Measurement and simulation of micromechanical properties of nanostructured aggregates via nanoindentation and DEM-simulation

Powder Technology 259 (2014) 1–13

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Measurement and simulation of micromechanical properties of nanostructured aggregates via nanoindentation and DEM-simulation Carsten Schilde ⁎, Christine Friederike Burmeister, Arno Kwade Institute for Particle Technology, TU Braunschweig, Volkmaroder Str. 5, 38104 Braunschweig, Germany

a r t i c l e

i n f o

Article history: Received 8 April 2013 Received in revised form 13 February 2014 Accepted 1 March 2014 Available online 22 March 2014 Keywords: Nanoindentation Silica DEM Micromechanical properties Stress Aggregate

a b s t r a c t During the production of nanoparticles in large scale processes, usually larger aggregates are synthesized and subsequently redispersed to obtain the desired product properties. Dispersion processes are energy intense and not always predictable or even unsuccessful. The resistance of aggregates against redispersion and thus, the micromechanical properties strongly depends on the aggregate structure and the primary particle properties as well as on the binding mechanisms. The aim of this study was the investigation of the deformation and breakage behavior of aggregated structures composed of nanoparticles. This was done by comparing experimental and computational results. The larger scale simulations were based on a dimensioning of interaction forces and geometry and calibrated using experimental nanoindentation results. As a result the nature and strength of particle–particle interactions can be described. Furthermore, the distribution of normal and radial forces within the aggregates was compared to theoretical models of aggregate breakage. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Nowadays, the application of nanoparticles is increasingly extended into industrial production processes, e.g. in paper, chemical and pharmaceutical industry, in order to improve product properties or to gain new properties. The enhancement of product properties due to the usage of nanoparticles is predominantly achieved by the size of the particles and their high surface to volume ratio. Usually larger aggregates are formed because of increased attractive interaction forces. The consequence for the production of colloidal systems with defined properties is an intense dispersion process. Especially, metal oxides were produced via flame pyrolysis or precipitation processes form compact aggregates with strong solid bonds between the primary particles. The performance of the dispersion process is affected by stress intensity, stress frequency and the mechanism of the dispersing, but is mainly determined by the strength and deformation behavior of the dispersed aggregates [1–4]. The strength of an aggregate depends on the aggregate size and elementary structure as well as on the particle– particle interaction forces [5,6]. Thus, it is extremely important to reduce the strength of the particle–particle interactions, i.e. the strength and amount of solid bonds within nanostructured aggregates [7]. Typically, the micromechanical properties of aggregated structures are studied by compression or impact tests. For the measurement of ⁎ Corresponding author at: Institute for Particle Technology, TU Braunschweig, Volkmaroder Str. 5, 38104 Braunschweig, Germany. Tel.: +49 531 3919604; fax: + 49 531 3919633. E-mail address: [email protected] (C. Schilde).

http://dx.doi.org/10.1016/j.powtec.2014.03.042 0032-5910/© 2014 Elsevier B.V. All rights reserved.

the micromechanical properties of nanostructured aggregates with aggregate sizes less than 50 μm, compression tests via nanoindentation are applicable. Basic research on plastic and elastic deformation as well as on aggregate breakage via compression tests using a nanoindenter were performed by Kendall and Weihs [10] and Raichman et al. [7]. The elastic–plastic deformation behavior depends on the different binding mechanisms between the primary particles such as solid bonds, interfacial forces (e.g. liquid bonds, capillary forces) and attractive particle–particle interaction forces (e.g. Van der Waals' forces, attractive electrostatic forces) [11–14]. The micromechanical properties depend strongly on the aggregate structure, the properties of the primary particles as well as on the rearrangement of particles during the indentation process [15,16]. Besides the deformation behavior, basic work on the breakage behavior of aggregates was carried out by Rumpf [11,17,18] and on the breakage of single particles by Schönert [19,20]. The breakage of aggregates or single particles is determined by the stress field which is caused by loading and by internal material failures. The highest stress level that arises in the contact region can be calculated by the model of Hertz [8] and Huber [9]. The breakage initiates near one contact regardless of the stress field. Thus, the highest stress level is created in the contact region. Internal cracks are only formed while large internal failures exist. The stress level decreases strongly with an increasing distance from the contact region. In addition to the purely experimental investigation of nanostructured aggregates, simulations are frequently used to improve the understanding of micromechanical properties or to verify experimental data. Experimental data on particle–particle interactions in the nanometer

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size range are often only available to a limited extent. For these investigations, the discrete element method (DEM) which is based on a timedependent numerical solution of the Newton's equations of motion, the so called “soft sphere” model of Cundall and Strack [21,22], can be used [23]. A description of the basic working principle of the discrete element method is summarized by several authors, e.g. Beinert et al. [24], Antonyuk et al. [25,26], and Zhu et al. [27]. In principle, particle–particle interactions and, thus, the micromechanical properties of aggregates (measurement and DEM simulation) depend on the particle morphology [28,29], the material properties [30,31] and the contact or field forces. Contact and field forces are gravity, electric and magnetic forces and forces transmitted via solid bridges [25,27,32–36]. The aim of this study is the determination of the micromechanical properties of aggregated structures composed of nanoparticles by comparing experimental and numerical methods. Since the measurement of contact, binding and interaction properties of aggregated nanoparticles is restricted, the establishment and validation of bulk calibration methods are suitable. Similar to other calibration methods for bulk solids the estimation of contact, binding and interaction properties is possible. Therefore, the deformation and breakage behavior of defined aggregated structures was measured via compression tests using a nanoindenter and was compared to DEM simulation results. Based on a dimensioning of the particle size and the particle–particle interaction forces and the calibration of the larger scale simulations using the experimental results, conclusions on the distribution of normal and radial forces within the aggregates can be drawn and compared to theoretical models.

2. Materials and methods 2.1. Sample material In this study, silica model aggregates from Fraunhofer ICT (Würzburg, Germany) with primary particle sizes of 400 nm were used as sample material. The primary particles were produced via modified Stöber synthesis using tetraalkoxysilane and water as educts and ethanol and ammonia as catalysts [37,38]. Afterwards, the almost monodisperse primary particles were spray dried at a drying temperature of 100 °C and separated by a cyclone (spray dryer, company Büchi) in order to generate aggregates with a narrow particle size distribution between 5 and 50 μm. Because of the narrow primary and secondary particle size distributions, a narrow distribution of the micromechanical aggregate properties was expected. Fig. 1 shows scanning electron microscope (SEM) images of the silica model aggregates. In addition to the primary and secondary particle sizes and morphologies, the structure of the aggregate such as coordination number of primary particles and aggregate porosity strongly influences the micromechanical aggregate properties [5,6]. The structure of the aggregated system was characterized via mercury porosity measurements (PoreMaster®, company Quantachrome). The measured value of the packing density of the aggregates (ε = 0.242) was similar to a closed-packing of spheres (εmin = 0.26). The values of the measured porosities were slightly smaller than the porosity of a close-packing due to the solid bonds

between the primary particles (see Fig. 1) [5]. The coordination number was estimated to be in the order of 10 to 12 as a result of close packing. 2.2. Measurement and simulation of the micromechanical properties The micromechanical aggregate properties (see Fig. 1) were measured via nanoindentation (TriboIndenter® TI 900, company Hysitron). The samples were stressed by a “Flat Punch” indenter tip of hardened steel with a diameter of 50 μm. The displacement was controlled by a maximum indentation displacement of 500 nm and a constant indentation velocity of 100 nm/s for the loading and unloading was chosen for both, the indentation measurements and the simulations. A displacement controlled load function was chosen in order to avoid measurement artifacts due to the substrate (more information on the measurement method is given by Schilde et al. [6]). Before compression the aggregates were separated on glass slides using a dry dispersing device (Rhodos, company Sympatec). Furthermore, the indenter tip was cleaned before and after each measurement in order to remove impurities of the previous sample, e.g. aggregate fragments. Fig. 2 shows the scheme of a stressed single aggregate via Flat Punch indentation and an exemplary force–displacement curve resulting from the indentation. An accurate description of the measurement method, a systematic study on the effect of solid bond diameter, bond strength and primary particle size on the micromechanical properties of the used model aggregates is summarized by Schilde and Kwade [5]. Based on the force–displacement curves, the plastic (Eplastic), elastic (Eelastic) and total (Etotal) deformation energies of the aggregates can be calculated as integral of the indentation force F to the displacement h [5,6,39,40]: Z Eelastic ¼

hmax 0

Z F unload ðhÞdhEplastic ¼

hmax 0

F load ðhÞ−F unload ðhÞdhEtotal

ð1Þ where hmax is the maximum displacement during indentation, Fload and Funload are the indentation forces during loading and unloading of the sample. 2.3. DEM models The discrete-element-method (DEM) was originally developed by Cundall and Strack [21] for the description of particulate systems consisting of discrete elements. Thus, it is predestined for aggregate structures. Most aggregate systems consist of a huge number of very small micro- or actually nanoparticles and an even higher number of particle–particle contacts. In extreme cases, for very dense and almost crystalline aggregated systems, the coordination number of the primary particles within these structures can increase up to a number of 12 for close packing. In this study silica model aggregates composed of several thousand primary particles with an aggregate diameter of 7 μm were used for the measurement and simulation of the nanoindentation

Fig. 1. SEM images of the model silica aggregates obtained from the Stöber synthesis and spray drying process: aggregates (left); single fractured aggregate (center); local aggregate structure of the fractured surface (right).

C. Schilde et al. / Powder Technology 259 (2014) 1–13

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Fig. 2. Schematic representation of the Flat Punch indentation (left) and force–displacement curve of the indentation of a single silica aggregate (right).

experiments. While the coordination number for the sample material was determined around 10, the simulated aggregates exhibit a coordination number of around 8. In the first step the aggregates were structured with a commercial simulation tool (EDEM®, company DEM Solutions) regarding aggregate and primary particle size, solid bond diameter, porosity and coordination number. In a spherical particle factory the primary particles were generated and an external force in the direction to the aggregate center and an additional attractive particle–particle interaction force were used for aggregation. Moreover, the rolling and static frictions of the standard DEM approach according to the model of Mindlin [41] were adjusted during aggregation to obtain different aggregate structures. Higher aggregate porosities were obtained by aggregation at higher friction values. The structured aggregates were positioned on a glass substrate and stressed with the hardened steel Flat Punch indenter (see Fig. 3, top). Besides the standard model of Mindlin [45] that describes the translational and rotational motion of each individual particle using Newton's second law of motion, additional contact and binding forces were introduced to the simulation for an accurate description of the particle– particle interaction forces within an aggregated system [25]. On the nanoscale, van der Waals, magnetic, electrostatic, molecular and entropic forces can be important. For dried silica aggregates with primary particle sizes of 400 nm magnetic forces, molecular forces and entropic forces are less significant [42–44]. Since the van der Waals attraction is direct proportional to the particle size and the gravitational forces are a function of the particle size to the power of three, the ratio of van der Waals attraction to gravitational forces increases strongly with decreasing particle size. For this reason, in the submicron size range gravitational forces are less important compared to solid bonds or van der Waals attraction [42]. Thus, according to Fig. 3 (bottom), the DEM contact model between the primary particles within the aggregate was described by combining three models: 1. Standard model of Mindlin [41]: Description of the behavior of the individual particles. 2. Contact model for solid bonds: The bonds were assumed as cylindrical mass less spring–dashpot elements with a virtual bond radius, to model the elastic–plastic deformation behavior of the aggregate system. 3. Additional attractive van der Waals interaction force. The first component of the contact model (see Fig. 3, bottom, a) acting on primary particles within an aggregate is normal and tangential contact forces, Fn and Ft, are calculated according to the following equations by the nonlinear contact model of Mindlin [41]: m

dv c d ¼ Fn þ Fn dt

ð2Þ

I dω c d r  ¼ Ft þ Ft þ Ft r dt

ð3Þ

where m and v are particle mass and velocity, I represents the polar moment of inertia, ω is the angular velocity and r the particle radius. The normal and tangential forces include reversible spring forces, Fc, irreversible damping forces, Fd, as well as the force induced by rolling friction, Frt. The normal spring and damping forces are defined as: c

Fn ¼

4  pffiffiffiffiffi 32  E  r  δn 3

rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5  β  Sn  m  vrel;n 6  pffiffiffiffiffiffiffiffiffiffiffiffiffi Sn ¼ 2  E  r   δn : d

F n ¼ −2 

ð4Þ

ð5Þ

The forces depend on the equivalent radius, r⁎, the equivalent Young's modulus, E⁎, the equivalent mass, m⁎, the normal overlap, δn, the relative bead velocity in normal direction, vrel,n and the damping coefficient, β, as well as on the stiffness in normal direction, Sn. The tangential spring and damping forces depend on the shear modulus, G⁎, the equivalent mass, m⁎, the tangential overlap, δt, the relative primary particle velocity in tangential direction, vrel,t, and the damping coefficient β and the stiffness in tangential direction, St. c

F t ¼ St  δt pffiffiffiffiffiffiffiffiffiffiffiffiffi  St ¼ 8  G  r   δn d

F t ¼ −2 

rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5  β  St  m  vrel;t : 6

ð6Þ

ð7Þ

Moreover, the tangential force depends on the static friction factor [24]. The material parameters for the DEM simulations are listed in Table 1. The equations of the damping coefficient, static friction and rolling friction as well as the equivalent values for the mass, radius and Young's modulus were described by Beinert et al. [24]. For the description of the solid bonds between the primary particles an additional solid bond model according to the solid bond model of Antonyuk [25] was implemented in the simulation as second component of the contact model (see Fig. 3, bottom, b). This model is based on the assumption of cylindrical solid bonds. Besides normal and tangential forces, the solid bonds between the primary particles are able to transfer higher forces than the particle contacts as well as an additional torque up to a maximum normal or tangential shear stress. The alteration of the acting normal and tangential forces, δFn and δFt, of the solid bonds as well as the bond torque in tangential and normal

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Fig. 3. Exemplary Flat Punch indentation of an aggregate with 3300 primary particles in the DEM simulation (top); components of the contact model (bottom): a. Hertz–Mindlin; b. Solid bonds; c. Van der Waals interaction force.

direction, δMt and δMn, are adjusted incrementally at each time step δt using the following equations [35]: δ F n ¼ −vn  N sf  kn  Ab  δt δ F t ¼ −vt  Nsf  kt  Ab  δt

ð8Þ

δM n ¼ −ωn  Nsf  kt  I  δt δM t ¼ −ωt  Nsf  kn  with : Ab ¼ π 

2 rb

I  δt 2

π 4 and I ¼  r b 2

ð9Þ

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Table 1 Material properties for the DEM simulations of the nanoindentation. Property

Symbol

Unit

Value

Poisson's ratio Shear modulus Coefficient of restitution Static friction factor Rolling friction factor Density primary particle Primary particle size Solid bridge radius Hamaker constant Scale factor in simulation Tangential solid bridge strength Normal solid bridge strength Tangential solid bridge stiffness (Cundall–Strack) Normal solid bridge stiffness (Cundall–Strack) Saint-Venant factor for glass Solid bridge strength proportionality Solid bridge stiffness proportionality

ν G e μs μr ρprimary xprimary Rb AH σt,max σn,max kt

[−] [Pa] [−] [−] [−] [kg/m3] [nm] [nm] [J] [−] [N/mm2] [N/mm2] [N/m3]

0.17 3.1 ∗ 1010 0.85 0.4 0.01 2200 400 0.05, 0.1, 0.5, 1, 5, 10, 15, 20, 25 0.01, 0.1, 1, 100, 10,000 1000000 1.1 · 109 0.05 · 109 kn · κSaint-Venant,glass

kn

[N/m3]

Nsf · π · Rb · E⁎

κSaint-Venant,glass N Nsf

[−] [−] [−]

0.886 0.0001, 0.0005, 0.001, 0.002, 0.003, 0.005, 0.01, 0.5 0.001, 0.01, 0.05, 0.1, 0.5, 1

where kn and kt are the normal and shear stiffness of the solid bond, Ab represents the cross section of the solid bond, δt is the incremental time step, vn and vt are the normal and tangential velocities, I represent the polar moment of inertia and ωn and ωt are the normal and tangential angular velocities. The velocities and angular velocities are based on relative values between the particles. The product of the velocities and angular velocities with the incremental time step results in the relative deformation and shift in the angle of the solid bridge. For the calculation of the absolute values of normal and tangential forces and torques the incremental values (see Eqs. (8)–(9)) are summed up with the forces and torques of the previous time step according to the following

equations: F n ðt Þ ¼ F n ðt−δt Þ þ δF n F t ðt Þ ¼ F t ðt−δt Þ þ δF t

ð10Þ

M n ðt Þ ¼ Mn ðt−δt Þ þ δMn ð11Þ M t ðt Þ ¼ Mt ðt−δt Þ þ δM t : The critical normal and tangential stresses for the breakage of the solid bridge, σn and σt, are calculated according to Eqs. (12) and (13) with Rb representing the radius of the solid bond. σn ¼

F n 2  M t  Rb þ Ab I

ð12Þ

Fig. 4. Schematic representation of the stress distribution in the area of contact (top) and characteristic sphere–plate contact pressure distribution during an elastic (bottom, a) and elastic– plastic deformation (bottom, b).

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Fig. 5. Schematic representation of the determination of the radial forces within the aggregate.

σt ¼

F τ M n  Rb : þ Ab I

ð13Þ

The breakage occurs when either the normal or the tangential stress is higher than the strength of the solid bond (normal strength, σn,max; tangential strength, σt,max): σ n NN  σ n;max

ð14Þ

σ t NN  σ t;max :

ð15Þ

The strength of the solid bonds was chosen as the strength value of silica multiplied by proportionality, N, due to inhomogeneities during solid bond formation (cf. Table 1). After a bond breakage has occurred, the contact forces are described by the standard Hertz–Mindlin contact model and attractive van der Waals particle–particle interaction. The parameters for the solid bond model are listed in Table 1. To reduce the number of contact model parameters in a first approximation the Saint-Venant shear correction factor of 0.886 (circulate cross section area, Poisson's ratio of around 0.3) according to Dong et al. [45] was used to describe the ratio of normal and tangential stiffness. The third component of the contact model (see Fig. 3, bottom, c) specifies a simplification of the attractive van der Waals interaction forces for small distances between primary particles in contact that are not linked by solid bonds. The van der Waals interaction force, FvdW, depends on the primary particle diameter, xprimary, the Hamaker constant, AH, and the distance between the particle surfaces, a, (see Eq. (14)). According to this equation, the van der Waals force would reach infinite values for two primary particles in contact, which is in contradiction to the implementation of the previous described Hertz– Mindlin model, where a particle overlap is required to calculate the normal and tangential forces of particles in contact. To overcome infinite values for two nanoparticles in contact, Sun et al. [43,44] calculate the Lennard–Jones force based on van der Waals attraction and Born repulsion for silica nanoparticles using molecular dynamics. For larger silica nanoparticles and bulk behavior a maximum attraction force at a distance of 0.4 to 0.5 nm was found. A similar distance was postulated by Rumpf [46] and Schubert [42]. Thus, as an approximation, the distance for the calculation of the van der Waals attraction between two primary particle in contact is set to a constant value of 0.4 nm (z = 0.4 nm, see Eq. (14)) when the contact distance, a, drops below 0.4 nm. The parameters for the van der Waals interaction force are listed

in Table 1.

F vdW F vdW

xprimary 2 ¼ 12  a2 xprimary AH  2 ¼ 12  z2 AH 

ða ≥zÞ

ð16Þ

ðabzÞ

In general, the experimental measurement of contact, binding and interaction parameters for the contact model of aggregated nanoparticles is hardly possible. Thus, the establishment and validation of bulk calibration methods such as the indentation measurement are appropriate. In this study a sensitivity analysis of contact model parameters was performed in order to identify the effect of various parameters on the micromechanical deformation behavior. 2.4. Dimensioning in the DEM simulation The investigation of indentation measurements of micro aggregates composed of nanoparticles is a major challenge, due to rounding errors and the small Rayleigh time steps for large process times during the computation of nanoparticles within DEM programs. Thus, it is advantageous to increase the geometrical dimensions of the examined system. However, related to the considerations of Feng et al. [47] the modulation must follow three principles of similarity: geometrical, mechanical and dynamical. The key issue of these principles is the constancy of stress and strain. The main requirement of this geometrical increase is the representation of the original physical problem and, thus, a constant ratio for the interaction forces. In this study all particle and domain sizes differ from the physical system by a constant scale factor of 106. Thus, based on the gravitational forces which are a function of the particle radius to the power of three (r3) all other forces were modulated with a constant factor to reach a value 1018 times higher than in the physical system. A more detailed view, e.g. of the van der Waals forces being proportional to the particle size, reveals that the increase in the geometrical dimensions of a particle normally decreases the influence of the van der Waals interactions. This leads to a changed aggregate behavior which does not represent the original physical problem. By modulation with a constant factor the effect of each interaction force is kept constant and the original behavior of the loaded aggregates can be retained. Physically, for the van der Waals interaction the modulation for increasing geometric dimension results in an expansion of the particle–particle pair interaction. This leads to a significantly exceeded Hamaker

Fig. 6. Simulation of the aggregate compression test via nanoindentation without breakage of the aggregate at high solid bond strength: Total forces acting on the primary particles (indication: Red colors indicate high forces, blue colors indicate low forces); solid bonds (indication: Existing solid bonds gray colored, broken solid bonds at which van der Waals interaction forces are acting white colored); Resulting force–displacement curves.

C. Schilde et al. / Powder Technology 259 (2014) 1–13

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C. Schilde et al. / Powder Technology 259 (2014) 1–13

Fig. 7. Simulation of the aggregate compression test via nanoindentation with breakage of the aggregate at low solid bond strength: Total forces acting on the primary particles (indication: Red colors indicate high forces, blue colors indicate low forces); solid bonds (indication: Existing solid bonds grey red colored, broken solid bonds at which van der Waals interaction forces are acting blue colored); Resulting force–displacement curves.

C. Schilde et al. / Powder Technology 259 (2014) 1–13 Table 2 Calculated relative deformation energies, Ep/Ep,max and Eel/Eel,max, quotient of plastic and elastic deformation energies, Ep/Eel, relative amount of broken solid bonds, n/n0, and relative breakage force, Fbreakage/Fmax, at various solid bond strengths. Strength proportionality

Ep/Ep,maxl

Eel/Eel,max

Epl/Eel

Fbreakage/Fmax

n/n0

1 (normalized) 0.5 0.1 0.05 0.01 0.005 0.003 0.002 0.001 0.0005 0.0001

1 0.94 0.93 0.92 0.93 1.04 1.15 1.31 1.06 0.57 0.13

1 1.01 1.01 1.01 1.01 0.91 0.72 0.41 0.07 0.01 0.001

0.81 0.75 0.74 0.74 0.75 0.92 1.29 2.58 11.76 39.79 117.93

No breakage No breakage No breakage No breakage No breakage 0.79 0.44 0.21 0.11 0.08 0.05

0.97 0.97 0.97 0.97 0.97 0.97 0.96 0.93 0.84 0.76 0.53

9

3.2. Simulation of the deformation and breakage behavior According to the concept of Rumpf [17,18] the elementary breaking stress of aggregates and, thus, their strength, σRumpf, depends on the nature and strength of particle–particle interactions, Fprimary, the primary particle surface, Sprimary, the average coordination number, k, and the aggregate porosity, ε (see Eq. (19)). Assuming spherical primary particles and solid bridges the equation of Rumpf can be adapted using the primary particle diameter, xprimary, and an expression according to a model of Bika et al. [12]. In this model the forces are transferred via solid bonds as function of the solid bond diameter, Rb, and strength of the solid bond, σsb (see Eq. (20)). In this equation Rb describes the narrowest portion of the solid bond (“bridge neck”). σ Rumpf ∝ð1−εÞ  k 

F primary Sprimary

ð19Þ

constant. The method of dimensional analysis or the use of dimensionless numbers is not applicable for the discrete numerical model [47]. 3. Results and discussion

σ Rumpf ∝

F primary ð1−εÞ  kðεÞ  2 π xprimary

3.1. Analysis of the DEM simulation

with :

F primary ¼ π  r sb  σ sb :

Similar to the experimental measurement the force acting on the Flat Punch indenter during the loading and unloading of the aggregates was determined based on the simulation data. Based on the force– displacement curves, the plastic, elastic and total deformation energies were calculated according to Eq. (1) through Eq. (3). Besides these conventionally used values for the description of the elastic–plastic deformation behavior of aggregated systems [15,16,39,40,47,49], the total force distribution in different horizontal layers of the aggregated systems, vertically to the moving direction of the indenter, was determined. Fig. 4 shows a schematic representation of the obtained stress distribution in horizontal direction (x–y section) from simulation data (top) as well as the characteristic sphere–plate contact pressure distribution during an elastic (Fig. 4a, bottom) and elastic–plastic deformation (Fig. 4b, bottom) according to Antonyuk [50]. The elastic–plastic deformation leads to a reduced stress, thus the linkage between the particles is damaged due to the irreversible dislocation of the particles. In the area of contact the pressure distribution was described by the total forces acting on the primary particles. Moreover, the radial forces between the primary particles which substantially affect the particle or the aggregate breakage at different horizontal layers within the aggregate were calculated (comparable to radial stresses according to Schönert [22]). Fig. 5 shows exemplarily one sectional plane (Fig. 5, left) and the geometric dependency (Fig. 5, right) for the determination of the radial forces. For the description of the radial stress of a single particle the force in tangential direction was calculated. This tangential force is based on the angles α and β which are independent of the particle position within the coordinate system. The angles are defined as function of the primary particle position (xp; yp), and the force components, Fx and Fy, in the x–y section using the following equations: ! yp α ¼ arctan xp   Fy β ¼ arctan : Fx

ð17Þ

Using the total force acting between the particles, Ftotal, absolute value of the force fraction in radial direction, Ft, can be calculated using the angles α and β: j F t j ¼ j F total j  cosðΦÞ  with : Φ ¼ β−α−90

ð18Þ

ð20Þ

2

There exist other concepts for the theoretical description of the strength, breakage and deformation behavior of aggregates, e.g. the fracture mechanics theory of Kendall [51]. However, Schilde and Kwade [5] and Schilde et al. [6] showed in previous studies that for similar particle material, the amount of surface groups and approximately isotropic aggregate behavior the equation of the elementary breaking stress according to Rumpf is adequate to describe the deformation behavior of nanostructured aggregates. In the previous experimental work, for the measurement of the aggregate compression, the strength and stiffness of the solid bonds within the aggregate were varied by tempering the aggregates at different temperatures between 100 and 600 °C [5,6]. In the simulations solid bond strength and stiffness according to Eq. (8) through Eq. (13) are varied. Due to the aggregate porosity, the coordination number and the primary particle diameter remain constant, the compression strength in Eq. (20) can be related to the diameter and the deformation behavior of the solid bonds. Figs. 6 and 7 show the results of the simulation of aggregate compression tests via nanoindentation of an aggregate at a maximum indentation displacement of 600 nm. Due to high solid bond strength breakage does not occur in Fig. 6. Because of a comparative low solid bridge strength in Fig. 7 an aggregate with a variety of fractured bonds is shown. In contrast to theoretical considerations of the normal and tensile stresses in single particles, e.g. by the continuum model of Huber [9], in Fig. 6 the distribution of total forces acting between the primary particles is given. Due to the comparatively small amount of particles in the area of contact and, thus, the uneven load transmission, the differentiation between tensile and normal stresses is of limited significance. However, if no aggregate breakage occurs (Fig. 6), the distribution of total forces can be compared qualitatively to the crack pattern inside single particles stressed via compression as shown by Schönert [20]. A conical force distribution in the areas of contact can be observed. An almost similar distribution but with significantly lower force values can be obtained in the case of aggregate breakage (see Fig. 7). Perpendicular to the meridian plane, cleavage cracks arise for aggregates with low solid bond strengths. The load–displacement curves of the aggregates with low solid bond strengths show a very different, more plastically deformation behavior (Fig. 7). The calculated deformation energies, the quotient of plastic and elastic deformation energy, the relative amount of broken solid bonds, nrel, as well as the breakage force, Fbreakage, at various solid bond strengths are listed in Table 2. The maximum force and breakage force values are normalized to the maximum indentation force without aggregate breakage. In the case of high

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Fig. 8. Force distribution in different layers in horizontal direction, vertically to the stress direction of the indenter (x–y section) without aggregate breakage (left) and with aggregate breakage (right) at a relative aggregate deformation of approximately 0.1.

strengths of solid bonds the values for the deformation energies, the relative amount of broken solid bonds and the maximum indentation force remain nearly constant. As expected, at a certain strength of solid bonds the aggregate breaks up and the plastic deformation energy, the quotient of plastic and elastic deformation energy, Ep/Eel, as well as the relative amount of solid bonds and the breakage force decrease strongly with decreasing strength of solid bonds (see Table 2). Thereby, the decrease in breakage force at constant aggregate structure is directly proportional to the strength of the solid bonds as proposed according to the equation of Rumpf (Eq. (20) and Table 2) [5,6]. In accordance with the previous studies where the equation of Rumpf is applicable with limitations to describe the deformation behavior of aggregates under compression, the used contact model in the DEM simulation seems to be able to describe the deformation behavior of these aggregates sufficiently [5,6]. During the elastic–plastic deformation of the aggregates by compression, first an elastic deformation occurs leading to the assumption that the pressure is smaller than the yield point [5]. This deformation behavior can be described by the Hertzian stress. In the case of higher pressures an additional plastic deformation in the center of the specimen takes place and the force–displacement curve as well as the stress distribution in the area of contact deviates from the theoretical model of Hertz (see Fig. 4 according to [50]). Fig. 8 shows the force distribution at various distances from the contact area (percentage relative to the aggregate radius) vertically to the stress direction of the indenter (x–y section) without aggregate breakage (left) and with aggregate breakage (right). Analogous to the maximum force and breakage force values, (see Table 2) the indentation forces in Fig. 8 are normalized to the maximum indentation force without aggregate breakage. The simulation data are in qualitatively good agreement with the theoretical considerations of Antonyuk for the normal stress distribution in the area of contact (see Fig. 4). In the case of aggregate breakage, plastic deformation occurs predominantly in the area of contact while the total forces acting on the primary particles are constant over a large range of the contact area. Due to the breakage of solid bonds especially directly below the indenter tip the contacts without bonds are not able to transfer high forces. Thus, the maximum load in this area is significantly lower and the location of the maximum forces is further away from the area of contact when aggregate breakage occurs (cp. Fig. 8). The crack that propagates through the aggregate as well as the distribution of the radial forces calculated according to Eqs. (17) and (18) is shown in Fig. 9 for a maximum indentation displacement of 600 nm (relative displacement approximately 0.1, as well shown in Figs. 7 and

8, right). The radial forces are averaged over the entire height of the aggregate in order to increase the confidential level. According to the theory of Schönert [20], the maximum stresses are predominantly located on the outer radius of the contact area (see Fig. 9). Here the cleavage cracks are usually initiated. Besides the maximum stresses at the outer radius of the contact area, the maximum radial forces are located in the same direction in which the first cleavage crack propagates through the entire aggregate. Thus, in accordance with Schönert, the radial forces (radial stresses) are responsible for the generation and propagation of the cleavage cracks during the compression of nanostructured aggregates via nanoindentation. 3.3. Comparison of the micromechanical properties obtained from measurement and simulation In consequence of the dimensioning of the interaction forces, the primary particle and aggregate dimensions as well as the unknown values of the contact model parameters, the absolute values of the micromechanical properties obtained via simulation differ considerably from the nanoindentation measurements. For the comparison of the measured and simulated nanoindentation experiments the calculated values of the indentation forces and displacements were normalized to maximum values for constant ratios of the elastic to plastic deformation energies. Fig. 10 shows exemplarily a simulated and a measured force–displacement curve for an almost constant ratio of the deformation energies with and without aggregate breakage. Due to differences in the aggregate structure, the distribution of primary particles, solid bond strengths and radii as well as defects within the aggregate structure, the force displacement curve for the measurement and simulation cannot be identical in the case of aggregate breakage. Apart from the absolute values the deformation behavior of the measured silica model aggregate is in a good agreement with the simulation. The used contact model for the DEM simulations is suitable to characterize the aggregate deformation behavior and the effect of various contact parameters. Since the measured and simulated micromechanical aggregate properties are related in a complex way to the aggregate structure and the particle–particle interactions, conclusions regarding the effect of nanoindentation measurements on the silica model aggregates and individual contact model parameters during simulation are a major challenge. Resulting from that only a few structural or particle–particle interaction parameters of the measured silica model aggregates can be influenced by post treatment without changing other important

Fig. 9. Distribution of solid bonds in the case of aggregate breakage in the x–y section (indication: Gray colors indicate the presence of solid bonds; white colors indicate broken solid bonds, i.e. acting van der Waals interaction forces) and the radial forces. (indication: Blue colors indicate high radial forces; red colors indicate low radial forces).

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Fig. 10. Comparison of measured and simulated force–displacement curves at constant ratio of the elastic and plastic deformation energies in measurement and simulation.

parameters. Hence, in this study, the effect of the solid bridge stiffness was investigated exemplarily in nanoindentation measurement and simulation. The strength and stiffness of the solid bridges within the silica models were varied by tempering the aggregates up to 600 °C. Afterwards, the aggregates in experiment and simulation were stressed without aggregate breakage. Thus, the increasing strength of solid bridges is of minor importance compared to the increasing stiffness of solid bridges. Fig. 11 shows the effect of the stiffness (simulation) and the tempering temperature (nanoindentation experiments) on the maximum indentation force and the ratio of the elastic and plastic deformation energies (without aggregate breakage). The values for the maximum indentation forces are normalized to the maximum indentation force of the silica aggregates without tempering whereas the ratio of the elastic and plastic deformation energies is nearly 1 (0.98 in the nanoindentation experiments and 0.92 for the DEM simulation). The normalization of the ratio of the elastic and plastic deformation energies is possible due to the assumption that the plastic and elastic deformation behavior is characterized by the stiffness of the solid bridges if no aggregate breakage occurs. Based on the results qualitative conclusions regarding the effect of increasing solid bridge stiffness with increasing tempering temperature can be drawn. However, due to the complex influence of the contact model parameters on the total interaction forces, the increase in maximum indentation force is not reproducible.

4. Conclusion The technological application of nanoparticulate products depends strongly on the micromechanical properties of the aggregates. These properties are a function of the aggregate structure, the properties of the primary particles as well as the binding mechanisms. In this study these micromechanical aggregate properties were investigated for silica model aggregates with a primary particle size of 400 nm via simulation using the discrete element method. The results were compared to experimental nanoindentation data. The simulations are in good agreement with the measured nanoindentation results. The force distribution in the area of contact and within the aggregates can be qualitatively compared to the theoretical considerations for predominantly elastic or elastic–plastic deformation behavior of aggregates according to Antonyuk et al. [25,50]. The solid bridge strength can be directly compared to the breakage force. Based on the calculation of the distribution of normal and radial forces within the aggregates, conclusions on the propagation of cleavage cracks as function of the nature and strength of solid bonds were drawn and compared to a theoretical model of Schönert [20]. The calculated radial forces are significant for the crack formation and propagation. Moreover, conclusions on the strength and stiffness of solid bonds based on the determination of the plastic and elastic deformation energy, the quotient of plastic and elastic deformation energies as well as the

Fig. 11. Effect of solid bond stiffness and tempering temperature on the maximum indentation force and ratio of the elastic and plastic deformation energies.

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