Mechanical strength of wet particle agglomerates

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Introduction
The agglomeration or granulation of solid particles is used in many sectors including powder metallurgy, chemical engineering, pharmaceutical industry, and iron-making processes to produce agglomerates or granules from small particles. The agglomerates are used to improve flow properties, enhance permeability for the interstitial gas or reduce segregation in the presence of several types of particles [1][2][3][4] . The binding material is generally a liquid, which is mixed with the primary particles in a granulator [5][6][7][8] . Hence, the mechanical strength of the 'raw granules' produced by the agglomeration of primary particles is ensured by the action of capillary and viscous forces due to the presence of liquid clusters in the pore space [9][10][11][12][13][14] .
The raw granules should be in a densely and homogeneously packed state in order to support the loads to which they are subjected during subsequent operations and give rise to strong solid granules upon sintering [15][16][17][18][19] . The granule strength is controlled by two types of parameters: 1) process parameters such as the method of mixing particles with the liquid (depending on the granulator) [20,21] and 2) material parameters such as the nature of the liquid and the size distribution of the particles [6,21,22] . The physical processes governing the growth of granules are complex due to the dynamic nature of granulation involving the collisions of particles and transport of the binding liquid inside a partially wet granular material [9,[23][24][25][26][27] . The agglomeration process in a fluidized bed can be correctly modeled by assuming binary collisions between wet particles [7,8,28] whereas in a rotating drum [29] , for example, the agglomeration occurs inside a dense cohesive granular flow whose rheology has only recently been studied by careful experiments [30][31][32][33] and simulations [20,34] .
The effect of material parameters on the granule strength reflects both the strength of cohesive bonds between particles and the granular texture, i.e. the organization of the primary particles inside the granule. For example, the strength of pharmaceutical tablets, measured by quasi-static compression between two platens, declines with porosity, which is a function of the consolidation pressure used to manufacture the tablet [35,36] . In the same way, the compressive strength of cohesive powder mixtures is an increasing function of the relative density [37] . The diametrical compression test is a simple way of measuring the tensile strength of powder compacts as the compressive strength (the stress at incipient failure of a granule) is proportional to the tensile strength of the granule [38] . This method has, however, been mostly used to study the fracture stress of brittle materials composed of particles glued via solid bonds [39][40][41] . In contrast, wet granules have been much less investigated. While their cohesive strength is mainly controlled by the Laplace pressure and surface energy of the liquid phase, the effects of granular texture resulting from the granulation process and material parameters are still poorly understood.
In this paper, we are interested in the influence of material parameters on the strength of wet spherical agglomerates in which the liquid is assumed to be distributed as binary bridges joining eligible particle pairs. We use the Discrete Element Method (DEM) with a capillary cohesion law in which the attraction force is an explicit function of the gap between particles and liquid-vapor surface tension, and the amount of liquid is mainly accounted for by a debonding distance. By simulating the diametrical compression of spherical agglomerates, we find that, due to particle rearrangements, they show a plastic behavior with a threshold that we analyze as a function of friction coefficient and size span of primary particles. We also introduce a model for the compressive granule strength that accounts for particle size distribution and we discuss the role of the class of fine particles for the plastic threshold.
In Section 2 , we introduce the numerical model and procedures used to prepare and simulate spherical agglomerates. In Section 3 , we discuss the evolution of the granule strength as a function of axial strain and the effects of particle size span and liquid volume. In Section 4 , we introduce an analytical model of granule strength in terms of texture parameters. We conclude in Section 5 with a short summary of salient results and routes to further research.

Numerical method and procedures
The Discrete Element Method (DEM) has been extensively used for the simulation of granular materials [42][43][44][45] . It is based on the step-wise integration of the equations of motion for all particles by taking into account the particle interactions. In advanced applications of the DEM, it is now possible to implement also the presence of an interstitial fluid or a solid binding matrix [11,46] . However, such applications require substantially more computation power and memory in order to discretize the degrees of freedom associated with the interstitial phase. For this reason, in DEM simulations of granular processes, it is necessary to set up a modeling strategy by making appropriate choices that allow for a balance between computational efficiency (large number of particles) and physical realism. In the case of unsaturated wet granular materials, it is found that the fluid phase can be correctly represented by its cohesive and viscous effect in the particle-particle interactions [47] . Hence, we rely on this approach to model the binding liquid in the granulation process.
On the other hand, the granule strength depends on its internal structure, which is controlled to some extent by the granulation device. Here, we are interested in the simpler case of 'ideal' granules of spherical shape where the primary particles and binding liquid are homogeneously distributed. This simplification allows us to investigate the effect of basic parameters such as the particle size distribution and friction between particles on the strength in the absence of specific granulation process parameters. Furthermore, in association with texture analysis, the results of this investigation can provide a reference behavior against which the effects of process parameters can be quantified in the next step. We use the DEM with a capillary force law and a simple algorithm for the construction of spherical granules. It is worth mentioning here Schematic drawing of the forces acting on particle i by a contacting particle j and by a non-contacting neighboring particle k . that this approach has been used for the simulation of an assembly of wet agglomerates in the pendular state in application to powder processes such as the coalescence of granules upon collision [48] and impact breakage of crystalline agglomerates [49][50][51][52] . In this section, we describe both the numerical method and the procedure that we used to create our 'ideal' granules.

Numerical method
In the DEM, the particles are modeled as rigid particles interacting via visco-elastic force laws relating the contact force to the local strain expressed from the relative particle displacements. The simulation of rigid particles requires a stiff repulsive potential and high time resolution. The motion of each particle i with radius R i is governed by Newton's second law: where f n and f t are the normal and tangential components, respectively, and n and t are the corresponding unit vectors pointing in the normal and tangential directions. m i and s i are the mass and position vector of particle i , respectively. In our simulations, we used the velocity-Verlet time-stepping scheme [45,53] . The force laws involve normal repulsion, normal damping, capillary cohesion and Coulomb friction. The normal force f n has three different sources: The first term in this equation is the normal repulsive contact force. The normal repulsion force is a linear function of the normal elastic deflection δ n approximated by the overlap between two particles [54,55] : where k n is the normal stiffness constant. The normal damping force f d n is assumed to be a viscous force proportional to the relative normal velocity ˙ δ n : f d n = γ n ˙ δ n , where γ n is the damping coefficient. Both these forces disappear when there is no overlap, i.e.
The last term in the Eq. (2) is the capillary cohesion force f c n due to the liquid bond between two particles. It depends on the gap, which we denote by δ n as for overlaps but with negative values (positive values representing a contact deflection), liquid volume V b , surface tension γ s , and the particle-liquid-gas contact angle θ ; see Fig. 1 [11,47,54] . The capillary cohesion force can be determined by integrating the Laplace-Young equations. Various solutions have been proposed for this equation including recent new analytical solutions [56,57] . We use the following explicit expression of the capillary force, which is in good agreement with experiments [28,58] : where R = R i R j is the geometrical mean of the particle radii R i and R j , and [58] κ = 2 πγ s cos θ (5) where d rupt is the debonding distance, defined as the distance beyond which the capillary bridge is unstable and the bond breaks. It is related to liquid volume by [54,55] : The length λ is the factor that controls the exponential falloff of the capillary attraction force in Eq. (4) . This factor is a function of liquid volume V b , the harmonic mean radius ( R = 2 R i R j / (R i + R j ) ) and the size ratio between two particles in contact r This form fits well the capillary force obtained from direct integration of the Laplace-Young equation by setting h (r) = r −1 / 2 and c 0.9 [54,58,59] . The capillary force declines in absolute value as a function of the gap δ n up to the debonding distance d rupt .
For the tangential force f t between particles in contact, we use a combination of a tangential elastic force k t δ t , where k t is the tangential stiffness constant, a Coulomb friction force threshold where μ is the friction coefficient, and a tangential damping term γ t ˙ δ t , where γ t is the tangential damping parameter and ˙ δ t is the contact tangential velocity [58,[60][61][62] :

Ideal granules
In order to create homogeneous agglomerates of particles of spherical shape, we first prepared large samples by means of isotropic compaction inside a box. The primary particles are spheres with their diameters defined in a range [ d min , d max ] with a given size ratio α = d max /d min . The size distribution is assumed to be uniform by particle volume fractions, i.e. with all size classes having the same volume. As the total volume of particles in each size class d i is proportional to d 3 i , the uniform distribution by volume fractions is defined by the condition that the product n i d 3 i is a constant and i P i = 1 , where P i is the numerical fraction of particles in class i . These conditions lead to the following distribution P of particles of diameters d : This distribution has the advantage of allowing the particles belonging to each size class to be correctly represented by their volume, i.e. a large number of small particles and a small number of large particles. This distribution leads to a dense packing as the pore space between large particles is filled by smaller particles [63,64] .
We constructed different samples with five different values of the size ratio α= 1, 2, 3, 4 and 5. Note that the mean particle diameter d is a function of d min and d max : The largest particle size was kept to a constant value d max = 10 μm, and d min was decreased from 10 μm to 2 μm. Eq. (10) shows that, since d max is fixed, the average diameter d declines by a factor 3 when α is increased from 1 to 5.
For isotropic compaction, the particles were introduced in a box and equal compressive stresses σ 0 were applied to the box walls without gravity until a packing in static equilibrium was achieved. During this step, the capillary force was set to zero and the friction coefficient to 0.1 at all contacts between particles in order to obtain a dense sample. Once all particles reached a state of static equilibrium, a spherical probe was placed in the center of the box and its radius was increased until exactly 50 0 0 particles were inside the probe. These particles were then extracted and allowed to relax with the capillary force law activated. The common numerical parameters in these simulations are θ = 0 , γ n = γ t = 5 × 10 −5 Ns/m and γ s = 0 . 072 N/m (water).
We subjected the granules prepared by the above procedure to axial compression between two platens, as illustrated in Fig. 2 (a). The bottom platen is fixed and a downward motion is applied to the top platen with a constant velocity v 0 = 0 . 1 ms − 1 . Hence, with time step δt = 10 −9 s used in our simulations, the total downward displacement during one time step is v 0 δt = 10 −10 m, which is 10 −5 times the size of the primary particles. This means that the diametrical deformation applied to the granules is slow enough to allow for a quasi-static compression test.
To see how quasi-static is the compression, one may also compare the average elastic force increment δf e between particles with the cohesive force f c = πγ s d. The incremental force between particles is simply given by the contact normal stiffness k n multiplied by the average normal displacement δ n at the contact points between primary particles. The latter is given by the mean diametral deformation v 0 δt / D g times the primary particle size d . We thus can define the dimensionless index: For monodisperse particles, from our parameter values we get I e 2 . 5 × 10 −3 , which means that the force increments are generally small compared to the cohesion force. Finally, the time associated with the dissipation of kinetic energy is given by the ra- with three different values of α before diametrical compression. In Fig. 3 , we display a snapshot of a granule for α = 1 at the end of diametrical compression. At the beginning of the test, at most three primary particles of the granule are in contact with either of the two platens. As the compression proceeds, the granule spreads without breaking between the two platens and the number of contacts between the granule and each platen increases. The mechanical response of the granule under diametrical compression requires a measure of the axial force and vertical deformation. The vertical deformation in compression is given by where h = v 0 t is the total downward displacement of the top platen (the lower platen being fixed). We also measure the vertical force component F between the granule and the top platen by summing up the normal forces between the primary particles and the platen. Let us note that, as the granule is in static equilibrium, all forces at each horizontal layer of the granule are balanced so that the vertical force acting between all horizontal layers is equal to F . By dividing this force by the sectional area π a 2 of the granule, where a is the radius of the central section of the granule perpendicular to the compression axis, we get the average vertical stress σ zz = F / (π a 2 ) in the center of the granule. The value of a is estimated from the positions of the particles located at the boundary of the actual central section.
The stress can also be obtained from the values of normal forces and branch vectors (vectors joining particle centers) using [65][66][67][68][69] where V g is the volume of the granule, N b is the number of bonds, n b = N b /V g is the number density of bonds, f k z and k z are the zcomponents of the bond force vector and branch vector, respectively, at the contact k , including all internal contacts as well as the contacts with the platens. The symbol k denotes averaging over all contacts k in the volume. We find that σ zz σ zz during vertical compression. Note that we can define a characteristic cohesive stress σ c from the capillary force and the mean particle diameter d : This stress depends on d and, by virtue of Eq. (10) , varies linearly with α as Its value increases in our simulations by a factor 3 as α is increased from 1 to 5 by reducing d min . Fig. 4 shows the evolution of the mean vertical stress σ zz normalized by the reference cohesive stress σ c as a function of the vertical strain ε for different values of granule parameters. In all cases, σ zz first increases with strain and reaches a plateau more or less fast depending on the values of parameters. It then declines smoothly as a result of the gradual loss of cohesive contacts. The stress plateau is a signature of plastic deformation due to particle rearrangements. However, this plateau does not persist as the cohesive contacts break apart and do not heal. This irreversible character of cohesive contacts is assumed to reflect the fact that the liquid contained in a capillary bridge between two particles is not physically available (e.g. due to evaporation or drainage) once the bridge disappears. This may not always be the case as liquid drops may survive at the surface of the particles and migrate through the vapor phase or by diffusion at the surface of the particles to the newly-formed contacts during a continuous deformation of the granule.

Granule strength
In this way, the observed behavior can globally be qualified as ductile with a well-defined plastic plateau before the beginning of a progressive loss of cohesion at strains above 0.2. This means that the debonding events between primary particles do not lead to spontaneous formation of a fracture surface. We also see that the plastic stress threshold σ p is of the order of 0.4 σ c for monodisperse particles and 0.3 σ c for α = 5 . On the plateau, the particles are well-connected with one another, and the loss of one or two cohesive bonds of a particle does not lead to macroscopic rupture. The values of parameters affect not only the plastic threshold but also the initial build-up and later fall-off of the stress.
The effect of size span α on the plastic strength is displayed in Fig. 5 for several values of the debonding distance d rupt . We see that the ratio σ p / σ c declines by nearly the same amount in all cases as α increases from 1 to 5. This is, however, only a small relative loss of strength with respect to σ c given that, according to Eq. (15) , σ c is multiplied by 3 when α increases from 1 to 5.
Hence, in absolute value, the cohesive strength of the granule increases by nearly a factor 3 as shown in the inset to Fig. 5 . Fig. 6 displays the cohesive (plastic) strength as a function of d rupt for different values of α. We see that this dependence is linear and quite weak for all values of α. As we shall see in the next section, this increase of cohesive strength reflects that of the connectivity  of primary particles by liquid bonds as the debonding distance increases.

Analytical model
The granule strength under diametrical compression reflects the microstructure of the granule, which depends in our simulations on the effects of size span and debonding distance (related to the amount of liquid). The granular microstructure can be described in terms of various scalar and tensorial variables such as the coordination number, packing fraction and fabric tensor [70] . Fig. 7 shows that the initial value of the wet coordination number Z 0 is an increasing function of both α and the debonding distance d rupt .
The wet coordination number is defined as the number of capillary bonds per particle. This is slightly above the "dry" coordination number, which accounts for only the geometrically touching particles. We see that the increase of Z 0 is more significant with d rupt than with α. Its increase is nonlinear with d rupt and it levels off around Z 0 = 12 .
In order to get a more clear understanding of the relation between the connectivity of particles and the plastic strength, we may use the expression (13) of the stress tensor applied in our case to the whole volume V g of the granule. To obtain an analytical expression of the cohesive stress, we consider the cohesive forces f i j c = πγ s d i d j between particles of diameters d i and d j . Most of the cohesive strength is carried by this cohesive force acting between particles that are in contact. But many particles are connected by capillary bridges with a nonzero gap, where the cohesive force is below f i j c [59] . At plastic threshold, a large number of bonds along the directions perpendicular to the compression axis are tensile whereas many others along the compression axis are compressive.
The ductility of the particle at failure is induced by the effect of tensile bonds that prevent the primary particles from sudden rupture. We will account below for these bonds by a prefactor estimated from simulations. The vertical component at plastic threshold is given by where the summation runs over all contacts ( ij ) and we have The angle θ (i j) z is the angle between the contact normal and the vertical axis. This angle is assumed to be uncorrelated with f (i j) c (i j) . For a nearly isotropic distribution of contact orientations, we have cos 2 θ z = 1 / 3 , so that The number density (number per unit volume) n b of bonds can be estimated as the number of bonds per particle Z /2 divided by the free volume V f occupied by each particle. This volume is simply the particle volume divided by the mean packing fraction of the granule such that the sum of all particle free volumes is equal to the granule volume V g . Hence, so that Introducing this expression in Eq. (18) , we get The geometric factor (d i + d j ) d i d j can be evaluated as [54] ( with the underlying assumption that there is no size segregation so that d i and d j are not correlated. Hence, we finally get The prefactor η is introduced here to account for the bonds with a nonzero gap where the cohesive force is below f i j c . Since Z denotes the coordination number for capillary bonds, Eq. (23) with η = 1 can be considered as an upper bound for the plastic strength. In the same way, for Z equal to the coordination number only for contacts, Eq. (23) provides a lower bound of the plastic strength.
As Z varies between 8 and 12 (see Fig. 7 ), we thus expect that η is generally below but close to 1. Fig. 8 displays the values of η computed from our simulations for all values of α and d rupt . We see that η declines slightly with α and d rupt . For small values of d rupt , its value is 0.7 for monodisperse granules and 0.6 for polydisperse granules. With this prefactor, Eq. (23) predicts that the dependence of the normalized strength σ p / σ c with respect to α and d rupt is mediated by that of sZ . Fig. 9 displays sZ as a function of d rupt /d max for different values of α. Interestingly, up to insignificant statistical fluctuations, sZ is independent of α and a nearly linear function of d rupt /d max as that of σ p in Fig. 6 .
We see that the analytical model presented in this section correctly links the microstructure to the overall strength of the agglomerate. The trends are well predicted by the model up to the prefactor η, which appears to be weakly dependent on the material parameters. The physical interpretation of its value may be related to the presence of the large proportion of capillary bonds with nonzero gap, as briefly discussed previously. However, it may also be a consequence of the inhomogeneous stress transmission inside the agglomerate.

Conclusions
In this paper, we used a 3D particle dynamics algorithm together with a capillary force law to analyze the cohesive strength and microstructure of spherical agglomerates. The agglomerates were constructed by extraction of spherical samples from a granular bed prepared by compaction for different values of particle size span, and subjected to diametrical compression between two platens for different values of debonding distance, which accounts for the amount of liquid in the capillary bonds. Despite the irreversible nature of cohesive bonds (i.e. no new cohesive bond), we observe a plastic plateau before the onset of failure. We showed that the plastic strength is proportional to the characteristic capillary stress γ s / d with a multiplicative factor that is a linear function of the debonding distance, increases with the size ratio and is nearly independent of the friction coefficient (not shown here). We also introduced an analytical expression of the cohesive strength in terms of the packing fraction, wet coordination number, size polydispersity and debonding distance. This model is in excellent agreement with the observed trends up to a prefactor that we estimated from the numerical data, and which is weakly dependent on the material parameters.
As previously discussed in this paper, our results provide the behavior of an 'ideal' granule in the sense that the granules were not created by an agglomeration process. We presently work on the simulation of the granulation process in a rotating drum that will allow us to investigate the important complementary problem of predicting the agglomerate microstructure from the process parameters such as rotating speed and filling rate. The ideal granule can then be used as a reference system with which we will compare part of our results. Given the broad applications of the agglomeration process, it is also desirable to validate the simulations by comparison with experimental observations. Systematic diametrical compression tests are presently underway in order to determine the effects of material parameters on the granule strength allowing for the validation of numerical results. Our experiments are in good agreement with the order of magnitude of the cohesive strength of agglomerates for η 0.4. In these experiments, particles of the same typical size as in simulations were used. They were mixed with water and agglomerated into granules in a rotating drum. Preliminary experimental results were presented in [71] .