Analytical interpretation of powder compaction during the loading phase

Powder Technology 90 (1997) 173–178

Letter

Analytical interpretation of powder compaction during the loading phase a

Nicholas G. Lordi a,U, Harry Cocolas b, Hiro Yamasaki c Laboratory for Pharmaceutical Compaction Research, Pharmaceutics Department, Rutgers University, New Brunswick, NJ 08855-0759, USA b Whitehall-Robins, Hammonton, NJ 08037, USA c Fujisawa Pharmaceutical Co. Ltd., Osaka 541, Japan

Received 29 February 1996; revised 8 June 1996

Abstract

A new powder compaction equation which describes compaction data during the loading phase has been proposed. The equation takes the form PVsK[1yexp(ybP)]qVdP where V is the specific volume at applied pressure P, Vd is the dynamic limiting specific volume of the material, 1/b is defined as the initial yield pressure (Po) and K is a measure of the total work of compaction. At pressures )Pd, the dynamic yield pressure of the material, a linear relation is observed between the PV and P. Pressure–volume data for high density polyethylene, polyethylene glycol 8000, starch 1500 and sodium chloride is presented to illustrate the applicability of the proposed powder compaction equation. Vd values were ca. 5% higher than values calculated from measured true densities. Pd values varied from 14.6 (polyethylene) to 93.4 (starch 1500) MPa. Starch 1500 showed the largest K value (16.3 MPa cc/g) as compared to polyethylene (7.25 MPa cc/g). Keywords:

Compaction equations; Pharmaceutical materials

1. Introduction

The compaction of pharmaceutical materials during the loading phase in a tabletting operation has been largely interpreted using the Athy–Heckel equation [1] which suggests that the log of the porosity should be a linear function of pressure, the reciprocal of the slope being defined as the average yield pressure. However, the first-order behavior predicted by the Athy–Heckel equation is observed for most materials only over a limited pressure range, failing to describe pressure–volume compaction data at both low and high pressures. Carstensen et al. [2], have proposed a modification of the Heckel relation which extends the range for linearity, especially at low pressures, by assuming thataresidual porosity will be present even at the highest pressures. More recently, Oates and Mitchell [3] have suggested a modification of the Kawakita Equation [4], which takes the form: VsV 9q(Vo9yV 9)P9/(PqP9) (1) where V is the observed specific volume at P, V9 is the apparent limiting specific volume at infinite pressure, Vo9 is the apparent specific volume at Ps0 and P9 is a constant which represents the pressure at which Vs(Vo9qV9)/2. Oates and Mitchell validated their model using peak compression data U

Corresponding author.

obtained on an instrumented rotary press over a 25–250 MPa range for a variety of materials. Their objective was to develop a model for estimating volume from pressure–time data. We have observed that Eq. (1) effectively fits compaction data over a wide pressure range; however, the values of P9, V9 and Vo9 depend on the specific range of data used for the calculation. We have inferred, from empirical observation of pressure– volume compaction data of different materials compressed under different conditions at pressures up to 400 MPa, that compaction data during the loading phase may be described by the following relation: PVsK[1yexp(ybP)]qVdP

(2)

where V is the specific volume at applied pressure P and Vd is defined as the dynamic limiting specific volume. K and b are parameters dependent on loading conditions (e.g., compression rate and form of the displacement profile) as well as material characteristics. When the upper punch pressure exceeds an apparent material-dependent minimum value, the dynamic yield pressure (Pd), the following linear relation holds: PVsKqVdP

or equivalently,

VsVdqK/P

(3)

Our objective in this communication is to demonstrate the application of Eq. (2) to the analysis of pressure–volume

0032-5910/97/$17.00 q 1997 Elsevier Science S.A. All rights reserved PII S0032-5910(96)03211-1

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Fig. 1. Representative examples of PV–P plots showing linear and non-linear components: high density polyethylene, polyethylene 8000, starch 1500 and sodium chloride.

data, using high density polyethylene, polyethylene glycol, starch 1500 and sodium chloride as examples.

2. Experimental

Polyethylene glycol 8000 (PEG 8000, Union Carbide), sodium chloride (Fisher Scientific) and starch 1500 (Colorcon) were used as-received. Particle size distributions were evaluated from sieve analysis data (Sonic Sifter, ATM Corp.). The polyethylene glycol was a granular, spray congealed material with a 160 mm median particle size (range: 53–470 mm). The median particle size of sodium chloride and starch 1500 was 430 mm (range: 260–700 mm) and 68 mm (range: 24–180 mm). The granular high density polyethylene (HDPE, Schaettifix 1820, Schaetti) had a specified particle size range of 200–400 mm. Material densities were measured using a helium pycnometer (Quantachrome Multipycnometer MVP-1). Compaction data were obtained using the Rutgers Integrated Compaction Research System (aka, compaction simulator) with 10.3 mm flat-faced BB-tooling. The operation and characteristics of this equipment are detailed elsewhere [5]. The transducer calibrated ranges were 12.5"0.0014 mm for displacement and 0–50"0.04 kN for the load cell. The standard waveform used for theexperiments reported herein was a single-ended sawtooth profile with initial slopes of 50 or 10 mm/s under position control. Compact thickness data was smoothed using a 5-point moving average and corrected for system deformation determined using the technique described by Holman and Marshall [6]. The elastic system deformation correction was 0.006 mm/kN (std. dev. 0.00025 mm/kN) based on 21 measurements at intervals over a 14 month period. This corresponds to 0.24 mm at 40 kN. Sample weights were 300 or 500 mg. All materials were

unlubricated, except for sodium chloride which was run both unlubricated and lubricated with 0.5% magnesium stearate. Linear and non-linear regression of pressure–specific volume data, ranging from 2–400 MPa, was accomplished using StatMostTM Statistical Analysis Software (DataMost Corp.). Only data obtained during the loading phase was analyzed. Three sets of duplicate runs with different peak pressures were combined for analysis. Combined datasets included from 400–700 experimental values. Minimum estimated uncertainties in measurement of pressure and volume for the 10.3 mm tooling were "0.5 MPa and "0.0015 cc, based on the precision of the 12-bit analog–digital converters used to acquire the data. 3. Results and discussion

Fig. 1 shows representative plots of PV versus upper punch pressure for each material, which suggest the existence of a transition pressure above which a linear relation is observed. Specifying the transition pressure as a single number is not possible, since the non-linear segment is asymptotic to the linear segment. The following technique was used to resolve this problem. Experimental data was fit to Eq. (2) using nonlinear regression, with initial estimates of K and Vd determined from linear regression (Eq. (3)) of the high pressure segment. Figs. 2, 3 and 4 show a representative set of results for 0.3 g starch 1500 compacts at 10 mm/s to illustrate the technique. Fig. 2 shows the curvature in the low pressure data upon which is superimposed a plot of KqVdP, where Ks16.13 MPa cc/g and Vds0.655 cc/g as determined by non-linear regression. Fig. 3 plots the difference (d(PV)) between PV calculated from Eq. (3) and the actual experimental data, showing the exponential decay of d(PV). Fig. 4 plots ln[d(PV)] versus pressure and ln[K exp(ybP)] which rep-

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Fig. 2. PV–P plot of 0.3 g starch 1500 compact at 10 mm/s: experimental values compared to calculated linear component (16.13q0.655P).

resents the initial, non-linear component of the data. Table 1 summarizes the results of fitting Eq. (2) to datarepresentative of all four materials, as well as listing parameters which can be calculated from K, b and Vd. Results are included for calculations based on both upper and lower punch pressures, as a check on the validity of the calculations. For example, Vd values should be independent of the punch used as the basis for calculation. Excellent fits were observed in all instances with r2)0.999. The dynamic limiting density (rd), defined as the reciprocal of Vd, has values similar to but greater than the pycnometer measured true or static limiting density (rt). We suggest that the dynamic density is the operational value which should be used to calculate apparent porosities under pressure. Negative porosities are calculated for HDPE and PEG 8000 using static densities at pressures )150 MPa. Ponchel and Duchene [7] made similar observations in compaction studies of poly-e-caprolactone. This phenomenon is

explained as a pressure induced increase in limiting density of amorphous materials due to an increase in crystallinity [8]. We also note that no significant difference exists in dynamic densities calculated from upper or lower punch pressures, which supports our view that Vd is a material constant. Consistency in estimation of the dynamic specific volume (Vd or V9) was used as a test of the relative merits of Eqs. (1) and (2) in fitting pressure–volume data. Comparison of the residuals in the calculated specific volumes for the two models (Fig. 5), as represented by starch 1500, reveals that Eq. (2) predicts a lower specific volume than Eq. (1) at P-10 MPa. The measured specific volume at 2.4 MPa is 1.3"0.01 cc/g; the calculated value is 1.21 cc/g. Table 2 lists parameters calculated for representative data sets fitted by non-linear regression to Eq. (1). With the exception of sodium chloride, V9 values are less than Vd values, but more important, significant differences exist between V9 values calculated from upper and lower punch pressures, e.g., starch

Fig. 3. Plot of the difference between the linear component and experimental PV values for starch 1500, showing the exponential decay of d(PV) with pressure.

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Fig. 4. Comparison of log plots of calculated and experimental d(PV) values vs. pressure for starch 1500. Intersection of linear function at ln[d(PV)]s0 and y1 defines the dynamic yield pressure range. Table 1 Examples of parameters derived from Eq. (2) Material

rt a

(g/cc)

Punch b (U/L)

Rate c (mm/s)

Weight (g)

HDPE

0.96

U

10

0.5

L HDPE

0.96

U

50

0.5

L PEG 8000

1.22

U

10

0.5

L PEG 8000

1.22

U

50

0.5

L Starch

1.50

U

10

0.3

L Starch 1500

1.50

U

50

0.3

L NaCl

2.16

U

50

0.5

L NaCl (lubricant)

2.16

U

50

L

0.5

Kd

b

(MPay1)

(cc/g)

(g/cc)

8.26 (0.027) 7.11 (0.025) 7.25 (0.018) 6.02 (0.015) 8.66 (0.076) 5.50 (0.047) 7.52 (0.021) 4.79 (0.011) 16.13 (0.08) 11.66 (0.06) 18.61 (0.157) 13.82 (0.10) 9.77 (0.066) 8.86 (0.084) 7.81 (0.047) 7.24 (0.043)

0.145 (0.0013) 0.167 (0.0017) 0.126 (0.00078) 0.151 (0.00096) 0.072 (0.0016) 0.121 (0.0029) 0.0795 (0.00048) 0.133 (0.00076) 0.0362 (0.00025) 0.050 (0.00035) 0.0313 (0.00033) 0.0425 (0.00037) 0.034 (0.00035) 0.037 (0.0005) 0.037 (0.00042) 0.040 (0.00045)

0.982 (0.0002) 0.984 (0.0002) 0.984 (0.00016) 0.987 (0.000015) 0.774 (0.00045) 0.781 (0.0024) 0.753 (0.00015) 0.760 (0.00016) 0.655 (0.0006) 0.654 (0.0006) 0.654 (0.0011) 0.648 (0.001) 0.437 (0.0004) 0.432 (0.0006) 0.434 (0.0002) 0.434 (0.00022)

1.02

(MPa cc/g)

Vd

rd

Pd

n

6.9

14.6

1.22

1.02

6.0

11.8

1.21

1.02

7.9

15.7

0.93

1.01

6.6

11.9

0.93

1.29

13.9

30.0

0.81

1.28

8.3

14.1

0.85

1.33

10.3

20.7

0.79

1.32

7.5

11.8

0.84

1.53

27.6

76.8

0.89

1.53

20.0

49.1

0.89

1.53

30.0

93.4

0.89

1.54

23.5

61.8

0.91

2.29

29.4

67.0

0.76

2.31

27.0

59

0.76

2.3

27.0

55.6

0.67

2.3

25.0

49.5

0.67

Po

(Mpa)

(Mpa)

Helium pycnometer measured densities. Pressure measured at upper (U) or lower (L) punch. c Initial compression rate. d Figures in () are standard deviations. a

b

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Fig. 5. Comparison of residuals in specific volume (experimentalycalculated) for starch 1500 fitted to Eqs. (1) and (2).

1500 where V9s0.645 cc at the upper and 0.581 cc at the lower punch. Truncation of the data at lower pressures results in different parameter estimates for Eq. (1), while Eq. (2) gives consistent parameter estimates as long as the dataset includes samples at pressures -Po and )Pd. Holman [9], in his description of the percolation model of powder compaction, defines at least three percolation thresholds for ductile materials: the formation of a continuous network of particles, ‘‘« the threshold where the continuous percolating structure becomes rigid’’ and the cessation of a continuous network in the system porosity. Using values calculated from upper punch pressures, we propose the following hypotheses. (i) The initial yield pressure, Pos1/b, defines the second threshold, the pressure above which particle deformation dominates particle rearrangement. (ii) The dynamic yield pressure Pd defines the third threshold, the pressure at which all particles which form the compact matrix are fixed in relation to one another. As the applied pressure Table 2 Examples of paramaters calculated using Oates–Mitchell model (Eq. (1)) Material

Rate (mm/s)

Punch (U/L)

HDPE

10

U L

PEG 8000

50

U L

Starch 1500

10

U L

NaCl (lubricant)

50

U L

Vo9

V9

P9

2.18 (0.0076) 2.18 (0.006) 1.51 (0.0016) 1.34 (0.01) 1.33 (0.001) 1.26 (0.007) 0.794 (0.0009) 0.793 (0.0008)

0.933 (0.0037) 0.945 (0.0028) 0.751 (0.0009) 0.679 (0.0084) 0.645 (0.0013) 0.581 (0.014) 0.432 (0.0007) 0.431 (0.0006)

10.56 (0.2) 9.2 (0.14) 12.16 (0.088) 13.9 (0.87) 33.29 (0.31) 33.2 (2.28) 27.09 (0.29) 25.32 (0.24)

(cc/g)

(cc/g)

(MPa)

exceeds Pd, the resultant volume reduction, VyVd, is proportional to 1/P. Pd represents the minimum pressure required to form a compact of sufficient strength, if bonding takes place. In the percolation theory, it is more appropriate to define a transition pressure range, i.e., the critical region corresponding to the threshold. This critical region can be precisely defined as the intersection of ln[d(PV)] at 0 and y1, as shown in Fig. 4. Therefore, the lower limit (Pd in Table 1) is set at d(PV)s1, i.e., ln(K/b) and the upper limit is set at d(PV)s0.368, i.e., PdqPo. Threshold pressures reported by Holman (Table 4 in Ref. [9]) for starch 1500 correspond to our 10 mm/s data: 27 MPa (Pos27.6) and 106.5 MPa (PdqPos104.4). Following Oates and Mitchell [2], the work of compaction between P1sPd and P2)Pd can be defined as: P2

W12s P dVsK ln(P2/Pd)

|

(4)

P1

where KsW12 for P2s2.72 Pd, the work required to compress a compact once formed. At P-Pd, W12 is a function of K and b. The limited data summarized in Table 1 show an increase in work of compaction for starch 1500 (Ks16.13 to 18.61 MPa cc/g) and a decrease in work for HDPE (Ks8.26 to 7.25 MPa cc/g) and PEG 8000 (Ks8.66 to 7.52 MPa cc/g) on increasing compression rate, as well as the expected decrease in work on lubricating sodium chloride (Ks9.77 to 7.81 MPa cc/g). An important observation is that the product bK calculated using upper or lower punch pressures has the same value, suggesting that bK can be interpreted as the maximumvolume reduction which can be attained for a material under pressure. The apparent initial volume is VdqbK. We define the material compressibility, relative to the dynamic limiting volume, as: (5)

n sbK/Vd

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Except for HDPE, no significant difference in n was observed on changing compression rates from 10 to 50 mm/s. The HDPE was a low friction, free-flowing powder. At the higher compression rate, particle densification to a lower volume resulted in a greater Po coupled with a lower work of compaction. A similar but smaller decreaseincompressibility was also noted upon lubricating sodium chloride.

Knowledge of the initial volume or true density of thematerial is not required for data analysis, since the limiting density is intrinsic in pressure–volume data obtained in a powder compaction study. Consequently, practical tablet formulations which may be complex composite systems as well as pure materials can be analyzed. Eq. (2) provides an attractive alternative to the Heckel equation, which is limited by its dependence on limiting density values, for analyzing compaction data.

4. Conclusions

A valid powder compaction equation of state should meet the following criteria: (a) material independence; (b) calculated parameters have physical significance; (c) the model is applicable under different loading conditions; (d) the model predicts system behavior outside the range of data used for estimation; and, most importantly, (e) the equation of state should be derivable from first principles. We have proposed a three-parameter empirical model which is a potential candidate as a powder compaction equation of state, in that it meets the first two criteria stated before. Although Eq. (2) fails to adequately fit data at the lowest pressures (-10 MPa), where particle rearrangement dominates particle interaction, excellent agreement with experimental data is observed during the compact formation phase. The parameters calculated from a non-linear fit of PV versus P data are a measure of the work of compaction (K), the initial yield pressure (Po) and the dynamic specific volume (Vd). Additional derived parameters include the compressibility (K/PoVd) and the dynamic yield pressure (Pd).

Acknowledgements

The authors appreciate the comments and suggestions of Drs Metin Celik (Pharmaceutics) and Alberto Cuitino (Mechanical Engineering). This work was supported in part by a grant from ICI Pharma Inc. References

[1] R.W. Heckel, Trans. Metall. Soc. AIME, 221 (1961) 221. [2] J.T. Carstensen, J.M. Geoffroy and C. Dellamonica, Powder Technol., 62 (1990) 119. [3] R.J. Oates and A.G. Mitchell, J. Pharm. Pharmacol., 46 (1993) 270. [4] K. Kawakita and Y. Tsuteumi, Bull. Chem. Soc. Jpn., 39 (1966) 1364. [5] M. Celik and K. Marshall, Drug Dev. Ind. Pharm., 15 (1989) 759. [6] L.E. Holman and K. Marshall, Pharm. Res., 10 (1992) 976. [7] G. Ponchel and D. Duchene, 5th Int. Conf. on Pharmaceutical Technology (1989) 191. [8] F. Rodriguez, Principles of Polymer Systems, Hemisphere, Washington, DC, 1982, pp. 264–265. [9] L.E. Holman, Powder Technol., 66 (1991) 265.

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