strength ( p o / p . ) , Effective Range of Interparticle Farce ( t o ) ,and Hamaker Constant ( A ) to*
material Regal 300 R Regal 300 I Elftea 8 Monarch 900 Sterling R Vulcan 3 N299 N650 N660 AR060 N990
open the compact packing structure. The experimental data are compared to the model tbrough eq 20. In Figure 5 the reduced interparticle force is plotted against the apparent separation distance (eq 14) for AR060 and Regal 300 R. The rhs of eq 20 is represented as solid lines and the lhs by the points. The Hamaker constant is adjusted such that the experimental (UIS) and theoretical (rhs) data overlap. The extent to which the two can be superimposed is an indication of whether the separation distance dependence is the same for the interparticle force derived from the experimental data and that calculated for the van der Waals force. For both blacks this agreement is good, as it is with all the blacks studied except Monarch 900 and N990, which will be discussed separately. Values for po/p. to, and A for all the blacks are given in Table 11. The Hamaker constant derived from experimental data compares favorably with the literature value3 of 1.09 X J from which we conclude that for carbon black powders the dominant interparticle force is the van der Waals force. There are a number of possible explanations for the discrepancies among the derived values of the Hamaker constant, as well as between derived and literature values. The physical properties of the blacks are different, each
0.1 pm Figure 6. Micrograph showing the aggregate structure of Monarch 900. containing various trace amounts (less than 2 wt W )of impurities in the form of ash (salts and various oxides) and volatiles (chemisorbed oxygen). These impurities could lead to additional interparticle forces including a moisture bonding force, arising from adsorption of water on hydrophilic surface oxide impurities, and a solid bridging force, arising from the precipitation of salts between adjacent particles. The presence of impurities could also lead to an increase in surface roughness, which would increase the separation distance and thus cause a reduction in cohesiveness. Another source of these discrepancies is the experimentalerror in tensile strength measurements, which is typically 5-10%,8 simply because it is impossible to obtain identical compacts even though the compaction technique is itself reproducible. There is also some difficulty in the determination of po/ps. We have used a nonlinear regression analysis to alleviate the uncertainty in this extrapolation. However, the powders consist of structured aggregates rather than separate primary particles, and this is a fador of considerable importance. In applying the model we have used the primary particle diameter even though the particles are aggregated. The primary particles at the ends of the aggregate tentacles are not surrounded by other particles, so the calculation of the force should be based upon the diameter of the primary particle rather than some equivalent aggregate diameter. This reasoning is supported by Parfitt and Picton” who found that the primary particle dimension rather than that of the aggregate determined the dispersion stability of carbon blacks. This situation is similar to ours in that it is an interparticle force which is the controlling element. The two key factors that determine powder cohesive strength, namely, the packing and interparticle force, can be understood more clearly by examining the two blacks that do not fit the model, Monarch 900 and N990. The value of the Hamaker constant derived from experimental data for Monarch 900 is 0.07 X lo-’’ J,which is not in agreement with the literature value of 1.09 X l(r” J. Here we have used the primary particle diameter of 0.018 pm. However, electron microscopy reveals that the particles are tightly grouped into spherical aggregates of diameter 0.088 pm, which have relatively smooth surfaces with few discernible primary particles (Figure 6). All of the other blacks have structures that allow identification of primary particles at the outer surface of the aggregates. A Hamaker constant of 5.9 X J is obtained if the (29) Parfitt, G . D.;Pieton, N. H.Tram. Forodoy Soc.
Langmuir, Vol. 1 , No. 6,1985 657
Dispersion of Powders in Liquids 150
I 0 10
PACKING DENSITY, p, p ,
Figure 7. Dibutyl phthalate absorption (DBPA) vs. packing density at zero tensile strength p o / p s ; ( 0 ) fluffy and ).( depelletized. aggregate diameter is used and this value is in closer agreement with the values derived for the other blacks and the literature value. The data for Monarch 900 clearly demonstrate that the packing characteristics of the powder in terms of aggregate structure should not be overlooked. Medalia and cow o r k e r have ~ ~ ~studied ~ ~ ~ the morphology of carbon black aggregates and have found that the more structured an aggregate the higher its DBPA. In Figure 7 DBPA is plotted against po/ps. This particular packing density was chosen since DBPA determinations are made on powders at their bulk (noncompacted)value. As aggregate structure is increased (increasing DBPA) the packing density is reduced. Furthermore, this implies that at constant packing density compacts of more structured aggregates have a higher tensile strength, Le., are more cohesive. Also demonstrated in this plot is the difference between fluffy blacks and those that have been depelletized in a high shear grinder. At constant DBPA and tensile strength, the depelletized blacks are packed more tightly than the fluffy blacks. This may be because the depelletized blacks were deaggregated (severing of primary particles from the outer tentacles of the aggregate) in the grinding process. Another possibility is that the depelletization is incomplete leaving the powder to contain small pellet fragments of high packing density. The other black that does not conform to the model is N990 and this powder should give good agreement as it contains particles more like those assumed in the theory than any of the other blacks. This black is different from the others in three respects; it is the least aggregated, has the largest particle diameter, and is manufactured by a thermal rather than the furnace process used for the other blacks. The product from this process contains absorbed tar,24which consists of aromatic compounds including pyrene, coronene, and their derivatives, and can be removed by Soxhlet extraction with dichlorobenzene for 48 h followed by toluene for 24 h.30 In Figure 8 a plot of tensile strength vs. packing density is shown for both the (30)Medalia, A. I. Cabot Corporation, personal communication.
PACKING DENSITY, / J p s
Figure 8. Tensile strength vs. packing density; N990 ( 0 )nonextracted,).( extracted with toluene and dichlorobenzene. nonextracted and extracted forms of N990. Removal of the tar is seen to change both the packing and force characteristics of the compact as evidenced by the shift in pO/paand the nonparallel nature of the curves. The tar may act as a viscous binding agent thus causing the change in both the nature and strength of the overall interparticle force. Application of our model suggests that there is an additional force acting since the derived Hamaker constant for the nonextracted form is 130 X J while for the extracted form the value drops to 60 X J. Clearly neither is in accord with the literature value. Particle diameter is a sensitive factor in the model, especially through the separation distance parameters which are averaged quantities. In actuality there is a radial distribution of interparticle separation distances within a compact and in future refinements of the theory it may be necessary to account for this. Another refinement involves the form of the van der Waals force equation used. For particles separated by distances greater than -0.01 pm this force is retarded.31 For all blacks except N990 (and possibly N650 and Monarch 900 to a lesser degree) the effect is negligible. Calculation of the retarded van der Waals force is not simple, however, estimates based on a correction factor given by O ~ e r b e e kindicate ~~ that the van der Waals force would be reduced by a factor of 2 and possibly by as much as 5. Ten of the eleven carbon blacks studied in this work conform closely to our theoretical predictions which gives us confidence in the model. We can therefore conclude that the van der Waals force is responsible for the cohesiveness of carbon black powders and is the interparticle force that has to be overcome in the disagglomeration stage of the dispersion of these powders in liquid media. Acknowledgment. We thank the Cabot and Huber Corporations for the carbon black samples and Dr. Avrom Medalia for providing analytical data and challenging comments on this work. (31)Casimir, H.B. G.; Polder, D. Phys. Reu. 1948,73,360. (32)Overbeek, J. Th.G. In “Colloid Science”: Kruvt. H. R., Ed.: Elsewer: Amsterdam, 1962;Vol. l, p 271.