Derivation of the Extended Kawakita Equation for Estimating the Yield State of Powder in Die

Abstract

For powder compaction, the Kawakita equation has been used to estimate the powder behavior inside the die. The compression pressure exerted on powders is not homogeneous because of the friction on the die wall. However, the yield pressure and porosity estimated using the Kawakita equation are defined based on the assumption that homogeneous voids and compression pressure are distributed throughout the powder bed. In this study, an extended Kawakita equation was derived by considering the variation in the compression pressure as it corresponds to the distance from the loading punch surface. The yield time section estimated from the extended Kawakita equation was wider than that which was estimated via the classical equation. This result is consistent with the assumptions used to derive the extended Kawakita equation. Furthermore, a comparison of the porosity changes before and after the yield pressure was applied indicate that the direct cause of the yield is the spatial constraints of the powder particles. Equivalent stresses were defined to clarify the critical factor that constitutes the extended Kawakita equation. As a result, “taking into account the die wall friction” was considered to be the critical factor in the extended Kawakita equation. As these findings were theoretically determined by the extended Kawakita equation, a useful model was derived for a better understanding of powder compaction in die.

Introduction

In order to obtain a compact powder product, it is important to estimate the compaction characteristics of powders and their behavior in the die. Several powder compaction equations have been recognized as useful for understanding the powder compaction behavior in the die. Persson and Alderborn attempted to estimate the tensile strength of tablets from the maximum compression pressure during compaction for powders with both ductile and brittle properties using the parameters of the powder compaction equation.¹⁾ As a result, accurate tensile strength was estimated using the parameters of the Kawakita equation and the yield pressure from the Heckel plot.¹⁾ However, some reports suggested a limitation of the Heckel plot. Rashid et al. measured force-displacement curves of raw and roller-compacted materials containing six different active pharmaceutical excipients.²⁾ The Kawakita equation showed good levels of agreement with each curve measured, whereas the Heckel plots were only in agreement with limited regions.²⁾ Rasid et al. concluded that the Heckel plot was, thus, only valid where the region with highly accurate.²⁾ Denny theoretically showed that the Heckel plot and the Kawakita equation were synonymous for powders with an initial porosity of about 0.66, assuming that the yield pressure constant in the Heckel plot was pressure dependent.³⁾ Based on the above reports, this study focused on the improvement of the Kawakita equation.

Several studies have been published on the Kawakita equation. Adams et al. demonstrated that the pressure for powder yield, as estimated using the Kawakita equation, was nearly identical to the yield pressure measured from a single particle.⁴⁾ Subsequently, they proposed an equation analogous to the Kawakita equation—the so-called Adams equation—utilizing a micromechanical model centered around a single particle. Following on from this, they clarified the physical significance of the parameters of the Kawakita equation.⁵⁾ Nordström et al. prepared four types of model granules consisting of different compositions and porosities. They then investigated the physical significance of the parameters obtained from the Kawakita and Adams equations. Results indicated that the parameters of the Kawakita equation reflected the granule plasticity well, although those of the Adams equation reflected their initial cracking.⁶⁾ Persson et al. evaluated the compaction properties of two powder types with yield pressures of granules, applying powder compression.⁷⁾ They showed that the Kawakita and the Adams equations were able to accurately evaluate the differences in yield pressure of both powders.⁷⁾ Tofiq et al. also reported that differences in the compaction properties of powders can be evaluated by using the Kawakita and Adams equations.⁸⁾ They showed that the initial porosity and flow of particles affected the compaction properties of powders.⁸⁾ Other reports evaluated the compaction properties of individual components.^9–13) Nicklasson and Alderborn¹⁴⁾ and Macho et al.¹⁵⁾ reported their findings for powder mixtures. Frenning et al.¹⁶⁾ and Mazel et al.¹⁷⁾ indicated that the fitting results of the Kawakita equation for powder mixtures can be predicted from those of the individual components. The Kawakita equation provides valuable insights into the granulation process by evaluating compressive properties. Solomon et al. reported that co-spray drying of excipients and surfactants improved compressibility, using the Kawakita equation to evaluate compressibility.¹⁸⁾ Furthermore, Partheniadis et al. reported that tableting of extrudates prepared with a hot melt extrusion improved compressibility by heating to 40 °C during tableting.¹⁹⁾ Furthermore, they also applied the Kawakita equation to evaluate the compaction properties of powders in die.¹⁹⁾ Other previous studies using the Kawakita equation to evaluate the compressive properties of the multicomponent powders have been reported, such as Xu et al.,²⁰⁾ Rojas and Hernandez²¹⁾ and Nakamura et al.²²⁾

As reported by Wu et al., the compression pressure decreases from the loading punch surface to the fixed punch surface because of friction on the die wall during the process of powder compaction in a die.²³⁾ However, powder compaction in the Kawakita equation is based on the assumption that the compression pressure on the powder is evenly distributed. Adams and McKeown estimated the yield pressure using the Kawakita equation and found that the measured yield pressure was lower than the estimated value when the aspect ratio was 0.5.⁵⁾ However, the measured values showed a good level of agreement with the estimations when the aspect ratio was extrapolated to be zero.⁵⁾ These findings suggest that the heterogeneity of the compressive pressure affects the estimation of the Kawakita parameters. Denny pointed out that ignoring the radial stress in the powder compaction equation for uniaxial compression in a die caused an incomplete result.³⁾ Therefore, we attempted to derive the extended Kawakita equation, considering the variability of the compression pressure corresponding to the distance from the loading punch surface. The extended equation was validated by measuring the load-displacement curves of 10 granule types with various excipient compositions. The powder compaction behavior in the die, based on the extended Kawakita equation, is fully discussed.

Experimental

Preparation of Granules

Ten model formulations were prepared, similar to what was described in a previous report.²⁴⁾ The formulations primarily contained lactose (LAC), cornstarch (CS), and microcrystalline cellulose (MCC), with ingredient ratios as presented in Table 1. The ingredients were mixed for 1 min using a mixer (Model KM4005, DeLonghi Japan Co., Ltd., Tokyo, Japan), with purified water added to the powder (30% of the total amount of mixed powder) followed by granulation for 5 min. After drying in a heat dryer (DF411, Yamato Scientific Co., Ltd., Tokyo, Japan) at 75 °C for 60 min, Mg-St was added and mixed using a V-type mixer (Model S-3, Tsutsui Rikagaku Kikai Co., Ltd., Tokyo, Japan).

Table 1. Formulation of Model Tablets

Formulation	1	2	3	4	5	6	7	8	9, 10
Microcrystalline cellulose (MCC) [mg]	136.5	136.5	58.5	58.5	136.5	97.5	58.5	97.5	97.5
Cornstarch (CS) [mg]	58.5	0	136.5	0	29.25	97.5	68.25	0	48.75
Lactose (LAC) [mg]	0	58.5	0	136.5	29.25	0	68.25	97.5	48.75
Polyvinylpyrrolidone K-90 (PVP K-90) [mg]	4.0
Magnesium stearate (Mg-St) [mg]	1.0
Total [mg]	200.0

LAC (Tabletose^® 80), CS (Graflaw^® M), and MCC (CEOLUS^® PH-101) were purchased from Meggre Japan Co., Ltd. (Tokyo, Japan), Nippon Starch Chemical Co., Ltd. (Osaka, Japan), and Asahi Kasei Chemicals Corporation (Tokyo, Japan), respectively. PVP K-90 and Mg-St (plant-derived magnesium stearate) were purchased from FUJIFILM Wako Pure Chemical Corporation (Osaka, Japan).

Collection of Tableting Data

Tableting data recorders (TK-TB20kN. Tokusyu Keisoku Co., Ltd., Kanagawa, Japan) were used to measure the upper and lower punch surface loads, die wall loads, and upper and lower punch surface displacements over time. The measurement was conducted with flat punches of φ = 8 mm. The amount of powder filled into the die was 200 mg. The measurements were repeated three times with a targeted maximum load of 14 kN and a displacement speed of 8 mm/s. The powder height in the die was determined by calculating the difference between the upper and lower punch surface displacements.

Kawakita Equation

Equation 1 (the classical Kawakita equation) shows relationship between porosity, n, and compression pressure, σ_p.

  

(1)

where P_y is the yield stress and n₀ is the initial porosity. Furthermore, Eq. 1 can be rewritten as follows:

  

(2)

In addition, n can also be expressed using Eq. 3 under the assumption that V is the apparent powder volume, V_∞ is the powder volume corresponding to σ_p→∞ (i.e., infinite load), and V_b is the void volume.

  

(3)

As noted by Wu et al.,²³⁾ the loading punch surface pressure is greater than the fixed punch surface pressure when die wall friction is active. Equation 4 is induced under the assumption that the powder height is h, the arbitrary powder height from the lower punch surface is z (0 ≤ z ≤ h), the compression pressure at the arbitrary powder height is σ_z, and the lower and upper punch surface pressures are σ_B and σ_T, respectively.

  

(4)

If the powder in the die is divided into N equal layers, as indicated in Fig. 1, the whole powder porosity n_w can be calculated as:

  

(5)

where v is the apparent powder volume of each compartment, v_bi and n_i are the void volume and porosity of the i-th compartment, respectively. In addition, the compression pressure in the i-th compartment can be calculated using Eq. 4. As the porosity corresponding to the i-th compartment is calculated from Eq. 2, Eq. 6 can be given as:

  

(6)

When we assume N→∞, Eq. 6 is expressed as Eq. 7 (the extended Kawakita equation):

  

(7)

Furthermore, Eq. 7 can be rewritten as:

  

(8)

Fig. 1. Conceptual Diagram of the Extended Kawakita Equation

In this study, the Kawakita equation parameters were estimated using the measured upper and lower punch surface pressures at each time point, σ_Tm and σ_Bm, in Eq. 9 (the classical Kawakita equation) and in Eq. 10 (the extended Kawakita equation):

  

(9)

  

(10)

where M_max and M₀ are the maximum compression pressure and compression starting points, respectively.

Results and Discussion

Comparison of the Classical and the Extended Kawakita Equations

Figure 2 shows the measured values of V, n, and 1/n in Form. 1 together with those calculated using Eqs. 9 and 10. The mean relative absolute error for each curve fitting was calculated, and the values were equivalent for all tableting data (Table 2). The results of P_y, V_∞, and n₀, as estimated from the classical and the extended Kawakita equations are shown in Fig. 3. Tables 3 and 4 show the values of n₀, n_y (porosity corresponding to P_y), and n_f (porosity corresponding to maximum load) calculated using and , and and , respectively. The values of n_y and n_f in Table 4 indicate when the loading punch surface pressure reaches and the maximum pressure, respectively. Both and were in the range of 38–50 MPa (1.9–2.5 kN). Moreover, n₀ − n_y and n_y − n_f were comparable. From analyses of the random sphere packing model, the porosity was reported to be about 0.36 to 0.46.^25–27) The estimated n₀ in this study was greater than 0.6, n_y was approximately 0.4, and n_f was approximately 0.1. Therefore, it was concluded that particle rearrangement mainly occurred before P_y, resulting in a significant decrease in porosity. In addition, the estimated and in this study are considered reasonable because the results are consistent with reports on the random sphere packing model.^25–27)

Fig. 2. Curve Fitting Results of Form. 1

The gray dots, red dashed lines, and black solid lines are calculated from the measured tableting data, and extended and classical Kawakita equation. Results for (a) powder volume, (b) porosity, and the (c) inverse of porosity, corresponding to the lower punch pressure.

Table 2. Mean Absolute Relative Error (MARE) in Curve Fitting to Classical and Extended Kawakita Equation

	(Classical Kawakita equation)			(Extended Kawakita equation)
	V	n		V	n
Form. 1	0.0194 ± 0.0001	0.0184 ± 0.0004	0.0184 ± 0.0004	0.0208 ± 0.0007	0.0195 ± 0.0000	0.0195 ± 0.0000
Form. 2	0.0214 ± 0.0016	0.0218 ± 0.0014	0.0217 ± 0.0014	0.0222 ± 0.0012	0.0224 ± 0.0011	0.0223 ± 0.0011
Form. 3	0.0280 ± 0.0012	0.0281 ± 0.0005	0.0281 ± 0.0005	0.0295 ± 0.0020	0.0293 ± 0.0012	0.0292 ± 0.0012
Form. 4	0.0277 ± 0.0017	0.0344 ± 0.0011	0.0341 ± 0.0011	0.0281 ± 0.0025	0.0347 ± 0.0019	0.0344 ± 0.0018
Form. 5	0.0182 ± 0.0023	0.0177 ± 0.0019	0.0176 ± 0.0018	0.0193 ± 0.0015	0.0185 ± 0.0013	0.0184 ± 0.0013
Form. 6	0.0239 ± 0.0045	0.0232 ± 0.0036	0.0231 ± 0.0036	0.0253 ± 0.0048	0.0241 ± 0.0037	0.0241 ± 0.0036
Form. 7	0.0208 ± 0.0005	0.0226 ± 0.0001	0.0225 ± 0.0001	0.0216 ± 0.0005	0.0233 ± 0.0001	0.0232 ± 0.0001
Form. 8	0.0224 ± 0.0014	0.0245 ± 0.0015	0.0244 ± 0.0015	0.0232 ± 0.0006	0.0251 ± 0.0007	0.0250 ± 0.0007
Form. 9	0.0233 ± 0.0031	0.0235 ± 0.0029	0.0234 ± 0.0029	0.0242 ± 0.0024	0.0241 ± 0.0022	0.0240 ± 0.0022
Form. 10	0.0183 ± 0.0000	0.0186 ± 0.0002	0.0186 ± 0.0002	0.0194 ± 0.0002	0.0194 ± 0.0003	0.0194 ± 0.0003

Mean ± standard deviation (S. D.), n = 3.

Fig. 3. Comparison of (a) Yield Pressure P_y, (b) Powder Volume at Infinite Loading V_∞, and (c) Initial Porosity n₀ Estimated from Classical and Extended Kawakita Equation

Table 3. Initial Porosity n₀, Porosity at Yield n_y, Porosity at Maximum Compression n_f, and Their Differences Estimated from the Classical Kawakita Equation

	(Classical Kawakita equation)
	n0	ny	nf	n0−ny	ny−nf
Form. 1	0.7245 ± 0.0032	0.4195 ± 0.0010	0.1163 ± 0.0001	0.3050 ± 0.0022	0.3031 ± 0.0012
Form. 2	0.7089 ± 0.0031	0.4141 ± 0.0018	0.1269 ± 0.0016	0.2948 ± 0.0012	0.2871 ± 0.0002
Form. 3	0.6892 ± 0.0031	0.4072 ± 0.0013	0.1322 ± 0.0003	0.2820 ± 0.0018	0.2750 ± 0.0010
Form. 4	0.6385 ± 0.0035	0.3888 ± 0.0014	0.1425 ± 0.0017	0.2498 ± 0.0021	0.2463 ± 0.0030
Form. 5	0.7126 ± 0.0012	0.4157 ± 0.0005	0.1323 ± 0.0030	0.2969 ± 0.0007	0.2834 ± 0.0036
Form. 6	0.7061 ± 0.0043	0.4128 ± 0.0017	0.1242 ± 0.0010	0.2933 ± 0.0027	0.2886 ± 0.0027
Form. 7	0.6690 ± 0.0012	0.4003 ± 0.0009	0.1403 ± 0.0001	0.2687 ± 0.0003	0.2600 ± 0.0008
Form. 8	0.6833 ± 0.0009	0.4056 ± 0.0006	0.1397 ± 0.0014	0.2778 ± 0.0003	0.2659 ± 0.0009
Form. 9	0.7016 ± 0.0018	0.4121 ± 0.0008	0.1284 ± 0.0006	0.2895 ± 0.0010	0.2837 ± 0.0013
Form. 10	0.6929 ± 0.0000	0.4090 ± 0.0002	0.1408 ± 0.0006	0.2838 ± 0.0002	0.2683 ± 0.0008

Mean ± S. D., n = 3.

Table 4. Initial Porosity n₀, Porosity at Yield n_y, Porosity at Maximum Compression n_f, and Their Differences Estimated from the Extended Kawakita Equation

	(Extended Kawakita equation)
	n0	ny	nf	n0−ny	ny−nf
Form. 1	0.7240 ± 0.0031	0.4260 ± 0.0009	0.1144 ± 0.0006	0.2980 ± 0.0021	0.3116 ± 0.0015
Form. 2	0.7084 ± 0.0032	0.4202 ± 0.0019	0.1252 ± 0.0020	0.2882 ± 0.0013	0.2950 ± 0.0001
Form. 3	0.6886 ± 0.0028	0.4127 ± 0.0013	0.1303 ± 0.0005	0.2759 ± 0.0015	0.2825 ± 0.0017
Form. 4	0.6381 ± 0.0032	0.3956 ± 0.0011	0.1411 ± 0.0021	0.2425 ± 0.0021	0.2545 ± 0.0032
Form. 5	0.7120 ± 0.0016	0.4219 ± 0.0011	0.1301 ± 0.0023	0.2900 ± 0.0005	0.2918 ± 0.0033
Form. 6	0.7056 ± 0.0044	0.4197 ± 0.0020	0.1226 ± 0.0010	0.2859 ± 0.0024	0.2971 ± 0.0030
Form. 7	0.6685 ± 0.0013	0.4060 ± 0.0014	0.1387 ± 0.0003	0.2625 ± 0.0001	0.2673 ± 0.0011
Form. 8	0.6827 ± 0.0012	0.4124 ± 0.0005	0.1379 ± 0.0022	0.2703 ± 0.0007	0.2746 ± 0.0017
Form. 9	0.7012 ± 0.0021	0.4183 ± 0.0003	0.1269 ± 0.0001	0.2829 ± 0.0017	0.2913 ± 0.0002
Form. 10	0.6923 ± 0.0000	0.4150 ± 0.0006	0.1388 ± 0.0004	0.2773 ± 0.0006	0.2762 ± 0.0002

Mean ± S. D., n = 3.

In addition, in the classical and the extended Kawakita equations, the powder heights were compared when the compression pressure reached the estimated yield pressure — see Table 5 for the results. For the extended Kawakita equation, the powder height at reaching the yield pressure was estimated as an interval, provided that σ_B and σ_T reached . For any of the powder heights, there was no discrepancy as they were higher than the tablet thickness, h_tab, after die ejection. For the classical Kawakita equation, the two powder heights corresponding to each P_y were equivalent. Therefore, two yielding powder states estimated from Eq. 1 were considered to represent the powder at the same point in time, regardless of the definition of compressive pressure. Therefore, the extended Kawakita equation provides a more realistic estimation of powder conditions at the point of yield than the classical equation. Specifically, as shown in Fig. 1, when the loading punch surface pressure reaches P_y, the particles near the loading punch surface yield. As the fixed punch surface approaches, the compression pressure decreases and some regions may be existent, in which unyielding particles have been distributed. Yielding throughout the whole powder bed is then completed when the fixed punch surface pressure reaches P_y. The extended Kawakita equation can estimate the above continuous yielding behavior. Table 4 suggests that the behavior before reaching P_y is particle rearrangement. Furthermore, the porosity at the yield point was consistent with reports on random sphere packing.^25–27) In addition, the findings have been reported by Tofiq et al.,⁸⁾ suggesting that the initial porosity and particle flow of powders affected the compaction properties of powders. Based on these findings, the yield condition of powder particles filled in the die was assumed to be the onset time when the particle rearrangement reaches the limit due to spatial constraints. Therefore, yielding is suggested to not only generate because of compression pressure, but as a result of multiple factors — such as particles alignment, their size distribution, and their friction. The particles do not rearrange homogeneously throughout the powders, but rather intermittently in localized and simultaneous rearrangements. Therefore, both from the deductive viewpoint of modeling the powder, and from the inductive viewpoint of estimating from experimental results, the extended Kawakita equation is considered to provide more accurate estimation of powder conditions at yielding.

Table 5. Tablet Thickness h_tab and Powder Height at Each Definition of Compression Pressure Reaching Yield Pressure

	htab	(Extended Kawakita equation)	(Classical Kawakita equation)
		h|σB=Py* − h|σT=Py*	h|σB=Py	h|σT=Py*
Form. 1	2.991 ± 0.010	3.350 ± 0.064 − 3.280 ± 0.064	3.310 ± 0.056	3.304 ± 0.062
Form. 2	2.809 ± 0.022	3.274 ± 0.030 − 3.218 ± 0.017	3.244 ± 0.018	3.238 ± 0.012
Form. 3	3.207 ± 0.013	3.299 ± 0.001 − 3.245 ± 0.014	3.257 ± 0.008	3.256 ± 0.012
Form. 4	2.809 ± 0.023	3.195 ± 0.001 − 3.128 ± 0.008	3.153 ± 0.005	3.150 ± 0.008
Form. 5	2.937 ± 0.019	3.265 ± 0.060 − 3.210 ± 0.059	3.234 ± 0.057	3.226 ± 0.069
Form. 6	3.115 ± 0.021	3.338 ± 0.016 − 3.271 ± 0.016	3.298 ± 0.028	3.294 ± 0.026
Form. 7	2.929 ± 0.015	3.190 ± 0.015 − 3.132 ± 0.027	3.136 ± 0.033	3.127 ± 0.019
Form. 8	2.806 ± 0.028	3.226 ± 0.033 − 3.158 ± 0.059	3.196 ± 0.036	3.196 ± 0.033
Form. 9	2.954 ± 0.021	3.336 ± 0.010 − 3.265 ± 0.024	3.280 ± 0.033	3.265 ± 0.012
Form. 10	2.973 ± 0.008	3.236 ± 0.022 − 3.185 ± 0.003	3.208 ± 0.011	3.198 ± 0.002

Mean ± S. D., n = 3.

Essential Factors Constituting the Extended Kawakita Equation

As shown in Fig. 3, V_∞ and n₀ estimated from the classical and the extended Kawakita equations were equivalent to each other, because these parameters represented the infinite load and the start of compression, respectively. This may lead to the assumption that equivalent values can be estimated using these equations, although P_y was assumed to be expressed as  = a + b, where a and b are constants. The linear correlation between and suggested that the extended Kawakita equation can be approximated as a linear equation (Eq. 1). Considering Eq. 8, however, it is necessary to introduce an assumption that allows for an equational transformation into a linear equation. In addition, it should be considered that the σ_p value has been reflected by σ_T and σ_B. In this study, the equations relating the vertical elastic strain (Eq. 11) or elastic strain energy (Eq. 12) in the direction of the load axis and the compression pressure were derived in a similar manner to produce Eq. 7 in Fig. 1. The σ₁ value is assumed to be homogeneously distributed throughout the whole powder. Then, σ₁, which gives the total load axial deformation in each segment of Fig. 1, can be defined by Eq. 11. Similarly, the homogeneous compression pressure σ₂, which gives the total elastic strain energy in each of the segments of Fig. 1, can be defined by Eq. 12.

  

(11)

  

(12)

where h₀ is the initial powder height and E is Young’s modulus. In this study, σ₁ and σ₂ are equivalent to the first and second equivalent stresses, respectively. Furthermore, Eqs. 11 and 12 can be rewritten as Eq. 13, where σ₁ and σ₂ are defined as σ_k (k = 1, 2).

  

(13)

Table 6 lists the n₀ and P_y as estimated from Eq. 1, using σ_B, σ_T, σ₁, or σ₂ as σ_p for the value of n calculated using and the measured value of h. Consequently, n₀ was equivalent to the value estimated using the extended Kawakita equation in any case of σ_p. The value of P_y varied based on the definition of σ_p. When σ_p was equal to σ_B, P_y was estimated to be higher than , and when σ_p was equal to σ_T, P_y was estimated to be lower than . Moreover, when σ_p was equal to σ₁ and σ₂, P_y was estimated to be equivalent to . These results suggest that the extended Kawakita equation (Eq. 7) could be approximated to the equation defining the compression pressure, σ_p, in the classical Kawakita equation (Eqs. 1 and 2) as the equivalent stress, σ_k. More specifically, the inhomogeneous distribution of voids on the entire powder bed can be ignored, provided that die wall friction is considered. Following the approximation by equivalent stress, the extended Kawakita equation agrees with the classical Kawakita equation by adding the following approximation: the compression pressure distributed in the powder bed is equal to the loading or fixed punch pressure (i.e., die wall friction can be ignored). Figure 4 illustrates the relationship between these two types of Kawakita equations and their approximations.

Table 6. Linear Regression Results Corresponding to the Definition of Compression Pressure σ_p

	n0					Py
	(Extended)	(Classical Kawakita equation)				(Extended)	(Classical Kawakita equation)
	Comparator n0 (Table 4)	σp = σB	σp = σT	σp = σ1	σp = σ2	Comparator	σp = σB	σp = σT	σp = σ1	σp = σ2
Form. 1	0.7240 ± 0.0031	0.7237 ± 0.0049	0.7184 ± 0.0035	0.7210 ± 0.0042	0.7211 ± 0.0042	37.87 ± 0.32	39.50 ± 0.43	36.97 ± 0.45	38.22 ± 0.44	38.23 ± 0.44
Form. 2	0.7084 ± 0.0032	0.7032 ± 0.0007	0.6990 ± 0.0019	0.7011 ± 0.0013	0.7011 ± 0.0013	40.71 ± 0.69	42.55 ± 0.87	39.72 ± 0.81	41.12 ± 0.84	41.13 ± 0.84
Form. 3	0.6886 ± 0.0028	0.6865 ± 0.0027	0.6826 ± 0.0010	0.6845 ± 0.0018	0.6846 ± 0.0018	43.42 ± 0.51	45.58 ± 0.42	42.55 ± 0.38	44.05 ± 0.40	44.06 ± 0.40
Form. 4	0.6381 ± 0.0032	0.6272 ± 0.0026	0.6254 ± 0.0010	0.6263 ± 0.0018	0.6263 ± 0.0018	49.83 ± 1.13	52.82 ± 1.08	48.78 ± 0.96	50.77 ± 1.02	50.78 ± 1.02
Form. 5	0.7120 ± 0.0016	0.7065 ± 0.0003	0.7015 ± 0.0029	0.7040 ± 0.0016	0.7041 ± 0.0016	44.42 ± 0.96	46.54 ± 0.98	43.64 ± 0.94	45.07 ± 0.96	45.08 ± 0.96
Form. 6	0.7056 ± 0.0044	0.7043 ± 0.0020	0.7005 ± 0.0022	0.7024 ± 0.0021	0.7025 ± 0.0021	40.88 ± 0.47	42.72 ± 0.25	40.05 ± 0.24	41.37 ± 0.25	41.37 ± 0.25
Form. 7	0.6685 ± 0.0013	0.6644 ± 0.0011	0.6617 ± 0.0013	0.6631 ± 0.0012	0.6631 ± 0.0012	48.15 ± 0.37	50.66 ± 0.37	46.98 ± 0.37	48.80 ± 0.37	48.81 ± 0.37
Form. 8	0.6827 ± 0.0012	0.6746 ± 0.0002	0.6710 ± 0.0023	0.6728 ± 0.0012	0.6729 ± 0.0012	46.38 ± 0.89	49.08 ± 0.81	45.29 ± 0.86	47.16 ± 0.84	47.17 ± 0.84
Form. 9	0.7012 ± 0.0021	0.6986 ± 0.0008	0.6952 ± 0.0013	0.6969 ± 0.0003	0.6969 ± 0.0002	42.66 ± 0.05	44.68 ± 0.28	41.52 ± 0.25	43.08 ± 0.26	43.09 ± 0.26
Form. 10	0.6923 ± 0.0000	0.6863 ± 0.0004	0.6825 ± 0.0002	0.6844 ± 0.0001	0.6844 ± 0.0001	47.95 ± 0.53	50.54 ± 0.46	47.15 ± 0.43	48.82 ± 0.45	48.83 ± 0.45

Mean ± S. D., n = 3.

Fig. 4. A Diagram Representing the Approximation from the Extended Kawakita Equation to the Classical Kawakita Equation

In P_y, as described in Table 6, the approximation by equivalent stress (columns σ_p = σ₁ and σ_p = σ₂ in Table 6) was closer to than the approximation ignoring the die wall friction (columns σ_p = σ_B and σ_p = σ_T in Table 6). This result indicates that the critical essence in the extended Kawakita equation is “taking into account the die wall friction.” Furthermore, it suggests that die wall friction is a non-negligible factor in powder compaction and that the mixing of the lubricant and the transmission of compression pressure toward the die wall affect the behavior of the powder in the die.

Conclusion

The classical Kawakita equation was extended based on a model that reflects the non-homogeneity of voids and compression pressure during powder compaction. The yield time section estimated from the extended Kawakita equation was wider than that estimated from the classical equation. This result is consistent with the assumptions used to derive the extended Kawakita equation. Furthermore, the porosity changes before and after the yield pressure indicated that the direct cause of yielding was not compression pressure, but spatial constraints such as rearrangement limits.

In addition, by using the equivalent stresses newly defined in this study, an approximation from the extended Kawakita equation to the classical equation was considered possible. The equivalent stress is the compression pressure that reflects the effect of die wall friction. In other words, ignoring the non-homogeneity of voids in the powder bed was allowed, provided that the effect of die wall friction was considered. Following the approximation by equivalent stress, the extended Kawakita equation was approximated to the classical equation based on the assumption that die wall friction was ignored. Furthermore, Kawakita parameter values equivalent to the extended Kawakita equation were estimated only in the approximation using the equivalent stress. Therefore, “taking into account the die wall friction” was theoretically revealed to be the critical factor in the extended Kawakita equation and a crucial discrepancy with actual powder in the classical Kawakita equation. Ten formulations consisting mainly of MCC, CS and LAC were experimentally validated. However, a wide variety of excipients and active ingredients are actually used. Furthermore, only one tableting condition was conducted. Therefore, the applicable powder types and tableting conditions for the extended Kawakita equation should be addressed in future studies.

From the extended Kawakita equation derived in this study, we theoretically determined two points: the occurrence of yielding caused by spatial constraints and the contribution of die wall friction in powder compaction. Therefore, the extended Kawakita equation is considered a useful model for a better understanding of powder compaction in die.

Acknowledgments

This study was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP17K08252.

Conflict of Interest

The authors declare no conflict of interest.

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