# Modeling of High-Density Compaction of Pharmaceutical Tablets Using Multi-Contact Discrete Element Method

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## Abstract

**:**

^{®}PH 200 (FMC BioPolymer, Philadelphia, PA, USA) and Pharmacel

^{®}102 (DFE Pharma, Nörten-Hardenberg, Germany) subjected to high confining conditions was studied. The objectives of these simulations were: (1) to investigate the micromechanical behavior; (2) to predict the macroscopic behavior; and (3) to develop a methodology for the calibration of the model parameters needed for the MC-DEM simulations. A two-stage calibration strategy was followed: first, the model parameters were directly measured at the micro-scale (particle level) and second, a meso-scale calibration was established between MC-DEM parameters and compression profiles of the pharmaceutical powders. The new MC-DEM framework could capture the main compressibility characteristics of pharmaceutical materials and could successfully provide predictions on compression profiles at high relative densities.

## 1. Introduction

## 2. DEM’s Theoretical Background

#### 2.1. Classical Hertz–Mindlin Contact Model

#### 2.2. Multi-Contact Adhesive Elastic-Plastic Model

**v**), the contact area (

**A**) between interacting particles, and a material-dependent prefactor (

**β**). More information may be found here [24]. The new multi-contact law formulation yields to this equation:

## 3. Materials and Methods

#### 3.1. Materials

^{®}PH 200 (FMC BioPolymer, Philadelphia, PA, USA) and Pharmacel

^{®}102 (DFE Pharma, Nörten-Hardenberg, Germany) were studied in depth. Henceforth, the powders shall be referred to by the abbreviations MCC-A and MCC-P. A certain proportion of the MCC particles, particularly the larger particle sizes, are rounded agglomerates. Taking this into account, and due to the simplicity of this approach in simulation, spherical shapes are used in DEM simulations. The powder characteristics particle size distribution (PSD) and true density, are given in Table 1 and are available in the literature [41].

#### 3.2. Experimental Methods

#### 3.3. Numerical Methods

_{s(pp)}) and between particles and walls (μ

_{s(pw)}). Later, the tumbling drum test was used to determine the final frictional and rotational DEM related parameters μ

_{s(pw)}, μ

_{r(pp)}, and μ

_{r(pw)}. This test functioned as a calibration test for the rolling friction as well as a second and final iteration for μ

_{s(pw)}, starting from the values obtained from the ring shear tester and the Jenike wall test. The complete calibration method is described in depth in the Cabiscol et al. study [41]. The DEM input parameters for the material properties are summarized in Table 1. The calibration of the input parameters of the new multi-contact model will be discussed in Section 4.1.1.

## 4. Results and Discussion

#### 4.1. Determination of a Representative Volume Element (RVE)

^{3}(1stRVE), (0.8 mm)

^{3}(2nd RVE), (1.0 mm)

^{3}(3nd RVE), and (1.4 mm)

^{3}(4th RVE) (see Figure 5). To eliminate the wall boundary effect, periodic boundaries were used along the X and Y axes. The cubes contain a top and a bottom plate. A series of uniaxial compression simulations were performed using our new multi-contact DEM model. After calibration, as will be discussed later in Section 4.1.1, and based on the data shown in Figure 6 we can confirm the existence of an RVE since the results are converging. However, due to the small size of the sample, the results of the first RVE underestimated its macroscopic stress–strain response. There is excellent agreement between the results of the following RVEs, as well as with the experimental data. It is therefore decided to use the second RVE for practical reasons (less computational time) as follows.

#### 4.1.1. Calibration Method for the Input Parameters of the Multi-Contact Model

^{3}(2nd RVE in Figure 5) along x-y-z directions. The system under consideration contains 698 particles for the MCC-A material and 1193 particles for the MCC-P material with a particle size distribution for both materials given in Section 3.1 and Table 1. The particles are initially randomly positioned in a cubic system with periodic boundary constraints in order to minimize wall effects. After initial deposition, the particles are allowed to grow. Growth is terminated as soon as the desired packing density of approximately 59% is reached and is in line with experimental results reported in the literature [41,51]. The sample was then compressed uni-axially along the z-axis to a maximum target strain of 57% for MCC-A and 53% for MCC-P, then it was decompressed. A strain-driven simulation was used to achieve the maximum desired stress of 29 MPa for the MCC-A material and 25 MPa for the MCC-P material. The calibrated material parameters presented in Table 2 (Section 3.2) were used here. However, the input parameters for the multi-contact model were obtained using an iterative process to determine the optimum parameters that better fits the experimental results. A parameter optimization method was used based on a series of simulations, similar to the one presented by Gao et al. [16]. Given the experimental macroscopic stress and strain response the

**R**

^{2}value between the experimental and simulated data was calculated from:

**R**

^{2}was used to evaluate the accuracy with which the simulated input parameters fit the experimental data; a successful fit was attained when

**R**

^{2}was close to 1. Therefore, when

**R**

^{2}exceeded 0.95, the iterations needed for calibration were terminated. Figure 7 shows that experimental and simulated results are in excellent agreement, indicating that the calibration was successful. Table 3 summarizes the input parameters that were calibrated. For the dimensionless plasticity depth ${\phi}_{f}$, a high and constant value was selected (low contact stiffness) to achieve a high contact stiffness; when necessary, the prefactor β was tuned accordingly.

#### 4.1.2. Verification for Uni-Axial Compaction for MCC-A

#### 4.1.3. Verification for Uni-Axial Compaction for MCC-P

## 5. Conclusions

^{®}PH 200 (FMC BioPolymer, Philadelphia, PA, USA) and Pharmacel

^{®}102 (DFE Pharma, Nörten-Hardenberg, Germany) were successfully predicted. It was also shown that the multi-contact effect predominates for strains higher than 0.2. A calibration strategy to calibrate the input parameters, prefactor β, for the multi-contact model was presented here. The prefactor β was calibrated at low relative densities (low macroscopic stress) and subsequently used for high relative densities (high macroscopic stress). The new multi-contact model requires a separate calibration for each material as prefactor β is a material-dependent parameter. However, more research is needed to determine if this is also true for a mixture of other relevant materials.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Martin, N.L.; Schomberg, A.K.; Finke, J.H.; Abraham, T.G.; Kwade, A.; Herrmann, C. Process Modeling and Simulation of Tableting—An Agent-Based Simulation Methodology for Direct Compression. Pharmaceutics
**2021**, 13, 996. [Google Scholar] [CrossRef] - Wünsch, I.; Finke, J.H.; John, E.; Juhnke, M.; Kwade, A. Mathematical Approach to Consider Solid Compressibility in the Compression of Pharmaceutical Powders. Pharmaceutics
**2019**, 11, 121. [Google Scholar] [CrossRef][Green Version] - Diarra, H.; Mazel, V.; Busignies, V.; Tchoreloff, P. Comparative study between Drucker-Prager/Cap and modified Cam-Clay models for the numerical simulation of die compaction of pharmaceutical powders. Powder Technol.
**2017**, 320, 530–539. [Google Scholar] [CrossRef] - Ohsaki, S.; Kushida, K.; Matsuda, Y.; Nakamura, H.; Watano, S. Numerical study for tableting process in consideration of compression speed. Int. J. Pharm.
**2020**, 575, 118936. [Google Scholar] [CrossRef] - Gethin, D.T.; Lewis, R.W.; Ransing, R.S. A discrete deformable element approach for the compaction of powder systems, Modelling Simul. Mater. Sci. Eng.
**2003**, 11, 101–114. [Google Scholar] - Procopio, A.T.; Zavaliangos, A. Simulation of multi-axial compaction of granular media from loose to high relative densities. J. Mech. Phys. Solids
**2005**, 53, 1523–1551. [Google Scholar] [CrossRef] - Demirtas, A.; Klinzing, G.R. Understanding die compaction of hollow spheres using the multi-particle finite element method (MPFEM). Powder Technol.
**2021**, 39, 34–45. [Google Scholar] [CrossRef] - Stránský, J.; Jirásek, M. Open Source FEM–DEM Coupling. In Proceedings of the 18th International Conference Engineering Mechanics, Prague, Czech Republic, 18 May 2012; pp. 1237–1251. [Google Scholar]
- Frenning, G. An efficient finite/discrete element procedure for simulating compression of 3D particle assemblies. Comput. Methods Appl. Mech. Eng.
**2008**, 197, 4266–4272. [Google Scholar] [CrossRef] - Luding, S. Introduction to discrete element methods. Eur. J. Environ. Civ. Eng.
**2008**, 12, 785–826. [Google Scholar] [CrossRef] - Iacobellis, V.; Radhi, A.; Behdinan, K. Discrete element model for ZrB
_{2}-SiC ceramic composite sintering. Compos. Struct.**2019**, 229, 111373. [Google Scholar] [CrossRef] - Horabik, J.; Wiącek, J.; Parafiniuk, P.; Stasiak, M.; Bańda, M.; Kobyłka, R.; Molenda, M. Discrete Element Method Modelling of the Diametral Compression of Starch Agglomerates. Materials
**2020**, 13, 932. [Google Scholar] [CrossRef][Green Version] - Raji, A.O.; Favier, J.F. Model for the deformation in agricultural and food particulate materials under bulk compressive loading using discrete element method. I: Theory, model development and validation. J. Food Eng.
**2004**, 64, 359–371. [Google Scholar] [CrossRef] - Harthong, B.; Jérier, J.-F.; Dorémus, P.; Imbault, D.; Donzé, F.-V. Modeling of high-density compaction of granular materials by the Discrete Element Method. Int. J. Solids Struct.
**2009**, 46, 3357–3364. [Google Scholar] [CrossRef][Green Version] - Garner, S.; Strong, J.; Zavaliangos, A. Study of the die compaction of powders to high relative densities using the discrete element method. Powder Technol.
**2018**, 330, 357–370. [Google Scholar] [CrossRef] - Gao, Y.; de Simone, G.; Koorapaty, M. Calibration and verification of DEM parameters for the quantitative simulation of pharmaceutical powder compression process. Powder Technol.
**2021**, 378, 160–171. [Google Scholar] [CrossRef] - Luding, S. Cohesive, frictional powders: Contact models for tension. Granul. Matter
**2008**, 10, 235–246. [Google Scholar] [CrossRef][Green Version] - Fischmeister, H.F.; Arzt, E. Densification of Powders by Particle Deformation. Powder Metall.
**1983**, 26, 82–88. [Google Scholar] [CrossRef] - Olsson, E.; Larsson, P.-L. A numerical analysis of cold powder compaction based on micromechanical experiments. Powder Technol.
**2013**, 243, 71–78. [Google Scholar] [CrossRef][Green Version] - Mesarovic, S.D.; Fleck, N.A. Frictionless indentation of dissimilar elastic–plastic spheres. Int. J. Solids Struct.
**2000**, 37, 7071–7091. [Google Scholar] [CrossRef][Green Version] - Jonsson, H.; Gråsjö, J.; Frenning, G. Mechanical behaviour of ideal elastic-plastic particles subjected to different triaxial loading conditions. Powder Technol.
**2017**, 315, 347–355. [Google Scholar] [CrossRef] - Brodu, N.; Dijksman, J.A.; Behringer, R.P. Multiple-contact discrete-element model for simulating dense granular media. Phys. Rev. E
**2015**, 91, 32201. [Google Scholar] [CrossRef][Green Version] - Frenning, G. Towards a mechanistic model for the interaction between plastically deforming particles under confined conditions: A numerical and analytical analysis. Mater. Lett.
**2013**, 92, 365–368. [Google Scholar] [CrossRef][Green Version] - Giannis, K.; Schilde, C.; Finke, J.H.; Kwade, A.; Celigueta, M.A.; Taghizadeh, K.; Luding, S. Stress based multi-contact model for discrete-element simulations. Granul. Matter
**2021**, 23, 1–14. [Google Scholar] [CrossRef] - Rojek, J.; Zubelewicz, A.; Madan, N.; Nosewicz, S. The discrete element method with deformable particles. Int. J. Numer. Methods Eng.
**2018**, 114, 828–860. [Google Scholar] [CrossRef] - Rojek, J.; Nosewicz, S.; Thoeni, K. 3D formulation of the deformable discrete element method. Int. J. Numer. Methods Eng.
**2021**, 122, 3335–3367. [Google Scholar] [CrossRef] - Rojek, J. Contact Modeling in the Discrete Element Method. In Contact Modeling for Solids and Particles; Popp, A., Wriggers, P., Eds.; Springer International Publishing: Cham, Switzerland, 2018; pp. 177–228. [Google Scholar]
- Thakur, S.C. Mesoscopic Discrete Element Modelling of Cohesive Powders for Bulk Handling Applications; School of Engineering the University of Edinburgh: Edinburgh, UK, 2014. [Google Scholar]
- O’Sullivan, C. Particulate Discrete Element Modelling; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Thornton, C. Granular Dynamics, Contact Mechanics and Particle System Simulations: A DEM Study, 1st ed.; Springer International Publishing: Cham, Switzerland, 2015. [Google Scholar]
- Cundall, P.A.; Strack, O.D.L. A discrete numerical model for granular assemblies. Géotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - Hertz, H. Ueber die Berührung Fester Elastischer Körper. J. Reine Angew. Math.
**2009**, 1882, 156–171. [Google Scholar] - Mindlin, R.D. Compliance of Elastic Bodies in Contact. J. Appl. Mech.
**1949**, 16, 259–268. [Google Scholar] [CrossRef] - Mindlin, R.D.; Deresiewicz, H. Elastic Spheres in Contact under Varying Oblique Forces. J. Appl. Mech.
**1953**, 20, 327–344. [Google Scholar] [CrossRef] - Di Renzo, A.; Di Maio, F.P. Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chem. Eng. Sci.
**2004**, 59, 525–541. [Google Scholar] [CrossRef] - Di Renzo, A.; Di Maio, F.P. An improved integral non-linear model for the contact of particles in distinct element simulations. Chem. Eng. Sci.
**2005**, 60, 1303–1312. [Google Scholar] [CrossRef] - Shäfer, J.; Dippel, S.; Wolf, D.E. Force Schemes in Simulations of Granular Materials. J. Phys. I
**1996**, 6, 5–20. [Google Scholar] [CrossRef] - Silbert, L.E.; Ertaş, D.; Grest, G.S.; Halsey, T.C.; Levine, D. Geometry of frictionless and frictional sphere packings. Phys. Rev. E
**2002**, 65, 31304. [Google Scholar] [CrossRef][Green Version] - Ai, J.; Chen, J.-F.; Rotter, J.M.; Ooi, J.Y. Assessment of rolling resistance models in discrete element simulations. Powder Technol.
**2011**, 206, 269–282. [Google Scholar] [CrossRef] - Persson, A.-S.; Frenning, G. An experimental evaluation of the accuracy to simulate granule bed compression using the discrete element method. Powder Technol.
**2012**, 219, 249–256. [Google Scholar] [CrossRef][Green Version] - Cabiscol, R.; Finke, J.H.; Kwade, A. Assessment of particle rearrangement and anisotropy in high-load tableting with a DEM-based elasto-plastic cohesive model. Granul. Matter
**2019**, 21, 1–23. [Google Scholar] [CrossRef] - Cabiscol, R.; Finke, J.H.; Kwade, A. Calibration and interpretation of DEM parameters for simulations of cylindrical tablets with multi-sphere approach. Powder Technol.
**2018**, 327, 232–245. [Google Scholar] [CrossRef] - Coetzee, C.J. Review: Calibration of the discrete element method. Powder Technol.
**2017**, 310, 104–142. [Google Scholar] [CrossRef] - Simons, T.A.; Weiler, R.; Strege, S.; Bensmann, S.; Schilling, M.; Kwade, A. A Ring Shear Tester as Calibration Experiment for DEM Simulations in Agitated Mixers—A Sensitivity Study. Procedia Eng.
**2015**, 102, 741–748. [Google Scholar] [CrossRef][Green Version] - Paulick, M.; Morgeneyer, M.; Kwade, A. A new method for the determination of particle contact stiffness. Granul. Matter
**2015**, 17, 83–93. [Google Scholar] [CrossRef] - Gitman, I.M.; Askes, H.; Sluys, L.J. Representative volume: Existence and size determination. Eng. Fract. Mech.
**2007**, 74, 2518–2534. [Google Scholar] [CrossRef] - Rojek, J.; Karlis, G.F.; Malinowski, L.J.; Beer, G. Setting up virgin stress conditions in discrete element models. Comput. Geotech.
**2013**, 48, 228–248. [Google Scholar] [CrossRef] [PubMed][Green Version] - Drosopoulos, G.A.; Giannis, K.; Stavroulaki, M.E.; Stavroulakis, G.E. Metamodeling-Assisted Numerical Homogenization for Masonry and Cracked Structures. J. Eng. Mech.
**2018**, 144, 4018072. [Google Scholar] [CrossRef] - Montero, F.; Medina, F. Determination of the RVE Size of Quasi-Brittle Materials Using the Discrete Element Method. In Proceedings of the II International Conference on Particle-Based Methods-Fundamentals and Applications PARTICLES 2011, Berlin, Germany, 9–12 July 2011. [Google Scholar]
- Wiącek, J.; Molenda, M.; Ooi, J.Y.; Favier, J. Experimental and numerical determination of representative elementary volume for granular plant materials. Granul. Matter
**2012**, 14, 449–456. [Google Scholar] [CrossRef][Green Version] - Nordström, J.; Alderborn, G.; Frenning, G. Compressibility and tablet forming ability of bimodal granule mixtures: Experiments and DEM simulations. Int. J. Pharm.
**2018**, 540, 120–131. [Google Scholar] [CrossRef] [PubMed]

**Figure 3.**A non-linear hysteretic, adhesive force-displacement ($\delta $) relation in normal direction. The slope ${k}_{2}^{*}$ of the unloading and reloading branch interpolates between ${k}_{1}$ and a maximum stiffness ${k}_{2}$.

**Figure 7.**Calibration for the: (

**a**) MCC-A material under uni-axial compaction at maximum target stress of 29 MPa; (

**b**) MCC-P material under uni-axial compaction at maximum target stress of 25 MPa.

**Figure 8.**Verification for the MCC-A material under uni-axial compaction at maximum target stress of 180 MPa: (

**a**) without the multi-contact effect (prefactor β = 0.0); (

**b**) with the multi-contact effect (prefactor β = 1.3).

**Figure 9.**Verification for the MCC-P material under uni-axial compaction at maximum target stress of 185 MPa: (

**a**) without the multi-contact effect (prefactor β = 0.0); (

**b**) with the multi-contact effect (prefactor β = 1.3).

**Table 1.**Powder characteristics: PSD and densities [41].

Material | x_{10} (Q_{3}) (μm) | x_{50} (Q_{3}) (μm) | x_{90} (Q_{3}) (μm) | Span (-) | True Density (kg m^{−3}) |
---|---|---|---|---|---|

MCC-A | 82.9 | 224.6 | 379.3 | 1.32 | 1541.1 |

MCC-P | 28.3 | 86.5 | 173.8 | 1.68 | 1533.7 |

**Table 2.**The input parameters for single particles and walls [41].

Property | Symbol | Units | MCC-A | MCC-P |
---|---|---|---|---|

Young’s modulus—particle (p) | E | Nm^{−2} | 2.58 × 10^{8} | 1.34 × 10^{9} |

Young’s modulus—wall (w) | E | Nm^{−2} | 7.62 × 10^{10} | 7.62 × 10^{10} |

Poisson’s ratio—particle | ν | - | 0.30 | 0.30 |

Poisson’s ratio—wall | ν | - | 0.31 | 0.31 |

Coefficient of restitution particle | COR(p-p) | - | 0.352 | 0.346 |

Coefficient of restitutio—wall | COR(p-w) | - | 0.352 | 0.346 |

Coefficient of sliding fric—(p-p) | μ_{s(pp)} | - | 0.561 | 0.548 |

Coefficient of sliding f—(p-w) | μ_{s(pw)} | - | 0.707 | 0.715 |

Coefficient of rollin—(p-p) | μ_{r(pp)} | - | 0.3 | 0.3 |

Coefficient of rol—(p-w) | μ_{r(pp)} | - | 0.01 | 0.01 |

Density | ρ | kg/m^{3} | 1541.1 | 1533.7 |

Property | Symbol | Units | MCC-A | MCC-P |
---|---|---|---|---|

Unloading stiffness | k_{2}/k_{1} | - | 120 | 120 |

Adhesion stiffness ratio | K_{c}/k_{1} | - | 0.5 | 0.5 |

Dimensionless plasticity depth | φ_{f} | - | 0.99 | 0.99 |

Prefactor of the MC-dem | β | - | 1.3 | 1.5 |

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**MDPI and ACS Style**

Giannis, K.; Schilde, C.; Finke, J.H.; Kwade, A. Modeling of High-Density Compaction of Pharmaceutical Tablets Using Multi-Contact Discrete Element Method. *Pharmaceutics* **2021**, *13*, 2194.
https://doi.org/10.3390/pharmaceutics13122194

**AMA Style**

Giannis K, Schilde C, Finke JH, Kwade A. Modeling of High-Density Compaction of Pharmaceutical Tablets Using Multi-Contact Discrete Element Method. *Pharmaceutics*. 2021; 13(12):2194.
https://doi.org/10.3390/pharmaceutics13122194

**Chicago/Turabian Style**

Giannis, Kostas, Carsten Schilde, Jan Henrik Finke, and Arno Kwade. 2021. "Modeling of High-Density Compaction of Pharmaceutical Tablets Using Multi-Contact Discrete Element Method" *Pharmaceutics* 13, no. 12: 2194.
https://doi.org/10.3390/pharmaceutics13122194