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Exploring a new method to study the agglomeration of powders: Application to nanopowders
Powder Technology 250 (2013) 13–20
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Exploring a new method to study the agglomeration of powders: Application to nanopowders François Henry a,b,⁎, Jacques Bouillard a, Philippe Marchal b, Alexis Vignes a, Olivier Dufaud b, Laurent Perrin b a b
INERIS, Parc Technologique ALATA, B.P. 2, F-60550, Verneuil-en-Halatte, France Laboratoire Réactions et Génie des Procédés, Université de Lorraine, CNRS, 1 rue Grandville, B.P. 20451, F-54001 Nancy, France
a r t i c l e
i n f o
Article history: Received 3 October 2012 Received in revised form 31 July 2013 Accepted 8 August 2013 Available online 20 August 2013 Keywords: Nanoparticles Agglomeration Flow Rheology Shear Risk assessment
a b s t r a c t The increasing use of nanoparticles in the production of nanomaterials has led to the identification of nanoparticle agglomeration in air or in a given medium as a key problem, both on the standpoint of process performance (i.e. product homogeneity) and above all on inhalation risks. This paper suggests a new method to evaluate the structure and the dynamics of agglomeration of a nanopowder, based upon monitoring of the shear stress of a powder which is submitted to the mechanical solicitation of a rheometer. At low and increasing shear rates, the powder flow will evolve from a Newtonian regime (dense powder) to a Coulombic regime (slightly rheofluidized dense phase). At larger shear rates, the powder will be set in suspension, which characterizes a kinetic regime governed by particle collisions. The specific shear energy, related to the specific agglomeration energy, has been calculated for different nanopowders (carbon, aluminum, silica) and compared to non-cohesive micrometric powders (glass beads). From the measurements of shear rates and stresses at the frictional/kinetic transition, agglomerate diameters have been evaluated for carbonaceous material and for silica. Values of these agglomerates can range from 200 to 500 μm and are related to their propensity to break apart. The carbonaceous materials seem to be the more difficult to deagglomerate, whereas silica nanopowder agglomerates are more easily breakable. Measures of cohesiveness (or specific agglomeration energy) can be useful for assessing the dispersibility of nanopowders and their relationships to inhalation or explosion risks. © 2013 Published by Elsevier B.V.
1. Introduction The growing development of nanomaterials in various industrial sectors [1,2] raises new kind of risks that need to be assessed carefully. In order to achieve such goal, it is necessary to better understand the physico-chemical properties related to nanoparticles and among them, their tendency to agglomerate [3]. Agglomeration phenomenon can in fact modify dispersibility of nanoparticles, their reactivity [4] as well as their toxicity and ecotoxicity. Therefore, nanoparticle agglomeration needs to be further understood as it directly influences their hazards and related risks. In that context, the purpose of this paper is to propose a new method to characterize the specific agglomeration energy and its effect on the rheological properties of nanoparticles. After having described the theories of dense flow regimes, the rheological apparatus and its associated protocol, this paper explains how rheograms can be used to calculate the specific shear energy, which is related to the agglomeration energy. The agglomeration regime of a nanopowder can be described at the powder
⁎ Corresponding author at: INERIS, Parc Technologique ALATA, B.P. 2, F-60550, Verneuilen-Halatte, France. E-mail address: [email protected] (F. Henry). 0032-5910/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.powtec.2013.08.010
level, but rheograms can also provide information on estimates of agglomerate diameters as well as internal volume solid fraction and number of particles in agglomerates. This paper shows that from macroscopic rheological properties, microscopic features of agglomerates can be inferred.
2. An experimental approach to evaluate the agglomeration of nanopowders According to ISO [5], an agglomerate is a “collection of loosely bound particles or aggregates or mixtures of the two, where the resulting external surface area is similar to the sum of the surface areas of the individual components”. This standard adds that the “forces holding an agglomerate together are weak forces, for example Van der Waals forces, as well as simple physical entanglement”. The terms agglomerate and aggregate are often confused, but as specified in the previous standard, an aggregate is made of particles strongly bonded or fused. As a consequence, contrary to agglomeration, aggregation can be considered as a non-reversible phenomenon. The literature contains many models to understand the cohesivity of agglomerates such as the one proposed by Rumpf [6], who defined the agglomerate tensile strength Γagg as the stress (force per unit area)
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F. Henry et al. / Powder Technology 250 (2013) 13–20
required to break into two halves a system of particles which are linked by weak bond forces:
Γ agg
This section presents the apparatus used to investigate the flow of nanopowders and also the theoretical description of the flow regimes.
1−eagg F ¼ 1:1 dp2 eagg
ð1Þ 3.1. The powder rheometer
where F represents the weak interaction forces (Van der Waals forces), dp is the particle diameter and eagg is the inner porosity of the agglomerate. It should be highlighted that this description encounters a limitation as it does not take into account the structural details of aggregates that may compose the agglomerate. With this kind of model, it is in fact assumed that an agglomerate is only composed of uniformly sized spheres holding by weak forces. The effects of the aggregates on the cohesivity of the agglomerate cannot be highlighted by such models. However, in view of simplification, Rumpf relationship will be used here as it can provide an order of magnitude of the agglomerate tensile strength. By considering Mohr–Coulomb criterion [7], it can be stated that an agglomerate will break at a critical quantity of shear and normal stresses. It can then be stated that it is possible to better understand agglomeration through rheological study of a powder. Indeed, under a mechanical stress, a powder will flow under a certain velocity gradient with a given dynamic viscosity, only when the shear stress reaches a critical magnitude. According to the Mohr–Coulomb theory, the shear stress σ is the sum of the effects of the cohesiveness Γc and the friction: σ ¼ Γ c þ μN
ð2Þ
where μ is the coefficient of friction and the normal stress N is equivalent to a pressure exerted on the powder and can be expressed locally as: ð3Þ
N ¼ ρb gz
To obtain a vertical average normal stress in the z direction, one can calculate an averaged normal stress as: N¼
1 h
Z
h 0
ρb gz dz ¼ρb g
h 2
ð4Þ
where g is the gravity, with an average magnitude of 9.81 m∙s−2, and h the height of the total volume of powder, and ρb is the total density of the powder. Hence, the averaged shear stress becomes: h σ ¼ Γ c þ μ ρb g 2
3. Powder rheology
The experiments have been led with a powder rheometer (AR 2000, TA Instruments) specifically designed by P. Marchal [8] and is presented on Fig. 1. An induction motor supplies a torque C and an optical sensor Ä measures the angular rate Ωn. Both parameters are linked to shear stress σ and shear rate γ thanks to proportional coefficients. The four blade vane (Fig. 2), which turns in a cylindrical tank with baffles, has been designed by analogy with a Couette system in such a way that [9]: γ˙ ¼ K γ Ω˙
σ ¼ Kσ C
with Kσ = 3.858.104 m−3 and Kγ = 2.037 rad−1. It is then possible to define the viscosity as the ratio between the shear stress and the shear rate, as η ¼ σ=γ˙. Hence, submitted to a mechanical solicitation, a powder will flow under a certain velocity gradient or a shear stress with a given dynamic viscosity η The baffles and the vane prevent the sample from slipping at the wall and at the surface of the measuring tool. Consequently, this design only promotes the powder/powder friction interaction. To provide a granular reorganization of the powder, the cell is submitted to vertical mechanical vibrations, at variable frequencies and amplitudes. They are controlled by a mini-shaker connected to a signal generator and an amplifier. The vibrations are sinusoidal, with an amplitude Θ, a frequency f and an energy Ev: 2
2
Ev ¼ mð2πf Þ Θ :
ð8Þ
This vibrating device aims at providing an ergodic granular temperature to the powder and avoids specific locked-in rest configurations of the powder that would lead to biased results at low shear rates. Marchal et al. [8] proposed then to use vibrations to induce a Brownian-like granular motion within the powder in order to access to all powder configurations to ensure a granular temperature to the system. For AR 2000 apparatus, the torque varies from 100 mN∙m to 200 mN∙m and the angular rate from 10− 8 rad∙s− 1 to 300 rad∙s− 1. By considering the values of Kγ and Ω, it can be stated that the apparatus has a technical limitation in terms of maximum experimental
ð5Þ
It should be noted that for non cohesive powders with similar surface roughness and made of the same material, the shear stress is expected to be only dependant on the apparent density of the powder for a given sheared volume of powder. By considering the Eq. (5), it is then possible to define an average specific agglomeration energy Eagg as the ratio between the lean cohesion stress and the mean total density: Eagg ¼ Eshear −μg
h 2
ð6Þ
where Eshear is the mean specific shear energy (J∙kg− 1). The Mohr– Coulomb theory says nothing about how the powder will flow, it rather describes the conditions of the onset of yielding but it gives us a glimpse that it may be possible to better understand the agglomeration level of powders and nanopowders by using rheological study.
ð7Þ
Fig. 1. The powder rheometer [10].
F. Henry et al. / Powder Technology 250 (2013) 13–20
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As shown by this equation, when the shear rate γ˙ is much lower than the critical shear rate γ˙c , the shear stress is a linear function of the shear rate, for which η0 is the slope of the curve; it corresponds to the Newtonian regime. At shear rate larger thanγ˙c, the shear stress tends to a constant value σf where the Coulombic regime is reached (medium shear rates), until the powder is set in suspension at high shear rates, which corresponds then to the kinetic regime. The powder rheometer allows us theoretically to observe Newtonian (low shear rate) and Coulombic (medium shear rate) regimes for all the samples. 4. Experiments and results 4.1. Experimental protocol
Fig. 2. Measurement cell.
shear rate that can be obtained, namely a maximum of 611 s− 1. In the frame of our study, the tests were performed at a frequency f of 50 Hz. 3.2. Granular regimes As described schematically in Fig. 3, a powder submitted to shear forces follows three different regimes. The Newtonian regime describes a regime of a dense powder at low shear rates. The Coulombic frictional regime corresponds to a packed/dense slightly rheofluidified phase (similar to a viscous non-newtonian liquid) and the kinetic regime to a leaner dense phase submitted to particle collisions (rheofluidized regime) at high shear rates. Strictly speaking, the “no-flow” condition is not considered as it is the case in the Jenike shear cell. However, at very low shear rates, the dynamic powder behavior is very close to that of the near-static regime of Jenike shear cell. Indeed, the cohesiveness Γc measured by this technique is very close to that measured by Jenike at very low solids consolidations, or normal stresses. The evolution of the shear stress in Newtonian and Coulombic regimes is given by [10
σ¼
σf η0γ˙ c ¼ 1 þγ˙ c =γ˙ 1 þγ˙ c =γ˙
ð9Þ
The experiments have been conducted with nanometric and micrometric powders, whose properties are presented in Table 1. Two nanometric carbon black powders (Corax N115 and Printex XE2) and a hydrophilic nanometric silica powder (Aerosil A200) were provided by EVONIK, carbon nanotubes produced by Chemical Vapor Deposition (CVD) [11] and two aluminum powders of 120 and 200 nm were provided by a consortium of industrial partners in the framework of the EU NANOSAFE 2 project [12]. For comparison purposes, two micrometric aluminum powders provided by Goodfellow were also used as well as micrometric glass beads powders of 0.33, 0.5, 0.8 and 1 mm diameter, which are assumed not to agglomerate, given their particle size distribution. The same sheared cell volume was considered for each powder. The total volume of a cell is 88.4 cm3 but the central shearing volume of powder moving in a Couette cylindrical fashion is a volume constrained between the outer blade diameter and the inner baffle diameter. The sheared volume can be estimated in our case to be 54.4 cm3. The mass mb of the powder loaded in the cell depends on the material and on the agglomeration level of the powder and can be expressed as: mb ¼ V b :ρp :ð1−eb Þ ¼ V b :ρp :ϕb
ð10Þ
where eb is the total porosity of the powder and φb the total volume solid fraction of the powder. For non-cohesive powder, this porosity will be the porosity between the particles, but for agglomerates, it will be the sum of the porosity inter- and intra-agglomerates. Table 1 gives the value of dp, ρp, mb and φb for various powders. For each powder, the evolution of the viscosity (or shear stress) versus the shear rate has been monitored and σf, η0, and γ˙c have been experimentally determined. 4.2. Nanopowders rheograms The evolution of the shear stress with an increasing shear rate has been investigated for the selected powders. The kinetic regime has Table 1 Material and bulk properties of the tested powders.
Fig. 3. The three flow regimes.
Mean diameter
dp (nm)
ρp (kg · m−3)
mb (g)
φb
Aluminum Aluminum Aluminum Aluminum Corax N115 Printex XE2 Carbon Nanotubes Aerosil Glass beads Glass beads Glass beads Glass beads
120 200 1.5, 104 2.5, 104 25 30 17 12 3.3, 105 5.0, 105 8.0, 105 1.0, 106
2700 2700 2700 2700 2050 2350 2050 2200 2700 2700 2700 2700
23.9 39.0 76.4 86.7 34.7 11.7 11.2 4.18 134.6 134.9 140.0 146.1
0.10 0.17 0.33 0.37 0.19 0.057 0.06 0.022 0.57 0.57 0.60 0.63
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F. Henry et al. / Powder Technology 250 (2013) 13–20
Fig. 4. Evolution of shear stress for glass beads (a) and nanopowders (b).
been observed for the glass beads powders, which are non cohesive (Fig. 4 (a)) as well as carbon black nanopowders, carbon nanotubes and hydrophilic nanosilica (Fig. 4 (b)). The kinetic regime has not been observed for aluminum nanopowders due to the technical limitation of our apparatus which does not allow exploring shear rates greater than 611 s− 1. Table 2 presents the values of the shear stress and shear rate at the limit between frictional and kinetic regimes. It can already be outlined that similar densities for the different glass beads powders lead then to similar rheograms (in Newtonian and Coulombic regimes) which is rather in accordance with the Mohr–Coulomb law for non cohesive powder with similar surface roughness. It can also be stated for aluminum nanopowders that the smaller the particle size, the larger the shear stress it is. This illustrates the significant role of cohesion when the particle size decreases. Indeed, the particle cohesion acts as a barrier to the flow, which leads to a higher viscosity (or shear stress), as displayed in Fig. 5. 4.3. Shear specific energy The shear stress is linked to the normal stress, which is applied to the powder and depends on the sheared mass. As a consequence, in order to avoid the effect of the powder mass, it has been chosen to divide the shear stress by the apparent density of the powder. The resulting value is homogeneous to a specific shear energy (in J∙kg−1). This energy is directly linked to the specific agglomeration energy of a powder as highlighted by Eq. 6. Contrary to the agglomeration energy, the specific shear energy is not dependant of the normal stress exerted on the powder and then may be considered as an intrinsic property of the powder, enabling to compare a powder to another in a relevant way. Figs. 6, 7
and 8 show the evolution of the shear stress and specific energy as a function of the shear rate, respectively for aluminum, carbonaceous materials and glass beads. It can be outlined from these figures that some material can have specific energy lower than that of other materials (e.g. specific energy of 330 μm-glass beads (0.11 J∙kg−1) is lower than that of Printex (0.27 J∙kg−1) and that this trend is reversed by considering the shear stress (the shear stress of 330 μm-glass beads (165 Pa) is larger than that of Printex XE2 (35.6 Pa), because of the influence of the density of the powder. The concept of shear specific energy appear then full of interest to compare powders between them and as a function of their mean particle size (cf. Fig. 9). Different energy bands appear in Fig. 9. The lower one takes into account the values of specific energy below 0.1 J.kg−1, including the glass beads. This lets us think that this energy band corresponds to noncohesive powders, or to powders that can easily deagglomerate. The slight observed difference may be mainly due to differences in surface roughness of the particles and the representativity of the equivalent particle diameter to describe a given particle size distribution. The hydrophilic silica and the micrometric aluminum are also in this energy band. For such category, even though these powders can agglomerate, the cohesivity of the powder is low and a low amount of energy could break the agglomerates into primary particles. This category would rank the powder as a very dusty powder. Two other categories could
Table 2 Shear stress and rate of the carbonaceous materials and glass beads powders. Mean diameter
dp (nm)
σf (Pa)
γ˙c (s−1)
Aluminum Aluminum Aluminum Aluminum Corax N115 Printex XE2 Carbon Nanotubes Aerosil Glass beads (1) Glass beads (2) Glass beads (3) Glass beads (4)
120 200 1.5, 104 2.5, 104 25 30 17 12 3.3, 105 5.0, 105 8.0, 105 1.0, 106
99 76 64 57.6 116 33 23.6 4.7 163 158 201 175
N611 N611 N611 N611 160 300 300 193 198 170 90 60
Fig. 5. Evolution of the shear stress with increase of aluminum powder particle size.
F. Henry et al. / Powder Technology 250 (2013) 13–20
17
Fig. 6. Shear stress and specific energy for the different aluminum powders.
be identified: one includes values of shear specific energies between 0.10 and 0.20 J∙kg−1 and another between 0.20 and 0.40 J∙kg−1. It is difficult to compare the different trends, because the materials are different. But it can be assumed that for higher specific energies, the powder would be more agglomerated. Concerning the carbonaceous materials, the lower energy scale of these materials reflects the existence of aggregates, which are formed during the fabrication process of the carbon blacks. Finally, concerning aluminum, it is observed that the aluminum powders are more cohesive at smaller particle diameters. The values of Fig. 9 are consistent with the study of Orband and Geldart [13], who investigated the evolution of the powder cohesiveness, depending on the particular diameter. In their two studies, the borderline between cohesive and non-cohesive powders is about dozens of micrometers. Fig. 9 represents also the values for 36 μmcopper and 29 and 67 μm-glass beads, which are considered as free-flowing powders by Orband and Geldart. The use of rheology appears as an interesting way to evaluate the cohesion energy of nanopowders and by following their tendency to agglomerate. Classifying the powders by referring to their specific shear energy can then constitute a useful approach to assess the dispersion risk of nanopowders. In such an approach, the energy band proposed above would allow a quantification of an intrinsic dustiness index and then of the associated risk related to the powder. Rheological measurements can also provide a more
detailed description of the physical properties of nanopowders by allowing the determination of parameters such as the mean diameter of an agglomerate, its number of particles, its porosity, its density and its shape through its fractal dimension. 5. Determination of the physical characteristics of the nanopowders Based on information extracted from rheograms, this section aims at estimating respectively the following characteristics of the investigated powders: • • • •
The agglomerate diameter dagg of the powder; The fractal dimension Df of the agglomerate; The agglomerate porosity eagg; The friction coefficient μ of the powder.
5.1. Procedure to estimate the agglomerate diameter from rheograms Agglomerate diameter depends on the shear stress applied on it. The goal of this section is to propose a procedure to estimate the diameter reached by the agglomerate at the transition from the Coulombic regime to the kinetic one. We carried out a parameter analysis. Basically, the shear stress σ depends on the characteristics of the agglomerates (diameter dagg, apparent density ρagg), the bulk density of the powder (ρb, assumed as constant) and the shear rate γ.˙ The problem consists
Fig. 7. Evolution of shear stress and specific energy for the carbonaceous materials and silica.
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F. Henry et al. / Powder Technology 250 (2013) 13–20
Fig. 8. Evolution of shear stress and specific energy for the glass beads.
then of five parameters, depending on three physical basic units (length, mass, time). The Buckingham theorem [14] states then that the problem is equivalent to an equation involving 2 original variables: dagg, ρagg and: (
a1 b1 c1 A1 ¼ σ dagg ρagg γ˙ a2 b2 c1 A2 ¼ ρbdagg ρagg γ˙
Then:
Where φ0 is the maximum value of φapp reached when the powder is compact and not flowing. When the kinetic regime is reached during the rheological experiments, a pair of values γ˙cin ; σ f is measured, which corresponds to the limit between Coulombic frictional regime and kinetic one; the Eq. (11) can then be used. Knowing that ρagg = ρp*φagg and using Eq. (13), dagg can then be expressed as:
dagg
8 < A ¼ 1
σ ρagg dagg 2γ˙ 2 : A2 ¼ ρagg =ρb ¼ φapp φapp is the ratio between the solid fraction in an agglomerate φagg and the solid fraction in bulk φb. According to Buckingham theorem, A1 = f(A2), the shear stress can then be expressed as: 2 2 σ ∝ρagg dagg γ˙ f φapp :
ð11Þ
This relationship is consistent with kinetic theory of granular flows [15] and the function f can then be expressed as: f φapp ¼
φ0 φapp
1 1=3
−1
ð12Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i σf 1=3 2=3 ¼ φ0 φapp −φapp : 2 γ˙cin ρp φb
ð13Þ
For a given value of φapp, the agglomerate diameter at the transition between Coulombic and kinetic regime can be then estimated. The only unknown parameter is the parameter φapp and it was assumed that φapp could be estimated roughly to be equivalent to the bulk solid fraction of glass beads powder, i.e. we assumed that φapp = 0.59. For non cohesive powders, the primary characteristics of the particles should be considered (dp,ρp). φapp is then defined as the ratio between the bulk density and the density of the particles, i.e. the solid fraction of the powder. The diameter of non cohesive powder can then be expressed as:
dp ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=3
σf φ0 −1 : γ˙ cin 2 ρp φb
ð14Þ
5.2. Evaluation of the agglomerate diameter of the tested powders
Fig. 9. Particular diameter dependence of the specific energy.
When the agglomerate diameter is equal to the primary particle diameter of the powder, it means that the powder is non-cohesive. The glass beads are non cohesive powders. The total solid fraction is then equal to the apparent solid fraction of the powder. By taking into the values of Tables 2 and 3, the calculated diameters are respectively 212, 271, 507 and 741 μm from Eq. (14) whereas the primary particle size of the glass beads were respectively 300, 500, 800 and 1000 μm. These estimated particle sizes are lower than the primary diameter but of the same order of magnitude, confirming that the particles do not agglomerate. In spite of the experimental uncertainty in the determination of φ0, it can be stated that considering the kinetic theory of granular flow for non-cohesive powders is a relevant approach. On the contrary, for cohesive powders, the agglomerate diameter is much larger than the primary particle size of the powders. It was calculated an agglomerate diameter of 213 μm, 401 μm, 500 μm, 229 μm for hydrophilic silica, carbon nanotubes, Corax N115 and Printex XE2 respectively.
F. Henry et al. / Powder Technology 250 (2013) 13–20
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Table 3 Main characteristics of the glass beads, hydrophilic silica and carbonaceous materials.
Carbon NT Hydrophilic silica Corax N115 Printex XE2 Glass beads (1) Glass beads (2) Glass beads (3) Glass beads (4)
γ˙cin (s−1)
φagg
ρagg (kg · m−3)
dagg (μm)
nagg
Df
Eagg
23.6 4.7 116 33 163 158 201 175
300 193 160 300 198 170 90 60
0.10 0.04 0.33 0.10 – – – –
186 82.0 674 227 – – – –
1920 214 495 228 212 271 507 741
1.53, 1014 2.09, 1011 2.56, 1012 4.28, 1010 – – – –
2.81 2.66 2.89 2.74 – – – –
0.69 0.31 0.83 0.75 – – – –
From a dispersion point of view, the following statements can be made. The frictional regime corresponds to a dense powder with agglomerates at their equilibrium diameter. At the beginning of this regime, the cohesion is offset by the shear stress, which remains at a constant value at low shear rates. At high shear rate (kinetic regime), the granular temperature (related to the shear rate) is large enough (shear thickening) to disperse these agglomerates. It can be then assumed that the diameter of dispersed agglomerates can be approached by calculation. Such a statement will need to be confirmed further in the future by proper measurements in air of the particle size distribution of the dispersion near the source-term, i.e. the rheometer. Based on these results, fractal structure of the agglomerate, its porosity as well as the friction coefficient of the Mohr–Coulomb law can be estimated. 5.3. Fractal structure of an agglomerate The fractal dimension Df is a parameter describing the sphericity of the agglomerate. Agglomerates with a fractal dimension close to 1 will be linear, whereas a fractal dimension of 3 will correspond to spherical agglomerates [16]. Gmachowski [17] defined the fractal dimension as: ln nagg −lnðkgÞ Df ¼ ln dg =dp
ð15Þ
where nagg is the number of particles of primary diameter dp within an agglomerate and kg is a structural coefficient, which takes into account the porosity of the agglomerate. dg is the diameter of gyration of the agglomerate [18]. Hess [19] determined the relationship between the diameter of gyration and the diameter of the sphere circumscribed on the agglomerate dagg as: dagg =dg ¼ ½ð2 þ D f Þ=D f
1=2
ð16Þ
The number of particles in the agglomerate can then be related to its diameter and its fractal dimension as: nagg ¼ kg ½D f =ð2 þ D f Þ
D f =2
D f dagg =dp ¼ k f dagg =dp D f
ð17Þ
Where kg and kf are structure coefficients that can be expressed as: Kg ¼ kf ¼
.
σf (Pa)
2 1=2
1:56−ð1:728−D f =2Þ
2
1:56−ð1:728−D f =2Þ
1=2
−0:228
−0:228
D
D
f
f
½ð2 þ D f Þ=D f
D f =2
Eshear
i.e. the particle diameter, the agglomerate diameter and the fractal dimension of the agglomerate. 5.4. Agglomerate porosity The agglomerate porosity is important information to estimate the agglomerate tensile strength, as shown in Eq. (1). The agglomerate porosity can be expressed as:
eagg ¼ 1−φagg ¼ 1−
dagg dp
!D
f −3
ð20Þ
The expression (17) is relevant for nearly spherical agglomerates however agglomerate may have irregular shape. It is then necessary to discuss the relevance of the choice of dagg. To characterize agglomerates, various equivalent diameters are used in the literature [20], i.e.: • dv: the volume equivalent diameter, which is the diameter of the sphere having the same volume as the agglomerate; • da: the aerodynamic equivalent diameter, which is the diameter of the sphere having a density of 1000 kg.m− 3 and the same terminal velocity as the agglomerate; • dm: the electrical mobility equivalent diameter, which is the diameter of the sphere having the same electrical mobility as the agglomerate. The sphere of diameter dv is smaller than the agglomerate and will underestimate the porosity of the agglomerate. The sphere of diameter dm as well as the sphere of diameter da are bigger than the agglomerate and will overestimate the porosity of the agglomerate. It is then necessary to balance the value of these equivalent diameters of agglomerate. Choosing dagg, the diameter of the sphere circumscribed on the agglomerate, appears then relevant as dagg is between da and dv. Moreover, the structure factor kf will balance the overestimation of eagg, the porosity linked to the calculation of dagg. 5.5. Friction coefficient The coefficient of friction μ, as presented in the Mohr–Coulomb law, varies with the shape of the agglomerates, evaluated by the fractal dimension and the structure factor, and with the parameter φapp. In order to determine the friction coefficient, the correlation established by Shinohara [21] can be used: 5:76 −74:6K f 5 ð1−φapp Þ μ ¼ tan 64:7 exp
ð21Þ
ð18Þ Some calculated parameters were gathered in Table 3. ð19Þ 6. Conclusions
Knowing the agglomerate diameter, the diameter of gyration can be expressed as a function of the fractal dimension (Eq. 16). The Eq. (15) can then be expressed solely as a function of 3 parameters,
The effect of agglomeration on a nanopowder flow has been studied with the use of an original powder rheometer. It has been shown that the powder follows three different flow regimes. The first two regimes,
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Newtonian and Coulombic, strongly depend on the cohesion of the particles, and the third one, the kinetic regime, depends mainly on the particle size and the shear rate. In the Coulombic regime, the shear stress is used to calculate the specific shear energy, information linked to the cohesion energy of a powder. This cohesion energy strongly depends on the primary particle diameter of the powder and can provide useful information on the tendency of the powder to disperse or agglomerate. The tests presented in this paper have shown three specific energy bands of cohesion. The lower one (small specific shear energy), which includes the non-cohesive glass beads, corresponds to agglomerates which are easily breakable. It appears that the 25 nm carbon black and the 120 nm aluminum are the most cohesive powders among those tested. The energy required to break their agglomerates will be then larger than that for the other nanopowders. The values of the shear rate and shear stress at the frictional/kinetic regime transition have also allowed the evaluation the agglomerate diameters for carbonaceous materials and hydrophilic silica. Intrinsic information on agglomerates could then be given, such as internal volume solid fraction, agglomerate density or number of particles. It should also be noted that the powder that has the highest specific shear energy and as consequence the highest specific agglomeration energy, is not necessarily made up of the largest agglomerates. For example, the specific shear energy for carbon black Printex XE2 (0.27 J∙kg− 1) is higher than those for carbon nanotubes CNT (0.25 J∙kg− 1), whereas the carbon black seems to present agglomerates smaller (315 μm) compared to that of the CNT (408 μm). Finally, it has been shown that the solid fraction in the kinetic regime seems to decrease when the shear rates increases. The evolution of the solid fraction, and then the loss of powder in the rheometer cell, could then be used to evaluate the powder dustiness. This information would be essential to assess dispersion risks linked to nanopowders handling. To improve (nano)powder experimental dense phase kinetic studies, wider dynamic shear rate ranges would be necessary. Hence, future studies would require geometrical modifications of the rheometer cell and the four-blade vane impeller described in this paper in order to achieve higher shear rates. Nomenclature C torque (N.m) dagg agglomerate diameter (m) dg diameter of gyration (m) dp particle diameter (m) Df fractal dimension eagg inner porosity of the agglomerate eb total porosity of the powder Ev energy of the vibrations (J) f frequency of the vibrations (Hz) g gravity (m.s−2) h height of powder in the cell (m) kf structure factor of the agglomerate kg structural coefficient Kσ Couette-analogy stress coefficient Kγ Couette-analogy rate coefficient m mass of powder in the cell (kg) mb mass of sheared powder (kg) nagg number of particles by agglomerate nb total number of particles in the powder Vagg agglomerate volume (m3) Vb volume of sheared powder (m3)
Greek symbols γ shear rate (s−1) γc critic al shear rate (s−1) γcin shear rate at the start of kinetic state (s−1) γmax maximal shear rate delivered by the rheometer (s−1) Γagg agglomerate tensile strength (Pa) Γc cohesion stress (Pa) η powder viscosity (Pa.s) η0 constant viscosity in Newtonian state (Pa.s) φagg agglomerate solid fraction φ app Ratio between the solid fraction of agglomerates and the bulk solid fraction φb Bulk solid fraction φ0 Bulk solid fraction of the compact powder μ coefficient of friction Θ amplitude of the vibrations (m) ρp density of the particles (kg.m−3) ρb bulk density (kg.m−3) σ shear stress (Pa) σf constant shear stress in frictional regime (Pa) Ω angular rate (s−1) Ωmax maximal angular rate delivered (s−1)
References [1] A.D. Maynard, R.J. Aitken, T. Butz, V. Colvin, K. Donaldson, G. Oberdörster, Safe handling of nanotechnology, Nat. 16 (2006) 267–269. [2] Woodrow Wilson's Nanotechnology Consumer Products Inventory, www. nanotechproject.org(last accessed 01.05.13). [3] O. Gundogdu, P.M. Jenneson, Understanding nanoagglomerates, Adv. Sci. Lett. 1 (2008) 161–164. [4] J. Bouillard, A. Vignes, O. Dufaud, L. Perrin, D. Thomas, Ignition and explosion risks of nanopowders, J. Hazard. Mater. 181 (2010) 873–880. [5] ISO/TS 27687, Nanotechnologies — Terminology and Definitions for Nano-objects — Nanoparticle, Nanofibre and Nanoplate, www.iso.org2008. [6] H. Rumpf, The strength of Granules and Agglomerates, Interscience Publishers, Agglomeration, 1962. [7] J. Thomas, Fundamentals of cohesive powder consolidation and flow, Granul. Matter 6 (2004) 75–86. [8] P. Marchal, N. Smirani, L. Choplin, Rheology of dense-phase vibrated powders and molecular analogies, J. Rheol. 53 (2009) 1–29. [9] A. Ai-Kadi, P. Marchal, L. Choplin, A.S. Chrissemant, M. Bousmina, Quantitative analysis of mixer-type rheometers using Couette analogy, Can. J. Chem. Eng. 80 (2002) 1166–1174. [10] P. Marchal, L. Choplin, Eléments de physique statistique appliqués à la rhéologie des milieux granulaires non cohésifs: le modèle du château de sables mouvants, Rheol. 5 (2004) 10–26. [11] A. Vignes, O. Dufaud, L. Perrin, D. Thomas, J. Bouillard, A. Janès, C. Vallières, Thermal ignition and self-heating of carbon nanotubes: from thermokinetic study to process safety, Chem. Eng. Sci. 64 (2009) 4210–4221. [12] F. Schuster, J. Bouillard, NANOSAFE2: Safe Production and Use of Nanomaterials, European Project No. 515843-2, 2005–2009, www.nanosafe2.org. [13] J.L.R. Orband, D. Geldart, Direct measurement of powder cohesion using a torsional device, Powder Technol. 92 (1997) 25–33. [14] E. Buckingham, On physically similar systems. Illustrations of the use of dimensional equations, Phys. Rev. 4 (1914) 345–376. [15] B. Andreotti, Y. Forterre, O. Pouliquen, Les milieux granulaires: entre fluide et solide, EDP Sci. (2011). [16] F.X. Ouf, PhD Thesis Caractérisation des aérosols émis lors d'un incendie, Faculté des sciences et des techniques, Rouen, 2006. [17] L. Gmachowski, Calculation of the fractal dimension of aggregates, Colloids Surf., A Physicochem. Eng. Asp. 211 (2002) 197–203. [18] D. Stauffer, H.E. Stanley, A. Lesne, From Newton to Mandelbrot, Springer, 1999. [19] W. Hess, H.L. Frisch, R. Klein, On the hydrodynamic behavior of colloidal aggregates, Phys. B Condens. Matter. 64 (1986) 65–67. [20] S. Bau, PhD Thesis Etude des moyens de mesure de la surface des aérosols ultrafins pour l'évaluation de l'exposition professionnelle, Institut National Polytechnique de Lorraine, Nancy, 2008. [21] K. Shinohara, M. Oida, B. Golman, Effect of particle shape on angle of internal friction by triaxial compression test, Powder Technol. 107 (2000) 131–136.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Exploring a new method to study the agglomeration of powders: Application to nanopowders François Henry a,b,⁎, Jacques Bouillard a, Philippe Marchal b, Alexis Vignes a, Olivier Dufaud b, Laurent Perrin b a b
INERIS, Parc Technologique ALATA, B.P. 2, F-60550, Verneuil-en-Halatte, France Laboratoire Réactions et Génie des Procédés, Université de Lorraine, CNRS, 1 rue Grandville, B.P. 20451, F-54001 Nancy, France
a r t i c l e
i n f o
Article history: Received 3 October 2012 Received in revised form 31 July 2013 Accepted 8 August 2013 Available online 20 August 2013 Keywords: Nanoparticles Agglomeration Flow Rheology Shear Risk assessment
a b s t r a c t The increasing use of nanoparticles in the production of nanomaterials has led to the identification of nanoparticle agglomeration in air or in a given medium as a key problem, both on the standpoint of process performance (i.e. product homogeneity) and above all on inhalation risks. This paper suggests a new method to evaluate the structure and the dynamics of agglomeration of a nanopowder, based upon monitoring of the shear stress of a powder which is submitted to the mechanical solicitation of a rheometer. At low and increasing shear rates, the powder flow will evolve from a Newtonian regime (dense powder) to a Coulombic regime (slightly rheofluidized dense phase). At larger shear rates, the powder will be set in suspension, which characterizes a kinetic regime governed by particle collisions. The specific shear energy, related to the specific agglomeration energy, has been calculated for different nanopowders (carbon, aluminum, silica) and compared to non-cohesive micrometric powders (glass beads). From the measurements of shear rates and stresses at the frictional/kinetic transition, agglomerate diameters have been evaluated for carbonaceous material and for silica. Values of these agglomerates can range from 200 to 500 μm and are related to their propensity to break apart. The carbonaceous materials seem to be the more difficult to deagglomerate, whereas silica nanopowder agglomerates are more easily breakable. Measures of cohesiveness (or specific agglomeration energy) can be useful for assessing the dispersibility of nanopowders and their relationships to inhalation or explosion risks. © 2013 Published by Elsevier B.V.
1. Introduction The growing development of nanomaterials in various industrial sectors [1,2] raises new kind of risks that need to be assessed carefully. In order to achieve such goal, it is necessary to better understand the physico-chemical properties related to nanoparticles and among them, their tendency to agglomerate [3]. Agglomeration phenomenon can in fact modify dispersibility of nanoparticles, their reactivity [4] as well as their toxicity and ecotoxicity. Therefore, nanoparticle agglomeration needs to be further understood as it directly influences their hazards and related risks. In that context, the purpose of this paper is to propose a new method to characterize the specific agglomeration energy and its effect on the rheological properties of nanoparticles. After having described the theories of dense flow regimes, the rheological apparatus and its associated protocol, this paper explains how rheograms can be used to calculate the specific shear energy, which is related to the agglomeration energy. The agglomeration regime of a nanopowder can be described at the powder
⁎ Corresponding author at: INERIS, Parc Technologique ALATA, B.P. 2, F-60550, Verneuilen-Halatte, France. E-mail address: [email protected] (F. Henry). 0032-5910/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.powtec.2013.08.010
level, but rheograms can also provide information on estimates of agglomerate diameters as well as internal volume solid fraction and number of particles in agglomerates. This paper shows that from macroscopic rheological properties, microscopic features of agglomerates can be inferred.
2. An experimental approach to evaluate the agglomeration of nanopowders According to ISO [5], an agglomerate is a “collection of loosely bound particles or aggregates or mixtures of the two, where the resulting external surface area is similar to the sum of the surface areas of the individual components”. This standard adds that the “forces holding an agglomerate together are weak forces, for example Van der Waals forces, as well as simple physical entanglement”. The terms agglomerate and aggregate are often confused, but as specified in the previous standard, an aggregate is made of particles strongly bonded or fused. As a consequence, contrary to agglomeration, aggregation can be considered as a non-reversible phenomenon. The literature contains many models to understand the cohesivity of agglomerates such as the one proposed by Rumpf [6], who defined the agglomerate tensile strength Γagg as the stress (force per unit area)
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F. Henry et al. / Powder Technology 250 (2013) 13–20
required to break into two halves a system of particles which are linked by weak bond forces:
Γ agg
This section presents the apparatus used to investigate the flow of nanopowders and also the theoretical description of the flow regimes.
1−eagg F ¼ 1:1 dp2 eagg
ð1Þ 3.1. The powder rheometer
where F represents the weak interaction forces (Van der Waals forces), dp is the particle diameter and eagg is the inner porosity of the agglomerate. It should be highlighted that this description encounters a limitation as it does not take into account the structural details of aggregates that may compose the agglomerate. With this kind of model, it is in fact assumed that an agglomerate is only composed of uniformly sized spheres holding by weak forces. The effects of the aggregates on the cohesivity of the agglomerate cannot be highlighted by such models. However, in view of simplification, Rumpf relationship will be used here as it can provide an order of magnitude of the agglomerate tensile strength. By considering Mohr–Coulomb criterion [7], it can be stated that an agglomerate will break at a critical quantity of shear and normal stresses. It can then be stated that it is possible to better understand agglomeration through rheological study of a powder. Indeed, under a mechanical stress, a powder will flow under a certain velocity gradient with a given dynamic viscosity, only when the shear stress reaches a critical magnitude. According to the Mohr–Coulomb theory, the shear stress σ is the sum of the effects of the cohesiveness Γc and the friction: σ ¼ Γ c þ μN
ð2Þ
where μ is the coefficient of friction and the normal stress N is equivalent to a pressure exerted on the powder and can be expressed locally as: ð3Þ
N ¼ ρb gz
To obtain a vertical average normal stress in the z direction, one can calculate an averaged normal stress as: N¼
1 h
Z
h 0
ρb gz dz ¼ρb g
h 2
ð4Þ
where g is the gravity, with an average magnitude of 9.81 m∙s−2, and h the height of the total volume of powder, and ρb is the total density of the powder. Hence, the averaged shear stress becomes: h σ ¼ Γ c þ μ ρb g 2
3. Powder rheology
The experiments have been led with a powder rheometer (AR 2000, TA Instruments) specifically designed by P. Marchal [8] and is presented on Fig. 1. An induction motor supplies a torque C and an optical sensor Ä measures the angular rate Ωn. Both parameters are linked to shear stress σ and shear rate γ thanks to proportional coefficients. The four blade vane (Fig. 2), which turns in a cylindrical tank with baffles, has been designed by analogy with a Couette system in such a way that [9]: γ˙ ¼ K γ Ω˙
σ ¼ Kσ C
with Kσ = 3.858.104 m−3 and Kγ = 2.037 rad−1. It is then possible to define the viscosity as the ratio between the shear stress and the shear rate, as η ¼ σ=γ˙. Hence, submitted to a mechanical solicitation, a powder will flow under a certain velocity gradient or a shear stress with a given dynamic viscosity η The baffles and the vane prevent the sample from slipping at the wall and at the surface of the measuring tool. Consequently, this design only promotes the powder/powder friction interaction. To provide a granular reorganization of the powder, the cell is submitted to vertical mechanical vibrations, at variable frequencies and amplitudes. They are controlled by a mini-shaker connected to a signal generator and an amplifier. The vibrations are sinusoidal, with an amplitude Θ, a frequency f and an energy Ev: 2
2
Ev ¼ mð2πf Þ Θ :
ð8Þ
This vibrating device aims at providing an ergodic granular temperature to the powder and avoids specific locked-in rest configurations of the powder that would lead to biased results at low shear rates. Marchal et al. [8] proposed then to use vibrations to induce a Brownian-like granular motion within the powder in order to access to all powder configurations to ensure a granular temperature to the system. For AR 2000 apparatus, the torque varies from 100 mN∙m to 200 mN∙m and the angular rate from 10− 8 rad∙s− 1 to 300 rad∙s− 1. By considering the values of Kγ and Ω, it can be stated that the apparatus has a technical limitation in terms of maximum experimental
ð5Þ
It should be noted that for non cohesive powders with similar surface roughness and made of the same material, the shear stress is expected to be only dependant on the apparent density of the powder for a given sheared volume of powder. By considering the Eq. (5), it is then possible to define an average specific agglomeration energy Eagg as the ratio between the lean cohesion stress and the mean total density: Eagg ¼ Eshear −μg
h 2
ð6Þ
where Eshear is the mean specific shear energy (J∙kg− 1). The Mohr– Coulomb theory says nothing about how the powder will flow, it rather describes the conditions of the onset of yielding but it gives us a glimpse that it may be possible to better understand the agglomeration level of powders and nanopowders by using rheological study.
ð7Þ
Fig. 1. The powder rheometer [10].
F. Henry et al. / Powder Technology 250 (2013) 13–20
15
As shown by this equation, when the shear rate γ˙ is much lower than the critical shear rate γ˙c , the shear stress is a linear function of the shear rate, for which η0 is the slope of the curve; it corresponds to the Newtonian regime. At shear rate larger thanγ˙c, the shear stress tends to a constant value σf where the Coulombic regime is reached (medium shear rates), until the powder is set in suspension at high shear rates, which corresponds then to the kinetic regime. The powder rheometer allows us theoretically to observe Newtonian (low shear rate) and Coulombic (medium shear rate) regimes for all the samples. 4. Experiments and results 4.1. Experimental protocol
Fig. 2. Measurement cell.
shear rate that can be obtained, namely a maximum of 611 s− 1. In the frame of our study, the tests were performed at a frequency f of 50 Hz. 3.2. Granular regimes As described schematically in Fig. 3, a powder submitted to shear forces follows three different regimes. The Newtonian regime describes a regime of a dense powder at low shear rates. The Coulombic frictional regime corresponds to a packed/dense slightly rheofluidified phase (similar to a viscous non-newtonian liquid) and the kinetic regime to a leaner dense phase submitted to particle collisions (rheofluidized regime) at high shear rates. Strictly speaking, the “no-flow” condition is not considered as it is the case in the Jenike shear cell. However, at very low shear rates, the dynamic powder behavior is very close to that of the near-static regime of Jenike shear cell. Indeed, the cohesiveness Γc measured by this technique is very close to that measured by Jenike at very low solids consolidations, or normal stresses. The evolution of the shear stress in Newtonian and Coulombic regimes is given by [10
σ¼
σf η0γ˙ c ¼ 1 þγ˙ c =γ˙ 1 þγ˙ c =γ˙
ð9Þ
The experiments have been conducted with nanometric and micrometric powders, whose properties are presented in Table 1. Two nanometric carbon black powders (Corax N115 and Printex XE2) and a hydrophilic nanometric silica powder (Aerosil A200) were provided by EVONIK, carbon nanotubes produced by Chemical Vapor Deposition (CVD) [11] and two aluminum powders of 120 and 200 nm were provided by a consortium of industrial partners in the framework of the EU NANOSAFE 2 project [12]. For comparison purposes, two micrometric aluminum powders provided by Goodfellow were also used as well as micrometric glass beads powders of 0.33, 0.5, 0.8 and 1 mm diameter, which are assumed not to agglomerate, given their particle size distribution. The same sheared cell volume was considered for each powder. The total volume of a cell is 88.4 cm3 but the central shearing volume of powder moving in a Couette cylindrical fashion is a volume constrained between the outer blade diameter and the inner baffle diameter. The sheared volume can be estimated in our case to be 54.4 cm3. The mass mb of the powder loaded in the cell depends on the material and on the agglomeration level of the powder and can be expressed as: mb ¼ V b :ρp :ð1−eb Þ ¼ V b :ρp :ϕb
ð10Þ
where eb is the total porosity of the powder and φb the total volume solid fraction of the powder. For non-cohesive powder, this porosity will be the porosity between the particles, but for agglomerates, it will be the sum of the porosity inter- and intra-agglomerates. Table 1 gives the value of dp, ρp, mb and φb for various powders. For each powder, the evolution of the viscosity (or shear stress) versus the shear rate has been monitored and σf, η0, and γ˙c have been experimentally determined. 4.2. Nanopowders rheograms The evolution of the shear stress with an increasing shear rate has been investigated for the selected powders. The kinetic regime has Table 1 Material and bulk properties of the tested powders.
Fig. 3. The three flow regimes.
Mean diameter
dp (nm)
ρp (kg · m−3)
mb (g)
φb
Aluminum Aluminum Aluminum Aluminum Corax N115 Printex XE2 Carbon Nanotubes Aerosil Glass beads Glass beads Glass beads Glass beads
120 200 1.5, 104 2.5, 104 25 30 17 12 3.3, 105 5.0, 105 8.0, 105 1.0, 106
2700 2700 2700 2700 2050 2350 2050 2200 2700 2700 2700 2700
23.9 39.0 76.4 86.7 34.7 11.7 11.2 4.18 134.6 134.9 140.0 146.1
0.10 0.17 0.33 0.37 0.19 0.057 0.06 0.022 0.57 0.57 0.60 0.63
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F. Henry et al. / Powder Technology 250 (2013) 13–20
Fig. 4. Evolution of shear stress for glass beads (a) and nanopowders (b).
been observed for the glass beads powders, which are non cohesive (Fig. 4 (a)) as well as carbon black nanopowders, carbon nanotubes and hydrophilic nanosilica (Fig. 4 (b)). The kinetic regime has not been observed for aluminum nanopowders due to the technical limitation of our apparatus which does not allow exploring shear rates greater than 611 s− 1. Table 2 presents the values of the shear stress and shear rate at the limit between frictional and kinetic regimes. It can already be outlined that similar densities for the different glass beads powders lead then to similar rheograms (in Newtonian and Coulombic regimes) which is rather in accordance with the Mohr–Coulomb law for non cohesive powder with similar surface roughness. It can also be stated for aluminum nanopowders that the smaller the particle size, the larger the shear stress it is. This illustrates the significant role of cohesion when the particle size decreases. Indeed, the particle cohesion acts as a barrier to the flow, which leads to a higher viscosity (or shear stress), as displayed in Fig. 5. 4.3. Shear specific energy The shear stress is linked to the normal stress, which is applied to the powder and depends on the sheared mass. As a consequence, in order to avoid the effect of the powder mass, it has been chosen to divide the shear stress by the apparent density of the powder. The resulting value is homogeneous to a specific shear energy (in J∙kg−1). This energy is directly linked to the specific agglomeration energy of a powder as highlighted by Eq. 6. Contrary to the agglomeration energy, the specific shear energy is not dependant of the normal stress exerted on the powder and then may be considered as an intrinsic property of the powder, enabling to compare a powder to another in a relevant way. Figs. 6, 7
and 8 show the evolution of the shear stress and specific energy as a function of the shear rate, respectively for aluminum, carbonaceous materials and glass beads. It can be outlined from these figures that some material can have specific energy lower than that of other materials (e.g. specific energy of 330 μm-glass beads (0.11 J∙kg−1) is lower than that of Printex (0.27 J∙kg−1) and that this trend is reversed by considering the shear stress (the shear stress of 330 μm-glass beads (165 Pa) is larger than that of Printex XE2 (35.6 Pa), because of the influence of the density of the powder. The concept of shear specific energy appear then full of interest to compare powders between them and as a function of their mean particle size (cf. Fig. 9). Different energy bands appear in Fig. 9. The lower one takes into account the values of specific energy below 0.1 J.kg−1, including the glass beads. This lets us think that this energy band corresponds to noncohesive powders, or to powders that can easily deagglomerate. The slight observed difference may be mainly due to differences in surface roughness of the particles and the representativity of the equivalent particle diameter to describe a given particle size distribution. The hydrophilic silica and the micrometric aluminum are also in this energy band. For such category, even though these powders can agglomerate, the cohesivity of the powder is low and a low amount of energy could break the agglomerates into primary particles. This category would rank the powder as a very dusty powder. Two other categories could
Table 2 Shear stress and rate of the carbonaceous materials and glass beads powders. Mean diameter
dp (nm)
σf (Pa)
γ˙c (s−1)
Aluminum Aluminum Aluminum Aluminum Corax N115 Printex XE2 Carbon Nanotubes Aerosil Glass beads (1) Glass beads (2) Glass beads (3) Glass beads (4)
120 200 1.5, 104 2.5, 104 25 30 17 12 3.3, 105 5.0, 105 8.0, 105 1.0, 106
99 76 64 57.6 116 33 23.6 4.7 163 158 201 175
N611 N611 N611 N611 160 300 300 193 198 170 90 60
Fig. 5. Evolution of the shear stress with increase of aluminum powder particle size.
F. Henry et al. / Powder Technology 250 (2013) 13–20
17
Fig. 6. Shear stress and specific energy for the different aluminum powders.
be identified: one includes values of shear specific energies between 0.10 and 0.20 J∙kg−1 and another between 0.20 and 0.40 J∙kg−1. It is difficult to compare the different trends, because the materials are different. But it can be assumed that for higher specific energies, the powder would be more agglomerated. Concerning the carbonaceous materials, the lower energy scale of these materials reflects the existence of aggregates, which are formed during the fabrication process of the carbon blacks. Finally, concerning aluminum, it is observed that the aluminum powders are more cohesive at smaller particle diameters. The values of Fig. 9 are consistent with the study of Orband and Geldart [13], who investigated the evolution of the powder cohesiveness, depending on the particular diameter. In their two studies, the borderline between cohesive and non-cohesive powders is about dozens of micrometers. Fig. 9 represents also the values for 36 μmcopper and 29 and 67 μm-glass beads, which are considered as free-flowing powders by Orband and Geldart. The use of rheology appears as an interesting way to evaluate the cohesion energy of nanopowders and by following their tendency to agglomerate. Classifying the powders by referring to their specific shear energy can then constitute a useful approach to assess the dispersion risk of nanopowders. In such an approach, the energy band proposed above would allow a quantification of an intrinsic dustiness index and then of the associated risk related to the powder. Rheological measurements can also provide a more
detailed description of the physical properties of nanopowders by allowing the determination of parameters such as the mean diameter of an agglomerate, its number of particles, its porosity, its density and its shape through its fractal dimension. 5. Determination of the physical characteristics of the nanopowders Based on information extracted from rheograms, this section aims at estimating respectively the following characteristics of the investigated powders: • • • •
The agglomerate diameter dagg of the powder; The fractal dimension Df of the agglomerate; The agglomerate porosity eagg; The friction coefficient μ of the powder.
5.1. Procedure to estimate the agglomerate diameter from rheograms Agglomerate diameter depends on the shear stress applied on it. The goal of this section is to propose a procedure to estimate the diameter reached by the agglomerate at the transition from the Coulombic regime to the kinetic one. We carried out a parameter analysis. Basically, the shear stress σ depends on the characteristics of the agglomerates (diameter dagg, apparent density ρagg), the bulk density of the powder (ρb, assumed as constant) and the shear rate γ.˙ The problem consists
Fig. 7. Evolution of shear stress and specific energy for the carbonaceous materials and silica.
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F. Henry et al. / Powder Technology 250 (2013) 13–20
Fig. 8. Evolution of shear stress and specific energy for the glass beads.
then of five parameters, depending on three physical basic units (length, mass, time). The Buckingham theorem [14] states then that the problem is equivalent to an equation involving 2 original variables: dagg, ρagg and: (
a1 b1 c1 A1 ¼ σ dagg ρagg γ˙ a2 b2 c1 A2 ¼ ρbdagg ρagg γ˙
Then:
Where φ0 is the maximum value of φapp reached when the powder is compact and not flowing. When the kinetic regime is reached during the rheological experiments, a pair of values γ˙cin ; σ f is measured, which corresponds to the limit between Coulombic frictional regime and kinetic one; the Eq. (11) can then be used. Knowing that ρagg = ρp*φagg and using Eq. (13), dagg can then be expressed as:
dagg
8 < A ¼ 1
σ ρagg dagg 2γ˙ 2 : A2 ¼ ρagg =ρb ¼ φapp φapp is the ratio between the solid fraction in an agglomerate φagg and the solid fraction in bulk φb. According to Buckingham theorem, A1 = f(A2), the shear stress can then be expressed as: 2 2 σ ∝ρagg dagg γ˙ f φapp :
ð11Þ
This relationship is consistent with kinetic theory of granular flows [15] and the function f can then be expressed as: f φapp ¼
φ0 φapp
1 1=3
−1
ð12Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i σf 1=3 2=3 ¼ φ0 φapp −φapp : 2 γ˙cin ρp φb
ð13Þ
For a given value of φapp, the agglomerate diameter at the transition between Coulombic and kinetic regime can be then estimated. The only unknown parameter is the parameter φapp and it was assumed that φapp could be estimated roughly to be equivalent to the bulk solid fraction of glass beads powder, i.e. we assumed that φapp = 0.59. For non cohesive powders, the primary characteristics of the particles should be considered (dp,ρp). φapp is then defined as the ratio between the bulk density and the density of the particles, i.e. the solid fraction of the powder. The diameter of non cohesive powder can then be expressed as:
dp ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=3
σf φ0 −1 : γ˙ cin 2 ρp φb
ð14Þ
5.2. Evaluation of the agglomerate diameter of the tested powders
Fig. 9. Particular diameter dependence of the specific energy.
When the agglomerate diameter is equal to the primary particle diameter of the powder, it means that the powder is non-cohesive. The glass beads are non cohesive powders. The total solid fraction is then equal to the apparent solid fraction of the powder. By taking into the values of Tables 2 and 3, the calculated diameters are respectively 212, 271, 507 and 741 μm from Eq. (14) whereas the primary particle size of the glass beads were respectively 300, 500, 800 and 1000 μm. These estimated particle sizes are lower than the primary diameter but of the same order of magnitude, confirming that the particles do not agglomerate. In spite of the experimental uncertainty in the determination of φ0, it can be stated that considering the kinetic theory of granular flow for non-cohesive powders is a relevant approach. On the contrary, for cohesive powders, the agglomerate diameter is much larger than the primary particle size of the powders. It was calculated an agglomerate diameter of 213 μm, 401 μm, 500 μm, 229 μm for hydrophilic silica, carbon nanotubes, Corax N115 and Printex XE2 respectively.
F. Henry et al. / Powder Technology 250 (2013) 13–20
19
Table 3 Main characteristics of the glass beads, hydrophilic silica and carbonaceous materials.
Carbon NT Hydrophilic silica Corax N115 Printex XE2 Glass beads (1) Glass beads (2) Glass beads (3) Glass beads (4)
γ˙cin (s−1)
φagg
ρagg (kg · m−3)
dagg (μm)
nagg
Df
Eagg
23.6 4.7 116 33 163 158 201 175
300 193 160 300 198 170 90 60
0.10 0.04 0.33 0.10 – – – –
186 82.0 674 227 – – – –
1920 214 495 228 212 271 507 741
1.53, 1014 2.09, 1011 2.56, 1012 4.28, 1010 – – – –
2.81 2.66 2.89 2.74 – – – –
0.69 0.31 0.83 0.75 – – – –
From a dispersion point of view, the following statements can be made. The frictional regime corresponds to a dense powder with agglomerates at their equilibrium diameter. At the beginning of this regime, the cohesion is offset by the shear stress, which remains at a constant value at low shear rates. At high shear rate (kinetic regime), the granular temperature (related to the shear rate) is large enough (shear thickening) to disperse these agglomerates. It can be then assumed that the diameter of dispersed agglomerates can be approached by calculation. Such a statement will need to be confirmed further in the future by proper measurements in air of the particle size distribution of the dispersion near the source-term, i.e. the rheometer. Based on these results, fractal structure of the agglomerate, its porosity as well as the friction coefficient of the Mohr–Coulomb law can be estimated. 5.3. Fractal structure of an agglomerate The fractal dimension Df is a parameter describing the sphericity of the agglomerate. Agglomerates with a fractal dimension close to 1 will be linear, whereas a fractal dimension of 3 will correspond to spherical agglomerates [16]. Gmachowski [17] defined the fractal dimension as: ln nagg −lnðkgÞ Df ¼ ln dg =dp
ð15Þ
where nagg is the number of particles of primary diameter dp within an agglomerate and kg is a structural coefficient, which takes into account the porosity of the agglomerate. dg is the diameter of gyration of the agglomerate [18]. Hess [19] determined the relationship between the diameter of gyration and the diameter of the sphere circumscribed on the agglomerate dagg as: dagg =dg ¼ ½ð2 þ D f Þ=D f
1=2
ð16Þ
The number of particles in the agglomerate can then be related to its diameter and its fractal dimension as: nagg ¼ kg ½D f =ð2 þ D f Þ
D f =2
D f dagg =dp ¼ k f dagg =dp D f
ð17Þ
Where kg and kf are structure coefficients that can be expressed as: Kg ¼ kf ¼
.
σf (Pa)
2 1=2
1:56−ð1:728−D f =2Þ
2
1:56−ð1:728−D f =2Þ
1=2
−0:228
−0:228
D
D
f
f
½ð2 þ D f Þ=D f
D f =2
Eshear
i.e. the particle diameter, the agglomerate diameter and the fractal dimension of the agglomerate. 5.4. Agglomerate porosity The agglomerate porosity is important information to estimate the agglomerate tensile strength, as shown in Eq. (1). The agglomerate porosity can be expressed as:
eagg ¼ 1−φagg ¼ 1−
dagg dp
!D
f −3
ð20Þ
The expression (17) is relevant for nearly spherical agglomerates however agglomerate may have irregular shape. It is then necessary to discuss the relevance of the choice of dagg. To characterize agglomerates, various equivalent diameters are used in the literature [20], i.e.: • dv: the volume equivalent diameter, which is the diameter of the sphere having the same volume as the agglomerate; • da: the aerodynamic equivalent diameter, which is the diameter of the sphere having a density of 1000 kg.m− 3 and the same terminal velocity as the agglomerate; • dm: the electrical mobility equivalent diameter, which is the diameter of the sphere having the same electrical mobility as the agglomerate. The sphere of diameter dv is smaller than the agglomerate and will underestimate the porosity of the agglomerate. The sphere of diameter dm as well as the sphere of diameter da are bigger than the agglomerate and will overestimate the porosity of the agglomerate. It is then necessary to balance the value of these equivalent diameters of agglomerate. Choosing dagg, the diameter of the sphere circumscribed on the agglomerate, appears then relevant as dagg is between da and dv. Moreover, the structure factor kf will balance the overestimation of eagg, the porosity linked to the calculation of dagg. 5.5. Friction coefficient The coefficient of friction μ, as presented in the Mohr–Coulomb law, varies with the shape of the agglomerates, evaluated by the fractal dimension and the structure factor, and with the parameter φapp. In order to determine the friction coefficient, the correlation established by Shinohara [21] can be used: 5:76 −74:6K f 5 ð1−φapp Þ μ ¼ tan 64:7 exp
ð21Þ
ð18Þ Some calculated parameters were gathered in Table 3. ð19Þ 6. Conclusions
Knowing the agglomerate diameter, the diameter of gyration can be expressed as a function of the fractal dimension (Eq. 16). The Eq. (15) can then be expressed solely as a function of 3 parameters,
The effect of agglomeration on a nanopowder flow has been studied with the use of an original powder rheometer. It has been shown that the powder follows three different flow regimes. The first two regimes,
20
F. Henry et al. / Powder Technology 250 (2013) 13–20
Newtonian and Coulombic, strongly depend on the cohesion of the particles, and the third one, the kinetic regime, depends mainly on the particle size and the shear rate. In the Coulombic regime, the shear stress is used to calculate the specific shear energy, information linked to the cohesion energy of a powder. This cohesion energy strongly depends on the primary particle diameter of the powder and can provide useful information on the tendency of the powder to disperse or agglomerate. The tests presented in this paper have shown three specific energy bands of cohesion. The lower one (small specific shear energy), which includes the non-cohesive glass beads, corresponds to agglomerates which are easily breakable. It appears that the 25 nm carbon black and the 120 nm aluminum are the most cohesive powders among those tested. The energy required to break their agglomerates will be then larger than that for the other nanopowders. The values of the shear rate and shear stress at the frictional/kinetic regime transition have also allowed the evaluation the agglomerate diameters for carbonaceous materials and hydrophilic silica. Intrinsic information on agglomerates could then be given, such as internal volume solid fraction, agglomerate density or number of particles. It should also be noted that the powder that has the highest specific shear energy and as consequence the highest specific agglomeration energy, is not necessarily made up of the largest agglomerates. For example, the specific shear energy for carbon black Printex XE2 (0.27 J∙kg− 1) is higher than those for carbon nanotubes CNT (0.25 J∙kg− 1), whereas the carbon black seems to present agglomerates smaller (315 μm) compared to that of the CNT (408 μm). Finally, it has been shown that the solid fraction in the kinetic regime seems to decrease when the shear rates increases. The evolution of the solid fraction, and then the loss of powder in the rheometer cell, could then be used to evaluate the powder dustiness. This information would be essential to assess dispersion risks linked to nanopowders handling. To improve (nano)powder experimental dense phase kinetic studies, wider dynamic shear rate ranges would be necessary. Hence, future studies would require geometrical modifications of the rheometer cell and the four-blade vane impeller described in this paper in order to achieve higher shear rates. Nomenclature C torque (N.m) dagg agglomerate diameter (m) dg diameter of gyration (m) dp particle diameter (m) Df fractal dimension eagg inner porosity of the agglomerate eb total porosity of the powder Ev energy of the vibrations (J) f frequency of the vibrations (Hz) g gravity (m.s−2) h height of powder in the cell (m) kf structure factor of the agglomerate kg structural coefficient Kσ Couette-analogy stress coefficient Kγ Couette-analogy rate coefficient m mass of powder in the cell (kg) mb mass of sheared powder (kg) nagg number of particles by agglomerate nb total number of particles in the powder Vagg agglomerate volume (m3) Vb volume of sheared powder (m3)
Greek symbols γ shear rate (s−1) γc critic al shear rate (s−1) γcin shear rate at the start of kinetic state (s−1) γmax maximal shear rate delivered by the rheometer (s−1) Γagg agglomerate tensile strength (Pa) Γc cohesion stress (Pa) η powder viscosity (Pa.s) η0 constant viscosity in Newtonian state (Pa.s) φagg agglomerate solid fraction φ app Ratio between the solid fraction of agglomerates and the bulk solid fraction φb Bulk solid fraction φ0 Bulk solid fraction of the compact powder μ coefficient of friction Θ amplitude of the vibrations (m) ρp density of the particles (kg.m−3) ρb bulk density (kg.m−3) σ shear stress (Pa) σf constant shear stress in frictional regime (Pa) Ω angular rate (s−1) Ωmax maximal angular rate delivered (s−1)
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