# A Mathematical Approach to Consider Solid Compressibility in the Compression of Pharmaceutical Powders

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Model of Heckel

_{ax}is the axial compression stress and k and A are constants. Heckel found that the constant k correlates with the reciprocal value of yield strength σ0 for metals and is thus a measure of the plasticity of a material [32]. Hersey and Rees introduced the term mean yield pressure Py for this correlation [33]:

#### 2.2. Model of Cooper & Eaton

_{0}is the initial volume, V

_{∞}is the powder volume at infinite compression stress and a

_{1}, a

_{2}, k

_{1}and k

_{2}are constants, with the boundary condition of the sum of a

_{1}and a

_{2}equalling one. a

_{1}and a

_{2}are the fractions of theoretically possible compression and describe the maximum possible volume reduction attributed to each respective mechanism. k

_{1}and k

_{2}are characteristic compression values, which indicate the stress range, in which the particular mechanism dominates the volume change. In this study, the initial specific volume V

_{0}is calculated using the data of the first measurement point of the force-displacement curve. The specific solid volume determined by helium pycnometry is used as specific volume at infinite compression stress V

_{∞}.

## 3. Materials

^{®}102, JRS Pharma, Rosenberg, Germany), anhydrous lactose (Lac, Lactose Anhydrous NF DT, Sheffield Bio Science, Norwich, NY, USA), anhydrous dicalcium phosphate (DCPA, DI-CAFOS

^{®}A150, Chemische Fabrik Budenheim, Budenheim, Germany) and paracetamol (Para, Novartis Pharma AG, Basel, Switzerland) were selected as pharmaceutical materials while magnesium stearate (Faci, Carasco, Italy) was used as lubricant.

## 4. Methods

#### 4.1. Powder Characterization

#### 4.2. Determination of the Bulk Modulus

^{®}GT60 (Quantachrome Instruments, Boynton Beach, FL, USA). The penetrometer with a volume of 0.5 cm

^{3}was filled with an appropriate amount of powder. The low pressure operation (up to 344 kPa) was used for the filling of the penetrometer with mercury and is not considered for the determination of solid compressibility because of the probably incomplete enclosure of the single particles by mercury. The maximum applied pressure of the high pressure operation was 414 MPa. Four samples per material were characterized and high pressure operation was repeated five times per sample to ensure elastic deformation as the reason for volume changes. Before the analysis, several blank measurements were performed for the consideration of the volume change due to the elastic deformation of the penetrometer and of mercury.

_{0}is the volume at ambient pressure and p is the hydrostatic pressure and dp can be replaced by p−p

_{0}and dV by V−V

_{0}. p

_{0}is the atmospheric pressure and can be neglected for powder compression. This results in

_{S}is the specific solid volume dependent on the hydrostatic pressure and V

_{S,0}is the specific solid volume at atmospheric pressure. The bulk modulus is hence the reciprocal of the slope of the measured volume change during mercury intrusion and was determined by linear regression using the software Excel 2010with the solver add-in (Microsoft, Redmond, WA, USA). It has to be noted that not the whole pressure range is considered because of non-linearity at lower pressure (frequently < 150 MPa), which indicates the presence of still unfilled pores and the still insufficient enclosure of the single particles by mercury. The bulk modulus was determined for all performed measurements and the average value was calculated per material.

#### 4.3. Tableting

#### 4.4. Application of the Compression Equations

#### 4.5. Estimation of Radial Die Expansion

_{rad}is the elastic strain in radial direction, $\sigma $

_{rad}is the stress in radial direction, $\sigma $

_{tan}is the elastic stress in tangential direction, $E$ is the Young’s modulus and $\nu $ the Poisson’s ratio. The radial stress was measured using an instrumented die, while the tangential stress for a cylinder with a thick wall ($r$

_{a}/$r$

_{i}≥ 1.2 according to DIN 2413) can be estimated as follows [35]:

_{i}is the inner radius, r

_{a}is the outer radius and p is the inner pressure acting on the die wall. The estimation of die expansion was performed for Euro D compression tools with an inner radius of 5.64 mm and an outer radius of 19.05 mm using a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.28, which are typical values of stainless steel. The influence of the die holder and the groove are neglected.

## 5. Results and Discussion

#### 5.1. Compression Behavior of the Model Materials

#### 5.2. Reasons for Apparent in-Die Porosities Below Zero

^{3}) compared to the true density determined by X-ray diffraction (1.2936 ± 0.0027 g/cm

^{3}). The slightly lower density determined by helium pycnometry might be explained by the consideration of closed intra-particulate pores and flaws, while these are not taken into account by evaluation of X-ray diffraction. The applied high stress during powder compression causes particle deformation and fragmentation, which may lead to the disappearance of closed pores and flaws. Therefore, the usage of the true density determined by X-ray diffraction seems to be useful for powder compression. Since often the solid density determined by helium pycnometry is used, it is necessary to consider the influence of the differences between solid density and true density on the calculation of in-die porosities. The lower solid density determined by helium pycnometry causes a slight shift of the apparent in-die porosity ε

_{app,He}curve of Para to slightly lower porosities compared to the curve calculated using the true density determined by X-ray diffraction (ε

_{app,XRD}) (Figure 3). However, in both cases physically not reasonable negative apparent in-die porosities are reached at high compression stresses. The apparent in-die porosities are calculated based on the assumption of a constant solid density during powder compression, although solid compressibility causes the increase of solid density with rising compression stress [19,21], especially for organic materials. Therefore, the bulk modulus measured by mercury porosimetry (Table 2) is used for the calculation of a stress-dependent specific solid volume according to Equation (5) based on the specific solid volume determined by helium pycnometry. The resulting in-die compression curve ε

_{c,Hg}is clearly shifted to the positive porosity range, especially at compression stresses above 200 MPa, as Figure 3 shows exemplarily for Para. The same trend is found for MCC and Lac (Table 2). The bulk modulus of the inorganic DCPA could not be determined by mercury porosimetry because of reaching the detection limit. The bulk modulus of DCPA is hence to be expected clearly higher compared to the three organic materials. The bulk modulus of DCPA could be alternatively calculated using the Young’s modulus and Poisson’s ratio. Beam bending of compactates or double compaction with a fully instrumented compaction simulator can be used for the determination of Young’s modulus and Poisson’s ratio of compactates with defined porosities [38,39,40]. Extrapolation to zero porosity can then be applied for the derivation of properties of the solid, which are needed for the calculation of the bulk modulus. Additionally, high-pressure diffraction studies of single particles could be performed [24].

#### 5.3. Introduction of the Solid Compressibility Term

_{ax}is the stress in axial direction and σ

_{rad}is the stress in radial direction. The axial and radial stresses acting on the single particles during uniaxial powder compression are unknown and cannot be measured yet due to the highly complex stress state inside the powder. In reality, a stress distribution exists due to the bulk structure. Usually, instrumented tablet presses enable only the measurement of the overall axial stress acting on the powder. The overall radial stress can be determined using an instrumented die. However, the usage of instrumented dies is not common because of difficulties in design, operation and data evaluation [45]. The estimation of the stresses acting on the single particles based on the measured data is not possible, yet, because of the complexity of influencing parameters. Therefore, the hydrostatic pressure is approximated by the overall axial compression stress acting on the powder. Furthermore, investigations of the compressibility of single crystals by hydrostatic compression with high pressures up to several GPa show the decrease of specific solid volume with rising hydrostatic pressure according to the Murnaghan equation [46]. This equation implies a stress dependency of the bulk modulus. The stress range applied for tableting is considerably lower compared to the applied stresses for single crystal analysis. Thus, a constant bulk modulus is assumed within the context of this investigation. The bulk modulus K of Equation (5) is replaced by the elastic compressibility factor C, which mainly comprises the influence of solid compressibility. According to these assumptions Equation (5) is modified as follows:

_{ax}is the overall axial compression stress acting on the powder and C is here referred to as elastic compressibility factor. The in-die porosity under consideration of solid compressibility ε

_{c}is given by

_{c}

#### 5.4. Extension of the Model of Heckel by the Solid Compressibility Term

_{y}and P

_{y,c}is very small for DCPA because of the high elastic compressibility factor. The differentiation of the compression behaviour of the different materials by P

_{y,c}is successful. The finding of the highest mean yield pressure for DCPA, the lowest for MCC and a medium value for Lac correlates well with the structural observation (Section 5.1) and with literature [23,47,48]. Commonly, the yield pressure is used for the classification of the plasticity of materials. It is assumed that low mean yield pressures, which indicate a low resistance against compression, are an indicator for a high plasticity, while the plasticity is decreasing with increasing mean yield pressure [47]. According to the derived mean yield pressures, MCC and Para possess the highest plasticity followed by Lac, while DCPA has the lowest plasticity overall (Table 5).

#### 5.5. Extension of the Model of Cooper & Eaton by the Solid Compressibility Term

_{2}of Para clearly decreases. k

_{1}, significant in the low compression stress range, is the lowest for DCPA followed by Lac, MCC and Para. The resistance against filling of large pores thus increases in this order. However, the differences between the four materials are small. k

_{2}differs more clearly. In this stress range, MCC shows the lowest resistance against the filling of small pores by deformation followed by DCPA, Para and Lac. The lowest compression resistance was expected for MCC and the highest for DCPA based on the slopes of the compression curves. Furthermore, the inapplicability of the solid compressibility term for Lac and DCPA has to be taken into account. The differentiation of the materials based on the resistance values of the model of Cooper & Eaton is difficult as the parameters cannot be correlated to macroscopically reasonable rank orders of the materials, as found as well by Sonnergaard [49]. Therefore, an extended in-die compression function is developed, which is based on the models of Heckel and of Cooper & Eaton.

#### 5.6. Extended in-die Compression Function

_{l}is the bulk densification strength which is significant for the low stress process, while εh is the porosity change attributed to the high stress process and σ

_{h}is the material densification strength of the high stress process. The sum of εl and εh equals the initial porosity at 0 MPa.

_{l}of the four materials in the low compression stress range is visible (Table 7). σ

_{l}is most likely mainly dependent on the friction between particles and their Young’s modulus. σ

_{l}rises in the order Para < MCC < Lac < DCPA. This correlates well with the experimental findings. The lowest bulk densification strength for Para may be explained by a facilitated particle rearrangement due to the rectangular particle shape. σ

_{l}of MCC is also low because of a facilitated particle rearrangement due to the initially loose packing (initial porosity > 0.7) of the elongated particles. The particle rearrangement of Lac may be hindered due to a closer initial packing of the particles because of the very broad particle size distribution. The reasons for the highest σ

_{l}for DCPA might be the hindered particle rearrangement due to a probably already very close packing of the spherical particles and the low elasticity. These are certainly only hypothetical considerations, which have to be proved by further investigations.

_{h}is the lowest for MCC. Para shows a slightly higher resistance against compression followed by Lac. DCPA has the overall highest material densification strength. The material densification strength of Para and MCC can be explained by the mainly elastic and plastic deformation behaviour of these materials. The medium material densification strength of Lac may be traced back to the higher necessary compression stress for the breakage of the agglomerated/aggregated particles and the higher resistance of the primary particles against deformation due to their smaller particle size. The reasons for the considerably higher material densification strength of DCPA might be the high stiffness and low elasticity of this material.

#### 5.7. Comparison in- and out-of-die Compressibility

_{c}and the out-of-die compressibilities ε

_{T}(Figure 10). Therefore, the remaining differences provide information on the elastic deformation of the framework within the bulk. It can be deduced that the elastic relaxation of the bulk framework of Lac is least pronounced while the highest values are measured for MCC. It needs to be clarified whether this can be described by an intercorrelation between solid compressibility and deformation mechanisms of the material. Therefore, the consideration of the decompression curve is necessary. The out-of-die compressibility of Para cannot be determined because of the occurrence of tablet defects due to capping. The differences of the compressibility curves also cause differences of the mean yield pressures derived by in-die Heckel analysis considering solid compressibility P

_{y,c}compared to out-of-die Heckel analysis P

_{y,T}(Table 5). P

_{y,c}is lower compared to P

_{y,T}because of the consideration of the elastic deformation of the particle framework. The deformation behaviour of MCC is classified as brittle based on P

_{y,T}. This is not reasonable because of the evidently plastic deformation behaviour of MCC [23,48]. The high mean yield pressure P

_{y,T}can be traced back to the small slope of the out-of-die compressibility between 200 MPa and 400 MPa. Furthermore, the mean yield pressure is dependent on the considered compression stress range. The compression behaviour of a powder is not only affected by plastic deformation and particle fragmentation but rather by all acting micro-processes including particle rearrangement and elastic deformation. Therefore, the comprehensive characterization of the compression behaviour requires the consideration of the entire compression curves including the effect of all these micro-processes. The introduced in-die approach enables such a comprehensive characterization of powder compressibility even if tablet defects occur.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Particle images of the pharmaceutical powders MCC, Lac, DCPA and Para determined by scanning electron microscopy.

**Figure 3.**Apparent in-die compressibility of Para calculated using the solid density determined by helium pycnometry ε

_{app,He}and by X-ray diffraction ε

_{app,XRD}and correction of the apparent in-die compressibility ε

_{c,Hg}with the bulk modulus determined by mercury porosimetry. The filled areas mark the 95% confidence interval regarding density ε

_{app,He}and ε

_{app,XRD}or rather the bulk modulus ε

_{c,Hg.}

**Figure 4.**Expansion of the die diameter in dependence on the applied compression stress for Para, MCC, Lac and DCPA.

**Figure 5.**(

**a**) Heckel plots calculated using the apparent and the corrected in-die porosity; (

**b**) corrected Heckel plots and applied model of Heckel for Para, MCC, Lac and DCPA.

**Figure 6.**(

**a**) Apparent and corrected relative volume change for Para and MCC; (

**b**) Relative volume change and applied model of Cooper & Eaton for Para, MCC, Lac and DCPA.

**Figure 8.**(

**a**) Corrected in-die porosity curves and applied extended compression function; (

**b**) Residual plot for the fitted extended compression function for Para, MCC, Lac and DCPA.

**Figure 10.**Comparison of in-die porosity considering solid compressibility ε

_{c}with out-of-die compressibility ε

_{T}for DCPA, Lac and MCC.

Material | x_{10} [µm] | x_{50} [µm] | x_{90} [µm] | ρ_{s} [g/cm^{3}] |
---|---|---|---|---|

MCC | 36 | 125 | 261 | 1.5537 ± 0.0080 |

Para | 4 | 25 | 72 | 1.2863 ± 0.0001 |

Lac | 12 | 147 | 342 | 1.5524 ± 0.0077 |

DCPA | 77 | 190 | 302 | 2.7718 ± 0.0097 |

Material | Bulk Modulus K [GPa] | Min. Apparent in-die porosity ε_{app,He,min} [-] | Min. in-die porosity ε_{c,Hg,min} [-] |
---|---|---|---|

MCC | 10.22 ± 1.02 | −0.035 | 0.006 |

Para | 14.93 ± 3.67 | −0.023 | 0.005 |

Lac | 27.07 ± 7.13 | 0.001 | 0.016 |

DCPA | - | 0.256 | - |

Material | Stress Ratio [-] | Die Expansion Coefficient [GPa] | Difference of Porosity at 400 MPa [-] |
---|---|---|---|

MCC | 0.47–0.81 | 207 ± 4 | 3.7 × 10^{−6} |

Para | 0.50–0.85 | 196 ± 3 | 4.2 × 10^{−6} |

Lac | 0.45–0.71 | 237 ± 4 | 2.9 × 10^{−6} |

DCPA | 0.21–0.43 | 382 ± 5 | 1.1 × 10^{−6} |

**Table 4.**Bulk modulus determined by mercury porosimetry and elastic compressibility factors derived by extension of the models of Heckel and of Cooper & Eaton with the solid compressibility term and by the extended in-die compression function.

Material | Bulk Modulus K [GPa] | C_{Heckel} [GPa] | C_{Cooper &Eaton} [GPa] | C_{Extended function} [GPa] |
---|---|---|---|---|

MCC | 10.22 ± 1.02 | 11.51 ± 0.17 | 11.87 ± 0.12 | 11.74 ± 0.30 |

Para | 14.93 ± 3.67 | 15.22 ± 0.07 | 18.41 ± 0.74 | 15.97 ± 1.00 |

Lac | 27.07 ± 7.13 | 14.09 ± 0.03 | - | 14.80 ± 1.46 |

DCPA | - | 150.02 ± 17.24 | - | 277.06 ± 10.02 |

Material | P_{y} [MPa] | Fit Range | With Solid Compressibility Term | P_{y,T} [MPa] | ||
---|---|---|---|---|---|---|

P_{y,c} [MPa] | Fit Range | R^{2} [-]Entire Curve | ||||

MCC | 72.6 ± 1.0 | 10–50% σ_{max} | 89.3 ± 1.2 | 40–90% σ_{max} | 0.9860 | 543.5 |

Para | 93.5 ± 1.2 | 6–56% σ_{max} | 121.2 ± 2.7 | 30–80% σ_{max} | 0.9739 | - |

Lac | 145.6 ± 0.5 | 25–75% σ_{max} | 194.5 ± 0.6 | 40–90% σ_{max} | 0.9935 | 233.1 |

DCPA | 765.4 ± 1.5 | 38–88% σ_{max} | 771.2 ± 2.5 | 40–90% σ_{max} | 0.9313 | 1062.6 |

Material | V^{*}_{app} | With Solid Compressibility Term V^{*}_{c} | |||||||
---|---|---|---|---|---|---|---|---|---|

a_{1} [-] | k_{1} [MPa] | a_{2} [-] | k_{2} [MPa] | a_{1c} [-] | k_{1c} [MPa] | a_{2c} [-] | k_{2c} [MPa] | R^{2} [-] | |

MCC | 0.58 ± 0.01 | 1.46 ± 0.16 | 0.45 ± 0.01 | 20.2 ± 1.0 | 0.57 ± 0.01 | 1.43 ± 0.20 | 0.45 ± 0.01 | 18.7 ± 0.9 | 0.9998 |

Para | 0.70 ± 0.003 | 1.40 ± 0.10 | 0.35 ± 0.003 | 43.7 ± 2.4 | 0.69 ± 0.003 | 1.37 ± 0.10 | 0.34 ± 0.003 | 38 ± 2.0 | 0.9993 |

Lac | 0.52 ± 0.002 | 0.84 ± 0.00 | 0.54 ± 0.002 | 53.6 ± 0.4 | 0.9963 | ||||

DCPA | 0.45 ± 0.01 | 0.96 ± 0.10 | 0.42 ± 0.009 | 29.2 ± 1.9 | 0.9926 |

Material | ε_{l} [-] | σ_{l} [MPa] | ε_{h} [-] | σ_{h} [MPa] | C [GPa] | R^{2} [-] |
---|---|---|---|---|---|---|

MCC | 0.20 ± 0.00 | 8.5 ± 0.6 | 0.51 ± 0.01 | 79.0 ± 1.9 | 11.74 ± 0.30 | 0.9998 |

Para | 0.24 ± 0.00 | 4.3 ± 0.1 | 0.28 ± 0.00 | 99.2 ± 3.2 | 15.97 ± 1.00 | 0.9993 |

Lac | 0.14 ± 0.01 | 9.4 ± 0.5 | 0.32 ± 0.00 | 177.3 ± 5.6 | 14.80 ± 1.46 | 0.9997 |

DCPA | 0.21 ± 0.00 | 16.8 ± 0.2 | 0.47 ± 0.00 | 668.6 ± 3.5 | 277.06 ± 10.20 | 0.9993 |

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**MDPI and ACS Style**

Wünsch, I.; Finke, J.H.; John, E.; Juhnke, M.; Kwade, A.
A Mathematical Approach to Consider Solid Compressibility in the Compression of Pharmaceutical Powders. *Pharmaceutics* **2019**, *11*, 121.
https://doi.org/10.3390/pharmaceutics11030121

**AMA Style**

Wünsch I, Finke JH, John E, Juhnke M, Kwade A.
A Mathematical Approach to Consider Solid Compressibility in the Compression of Pharmaceutical Powders. *Pharmaceutics*. 2019; 11(3):121.
https://doi.org/10.3390/pharmaceutics11030121

**Chicago/Turabian Style**

Wünsch, Isabell, Jan Henrik Finke, Edgar John, Michael Juhnke, and Arno Kwade.
2019. "A Mathematical Approach to Consider Solid Compressibility in the Compression of Pharmaceutical Powders" *Pharmaceutics* 11, no. 3: 121.
https://doi.org/10.3390/pharmaceutics11030121