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Comparison of the micromechanical aggregate properties of nanostructured aggregates with the stress conditions during stirred media milling
Chemical Engineering Science 66 (2011) 4943–4952
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Review
Comparison of the micromechanical aggregate properties of nanostructured aggregates with the stress conditions during stirred media milling Carsten Schilde n, Stefan Beinert, Arno Kwade Institute for Particle Technology, TU Braunschweig, Germany
a r t i c l e i n f o
a b s ta c t
Article history: Received 25 February 2011 Received in revised form 18 June 2011 Accepted 4 July 2011 Available online 20 July 2011
In many cases nanosized particles are produced not as single primary particles but rather as particle collectives consisting of several primary particles. For many applications the particles must be available in liquid as separately dispersed primary particles or in a certain aggregate size. Especially the micromechanical properties of nanostructured aggregates, for example the breakage energy, have a strong impact on their breaking behaviour and, thus, on the dispersion process. For the determination of the micromechanical properties of nanostructured silica aggregates different measurements with a nanoindenter have been carried out. Comparing the measured micromechanical properties with dispersion results in a stirred media mill, conclusions concerning the influence of particle interactions and solid bridges between the primary particles and the strength of aggregates and their dispersibility can be drawn. The strength of the aggregates can be changed using different primary particle sizes. Generally, the maximum achievable product fineness and the efficiency of the dispersion process increases with decrease in aggregate strength and, thus, increasing primary particle size. With the help of the calculated stress energy distribution in the stirred media mill using the discrete element method and the measured fracture distribution of the aggregates measured via nanoindentation an effective dispersion fraction can be calculated. Comparing the effective dispersion fraction with the dispersion progress in the stirred media mill a linear correlation can be obtained. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Nanostructure Dispersion Simulation Product processing Nanoindentation Stirred media mill
Contents 1. 2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4943 Experimental set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4944 2.1. Particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4944 2.2. Measuring the micromechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4944 2.3. Dispersion process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4945 2.4. Simulation using the discrete-element-method (DEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4946 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4947 3.1. Measuring the micromechanical properties on ideal model aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4947 3.2. Effect of the micromechanical properties on dispersion experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4948 3.3. Comparison of the micromechanical properties and dispersion experiments with the simulation of the stress intensity distribution in the stirred media mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4949 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4950 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4951 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4951 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4952
1. Introduction In the past the importance of industrial mass production of high quality products and specialty chemicals raised rapidly. Depending on
n
Corresponding author. E-mail address: [email protected] (C. Schilde).
0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.07.006
the application of nanostructured particles, certain characteristics of the product quality, such as size, morphology, abrasion resistance, specific surface and tendency to agglomeration are important. They depend on the physicochemical properties of the material as well as the technical process control of the production process. Depending on the material system and the process parameters, very small particles, for example nanoparticles, can be produced. In many cases nanosized particles are produced via precipitation or pyrolysis processes not as
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single primary particles but rather as particle collectives consisting of several primary particles. Typically, in these processes the primary particles are bonded by strong solid bonds. In most applications these particles must be available in liquid as separately dispersed primary particles or in a certain aggregate size. Generally, the influencing factors for breaking up the aggregates depend on the grinding and dispersing process itself, as well as on the properties of the aggregates as shown in previous works (Schilde et al., 2007a). Especially the micromechanical properties, for example the breakage energy as well as plastic and elastic deformation energy, of nanostructured aggregates have a strong effect on their breaking behaviour and thus on the dispersing process and the application properties (Schilde et al., 2009d; Schilde and Kwade, 2009b). Basic research on plastic and elastic deformation, breakage of aggregates and deformation of aggregates by nanoindentation was done by Kendall and Weihs (1992) and Raichmann et al. (2006). Basically the material properties, the surface modification, the porosity, the structure, the particle size distribution and the particle–particle-interactions have a strong influence on the micromechanical properties. Especially for nanostructured aggregates the particle–particle interactions are very strong. For the determination of these micromechanical properties of the nanostructured aggregates different measurements with a nanoindenter were carried out. By comparing the measured micromechanical properties with dispersion results, conclusions about the influence of particle interactions and solid bridges between the primary particles and the strength of aggregates and their dispersibility can be drawn. With the help of the calculated stress energy distribution in the stirred media mill using the discrete element method and the measured fracture distribution of the aggregates measured via nanoindentation an effective dispersion fraction can be calculated and compared to the kinetics of the dispersion processes.
2. Experimental set-up 2.1. Particle system For the dispersion experiments and measurement of the micromechanical properties via nanoindentation, ideal silica model aggregates with different primary particle sizes between 50 and 400 nm are used. The monodisperse silica primary ¨ particles are produced by Stober synthesis followed by a spray drying process and separation of the generated aggregates (between 5 and 50 mm) by a cyclone. For this reason a narrow
primary and secondary particle size distribution and, thus, a narrow distribution of the micromechanical aggregate properties result. In addition low drying temperatures in the spray drying process lead to homogenous aggregates with less cavities and the possibility to increase the content of solid bonds between the primary particles and, thus, the strength of the aggregates by thermal treatment. Fig. 1 shows the silica model aggregates with different primary particles sizes: primary particle size of 50 nm (left), 200 nm (centre), 400 nm (right) in low SEM resolution for the representation of the global aggregate structure (top) and high SEM resolution for the local aggregate structure (bottom). 2.2. Measuring the micromechanical properties In this study the micromechanical properties of the silica aggregates were measured by nanoindentation using TriboIndenters TI 900 from Hysitron Inc. The samples were stressed by a Flat Punch specimen with known material properties and geometry. In this work the deep-sensitive normal force measurement was adopted, where the sample is stressed by a normal force and the resultant force–displacement curve is plotted. In the experiments the measurements were displacement controlled with a total indentation depth of 1000 nm. At this indentation depth a fracture of all measured aggregates could be guaranteed. A constant indentation velocity of the stress and discharge curves at 100 nm/s was chosen. In order to separate the silica aggregates, the dried samples were prepared on glass slides using a dry dispersing device (Rodos, company Sympatec). As an example, the schematic plot of stressing an individual aggregate by the Flat Punch method is shown in Fig. 2. Thereby, the Flat Punch requires precise calibration on the parallelism of the indenter and the glass slide (Fig. 2) and was cleaned before and after each measurement in order to remove defects such as aggregate fragments, dust or external impurities (Arfsten et al., 2008). For a good reproducibility of the micromechanical measurements and in order to avoid measurement artefacts due to different aggregate sizes, silica aggregates between 10 and 15 mm were stressed until aggregate breakage occurs. The resulting stress and discharge curves give information about the maximum normal force at certain indentation depth and the fracture energy of the aggregates. For the determination of the fracture force and fracture energy and, thus, the strength of the silica aggregates, a load function was selected, where the aggregates are stressed until breakage occurs. Fig. 3 shows a schematic representation of the
Fig. 1. SEM micrograph of silica model aggregates with different primary particle sizes: 50 nm (left), 200 nm (centre), 400 nm (right).
C. Schilde et al. / Chemical Engineering Science 66 (2011) 4943–4952
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Fig. 2. Schematic drawing of the Flat Punch method.
Fig. 3. Schematic representation of the indentation results with breakage of the aggregates.
yttria stabilized zirconia grinding media (company Sigmund Lindner). The filling ratio of the grinding media was set to 0.8. The stress intensity in the stirred media mill is proportional to the kinetic energy of the grinding media and inversely proportional to the stressed particle mass (Kwade et al., 1996, 1997; Kwade, 2003). Consequently, in case of stressing single particles at constant operating parameters of the mill, the stress intensity is inversely proportional to the third power of the product particle size x. Besides the particle size, the kinetic energy of the grinding media, Ekin, strongly influences the median stress intensity in stirred media mills and is proportional to the third power of the grinding media 3 diameter, dGM , the square of the tip speed, v2t , and the grinding media density, rGM (see Eq. (2)). The kinetic energy is also proportional to the stress energy of the grinding media SE. Due to friction losses, energy dissipation, deformation energy and resistance against fluid displacement during the approach of two grinding beads the mean values of stress intensity and stress energy are not equal but proportional to the kinetic energy of the grinding media: 3
indentation results with breakage of the aggregates. There the fracture energy Efracture can be calculated using: Z hðFmax Þ FðhÞdh ð1Þ Efracture ¼ 0
In order to increase the confidence level and get an impression of the distribution of the micromechanical properties, 40 aggregates of each sample were measured. Fig. 4 illustrates the breakage indentation forces, their standard deviation and percentage standard deviations as function of the number of measured aggregates for the used Flat Punch indenter geometry. Whereas the average maximum indentation force for the test series stays constant between 10 and 40 measured aggregates, the standard deviation decreases with increase in number of measured aggregates. However, the standard deviation of the measurement converge to a constant value, which is determined by the distribution of the micromechanical properties of the aggregated system (at approximately 30–40 measured aggregates). Therefore, a rough estimation of the distribution of the micromechanical properties for the measurement of approximately 40 aggregates can be calculated (Arfsten et al., 2009; Schilde et al., 2009d). 2.3. Dispersion process Due to the small grinding chamber volume of 0.011 l the stirred media mill of the type Picoliq (company Hosokawa Alpine) was used for the simulation and the dispersing experiments of the silica aggregates. The annular gap mill was operated in discontinuous mode at a temperature of 20 1C with a tip speed of 8 m/s and 400 mm
SIp
Ekin E d v2 r p kin p GM t3 GM 3 m x x 3
SEpdGM v2t rGM
ð2Þ
Certainly, numeric investigations show, that the stress energy and the number of stress events are not constant throughout the grinding chamber volume (Blecher and Schwedes, 1996; Theuerkauf and Schwedes, 1999). Thus, for an exact description of the mill a stress energy distribution must be known. A model for the calculation of a stress energy distribution in stirred media mills with different geometries was developed by Stender et al. (2004). However, simulations in stirred media mills using the discrete-element-method demonstrate a significant overestimation of the calculated stress energy distribution of Stender et al., especially for annular gap mills (Piechatzek, 2009; Stender et al., 2004). Up to now it is not possible to quantify the stress energy distribution for mills with different operating and geometry conditions experimentally. Thereby, the mean stress intensity SI and the total number of stress events, which lead to a successful breakage of the aggregates, can be described by the following dispersion kinetic (Schilde et al., 2010): t xðtÞ ¼ x0 þ ðxend x0 Þ t þ Kt with: xend ¼ f ðSI, sÞ ¼ f ðvt ,dMK , r,E, Z,TÞ N Þ ¼ f ðV,vt ,cV Þ Kt ¼ f ðSF, sÞ ¼ f ð tnparticle
ð3Þ
where x is the particle size as function of the dispersing time and x0 is the agglomerate/aggregate size of the feed particles at the beginning of the dispersion process. The parameter xend is a
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Fig. 4. Maximum indentation force and percentage standard deviation as function of the number of measured aggregates for a Flat Punch indenter geometry with and without aggregate breakage.
function of the stress intensity SI acting on the aggregate and the aggregate strength s and indicates the minimum achievable endparticle size for long dispersing times or high dispersion energies. For short dispersion times close to zero the dispersion progress according to Eq. (3) can be described by the quotient ((x0 xend)/ Kt). The parameter Kt is a function of the stress frequency SF in the dispersion process and has a significant influence when the dispersing time is short. The stress frequency SF is defined as the quotient of the number of stress events N and the product of the time t and the number of particles nparticle. The stress frequency depends on the volume V of the suspension, the tip speed vt, the volume concentration cv, the dynamic fluid viscosity Z and the proc ess temperature T. The parameter Kt describes the time after which the original particle size x0 is reduced half way down to the minimum reachable endparticle size (Eq. (4)), that means the parameter Kt is a kind of half-life. The values for xend and Kt are useful for comparing stress intensities and stress frequency at a certain time. A decrease of the maximum product fineness xend can be associated with an increase of the stress intensity. A decrease of the value Kt and, thus, an increase of the absolute value of the slope of the curve ((x0 xend)/Kt), can be associated with an increasing number of stress events which lead to an effective aggregate dispersion. xðKt Þ ¼
ðxend þ x0 Þ 2
ð4Þ
2.4. Simulation using the discrete-element-method (DEM) The annular gap mill ‘‘Picoliq’’ was used for the simulation and the dispersing experiments due to the small grinding chamber volume. For the calculation of the stress energy distribution in this annular gap mill actually a four way DEM–CFD approach would be sufficient to describe the motion and interaction of fluid and grinding media, e. g. the movement of neighbouring grinding beads. Although the grinding chamber of the used annular gap mill is one of the smallest worldwide, the number of grinding beads in the mill is still around 200.000. Moreover, the grinding media package density, cV, is very high (cV ¼0.8). Simulations of
such high number of particles via a four way DEM–CFD coupling are still beyond computational feasibility. This is why simplifications had to be introduced. On this account a DEM approach according to the model of Cundall and Strack (1979) was used with additional simplified fluid drag forces and under consideration of a damping effect by the fluid displacement between two approaching beads. According to the model of Cundall and Strack, every particle (grinding bead) in a considered system (grinding chamber) possesses translational and rotational motion, which can be described by Newton0 s second law of motion. Whether the grinding bead motion inside the grinding chamber is fluid or contact controlled, can be investigated by comparing the ratio of the momentum response time of a grinding bead to the time between collisions (Crowe et al., 1998). Thus, the two-phase flow inside the grinding chamber can be considered dense if nprGB d4GB vrel t ¼ 41 18Z tc
ð5Þ
where tc is the average time between bead–bead collisions and t the momentum response time, which corresponds to the time a grinding bead requires to respond to a change in fluid velocity. n is the number density of grinding beads, rGB is the density and dGB the diameter of a single grinding bead, Z is the fluid viscosity and vrel the relative velocity of grinding beads. Applying this equation for stirred media mills it follows that the flow inside the grinding chamber of wet operated stirred media mills can be considered as dense. Therefore, the grinding bead motion is controlled primarily by collisions (contact forces) and marginally by fluid forces (drag and lift). Under this condition the fluid primarily damps the collisions and lubricates the contact surfaces. The most important forces acting on a grinding bead inside a dry and wet operated stirred media mill are normal and tangential contact forces, which are calculated by a nonlinear contact model based upon the work of Mindlin and Deresiewicz (1953) (see Eq. (6)–(11)) mGM
dv ¼ Fnc þFnd dt
ð6Þ
C. Schilde et al. / Chemical Engineering Science 66 (2011) 4943–4952
I do ¼ Ftc þ Ftd þ Ftr r dt
ð7Þ
4 n pffiffiffiffinffi ð2=3Þ E r dn 3
ð8Þ
Fnc ¼
Fnd ¼ 2
rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffin b Sn m vrel,n 6
ð9Þ
Ftc ¼ St dt
ð10Þ
rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffinffi b St m vrel,t Ftd ¼ 2 6
R2 ¼0.99985 in Eq. (14) (Beinert et al., unpublished). Where r is the distance from the centre of the annular gap mill. The coefficients of the fit function in Eq. (14) for the used annular gap mill are determined to a0 ¼ 5.5913 104 (m/s), a1 ¼ 1.4075 107 (dimensionless), a2 ¼ 1.4169 109 (s/m), a3 ¼ 7.1301 1010 (s2/m2), a4 ¼1.7929 1012 (s3/m3) and a5 ¼ 1.8021 1013 (s4/m4). The difference between the grinding bead velocity and fluid velocity according to Eq. (14) can be used for calculating the fluid drag force (Eq. (13)). Fd ¼ cd rf
v2p
ð11Þ
where v and o are the linear and angular velocities, mGM is the mass of a single grinding bead, r the radius and I the moment of inertia. The indices n and t represent the forces in normal and tangential directions. The indices c, d and r characterize the reversible spring force, the irreversible damping force and the force induced by rolling friction. Thereby, the normal spring and damping forces depend on the equivalent radius, rn, equivalent Young0 s modulus, En, the equivalent mass, mn, the normal overlap, dn, the stiffness in normal direction, Sn, the relative bead velocity in normal direction, vrel,n and the damping coefficient b. Thereby, the equivalent values are calculated for different grinding bead mass or grinding bead/grinding chamber contacts (Beinert et al., unpublished). The tangential spring and damping forces depend on the equivalent mass, mn, the tangential overlap, dt, the stiffness in tangential direction, St, the relative bead velocity in tangential direction, vrel,t and the damping coefficient b. For the DEM-simulation the damping coefficient b was determined as function of the restitution coefficient e (see Eq. (12)) ln e
b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð12Þ
ðln eÞ2 þ p2
In this study the restitution coefficient for a dry contact of the zirconia grinding media was determined using the experimental drop test set-up presented in Fig. 3(left). The necessary material parameters for the DEM and CFD simulations are listed in Table 1. The values of yttrium stabilised zirconium oxide (as the grinding media and mill chamber) are taken from the producer0 s worksheets or experimental investigations (see Fig. 3). The material parameters as well as the equations for the calculation of the equivalent radius, rn, equivalent Young0 s modulus, En, the equivalent mass, mn are described by Beinert et al., (unpublished). In case of wet operation the effects of the flow field and fluid drag forces on the grinding bead motion cannot be neglected (Beinert et al., unpublished). Under the assumption of spherical particles, rotation-symetric flow field and no axial flow the drag force Fd on the grinding media can be calculated using Eq. (13), where cd is the drag coefficient, rf the fluid density, vp the relative velocity between grinding media and fluid and Ap the projected area of a grinding bead. Thereby, the flow field in the annular gap mill can be estimated via CFD simulations and described with the following 5th order polynomwith a correlation coefficient of Table 1 Material properties for the DEM and/or CFD simulations. Property
Symbol
Unit
Value
Poisson ratio Shear modulus Coefficient of restitution Static friction factor Rolling friction factor Density media Density water Viscosity water
n
– GPa – – – kg/m3 kg/m3 mPa s
0.31 76.34 0.92 0.15 0.01 6000 998.2 1.0021
G e
ms mr rGM rf Z
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2
Ap
vðrÞ ¼ a0 þ a1 r þ a2 r 2 þ a3 r 3 þ a4 r 4 þ a5 r 5
ð13Þ ð14Þ
Besides the additional drag force, the coefficient of restitution, e, of two grinding beads which was measured in dry phase is reduced due to the damping effect of the displaced viscous fluid phase. According to Davis and Serayssol (1986) and Knieke et al. (2010) the reduced coefficient of restitution ed can be described as function of the Stokes number St0. The calculation of the complete restitution coefficient can be simplified according to Eq. (15), where xs is the length ratio of the distance the displacement force starts to influence the grinding bead velocity and the distance the contact starts. The length ratio of the distance, at which the displacement force starts to influence the grinding bead velocity, and the distance at which the contact starts, xc ¼0.001, was estimated based on the kinetic gas theory (Beinert et al., unpublished). The complete derivation of the contact model used in the DEM simulation and a detailed discussion of the simulation result of the annular gap mill using DEM and CFD can be found at Beinert et al., (unpublished). Compared to a four way DEM–CFD coupling this model reflects only an estimation of the grinding media movement. However, since the bead motion is contact controlled and not controlled by the fluid forces due to the high bead concentration, for the design of a grinding or dispersion process (dependency between dispersion efficiency, product properties and stress energy distribution in the mill) such a simple estimation of the grinding media movement is practically usually sufficient Fig. 5. 1 lnðxs Þ 1 þe ed ¼ 1 þ lnðxs Þ eþ lnðxs Þ ð15Þ ¼ eþ St0 lnðxs Þ þ St0 St0 Adopting the Model of Hertz–Mindlin and the Eqs. (13)–(15) under the assumption that the system is contact dominated and the dissipated energy at a plastic contact is transferred to the product particle the stress energy can be calculated from DEM-simulations using the equation for a plastic impact with the following equation: SEpDE ¼
mGM n n 2 ðvi vj Þ 4
ð16Þ
where mGM is the average mass of a grinding bead and vni and vnj the normal component of the velocity of grinding bead i and grinding bead j. This equation can be applied for grinding beads with equal grinding bead mass. According to Eq. (16) the stress energies of all contacts between the grinding media in the entire annular gap mill are distributed (see Fig. 9). A detailed discussion of the simulated stress energy distributions at varying contact model modifications in the used annular gap mill can be found at Beinert et al., (unpublished).
3. Results and discussion 3.1. Measuring the micromechanical properties on ideal model aggregates Typically, in the field of comminution the fracture probability can be described due to Weibull statistics. Thereby, the Weibull
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Fig. 5. Experimental set-up for the determination of the coefficient of restitution (left) and the coefficient of restitution as function of the contact velocity (right).
Fig. 6. Cumulative distribution of the fracture energy as function of the primary particle size measured by nanoindentation.
statistics is based on the principle of the weakest link in a chain. It gives the probability for fracture of a chain, which consists out of a number of links of a certain strength when the load is applied (Vogel,. 2005). Typically, for most aggregated systems the primary particles are bonded by strong solid bonds. In the case of stressing aggregate by compression stresses, e.g. between two grinding beads, more than one solid bond in the aggregate has to break up for sufficient aggregate fracture. Moreover, the strength and radius of solid bonds between primary particles, the primary particles itself as well as defects inside the aggregate structure are partially log normal distributed. For this reason a log normal distribution instead of Weibull statistics are expected for aggregate fracture (Schilde et al., 2009d). As shown in Fig. 4 the standard deviation of the measurement reaches a constant value, which is determined by the distribution of the micromechanical properties of the aggregated system (at 40 measured aggregates). Because of this constant standard deviation and under the assumption of a log normal distribution a distribution of the micromechanical properties can be determined (Arfsten et al., 2009). Fig. 6 shows the cumulative distribution of the fracture energy as function of fracture energy for the three different primary particle sizes and 40 measured aggregates. In this experiment the aggregates are stressed with a high indentation
Fig. 7. Effect of the primary particle size on the dispersion of nanostructured silica aggregates in a wet operated stirred media mill measured by static light scattering.
velocity of 100 nm/s until breakage of the aggregates occurs. As expected the fracture energy increases strongly with decreasing primary particle size and, thus, increasing number of particle– particle-interactions and solid bonds. The resistance of the aggregates against fragmentation increases. The aggregate strength increases strongly with decreasing primary particle size. Generally, two of the three distributions in Fig. 6 fit well to a log normal distribution. 3.2. Effect of the micromechanical properties on dispersion experiments The effect of the micromechanical properties on the dispersion efficiency was studied using the nanostructured silica with different primary particle sizes and, thus, different particle– particle interactions in the aggregates. For the dispersing experiments the stirred media mill Picoliq was operated at the mentioned constant operating parameters. Fig. 7 presents the product fineness measured by static light scattering (Helos, company Sympatec) as function of the dispersion time for dispersing the nanostructured silica with different primary particles in the stirred media mill. Typically the median particle size decreases for the silica with increasing dispersion time. The larger
C. Schilde et al. / Chemical Engineering Science 66 (2011) 4943–4952
Table 2 Parameters of the dispersing model (Eq. (3)) for the dispersion process in a stirred media mill with varying primary particle sizes. Primary particle size
50 200 400
Parameter of the new model x0 [mm]
xend [nm]
Kt [s]
7.0 7.0 7.0
50 200 400
6265 1210 192
Fig. 8. Particle size distribution during dispersing nanostructured silica aggregates with a primaryparticle size of 400 nm in a wet operated stirred media mill.
the primary particle size, the higher is the product fineness in the dispersion process and, thus, the smaller is the strength of the aggregates. Dispersing the silica with a primary particle size of 50 nm, the product fineness and efficiency of the dispersion process is lower than dispersing the silica with primary particle sizes of 200 and 400 nm. Reason for this effect is the increase of the number of solid bonds with decreasing primary particle size as shown in the SEM pictures (Fig. 1). This leads to extremely higher fracture energies (Fig. 6) at small primary particle sizes. Moreover, the measurement data in Fig. 7 are fitted by the dispersing model described by Eq. (3). The particle sizes achieved in the dispersion processes are fitted using the least square method. The model fits well to the measurement data and the values of the model parameters xend (given by the primary particle size) and Kt (obtained by the least square method) are given in Table 2. The decrease of the fit parameters Kt and, thus, an increase of the absolute value of the slope of the curve ((x0 xend)/Kt), with increasing primary particle size (see table 2) can be associated with an increasing number of stress events, which lead to an effective aggregate dispersion. Reason for this effect is the decreasing median fracture energy and, thus decreasing aggregate strength with increasing primary particle size (Fig. 6). More precisely the content of the stress energy distribution of the stirred media mill, which is responsible for an effective dispersion of the aggregates, increases with increase in primary particle size. In order to characterize this effective content of the stress energy distribution, the stress energy distribution of the stirred media mill simulated via the DEM-method must be compared to the fracture energy distribution at different primary particle sizes. The minimum reachable end particle size xend is given by the primary particle size. Fig. 8 shows the progress in the particle size distribution during the dispersion process exemplary for a primary particle size of 400 nm measured via static light scattering. As expected, a bimodal particle size distribution can be observed with increasing dispersion time. The fine fraction of the
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particle size distribution and, thus, the expected dispersion limit is located between 400 and 500 nm, which is corresponds well to the primary particle size of 400 nm. The dispersion progress of the experiments in the dry operated mill was barely measurable and no dispersion results were obtained. 3.3. Comparison of the micromechanical properties and dispersion experiments with the simulation of the stress intensity distribution in the stirred media mills In most cases the estimation of the stress intensity distribution in stirred media mills is described by different authors by varying stress models (Becker et al., 2001; Blecher et al., 1996; Davis et al., 2002; Knieke et al., 2010; Kwade, 2003; Kwade et al., 1996, , 1997; Kwade and Schwedes, 2002; Stender et al., 2004). However, these models deliver a rough estimation of the stress energy distribution. Especially for annular gap mills, which are strongly affected by energy losses due to friction between the grinding media and the mill chamber, the estimation of the stress energy distribution vary across a wide size range. In order to receive a more precise stress energy distribution in this study a discrete element simulation was carried out by simulating the annular gap mill used in the dispersion experiments. Thereby, the stress energy at each grinding bead–grinding bead or grinding bead–wall contact was calculated according to a plastic impact (Eq. (16)). Fig. 9 shows the calculated cumulated fraction of the stress energy distribution of the simulation of the used annular gap mill for dry and wet operation mode. For the analysis of the simulation data the stress energy distribution in the simulation must be independent on the simulation time, i.e. the movement of the grinding media in the simulation process have to reach a steady state condition. The mean stress energy in the wet operated annular gap mill is about one and a half decade higher than in dry operation mode. Thereby, the proportionality factor for the calculation of the stress energy in wet operated mills (Eq. (2)) depends on the resistance against fluid displacement between the grinding beads. This resistance against fluid displacement due to the fluid viscosity reduces the stress energy strongly with increasing fluid viscosity (Davis et al., 2002; Knieke et al., 2010). For this reason, the stress energy should theoretically decrease strongly in wet operation mode. However, Fig. 9 shows an increase in the stress energy distribution. Reason for higher stress energies between the grinding beads in wet operation mode are the high flow forces due to the calculated and superimposed flow profile using the finite element method (Beinert et al., unpublished). Higher fluid forces lead to an increase in the relative velocities of the grinding media and, thus, to higher stress
Fig. 9. Simulated stress energy distribution in the used annular gap mill Picoliq (Hosokawa Alpine AG) for dry and wet operation mode.
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Fig. 10. Particle velocities in dry (a) and wet (b) operation mode.
intensities. Moreover, the width of the stress energy distribution in wet operation mode is broader than in dry operation mode. Therefore, totally different velocity distributions of the grinding media in the two different operating modes occur (see Fig. 10). Fig. 10 shows that the amount of beads with velocities higher than 0.5 m/s is significantly higher for wet operation mode. Reason for this effect is again the increase in stress energy due to the high fluid forces in wet operation. Due to the rotor slip through the grinding media the stress energies in dry operation are very low. These stress energies can be increased by the use of different rotor geometries, e.g. by the use of a disc rotor or pin to pin rotor. The mean stress energy of the wet annular gap mill is in the range of 40 pJ, while the mean stress energy of the dry operation mode is in the range of 1.5 pJ. Comparing the stress energy distribution of the stirred media mill (Fig. 9) with the fracture energy distribution of the aggregates (Fig. 6), it can be concluded that predominantly stress energies above 400 pJ are able to disperse the strong silica aggregates. This means that in case of the dry operation only 4.7% of the collisions lead to a successful dispersion of the aggregates. However, calculating the net power input from the simulation, the 4.7% of collisions, which lead to a successful dispersion transfer 94% (effective dispersion energy) of the overall net energy input. In wet operation mode 30% of the collisions lead to a succesful dispersion of the aggregates. Weighing the breakage energy, Efracture, of the aggregate breakage energy distribution, Q0,aggregate, with the incremental fraction, dSEmill, of the stress energy distribution of the mill, Q0,mill, an effective dispersion fraction DFeff can be calculated (see Eq. (18)). The effective dispersion fraction is the portion of the stress events at which aggregates with a certain breakage energy distribution break in a mill with a certain stress energy distribution. Typically, the aggregate strength increases with decrease in aggregate size. For this reason the effective dispersion fraction calculated on basis of the breakage energy distribution of the original/feed aggregates can only be compared at the beginning of a dispersion process. The value of the effective dispersion fraction determined using Eq. (18) can directly be correlated to the effective stress number ESN at the beginning of the dispersion process (Eq. (19)), i.e. the number of stress events at which a breakage of the aggregates occur. Thus, the effective dispersion fraction can be plotted as function of the quotient of the reduction in particle size, (xstart xend), and the parameter Kt (Fig. 11). According to Eq. (20) the stress number, SNdeagg, of a dispersion process is a function of porosity of the grinding media, e, the filling ratio of the grinding media, jGM, the product volume concentration, cv, the grinding media diameter, dGM, the number of revolutions, n and the dispersion time, t (Kwade, 1996). The effective dispersion fraction (Eq. (18)) is plotted in Fig. 11 as function of the stress frequency of the dispersion process (Eq. (3) and Table 2) for aggregates with different primary particle sizes. As expected, a linear
Fig. 11. Comparison of the effective dispersion fraction calculated with the fracture energy distribution of the aggregates and the stress energy distribution of the annular gap mill (Eq. (9)) with the stress frequency received from the model of Schilde et al. (2010)).
trend between the effective dispersion fraction DFeff and the effective stress frequency (xstart xend)/Kt is received (Eq. (17)) y ¼ ax þ b with : a ¼ 0:00137, b ¼ 0:1592, R2 ¼ 0:985
ð17Þ
This means that based on the micromechanical measurements of the breakage energies of aggregates and the stress energy distribution of a stirred media mill, determined for example by DEMsimulation, the effective number of stress events per time and, thus, the efficiency of the dispersing process can be determined. Z SEmax DFeff ¼ ð1Q0,mill ðSEmill ÞÞQ0,aggregate ðEfracture ÞdSE ð18Þ Efracture,min
ESN ¼ DFeff SNdeagg
ð19Þ
jGM ð1eÞ nt ESNpDFeff 1jGM ð1eÞ cv dGM
ð20Þ
4. Conclusion The application properties of nanoparticles are a function of the physicochemical properties of the nanostructured material. In most applications the particles must be available in liquid as separately dispersed primary particles or in a certain aggregate
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size. Thereby, the success of a dispersion process strongly depends on the micromechanical properties of the nanostructured aggregates. These micromechanical properties can be measured by nanoindentation. In this study sol–gel produced silica aggregates with varying primary particle size were used for dispersion experiments and measurement of the micromechanical properties. The measurement of the micromechanical properties such as breakage force as well as deformation and fracture energy give informations about the product properties, e.g. the efficiency of the dispersion process of nanostructured aggregates. An increase in the number of the solid bonds in the aggregates by decreasing the primary particle size leads to an increase of the fracture energy. By comparing the breakage energy distribution measured via nanoindentation with the stress energy distribution of the stirred media mill simulated via the discrete element method, an effective dispersion fraction can be calculated. This effective dispersion fraction is a measure for the efficiency of the dispersion process and can well be correlated to the effective number of stress events at the start of the process. Especially, the dispersion progress at the beginning of the dispersion process, which is a function of the stress energy distribution of the mill and the breakage energy distribution of the aggregates, can be described by a linear correlation with the calculated effective dispersion fraction. At constant operating parameters the stress energy distribution of the stirred media mill stays constant and the dispersion progress is only described as a function of the breakage energy distribution of the aggregates. Therefore, the stress energy distribution in a stirred media mill simulated via DEM and the measurement of the micromechanical properties via nanoindentation can be quantitatively correlated with the dispersion process.
Nomenclature a0 a1
coefficient of the fit function of the flow field (m/s) coefficient of the fit function of the flow field (dimensionless) a2 coefficient of the fit function of the flow field (s/m) a3 coefficient of the fit function of the flow field (s2/m2) a4 coefficient of the fit function of the flow field (s3/m3) a5 coefficient of the fit function of the flow field (s4/m4) Ap projected area (m2) cd drag coefficient (dimensionless) cm mass concentration (dimensionless) cv volume concentration (dimensionless) dGB diameter of a single grinding bead (mm) dGM median grinding media diameter (mm) DFeff effective dispersion fraction (dimensionless) e coefficient of restitution (dimensionless) ed reduced coefficient of restitution (dimensionless) E Young0 s modulus (N/m2) En equivalent Young0 s modulus (N/m2) Efracture fracture energy (pJ) Efracture,min minimum fracture energy (pJ) Ekin kinetic energy of the grinding media (J) ESN effective stress number (dimensionless) F indentation force (mN) Fd drag force (N) Fmax maximum indentation force (mN) Fcn spring force in normal direction (N) F dn damping force in normal direction (N) Fct spring force in tangential direction (N) Fdt damping force in tangential direction (N) Fr force induced by rolling friction (N)
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G shear modulus (GPa) h displacement (nm) Kt dispersion kinetic parameter (s) m product particle mass (kg) mn equivalent mass of an grinding bead (kg) mGM average mass of a grinding bead (kg) n number density of grinding beads (dimensionless) nparticles number of particles (dimensionless) q3 particle size distribution (1/mm) Q0,aggregate aggregate breakage energy distribution (dimensionless) Q0,mill stress energy distribution of the mill (dimensionless) N number of stress events (dimensionless) r distance from the centre of the mill (m) r grinding bead radius (m) rn equivalent grinding bead radius (m) Sn, St normal, tangential stiffness (N/mm2) SE stress energy of the grinding media (J) SEmill stress energy of the mill (pJ) SEmax maximum stress energy of the mill (pJ) SF stress frequency (1/s) SNdeagg stress number of a dispersion process (dimensionless) SI stress intensity in the stirred media mill (Nm) St0 Stokes number (dimensionless) t dispersion time [s] T temperature [1C] v(r) flow field in the annular gap mill [m/s] vni normal component of the velocity of grinding bead i (m/s) vnj normal component of the velocity of grinding bead j (m/s) vp relative velocity between grinding bead and fluid (m/s) vrel, n, vrel,t relative velocity between two grinding beads (m/s) vt tip speed (m/s) V suspension volume (m3) x product particle size (mm), (nm) x0 aggregate/aggregate size of the feed particles (mm) xend minimum achievable end-particle size (nm) xs length ratio—force starts to influence the grinding bead velocity (dimensionless) b damping coefficient (dimensionless) e porosity of the grinding media (dimensionless) dn normal overlap (m) dt tangential overlap (m) rGB density of a single grinding bead (kg/m3) jGM filling ratio of the grinding media (dimensionless) rf fluid density (kg/m3) rGM grinding media density (kg/m3) s aggregate/agglomerate strength (n/m2) Z fluid viscosity (mPa s) ms static friction factor (dimensionless) mr rolling friction factor (dimensionless) n Poisson ratio (dimensionless) tc average time between bead–bead collisions (s) t momentum response time (s) o angular velocity of a grinding bead (rad/s)
Acknowledgment The authors gratefully acknowledge the financial support by the DFG within the SPP ‘‘colloid technology’’. The SEM pictures were kindly taken by the PTB, Braunschweig. The Fraunhofer ¨ Silicatforschung, Wurzburg ¨ Institut fur is acknowledged for the supply of the model silica aggregates produced by sol–gel synthesis.
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References ¨ Arfsten, J., Bradtmoller, C., Kampen, I., Kwade, A., 2008. Compressive testing of single yeast cells in liquid environment using a nanoindentation system. J. Mater. Res. 23, 3153–3160. Arfsten, J., Kampen, I., Kwade, A., 2009. Mechanical testing of single yeast cells in liquid environment: Effect of the extracellular osmotic conditions on the failure behavior. Int. J. Mater. Res. 100, 978–983. Becker, M., Kwade, A., Schwedes, J., 2001. Stress intensity in stirred media mills and its effect on specific energy requirement. Int. J. Miner. Process. 61, 189–208. Beinert, S., Schilde, C., Kwade, A. Simulation of stress energy and grinding media movement within a wet operated annular gap mill using the discrete element method. Unpublished. Blecher, L., Kwade, A., Schwedes, J., 1996. Motion and stress intensity of grinding beads in a stirred media mill part I: Motion of single grinding beads. Powder Technol. 86, 59–68. Blecher, L., Schwedes, J., 1996. Energy distribution and particle trajectories in a grinding chamber of a stirred ball mill. Int. J. Miner. Process. 44-45, 617–627. Crowe, C.T., Sommerfeld, M., Tsuji, Y., 1998. Multiphase Flows with Droplets and Particles. CRC Press. Cundall, P.A., Strack, O.D.L., 1979. A distinct element model for granular assemblies. Geotechnique 29, 47–65. Davis, R.H., Rager, D.A., Good, B.T., 2002. Elastohydrodynamic rebound of spheres from coated surfaces. J. Fluid Mech. 468, 107–119. Davis, R.H., Serayssol, J.-M., 1986. The elastohydrodynamic collision of two spheres. J. Fluid Mech. 163, 479–497. Kendall, K., Weihs, T.P., 1992. Adhesion of nanoparticles within spray dried agglomerates. Appl. Phys. A 25, 3–8. Knieke, C., Steinborn, C., Romeis, S., Peukert, W., Breitung-Faes, S., Kwade, A., 2010. Nanoparticle production with stirred-media mills: opportunities and limits. Chem. Eng. Technol. 33, 1401–1411. ¨ ¨ Kwade, A., 1996. Autogenzerkleinerung von Kalkstein in Ruhrwerkskugelm uhlen. Institute of Mechanical Engineering, TU Braunschweig, Braunschweig, p. 219.
Kwade, A., 2003. A stressing model for the description and optimization of grinding processes. Chem. Eng. Technol 26, 199–205. Kwade, A., Blecher, L., Schwedes, J., 1996. Motion and stress intensity of grinding beads in a stirred media mill part II: stress intensity and its effect on comminution. Powder Technol. 86, 69–76. ¨ und BeweKwade, A., Blecher, L., Schwedes, J., 1997. Beanspruchungsintensitat ¨ ¨ ¨ gung der Mahlkorper in Ruhrwerkm uhlen. Chem.-Ing. Tech. 69, 836–839. Kwade, A., Schwedes, J., 2002. Breaking charakteristics of different materials and their effect on stress intensity and stress number in stirred media mills. Powder Technol. 122, 109–121. Mindlin, R.D., Deresiewicz, H., 1953. Elastic speres in contact under varying oblique forces. J. Appl. Mech. 20, 327–344. ¨ ¨ Piechatzek, T., 2009. Charakterisierung von Ruhrwerkskugelm uhlen auf Basis der ¨ Partikeltechnik, Technische Diskrete Elemente Methode (DEM). Institut fur ¨ Braunschweig, Braunschweig. Universitat Raichmann, Y., Kazakevich, M., Rabkin, E., Tsur, Y., 2006. Inter-nanoparticle bonds in agglomerates studied by nanoindentation. Adv. Mater. 18, 2028–2030. Schilde, C., Breitung-Faes, S., Kwade, A., 2007a. Dispersing and grinding of alumina nano particles by different stress mechanisms. Ceram. Forum Int. 84, 12–17. Schilde, C., Gothsch, T., Quarch, K., Kind, M., Kwade, A., 2009d. Effect of important process parameters on the redispersion process and the micromechanical properties of precipitated silica. Chem. Eng. Technol. 32, 1078–1087. Schilde, C., Kampen, I., Kwade, A., 2010. Dispersion kinetics of nano-sized particles for different dispersing machines. Chem. Eng. Sci. 65, 3518–3527. Schilde, C., Kwade, A., 2009b. Mikromechanische eigenschaften und dispergier¨ barkeit unterschiedlich gefallter nanostrukturierter aggregate. Chem. Ing. Tech. 81, 1155. Stender, H.-H., Kwade, A., Schwedes, J., 2004. Stress energy distribution in different stirred media mill geometries. Int. J. Miner. Process. 74, 103–117. Theuerkauf, J., Schwedes, J., 1999. Theoretical and experimental investigation on particle and fluid motion in stirred media mills. Powder Technol. 105, 406–412. Vogel, L., W.P., 2005. From single particle impact behaviour to modelling of impact mills. Chem. Eng. Sci. 60, 5164–5176.
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Review
Comparison of the micromechanical aggregate properties of nanostructured aggregates with the stress conditions during stirred media milling Carsten Schilde n, Stefan Beinert, Arno Kwade Institute for Particle Technology, TU Braunschweig, Germany
a r t i c l e i n f o
a b s ta c t
Article history: Received 25 February 2011 Received in revised form 18 June 2011 Accepted 4 July 2011 Available online 20 July 2011
In many cases nanosized particles are produced not as single primary particles but rather as particle collectives consisting of several primary particles. For many applications the particles must be available in liquid as separately dispersed primary particles or in a certain aggregate size. Especially the micromechanical properties of nanostructured aggregates, for example the breakage energy, have a strong impact on their breaking behaviour and, thus, on the dispersion process. For the determination of the micromechanical properties of nanostructured silica aggregates different measurements with a nanoindenter have been carried out. Comparing the measured micromechanical properties with dispersion results in a stirred media mill, conclusions concerning the influence of particle interactions and solid bridges between the primary particles and the strength of aggregates and their dispersibility can be drawn. The strength of the aggregates can be changed using different primary particle sizes. Generally, the maximum achievable product fineness and the efficiency of the dispersion process increases with decrease in aggregate strength and, thus, increasing primary particle size. With the help of the calculated stress energy distribution in the stirred media mill using the discrete element method and the measured fracture distribution of the aggregates measured via nanoindentation an effective dispersion fraction can be calculated. Comparing the effective dispersion fraction with the dispersion progress in the stirred media mill a linear correlation can be obtained. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Nanostructure Dispersion Simulation Product processing Nanoindentation Stirred media mill
Contents 1. 2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4943 Experimental set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4944 2.1. Particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4944 2.2. Measuring the micromechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4944 2.3. Dispersion process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4945 2.4. Simulation using the discrete-element-method (DEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4946 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4947 3.1. Measuring the micromechanical properties on ideal model aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4947 3.2. Effect of the micromechanical properties on dispersion experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4948 3.3. Comparison of the micromechanical properties and dispersion experiments with the simulation of the stress intensity distribution in the stirred media mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4949 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4950 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4951 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4951 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4952
1. Introduction In the past the importance of industrial mass production of high quality products and specialty chemicals raised rapidly. Depending on
n
Corresponding author. E-mail address: [email protected] (C. Schilde).
0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.07.006
the application of nanostructured particles, certain characteristics of the product quality, such as size, morphology, abrasion resistance, specific surface and tendency to agglomeration are important. They depend on the physicochemical properties of the material as well as the technical process control of the production process. Depending on the material system and the process parameters, very small particles, for example nanoparticles, can be produced. In many cases nanosized particles are produced via precipitation or pyrolysis processes not as
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single primary particles but rather as particle collectives consisting of several primary particles. Typically, in these processes the primary particles are bonded by strong solid bonds. In most applications these particles must be available in liquid as separately dispersed primary particles or in a certain aggregate size. Generally, the influencing factors for breaking up the aggregates depend on the grinding and dispersing process itself, as well as on the properties of the aggregates as shown in previous works (Schilde et al., 2007a). Especially the micromechanical properties, for example the breakage energy as well as plastic and elastic deformation energy, of nanostructured aggregates have a strong effect on their breaking behaviour and thus on the dispersing process and the application properties (Schilde et al., 2009d; Schilde and Kwade, 2009b). Basic research on plastic and elastic deformation, breakage of aggregates and deformation of aggregates by nanoindentation was done by Kendall and Weihs (1992) and Raichmann et al. (2006). Basically the material properties, the surface modification, the porosity, the structure, the particle size distribution and the particle–particle-interactions have a strong influence on the micromechanical properties. Especially for nanostructured aggregates the particle–particle interactions are very strong. For the determination of these micromechanical properties of the nanostructured aggregates different measurements with a nanoindenter were carried out. By comparing the measured micromechanical properties with dispersion results, conclusions about the influence of particle interactions and solid bridges between the primary particles and the strength of aggregates and their dispersibility can be drawn. With the help of the calculated stress energy distribution in the stirred media mill using the discrete element method and the measured fracture distribution of the aggregates measured via nanoindentation an effective dispersion fraction can be calculated and compared to the kinetics of the dispersion processes.
2. Experimental set-up 2.1. Particle system For the dispersion experiments and measurement of the micromechanical properties via nanoindentation, ideal silica model aggregates with different primary particle sizes between 50 and 400 nm are used. The monodisperse silica primary ¨ particles are produced by Stober synthesis followed by a spray drying process and separation of the generated aggregates (between 5 and 50 mm) by a cyclone. For this reason a narrow
primary and secondary particle size distribution and, thus, a narrow distribution of the micromechanical aggregate properties result. In addition low drying temperatures in the spray drying process lead to homogenous aggregates with less cavities and the possibility to increase the content of solid bonds between the primary particles and, thus, the strength of the aggregates by thermal treatment. Fig. 1 shows the silica model aggregates with different primary particles sizes: primary particle size of 50 nm (left), 200 nm (centre), 400 nm (right) in low SEM resolution for the representation of the global aggregate structure (top) and high SEM resolution for the local aggregate structure (bottom). 2.2. Measuring the micromechanical properties In this study the micromechanical properties of the silica aggregates were measured by nanoindentation using TriboIndenters TI 900 from Hysitron Inc. The samples were stressed by a Flat Punch specimen with known material properties and geometry. In this work the deep-sensitive normal force measurement was adopted, where the sample is stressed by a normal force and the resultant force–displacement curve is plotted. In the experiments the measurements were displacement controlled with a total indentation depth of 1000 nm. At this indentation depth a fracture of all measured aggregates could be guaranteed. A constant indentation velocity of the stress and discharge curves at 100 nm/s was chosen. In order to separate the silica aggregates, the dried samples were prepared on glass slides using a dry dispersing device (Rodos, company Sympatec). As an example, the schematic plot of stressing an individual aggregate by the Flat Punch method is shown in Fig. 2. Thereby, the Flat Punch requires precise calibration on the parallelism of the indenter and the glass slide (Fig. 2) and was cleaned before and after each measurement in order to remove defects such as aggregate fragments, dust or external impurities (Arfsten et al., 2008). For a good reproducibility of the micromechanical measurements and in order to avoid measurement artefacts due to different aggregate sizes, silica aggregates between 10 and 15 mm were stressed until aggregate breakage occurs. The resulting stress and discharge curves give information about the maximum normal force at certain indentation depth and the fracture energy of the aggregates. For the determination of the fracture force and fracture energy and, thus, the strength of the silica aggregates, a load function was selected, where the aggregates are stressed until breakage occurs. Fig. 3 shows a schematic representation of the
Fig. 1. SEM micrograph of silica model aggregates with different primary particle sizes: 50 nm (left), 200 nm (centre), 400 nm (right).
C. Schilde et al. / Chemical Engineering Science 66 (2011) 4943–4952
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Fig. 2. Schematic drawing of the Flat Punch method.
Fig. 3. Schematic representation of the indentation results with breakage of the aggregates.
yttria stabilized zirconia grinding media (company Sigmund Lindner). The filling ratio of the grinding media was set to 0.8. The stress intensity in the stirred media mill is proportional to the kinetic energy of the grinding media and inversely proportional to the stressed particle mass (Kwade et al., 1996, 1997; Kwade, 2003). Consequently, in case of stressing single particles at constant operating parameters of the mill, the stress intensity is inversely proportional to the third power of the product particle size x. Besides the particle size, the kinetic energy of the grinding media, Ekin, strongly influences the median stress intensity in stirred media mills and is proportional to the third power of the grinding media 3 diameter, dGM , the square of the tip speed, v2t , and the grinding media density, rGM (see Eq. (2)). The kinetic energy is also proportional to the stress energy of the grinding media SE. Due to friction losses, energy dissipation, deformation energy and resistance against fluid displacement during the approach of two grinding beads the mean values of stress intensity and stress energy are not equal but proportional to the kinetic energy of the grinding media: 3
indentation results with breakage of the aggregates. There the fracture energy Efracture can be calculated using: Z hðFmax Þ FðhÞdh ð1Þ Efracture ¼ 0
In order to increase the confidence level and get an impression of the distribution of the micromechanical properties, 40 aggregates of each sample were measured. Fig. 4 illustrates the breakage indentation forces, their standard deviation and percentage standard deviations as function of the number of measured aggregates for the used Flat Punch indenter geometry. Whereas the average maximum indentation force for the test series stays constant between 10 and 40 measured aggregates, the standard deviation decreases with increase in number of measured aggregates. However, the standard deviation of the measurement converge to a constant value, which is determined by the distribution of the micromechanical properties of the aggregated system (at approximately 30–40 measured aggregates). Therefore, a rough estimation of the distribution of the micromechanical properties for the measurement of approximately 40 aggregates can be calculated (Arfsten et al., 2009; Schilde et al., 2009d). 2.3. Dispersion process Due to the small grinding chamber volume of 0.011 l the stirred media mill of the type Picoliq (company Hosokawa Alpine) was used for the simulation and the dispersing experiments of the silica aggregates. The annular gap mill was operated in discontinuous mode at a temperature of 20 1C with a tip speed of 8 m/s and 400 mm
SIp
Ekin E d v2 r p kin p GM t3 GM 3 m x x 3
SEpdGM v2t rGM
ð2Þ
Certainly, numeric investigations show, that the stress energy and the number of stress events are not constant throughout the grinding chamber volume (Blecher and Schwedes, 1996; Theuerkauf and Schwedes, 1999). Thus, for an exact description of the mill a stress energy distribution must be known. A model for the calculation of a stress energy distribution in stirred media mills with different geometries was developed by Stender et al. (2004). However, simulations in stirred media mills using the discrete-element-method demonstrate a significant overestimation of the calculated stress energy distribution of Stender et al., especially for annular gap mills (Piechatzek, 2009; Stender et al., 2004). Up to now it is not possible to quantify the stress energy distribution for mills with different operating and geometry conditions experimentally. Thereby, the mean stress intensity SI and the total number of stress events, which lead to a successful breakage of the aggregates, can be described by the following dispersion kinetic (Schilde et al., 2010): t xðtÞ ¼ x0 þ ðxend x0 Þ t þ Kt with: xend ¼ f ðSI, sÞ ¼ f ðvt ,dMK , r,E, Z,TÞ N Þ ¼ f ðV,vt ,cV Þ Kt ¼ f ðSF, sÞ ¼ f ð tnparticle
ð3Þ
where x is the particle size as function of the dispersing time and x0 is the agglomerate/aggregate size of the feed particles at the beginning of the dispersion process. The parameter xend is a
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Fig. 4. Maximum indentation force and percentage standard deviation as function of the number of measured aggregates for a Flat Punch indenter geometry with and without aggregate breakage.
function of the stress intensity SI acting on the aggregate and the aggregate strength s and indicates the minimum achievable endparticle size for long dispersing times or high dispersion energies. For short dispersion times close to zero the dispersion progress according to Eq. (3) can be described by the quotient ((x0 xend)/ Kt). The parameter Kt is a function of the stress frequency SF in the dispersion process and has a significant influence when the dispersing time is short. The stress frequency SF is defined as the quotient of the number of stress events N and the product of the time t and the number of particles nparticle. The stress frequency depends on the volume V of the suspension, the tip speed vt, the volume concentration cv, the dynamic fluid viscosity Z and the proc ess temperature T. The parameter Kt describes the time after which the original particle size x0 is reduced half way down to the minimum reachable endparticle size (Eq. (4)), that means the parameter Kt is a kind of half-life. The values for xend and Kt are useful for comparing stress intensities and stress frequency at a certain time. A decrease of the maximum product fineness xend can be associated with an increase of the stress intensity. A decrease of the value Kt and, thus, an increase of the absolute value of the slope of the curve ((x0 xend)/Kt), can be associated with an increasing number of stress events which lead to an effective aggregate dispersion. xðKt Þ ¼
ðxend þ x0 Þ 2
ð4Þ
2.4. Simulation using the discrete-element-method (DEM) The annular gap mill ‘‘Picoliq’’ was used for the simulation and the dispersing experiments due to the small grinding chamber volume. For the calculation of the stress energy distribution in this annular gap mill actually a four way DEM–CFD approach would be sufficient to describe the motion and interaction of fluid and grinding media, e. g. the movement of neighbouring grinding beads. Although the grinding chamber of the used annular gap mill is one of the smallest worldwide, the number of grinding beads in the mill is still around 200.000. Moreover, the grinding media package density, cV, is very high (cV ¼0.8). Simulations of
such high number of particles via a four way DEM–CFD coupling are still beyond computational feasibility. This is why simplifications had to be introduced. On this account a DEM approach according to the model of Cundall and Strack (1979) was used with additional simplified fluid drag forces and under consideration of a damping effect by the fluid displacement between two approaching beads. According to the model of Cundall and Strack, every particle (grinding bead) in a considered system (grinding chamber) possesses translational and rotational motion, which can be described by Newton0 s second law of motion. Whether the grinding bead motion inside the grinding chamber is fluid or contact controlled, can be investigated by comparing the ratio of the momentum response time of a grinding bead to the time between collisions (Crowe et al., 1998). Thus, the two-phase flow inside the grinding chamber can be considered dense if nprGB d4GB vrel t ¼ 41 18Z tc
ð5Þ
where tc is the average time between bead–bead collisions and t the momentum response time, which corresponds to the time a grinding bead requires to respond to a change in fluid velocity. n is the number density of grinding beads, rGB is the density and dGB the diameter of a single grinding bead, Z is the fluid viscosity and vrel the relative velocity of grinding beads. Applying this equation for stirred media mills it follows that the flow inside the grinding chamber of wet operated stirred media mills can be considered as dense. Therefore, the grinding bead motion is controlled primarily by collisions (contact forces) and marginally by fluid forces (drag and lift). Under this condition the fluid primarily damps the collisions and lubricates the contact surfaces. The most important forces acting on a grinding bead inside a dry and wet operated stirred media mill are normal and tangential contact forces, which are calculated by a nonlinear contact model based upon the work of Mindlin and Deresiewicz (1953) (see Eq. (6)–(11)) mGM
dv ¼ Fnc þFnd dt
ð6Þ
C. Schilde et al. / Chemical Engineering Science 66 (2011) 4943–4952
I do ¼ Ftc þ Ftd þ Ftr r dt
ð7Þ
4 n pffiffiffiffinffi ð2=3Þ E r dn 3
ð8Þ
Fnc ¼
Fnd ¼ 2
rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffin b Sn m vrel,n 6
ð9Þ
Ftc ¼ St dt
ð10Þ
rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffinffi b St m vrel,t Ftd ¼ 2 6
R2 ¼0.99985 in Eq. (14) (Beinert et al., unpublished). Where r is the distance from the centre of the annular gap mill. The coefficients of the fit function in Eq. (14) for the used annular gap mill are determined to a0 ¼ 5.5913 104 (m/s), a1 ¼ 1.4075 107 (dimensionless), a2 ¼ 1.4169 109 (s/m), a3 ¼ 7.1301 1010 (s2/m2), a4 ¼1.7929 1012 (s3/m3) and a5 ¼ 1.8021 1013 (s4/m4). The difference between the grinding bead velocity and fluid velocity according to Eq. (14) can be used for calculating the fluid drag force (Eq. (13)). Fd ¼ cd rf
v2p
ð11Þ
where v and o are the linear and angular velocities, mGM is the mass of a single grinding bead, r the radius and I the moment of inertia. The indices n and t represent the forces in normal and tangential directions. The indices c, d and r characterize the reversible spring force, the irreversible damping force and the force induced by rolling friction. Thereby, the normal spring and damping forces depend on the equivalent radius, rn, equivalent Young0 s modulus, En, the equivalent mass, mn, the normal overlap, dn, the stiffness in normal direction, Sn, the relative bead velocity in normal direction, vrel,n and the damping coefficient b. Thereby, the equivalent values are calculated for different grinding bead mass or grinding bead/grinding chamber contacts (Beinert et al., unpublished). The tangential spring and damping forces depend on the equivalent mass, mn, the tangential overlap, dt, the stiffness in tangential direction, St, the relative bead velocity in tangential direction, vrel,t and the damping coefficient b. For the DEM-simulation the damping coefficient b was determined as function of the restitution coefficient e (see Eq. (12)) ln e
b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð12Þ
ðln eÞ2 þ p2
In this study the restitution coefficient for a dry contact of the zirconia grinding media was determined using the experimental drop test set-up presented in Fig. 3(left). The necessary material parameters for the DEM and CFD simulations are listed in Table 1. The values of yttrium stabilised zirconium oxide (as the grinding media and mill chamber) are taken from the producer0 s worksheets or experimental investigations (see Fig. 3). The material parameters as well as the equations for the calculation of the equivalent radius, rn, equivalent Young0 s modulus, En, the equivalent mass, mn are described by Beinert et al., (unpublished). In case of wet operation the effects of the flow field and fluid drag forces on the grinding bead motion cannot be neglected (Beinert et al., unpublished). Under the assumption of spherical particles, rotation-symetric flow field and no axial flow the drag force Fd on the grinding media can be calculated using Eq. (13), where cd is the drag coefficient, rf the fluid density, vp the relative velocity between grinding media and fluid and Ap the projected area of a grinding bead. Thereby, the flow field in the annular gap mill can be estimated via CFD simulations and described with the following 5th order polynomwith a correlation coefficient of Table 1 Material properties for the DEM and/or CFD simulations. Property
Symbol
Unit
Value
Poisson ratio Shear modulus Coefficient of restitution Static friction factor Rolling friction factor Density media Density water Viscosity water
n
– GPa – – – kg/m3 kg/m3 mPa s
0.31 76.34 0.92 0.15 0.01 6000 998.2 1.0021
G e
ms mr rGM rf Z
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2
Ap
vðrÞ ¼ a0 þ a1 r þ a2 r 2 þ a3 r 3 þ a4 r 4 þ a5 r 5
ð13Þ ð14Þ
Besides the additional drag force, the coefficient of restitution, e, of two grinding beads which was measured in dry phase is reduced due to the damping effect of the displaced viscous fluid phase. According to Davis and Serayssol (1986) and Knieke et al. (2010) the reduced coefficient of restitution ed can be described as function of the Stokes number St0. The calculation of the complete restitution coefficient can be simplified according to Eq. (15), where xs is the length ratio of the distance the displacement force starts to influence the grinding bead velocity and the distance the contact starts. The length ratio of the distance, at which the displacement force starts to influence the grinding bead velocity, and the distance at which the contact starts, xc ¼0.001, was estimated based on the kinetic gas theory (Beinert et al., unpublished). The complete derivation of the contact model used in the DEM simulation and a detailed discussion of the simulation result of the annular gap mill using DEM and CFD can be found at Beinert et al., (unpublished). Compared to a four way DEM–CFD coupling this model reflects only an estimation of the grinding media movement. However, since the bead motion is contact controlled and not controlled by the fluid forces due to the high bead concentration, for the design of a grinding or dispersion process (dependency between dispersion efficiency, product properties and stress energy distribution in the mill) such a simple estimation of the grinding media movement is practically usually sufficient Fig. 5. 1 lnðxs Þ 1 þe ed ¼ 1 þ lnðxs Þ eþ lnðxs Þ ð15Þ ¼ eþ St0 lnðxs Þ þ St0 St0 Adopting the Model of Hertz–Mindlin and the Eqs. (13)–(15) under the assumption that the system is contact dominated and the dissipated energy at a plastic contact is transferred to the product particle the stress energy can be calculated from DEM-simulations using the equation for a plastic impact with the following equation: SEpDE ¼
mGM n n 2 ðvi vj Þ 4
ð16Þ
where mGM is the average mass of a grinding bead and vni and vnj the normal component of the velocity of grinding bead i and grinding bead j. This equation can be applied for grinding beads with equal grinding bead mass. According to Eq. (16) the stress energies of all contacts between the grinding media in the entire annular gap mill are distributed (see Fig. 9). A detailed discussion of the simulated stress energy distributions at varying contact model modifications in the used annular gap mill can be found at Beinert et al., (unpublished).
3. Results and discussion 3.1. Measuring the micromechanical properties on ideal model aggregates Typically, in the field of comminution the fracture probability can be described due to Weibull statistics. Thereby, the Weibull
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Fig. 5. Experimental set-up for the determination of the coefficient of restitution (left) and the coefficient of restitution as function of the contact velocity (right).
Fig. 6. Cumulative distribution of the fracture energy as function of the primary particle size measured by nanoindentation.
statistics is based on the principle of the weakest link in a chain. It gives the probability for fracture of a chain, which consists out of a number of links of a certain strength when the load is applied (Vogel,. 2005). Typically, for most aggregated systems the primary particles are bonded by strong solid bonds. In the case of stressing aggregate by compression stresses, e.g. between two grinding beads, more than one solid bond in the aggregate has to break up for sufficient aggregate fracture. Moreover, the strength and radius of solid bonds between primary particles, the primary particles itself as well as defects inside the aggregate structure are partially log normal distributed. For this reason a log normal distribution instead of Weibull statistics are expected for aggregate fracture (Schilde et al., 2009d). As shown in Fig. 4 the standard deviation of the measurement reaches a constant value, which is determined by the distribution of the micromechanical properties of the aggregated system (at 40 measured aggregates). Because of this constant standard deviation and under the assumption of a log normal distribution a distribution of the micromechanical properties can be determined (Arfsten et al., 2009). Fig. 6 shows the cumulative distribution of the fracture energy as function of fracture energy for the three different primary particle sizes and 40 measured aggregates. In this experiment the aggregates are stressed with a high indentation
Fig. 7. Effect of the primary particle size on the dispersion of nanostructured silica aggregates in a wet operated stirred media mill measured by static light scattering.
velocity of 100 nm/s until breakage of the aggregates occurs. As expected the fracture energy increases strongly with decreasing primary particle size and, thus, increasing number of particle– particle-interactions and solid bonds. The resistance of the aggregates against fragmentation increases. The aggregate strength increases strongly with decreasing primary particle size. Generally, two of the three distributions in Fig. 6 fit well to a log normal distribution. 3.2. Effect of the micromechanical properties on dispersion experiments The effect of the micromechanical properties on the dispersion efficiency was studied using the nanostructured silica with different primary particle sizes and, thus, different particle– particle interactions in the aggregates. For the dispersing experiments the stirred media mill Picoliq was operated at the mentioned constant operating parameters. Fig. 7 presents the product fineness measured by static light scattering (Helos, company Sympatec) as function of the dispersion time for dispersing the nanostructured silica with different primary particles in the stirred media mill. Typically the median particle size decreases for the silica with increasing dispersion time. The larger
C. Schilde et al. / Chemical Engineering Science 66 (2011) 4943–4952
Table 2 Parameters of the dispersing model (Eq. (3)) for the dispersion process in a stirred media mill with varying primary particle sizes. Primary particle size
50 200 400
Parameter of the new model x0 [mm]
xend [nm]
Kt [s]
7.0 7.0 7.0
50 200 400
6265 1210 192
Fig. 8. Particle size distribution during dispersing nanostructured silica aggregates with a primaryparticle size of 400 nm in a wet operated stirred media mill.
the primary particle size, the higher is the product fineness in the dispersion process and, thus, the smaller is the strength of the aggregates. Dispersing the silica with a primary particle size of 50 nm, the product fineness and efficiency of the dispersion process is lower than dispersing the silica with primary particle sizes of 200 and 400 nm. Reason for this effect is the increase of the number of solid bonds with decreasing primary particle size as shown in the SEM pictures (Fig. 1). This leads to extremely higher fracture energies (Fig. 6) at small primary particle sizes. Moreover, the measurement data in Fig. 7 are fitted by the dispersing model described by Eq. (3). The particle sizes achieved in the dispersion processes are fitted using the least square method. The model fits well to the measurement data and the values of the model parameters xend (given by the primary particle size) and Kt (obtained by the least square method) are given in Table 2. The decrease of the fit parameters Kt and, thus, an increase of the absolute value of the slope of the curve ((x0 xend)/Kt), with increasing primary particle size (see table 2) can be associated with an increasing number of stress events, which lead to an effective aggregate dispersion. Reason for this effect is the decreasing median fracture energy and, thus decreasing aggregate strength with increasing primary particle size (Fig. 6). More precisely the content of the stress energy distribution of the stirred media mill, which is responsible for an effective dispersion of the aggregates, increases with increase in primary particle size. In order to characterize this effective content of the stress energy distribution, the stress energy distribution of the stirred media mill simulated via the DEM-method must be compared to the fracture energy distribution at different primary particle sizes. The minimum reachable end particle size xend is given by the primary particle size. Fig. 8 shows the progress in the particle size distribution during the dispersion process exemplary for a primary particle size of 400 nm measured via static light scattering. As expected, a bimodal particle size distribution can be observed with increasing dispersion time. The fine fraction of the
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particle size distribution and, thus, the expected dispersion limit is located between 400 and 500 nm, which is corresponds well to the primary particle size of 400 nm. The dispersion progress of the experiments in the dry operated mill was barely measurable and no dispersion results were obtained. 3.3. Comparison of the micromechanical properties and dispersion experiments with the simulation of the stress intensity distribution in the stirred media mills In most cases the estimation of the stress intensity distribution in stirred media mills is described by different authors by varying stress models (Becker et al., 2001; Blecher et al., 1996; Davis et al., 2002; Knieke et al., 2010; Kwade, 2003; Kwade et al., 1996, , 1997; Kwade and Schwedes, 2002; Stender et al., 2004). However, these models deliver a rough estimation of the stress energy distribution. Especially for annular gap mills, which are strongly affected by energy losses due to friction between the grinding media and the mill chamber, the estimation of the stress energy distribution vary across a wide size range. In order to receive a more precise stress energy distribution in this study a discrete element simulation was carried out by simulating the annular gap mill used in the dispersion experiments. Thereby, the stress energy at each grinding bead–grinding bead or grinding bead–wall contact was calculated according to a plastic impact (Eq. (16)). Fig. 9 shows the calculated cumulated fraction of the stress energy distribution of the simulation of the used annular gap mill for dry and wet operation mode. For the analysis of the simulation data the stress energy distribution in the simulation must be independent on the simulation time, i.e. the movement of the grinding media in the simulation process have to reach a steady state condition. The mean stress energy in the wet operated annular gap mill is about one and a half decade higher than in dry operation mode. Thereby, the proportionality factor for the calculation of the stress energy in wet operated mills (Eq. (2)) depends on the resistance against fluid displacement between the grinding beads. This resistance against fluid displacement due to the fluid viscosity reduces the stress energy strongly with increasing fluid viscosity (Davis et al., 2002; Knieke et al., 2010). For this reason, the stress energy should theoretically decrease strongly in wet operation mode. However, Fig. 9 shows an increase in the stress energy distribution. Reason for higher stress energies between the grinding beads in wet operation mode are the high flow forces due to the calculated and superimposed flow profile using the finite element method (Beinert et al., unpublished). Higher fluid forces lead to an increase in the relative velocities of the grinding media and, thus, to higher stress
Fig. 9. Simulated stress energy distribution in the used annular gap mill Picoliq (Hosokawa Alpine AG) for dry and wet operation mode.
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Fig. 10. Particle velocities in dry (a) and wet (b) operation mode.
intensities. Moreover, the width of the stress energy distribution in wet operation mode is broader than in dry operation mode. Therefore, totally different velocity distributions of the grinding media in the two different operating modes occur (see Fig. 10). Fig. 10 shows that the amount of beads with velocities higher than 0.5 m/s is significantly higher for wet operation mode. Reason for this effect is again the increase in stress energy due to the high fluid forces in wet operation. Due to the rotor slip through the grinding media the stress energies in dry operation are very low. These stress energies can be increased by the use of different rotor geometries, e.g. by the use of a disc rotor or pin to pin rotor. The mean stress energy of the wet annular gap mill is in the range of 40 pJ, while the mean stress energy of the dry operation mode is in the range of 1.5 pJ. Comparing the stress energy distribution of the stirred media mill (Fig. 9) with the fracture energy distribution of the aggregates (Fig. 6), it can be concluded that predominantly stress energies above 400 pJ are able to disperse the strong silica aggregates. This means that in case of the dry operation only 4.7% of the collisions lead to a successful dispersion of the aggregates. However, calculating the net power input from the simulation, the 4.7% of collisions, which lead to a successful dispersion transfer 94% (effective dispersion energy) of the overall net energy input. In wet operation mode 30% of the collisions lead to a succesful dispersion of the aggregates. Weighing the breakage energy, Efracture, of the aggregate breakage energy distribution, Q0,aggregate, with the incremental fraction, dSEmill, of the stress energy distribution of the mill, Q0,mill, an effective dispersion fraction DFeff can be calculated (see Eq. (18)). The effective dispersion fraction is the portion of the stress events at which aggregates with a certain breakage energy distribution break in a mill with a certain stress energy distribution. Typically, the aggregate strength increases with decrease in aggregate size. For this reason the effective dispersion fraction calculated on basis of the breakage energy distribution of the original/feed aggregates can only be compared at the beginning of a dispersion process. The value of the effective dispersion fraction determined using Eq. (18) can directly be correlated to the effective stress number ESN at the beginning of the dispersion process (Eq. (19)), i.e. the number of stress events at which a breakage of the aggregates occur. Thus, the effective dispersion fraction can be plotted as function of the quotient of the reduction in particle size, (xstart xend), and the parameter Kt (Fig. 11). According to Eq. (20) the stress number, SNdeagg, of a dispersion process is a function of porosity of the grinding media, e, the filling ratio of the grinding media, jGM, the product volume concentration, cv, the grinding media diameter, dGM, the number of revolutions, n and the dispersion time, t (Kwade, 1996). The effective dispersion fraction (Eq. (18)) is plotted in Fig. 11 as function of the stress frequency of the dispersion process (Eq. (3) and Table 2) for aggregates with different primary particle sizes. As expected, a linear
Fig. 11. Comparison of the effective dispersion fraction calculated with the fracture energy distribution of the aggregates and the stress energy distribution of the annular gap mill (Eq. (9)) with the stress frequency received from the model of Schilde et al. (2010)).
trend between the effective dispersion fraction DFeff and the effective stress frequency (xstart xend)/Kt is received (Eq. (17)) y ¼ ax þ b with : a ¼ 0:00137, b ¼ 0:1592, R2 ¼ 0:985
ð17Þ
This means that based on the micromechanical measurements of the breakage energies of aggregates and the stress energy distribution of a stirred media mill, determined for example by DEMsimulation, the effective number of stress events per time and, thus, the efficiency of the dispersing process can be determined. Z SEmax DFeff ¼ ð1Q0,mill ðSEmill ÞÞQ0,aggregate ðEfracture ÞdSE ð18Þ Efracture,min
ESN ¼ DFeff SNdeagg
ð19Þ
jGM ð1eÞ nt ESNpDFeff 1jGM ð1eÞ cv dGM
ð20Þ
4. Conclusion The application properties of nanoparticles are a function of the physicochemical properties of the nanostructured material. In most applications the particles must be available in liquid as separately dispersed primary particles or in a certain aggregate
C. Schilde et al. / Chemical Engineering Science 66 (2011) 4943–4952
size. Thereby, the success of a dispersion process strongly depends on the micromechanical properties of the nanostructured aggregates. These micromechanical properties can be measured by nanoindentation. In this study sol–gel produced silica aggregates with varying primary particle size were used for dispersion experiments and measurement of the micromechanical properties. The measurement of the micromechanical properties such as breakage force as well as deformation and fracture energy give informations about the product properties, e.g. the efficiency of the dispersion process of nanostructured aggregates. An increase in the number of the solid bonds in the aggregates by decreasing the primary particle size leads to an increase of the fracture energy. By comparing the breakage energy distribution measured via nanoindentation with the stress energy distribution of the stirred media mill simulated via the discrete element method, an effective dispersion fraction can be calculated. This effective dispersion fraction is a measure for the efficiency of the dispersion process and can well be correlated to the effective number of stress events at the start of the process. Especially, the dispersion progress at the beginning of the dispersion process, which is a function of the stress energy distribution of the mill and the breakage energy distribution of the aggregates, can be described by a linear correlation with the calculated effective dispersion fraction. At constant operating parameters the stress energy distribution of the stirred media mill stays constant and the dispersion progress is only described as a function of the breakage energy distribution of the aggregates. Therefore, the stress energy distribution in a stirred media mill simulated via DEM and the measurement of the micromechanical properties via nanoindentation can be quantitatively correlated with the dispersion process.
Nomenclature a0 a1
coefficient of the fit function of the flow field (m/s) coefficient of the fit function of the flow field (dimensionless) a2 coefficient of the fit function of the flow field (s/m) a3 coefficient of the fit function of the flow field (s2/m2) a4 coefficient of the fit function of the flow field (s3/m3) a5 coefficient of the fit function of the flow field (s4/m4) Ap projected area (m2) cd drag coefficient (dimensionless) cm mass concentration (dimensionless) cv volume concentration (dimensionless) dGB diameter of a single grinding bead (mm) dGM median grinding media diameter (mm) DFeff effective dispersion fraction (dimensionless) e coefficient of restitution (dimensionless) ed reduced coefficient of restitution (dimensionless) E Young0 s modulus (N/m2) En equivalent Young0 s modulus (N/m2) Efracture fracture energy (pJ) Efracture,min minimum fracture energy (pJ) Ekin kinetic energy of the grinding media (J) ESN effective stress number (dimensionless) F indentation force (mN) Fd drag force (N) Fmax maximum indentation force (mN) Fcn spring force in normal direction (N) F dn damping force in normal direction (N) Fct spring force in tangential direction (N) Fdt damping force in tangential direction (N) Fr force induced by rolling friction (N)
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G shear modulus (GPa) h displacement (nm) Kt dispersion kinetic parameter (s) m product particle mass (kg) mn equivalent mass of an grinding bead (kg) mGM average mass of a grinding bead (kg) n number density of grinding beads (dimensionless) nparticles number of particles (dimensionless) q3 particle size distribution (1/mm) Q0,aggregate aggregate breakage energy distribution (dimensionless) Q0,mill stress energy distribution of the mill (dimensionless) N number of stress events (dimensionless) r distance from the centre of the mill (m) r grinding bead radius (m) rn equivalent grinding bead radius (m) Sn, St normal, tangential stiffness (N/mm2) SE stress energy of the grinding media (J) SEmill stress energy of the mill (pJ) SEmax maximum stress energy of the mill (pJ) SF stress frequency (1/s) SNdeagg stress number of a dispersion process (dimensionless) SI stress intensity in the stirred media mill (Nm) St0 Stokes number (dimensionless) t dispersion time [s] T temperature [1C] v(r) flow field in the annular gap mill [m/s] vni normal component of the velocity of grinding bead i (m/s) vnj normal component of the velocity of grinding bead j (m/s) vp relative velocity between grinding bead and fluid (m/s) vrel, n, vrel,t relative velocity between two grinding beads (m/s) vt tip speed (m/s) V suspension volume (m3) x product particle size (mm), (nm) x0 aggregate/aggregate size of the feed particles (mm) xend minimum achievable end-particle size (nm) xs length ratio—force starts to influence the grinding bead velocity (dimensionless) b damping coefficient (dimensionless) e porosity of the grinding media (dimensionless) dn normal overlap (m) dt tangential overlap (m) rGB density of a single grinding bead (kg/m3) jGM filling ratio of the grinding media (dimensionless) rf fluid density (kg/m3) rGM grinding media density (kg/m3) s aggregate/agglomerate strength (n/m2) Z fluid viscosity (mPa s) ms static friction factor (dimensionless) mr rolling friction factor (dimensionless) n Poisson ratio (dimensionless) tc average time between bead–bead collisions (s) t momentum response time (s) o angular velocity of a grinding bead (rad/s)
Acknowledgment The authors gratefully acknowledge the financial support by the DFG within the SPP ‘‘colloid technology’’. The SEM pictures were kindly taken by the PTB, Braunschweig. The Fraunhofer ¨ Silicatforschung, Wurzburg ¨ Institut fur is acknowledged for the supply of the model silica aggregates produced by sol–gel synthesis.
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