- Email: [email protected]
Modeling the Uniaxial Compaction of Pharmaceutical Powders Using the Mechanical Properties of Single Crystals. I: Ductile Materials
Modeling the Uniaxial Compaction of Pharmaceutical Powders Using the Mechanical Properties of Single Crystals. I: Ductile Materials WENDY
c. DUNCAN-HEWITT*’AND GEORGEc. WEATHERLY’
Received February 6, 1989, from the ‘Faculty of Pharmacy, University of Toronto, 19 Russell Street, Toronto, Ontario, M5S 1Al Canada and the ‘Department of Metallurgy and Materials Science, University of Toronto, Toronto, Ontario, M5S 1A4 Canada. Accepted for publication May 31, 1989. - -__~-
Abstract 0A model is presented which uses the Vickers microinden-
tation hardness of ductile crystals such as sodium chloride to predict the uniaxial compaction behavior of compacts. A general approach first developed in the materials science field to predict the densification of particulate matter under hydrostatic loading was followed. However, modifications to account for the effects of particle geometry and the closed-die loading conditions were considered. Using the standard microindentation hardness value of sodium chloride,the model predicted the densification behavior of this material at a punch displacement rate of 1 mmlmin. Densification at higher compaction rates was predicted by considering the effect of deformation kinetics on the hardness. Secondary factors which affect compaction, such as particle size effects and die-wall friction, are also briefly discussed.
1
313
05! 0.0
.
I
0.1
.
, 02
.
,
.
0.3
nomalizcd mean compaction rtreu, ;I
It is difficult to make tablets which are sufficiently strong to survive, intact, until they are ingested by a patient. Tablet strength is a strong function of the interparticulate contact area and the primary function of tablet compaction is to increase this area of contact. Therefore, an understanding of the compaction process will provide considerable insight into the origins of tablet strength, an insight which should facilitate the subsequent production of tablets having controlled strength characteristics. Many investigations of tablet compaction have attempted to elucidate putative correlations between measurable tableting parameters such as compaction rate, punch and die wall stresses, stress relaxation, and tablet porosity to evaluate tablet “compactability” (the relative ease with which a given material consolidates). The simplest set of these parameters consists of the maximum force applied to the punch and the relative density of the resultant tablet. Although many empirical equations have been developed to describe the relationship between these compaction parameters,’ the most popular of these is the Heckel relationship (eq l),the form of which can yield useful information about the nature of the materials being compacted:
ln(l/(l - p)) = KO + k
~
q
(1)
where pis the relative density of the compact, KOis a constant, K, is the slope of Heckel plot (“inverse mean yield pressure”), and a, is the compaction stress (punch stress). The primary factors which are believed to influence the compaction behavior of a given material are its yield stress (or hardness) and its fracture toughness. The magnitude of the “mean yield pressure” in the Heckel relationship indicates the relative ease with which the material in question may be compacted and must be a function of the hardness of single particles. The effect of fracture toughness is reflected by the shape of the Heckel plot: if it is linear (Figure la) then oO22-3~9/90/02OO-0 147$0 1 .OO/O 0 1990, American Pharmaceutical Association
, 0.4
.
I
05
.
4
0.6
Y
Flgure 1-Heckel plots for sodium chloride, a model ductile matela1 (a) and sucrose, a brittle material (b). The mean compaction stress (u), the average of the punch and die-wall stresses, is normalized by dividing by the plastic yield stress characteristic of the material. In many instances, the yield stress is approximately one-third of the hardness.
essentially ductile behavior is indicated, while a curvilinear plot (Figure lb) is believed to indicate brittle behavior. An alternate method, adopted in the present study, models the process of tablet compaction and attempts to provide an explanation for the relationship between measurable variables in terms of the microscopic properties of the system. In the case of tablet compaction, the compaction stress and relative density relationship is modeled using the mechanical properties of the constituent crystals. Two models of tablet compaction, one for ductile materials and one for brittle materials, are required. However, it is necessary to find a criterion a priori to establish which mode of compaction behavior a given material will exhibit. “Brittleness” factors (e.g., the ratio of fracture toughness to hardness) have been used extensively in the wear literature: and appear to be reasonable criteria to differentiate compaction behavior.3 The ratio of the fracture toughness and hardness for sodium chloride is an order of magnitude larger than those of brittle materials such as sucrose and acetaminophen.3 This effectively divides the materials into two groups and correctly reflects the Heckel plot and microscopic behavior of these materials during compaction. Models of Compaction Behavior-Models of densification of ductile metal powders under hydrostatic loading have been developed by Fischmeister and Arzt4.6 and Swinkels et al.6 We will follow their approach but will consider, where appropriate, modifications to account for particle geometry and the uniaxial closed-die loading conditions of our experiments. Four factors must be considered in any model of powder compaction: (1) the change in the number of particle contacta (coordination number = Z) as densification proceeds; (2) the Journal of Pharmaceutical Sciences I 147 Vol. 79, No.2, February 1990
condition for local yielding at individual contacts; (3) the average area of each contact as a function of the particle geometry and relative density of the compact; and (4) the relationship between the far-field stresses and the local stresses a t each contact. The role that each of these factors plays in the models of tablet compaction is described below for ductile materials and in a companion paper for brittle materials. Expression for Z-Several relationships between relative density and coordination number have been described in the literature, ranging from lists of relative densities associated with various perfect packing of spheres to descriptions of random packings of spheres.6.7.8The relationship proposed by Ant6 and which was used throughout the present study is expressed by the following set of equations:
p 2 0.85
(2b)
where p is the relative density and Z is the coordination number. These equations were derived from the packing geometry associated with a random packing of spheres; since the packing of particles in a powder bed is random, the A n t 6 relationship is preferred. While the coordination number relationships given by A n t are strictly correct only for relative densities h 0.64, Figure 2 indicates that the actual values which result from back-extrapolation using eq 2 are similar to the values proposed by Ridgway and Tarbuckle.7 Therefore, for the sake of simplicity, the relationship given in eq 2a was used for relative densities less than 0.64. Although many pharmaceutical compacts are made from angular particles, the particles chosen for this study possessed shape factors which differed from that of a sphere by two or less. (Shape factors are used to indicate the degree to which the shape of a particle departs from sphericity, and are defined in terms of factors such as relative surface area and relative length.) Furthermore, after a little compaction, any angular asperities will become rounded by plastic deformation and the particle shape will approximate a sphere. The distribution function was derived for a bed of equalsized spheres. Large deviations from this condition will give rise to other relationships. To avoid this possibility it is necessary experimentally to limit the range of the particle size distribution (e.g., by sieving).
7
12 L
$
11:
E,
10.
C
c
0 .c m .-c E 0 0 0
/
I
’
9.
8.
7. 6. 5.
4.
31 .2
. , . , . , . . . , . . . . . . .3
.4
.5
.6
.7
.a
.9
uc=
H
=
FJA
(3)
where a, is the interparticulate contact stress, H is the hardness of the crystal, F, is the contact force, and A is the contact area. For angular particles, this relationship should be modified by a geometric factor that accounts for the particular geometry of the contact. However, numerous studies”12 have shown that provided the angle of the cone of the indenter is ?goo, the correction factor is less than 8%and eq 3 should give an adequate representation of the limiting area of a given contact under hydrostatic loading. However, the non-hydrostatic loading conditions in a closed die lead to substantial shear stresses a t the contact points, enhancing the densification. Schwartz and Holland13 found that the slope of the Heckel plot for iron (l/mean yield value) derived from isostatic compaction was 4 . 9 ~ 1 0 -MPa-l, ~ while those quoted by Heckel for iron powders undergoing uniaxial compaction, ranged from l.l.10-7 to 1.4-10-7 MPa-’. While they attributed this effect to the presence of fines, it is more likely that shear, while not causing densification, facilitates it. Using a die which permitted the direct application of shear stresses, Hardman and Lilley14 found that shear facilitated the densification of sodium chloride. While the contact area in the absence of shear stresses will be governed by the hardness, any tangential movement will enlarge the area of contact if the material is ductile.16 Johnson16 explored the effect of combined shear and normal pressure when a plastic wedge is deformed by a flat, rigid die. This geometry may be used to represent asperity contact during compaction,and provides a simple means of comparing the modes of deformation in the two limiting conditions. Johnson16 found that the contact area increased from a value of FJH if no shear stress is present to a value of 5.2FJH when the shear stress reaches the same value as the normal stress. In the latter case the area of contact is determined by the shear strength of the material. The problem now is to relate these values to the behavior of an average contact in closed-die pressing. The following approach was used to calculate the proportionality between the mean contact stress in the die and the hardness of the material. For closed-die pressing, the punch to die-wall stress ratio is typically observed to be 2.5:l for pharmaceutical materia1~.1~-~0 We assume that the three principal stresses are then crx, 0.4 u.,,and 0.4 a, and that they are aligned parallel and perpendicular to the loading direction. For contacts aligned parallel to principal stresses the shear stress is zero and the contact area would be given by FJH. For orientations where the shear stress is greater than or equal to the normal stress, the contact area is given by 5.2FJH. Between these two limiting cases, the proportionality factor (linking the contact area and the hardness) was assumed to vary linearly from 1 to 5.2, as discussed by Johnson. When these factors are averaged over the range of possible orientations of interparticulate contacts, we find that the proportionality factor is 3.421 and eq 3 should be replaced by:
1
Relative Density Flgure 2-Relationships between coordination number and relative density. That proposed by Ridgway (1) closely approximates the relationship describing a progression of perfect packing configurations of spheres while that proposed by Tarbuckle and Arzt (2) arises from the packing distributionfunction of a random close-packedarray of spheres. 148 I Journal of Pharmaceutical Sciences Vol. 79, No. 2, Februaty 1990
Local Yielding-The problem of local yielding a t particleparticle contacts is closely related to the hardness indentation test or the mutual indentation of two crystals. For hydrostatic loading, it is usually assumed that the limiting contact area is given by an expression of the type:
a, = W3.4 = FJA
(4)
This equation applies for uniaxial compaction with die wall stresses equal to 0.4 times the punch stress. The following argument suggests that a proportionality factor of 3.4 is about right. Using the ratio between the
hydrostatic (normal) stress and deviatoric (shear) stress within the die, the yield criterion of Green15 predicts that yielding should commence when the ratio between the mean normal stress and the shear strength of the material is approximately 3.2. If it is assumed that the hydrostatic part of the total stress causes consolidation, these values correspond closely to the ratio which may be calculated from the experimental results reported by Schwartz and Holland13 for ion (-3.9),as discussed above. Two additional factors make the hypothesis that shear strain facilitates consolidation attractive. It provides a means of explaining the effect of lubricants and glidants on the densification process. These materials facilitate densification by decreasing the interparticulate shear stress, but weaken the resultant compact. Since very thin layers of lubricants are present on the surfaces of the crystals, they would play a minor role if only normal stresses caused densification. It also provides a mechanism of consolidation which should be more rate dependent than indentation, since shear at the interface is inhibited by the roughness of the contacting surfaces. Friction is especially rate dependent in the presence of surface contaminants.22 Average Area of ParticlelParticle Contact as a Function of Relative Density-The crystals used in this study possessed relatively compact habits, and it is assumed that their behavior resembles that of spheres. As noted earlier, while the asperities of the particles are initially angular, small amounts of deformation will tend to round them so that on average, the contacts will deform as sections of spheres. A n t 6 developed an elegant model to describe the densification of spherical powders based on the radial distribution function described above. A spherical particle with an initial radius of unity (R;Figure 3) is assumed to grow concentrically within a polyhedral cell (Voronoi ce1l29, the dimensions of which are defined by the packing. The cell is constructed from the bisectors of the lines which join the particle being considered and its nearest neighbors. If the particle contacts a side of its cell, this indicates that the particle is contacting its neighbor in that direction. When the particle contacts and Figure 3), the volume of overlaps a face of the polyhedron (R’; the material removed by the contact (Vex) is assumed to be
distributed evenly across the remaining free surface (S,) of the particle which increases the particle radius further (R). Using a geometrical argument based on the radial distribution function, A n t derived the following relationships which provide an expression for the average contact area in terms of the relative density:
+ (15.5.~/12).(R‘ - 1)3.(3.R’+ 1)
(5c)
+ 15.5 + R2.15.5.(2-R-3)]
(5e)
where pis the present relative density, p I is the initial relative density, Sf is the free surface remaining on sphere, R is the present radius of sphere with deposited material, R’ is the enlarged radius of sphere before material deposition, Vex is the volume of material cut off by face of Voronoi cell, Zi is the initial coordination number, Z is the present coordination number, and A is the average contact area. The constant 15.5 is derived from the radial distribution function. Local Stresses at Particle Contacts as a Function of Far-Field Stresses-Relationship between Uniaxial Compaction Stress and the Mean Stress Which Causes Densification-Die compaction differs from isostatic pressing since the load is applied from one direction only. It is assumed that densification is caused by the hydrostatic component of the stress within the die, which is smaller than the punch stress, while the shear component of the stress causes changes in shape and helps to enlarge the contact areas.24 The hydrostatic component is the mean of the principal stresses within the die. This calculation is not trivial, since the effects of friction and the discontinuous nature of the powder compact will cause rotation of the direction of the principal stresses with increasing depth and radius within the compact, and in fact will probably change from particle to particle. Therefore, to simplify the problem, it was assumed that the first principal direction is parallel to the axis of the die; the others are perpendicular to this, but otherwise are arbitrarily directed (uy= uz). The ratio’of the die-wall stress (a to the punch stress (ux)has been measured for sodium ch?oride17-19 and sucrose,~7~~9~20 and appears to be essentially constant (-0.4) for uwvalues between 0 and 197 MPa, which is greater than the range of stre_ssesemployed in the present experiments. The mean stress (u), assumed to be equal to the hydrostatic component of stress, is then:
-
u = (ux+ cry i- uz)/3
Flgure &View of the densification model. The central particle is assumed to possess the initial radius (R) and is located within a polygonal cell (Voronoi cell). During densification the patticle is assumed to grow concentrically within its cell and impinge upon other nearby particles. Material which overlaps the boundaries of the Voronoi cell is redistributed across the remaining free surface of the particle.
(6)
where ax is the punch stress and uy and uzare the die wall stresses. Rezationshipbetween the Mean Stress and the Averwe Force at One Cont~t-Equation 7, Proposed by M o l e r ~ s expressed ,~~ the relationship between the loads at the contacts between spherical particles and the relative density of a powder mass: Journal of Pharmaceutical Sciences I 149 Vol. 79, No. 2, February 1990
(7) where o is the hydrostatic external stress (-3, F, is the contact force, Z is the coordination number, and r is the particle radius. A n t 5 and Swinkels et a1.6 used this relationship to complete their models of densification. This relationship assumes that the particles are spherical and that the load is transmitted through point contacts. This assumption is not generally true during compaction. Therefore, the following is proposed to replace eq 7. Since the hydrostatic stress gives rise to densification, it is assumed that the mean compaction stress is distributed over the surface of a sphere. The average relative density of the contents of the sphere is equal to the present relative density of the compact. It is assumed that the size of the Voronoi cell (or equivalently, the reference sphere over which the external stress is distributed) is constant, while the particle within the reference sphere grows. Then the surface area of the reference sphere also remains constant and may be calculated from the initial relative density as follows: fi =
SA
=
v,/v, 4.rr.r:
where pi is the initial relative density, V, is the particle volume, V, is the volume of reference cell, rr is the radius of reference cell, and SA is the surface area of the external reference. The local contact stress is then related to the far-field stresses as follows:
F,
=
r,*A
where is given by eq 6, Z is defined by eq 2, and A, the area of contact, can be found from the relationship presented in eq 5. Application of the Model-The model is applied as follows: (1)the initial relative density is that of the powder before consolidation begins; (2)a set of increasing relative densities is prescribed; (3) the coordination number for each relative density is calculated using eq 2; (4) the contact area per asperity is calculated as a function of the relative density using eq 5; (5) the total force per particle is calculated using eq 4; (6) the hydrostatic or mean stress is calculated using eq 9; and (7) the punch stress is calculated from eq 6. In summary, this procedure calculates the punch stress a t each prescribed relative density during uniaxial compaction from the mechanical properties of the crystals being considered.
Experimental Section Sodium chloride was chosen as the model “ductile” material for study. The sodium chloride employed (Fisher, lot 740662A, density = 2.165 glmL, Mellor, 19643was recrystallized from water according to the method described by Duncan-Hewitt and Grant.26 The cubic crystals thus obtained were sieved, and the fractions ranging from 500-710 pm were collected for testing. Twenty crystals were prepared and indented (1 or 2 indentations per crystal) at room temperature with a load of 147 mN using a Leitz-Wetzlar Miniload hardness tester which was equipped with a Vickers dlamond indenter as described by Duncan-Hewitt and Weatherly.3 The indentation time was 10 s. For the compaction studies a 1.3cm diameter die was brushed with a slurry of 10% magnesium stearate in ethanol and allowed to dry for 150 I Journal of Pharmaceutical Sciences Vol. 79, No. 2, February 7990
5 min. Approximately 3.0 g of the crystals was weigned accuretelv using a Mettler balance (model 1-911) and poured into the die. A Euny dense tablet would then possess a thickness of 1am. The die was then clamped onto the table of a n electronically controlled shaker (Janke and Kunkel IKA-Vibrax-VXR) and vibrated a t 35 Hz for 1min. The top flat faced punch was inserted into the die and the assembly was shaken for an additional 1 min to minimize the variability of the density distribution in the green compact. The punch was guided by a face plate which was equipped with a rubber ring. Complete insertion of the upper punch required pushing the face plate to meet the top of the die. This resulted in precompression loads ranging from 50-200 N, depending upon the amount of lubricant which penetrated under the rubber ring. As a result, the initial relative density of the bed of crystals ranged from 0.55 to 0.58. Compacts were prepared using a n Instron Universal Testing Apparatus (Model 4206). The assembled die was placed on the lower platform of the Instron, and the crosshead was lowered until a compressive force of 0.01 N was measured. The initial position of the crosshead relative to the bottom of the die cavity was then recorded and the initial relative density of the bed of crystals within the die was calculated. The crosshead was lowered a t controlled rates of 1,10, and 50 m d m i n and the maximum compressive force (up to 25 kN) was recorded. The force required to overcome the friction exerted by the rubber ring, estimated as the apparent yield value at the beginning of the test, was subtracted from the value of the punch force. After compaction, the tablets were removed from the die, and their dimensions were measured using a micrometer. These values were employed to calculate the relative density of the compacts. This calculation is based on the assumption that the compacts possess a uniform density distribution; the validity of the assumption will be discussed below. Some compacts were cut into halves either vertically or horizontally. The surfaces of these pieces were ground and polished to produce a relatively smooth finish and examined microscopically (Leitz Microscope).
Results and Discussion Hardness of Sodium Chloride Crystals-The sodium chloride crystals were difficult to test, as the crystals sometimes adhered to the indenter, invalidating the results of that experiment. Chipping was not observed. The Vickers Hardness Number (H) was calculated from the mean diagonal length as follows:3
H
=
loadlarea of contact = 1.854.P/d2
(10)
where P is the load and d is the mean length of the indentation diagonal. The hardness of the sodium chloride crystals used in this study was found to be 213 2 50 MPa (95% confidence interval, n = 15). Appearance of Compacts-When the sodium chloride tablets were examined microscopically,the crystals exhibited very little fracture. This observation has also been reported in the literature.27 A t high relative densities, the crystals were usually oriented in columns. These columns tended to follow the contours of stress which are interpolated from the density distribution in a tablet produced by uniaxial compression.= The crystals retained their essentially cubical shape, but the contact areas were rounded. Compaction Stress versus Relative Density Relations h i p T h e effect of strain rate on the compaction of the sodium chloride compacts is shown in Figure 4. A t a compaction rate of 1 m d m i n , the experimental data appear to agree with those reported by Hersey et al.27 for crystals in the 40-80 mesh size range compressed a t the same rate in an unlubricated die (diameter = 33 mm) using an Instron Universal testing apparatus. A t any given value of compaction stress, the compacts were less dense a t a higher rate of compaction (Figure 4). Fit of Model to Experimental Results-The average ratio of punch stress to die wall stress was taken to be 0.4, the
-. P
1.70
,
1.50
I
m o
v-
D'
1.30
-E
1.10
g
0.90
$ 1 I
=
m o
c
i
'
m o
9
0.70) 0
a1 mm/min 0 1 0 mmlmin
m50 mmlmin '
' 10
'
' 20
.
'
"
.
40
30
'
'
'
60
50
Punch Stress, MPa
Flgure +Heckel plots for sodium chloride tablets compacted at punch displacement rates of 1, 10, and 50 mm/min.
average value found in the literature.17-19 Using this value, and the hardness value of 213 MPa, the model predicted the densification of sodium chloride at a compaction rate of 1 m d m i n (Figure 5). This conclusion is justified on the basis that the predicted and experimental curves coincide within the range of the experimental variability associated with the hardness values and the compaction data. Figure 5 also indicates that accuracy of the hardness is crucial to the agreement between theory and experiment. Effect of Rate of Compaction-The plastic yield stress is a function of the strain-rate; therefore the use of hardness values, or values calculated from them, is strictly valid only if the contact times during hardness testing and compaction are comparable. As stated above, a hardness value of 213 MPa could predict densification only for crystals compacted at a rate of 1m d m i n . At higher compaction rates, larger stresses were required to produce tablets of a given relative density. Assuming that the increased strain-rate causes the apparent interparticulate contact stress to increase concurrently, the following approach was employed to estimate the contact stress at higher compaction rates. Verrall et 81.29 constructed a deformation mechanism map for sodium chloride, which shows that the hardness will increase with strain rate. Assuming that increasing strainrate has a similar effect on both the indentation hardness and the resistance to compaction, the contact stresses at 10 and 50 m d m i n were interpolated from the deformation mechanism map, by finding strain rates 10 and 50 times greater than that 2.0 1
corresponding to a hardness value of 213 MPa. The resultant hardnesses were found to be 278 MPa and 335 MPa, respectively. The ability of these values to predict compaction at 10 and 50 m d m i n is shown in Figure 6. It may be concluded that the predictive capability of the model is improved if the effect of strain-rate is considered in the manner described above. This suggests that indentation creep experiments are wellsuited to the characterization of materials which are to be compacted. It must be emphasized that the foregoing analysis provides only a rough approximation: the correspondence between a compaction rate of 1m d m i n used in the compaction tests and a standard Vickers Hardness Test (i.e., 10 s indentation time) was found empirically. Furthermore, the strain-rate is not constant during compaction: both strain-hardening, and undetermined changes in the contact geometry may cause additional complications which have not been addressed in this study. Other Factors Affecting Compaction-Other variables which are believed to have a predictable effect on the densification behavior of pharmaceutical materials include elasticity, particle size, initial packing of the powder in the die, and the range of stresses employed.SG33 Elastic effects may in part be related to the rate dependence of compaction through the concept of viscoelasticity:34 as the rate of compaction increases, the amount of reversible (elastic) deformation increases relative to the amount of irreversible (plastic) deformation. In this study, the complications that the consideration of elastic deformation entails were avoided by relating the compaction stress to the density of the compacts once they have been removed from the die (i.e,,after elastic recovery has occurred). The consequence of this approximation is important: tablets of some materials are known to fail by capping and laminating, either during decompression or as the tablet is ejected from the die. Neglecting elasticity effects will compromise the ability of the models to predict the strength of such compacts. In future studies it will be essential to address the elastic and plastic properties of individual crystals in much more detail since the amount of stored elastic strain determines the stress available to break the bonds which are formed while the compact is still under pressure.35 There appears to be considerable disagreement in the literature about whether an increase in particle size facilitates densification or whether it renders densification more difficult, and opposite trends have been reported.3" The effect of particle size is obviously complex, and may arise from size-dependent differences in particle shape, size distribution,
1.8
I
10 mm/mln, pledictcd using H = 278 MPa
130 125
(expt'l. if H = 150 MPa) 120
0.8
0a
experimental data, H = 213 MPa 0.1
02
03
0.4
-
1.15 05
0.6
1.10
normalized mean compaction stress, 0 I Y
Flgure %Fit of the densification model to the experimental Heckel plot for sodium chloride compacted at a rate of 1 mmimin. The effect of
varying the hardnesgvaluesused in the model is also shown. The mean compaction stress (u), the average of the punch and die-wall stresses, is normalized by dividing by the plastic yield stress characteristic of the material. Because the punch stress value is normalized, the position of the experimental curve shifts as the hardness changes.
/
.a
1
.21
.25
M mm/rnin, predicted using H = 33.5 MPa I
.29
.
,
33
.
,
.
normalized mean compaction stress,
,
41
37
.
,
.45
IY
Flgure &Predicting the densification curves for sodium chloride cornpacted at punch displacement rates of 10 and 50 mm/min. The hardness values are derived by considering the effect of strain rate on plastic
yielding.
Journal of Pharmaceutical Sciences I 151 Vol. 79, No. 2,February 7990
or mechanical properties or the effect may arise from differences in particle packing in the die. It was felt that the effects of particle size could be ignored in this study because the hardness tests were performed on the same crystals which were employed in the compaction studies. Extrapolation of the results of this study to much smaller particle such as occur in pharmaceutical powders is not a priorz justified, and requires careful theoretical consideration. Fedion between the contents of the die and the die wall causes the effective compaction stress to vary through the powder bed, and as a result, the density of the compact is not homogeneous. The relative effects ofdie-wall friction are greater the smaller the tablet; therefore, the pressure required to produce a tablet of a given relative density increases concurrently.36 Application to Other Materials+ would be useful to determine whether this mddel predicts the behavior of materials with mechanical properties similar to those of sodium chloride. Heckel plots for potassium chloride are available in the literature (particle size fraction 60-80 mesh).*7 The literature provided hardness values ranging from 92 MPa37 to 220 MPa.38 A hardness of 125 MPa, which is close to the median value, fits the data best (Figure 7). It is remarkable that the range of hardness values reported in the literature is quite large for both sodium chlonde (150-314 MPa) and potassium chloride (92-220 MPa). Brace39 observed that the hardness value of sodium chloride may be doubled if work-hardening has occurred. Rapid crystallization may also give rise to a large number of defects in the crystal, with a concurrent increase in hardness. Crystals which are employed to form compacts are not annealed, and handling procedures probably introduce some surface defects. It appears reasonable, therefore, that in the absepce of experimental data, the median hardness ought to predict the densification behavior reasonably well, as observed for potassium chloride.
Conclusions A model has been developed which predicts the densification behavior of sodium chloride during uniaxial compaction to within experimental error. The effects of compaction rate were considered explicitly and other factors which affect compaction were discussed briefly. Further development of the model should consider the effects of elastic strain, which may cause tablet failure due to capping or lamination. Ultimately, a model is required which can predict the strength of pharmaceutical compacts from the mechanical (expt'l. if H = 220 MPa) \
0.0
01
0.2
/
03
0.4
normalized mean compaction stress, / Y
05
0.6
0.7
(T
Flgure 7-Fit of the densification model to the experimental Heckel plot for potassium chloride compacted at a rate of 1 mmlmin. The effect of varying the hardnessvaluesused in the model IS also shown. The mean compaction stress (o),the average of the punch and die-wall stresses, is normalized by dividing by the plastic yield stress characteristic of the material. Because the punch stress value is normalized, the position of the experimental curve shifts as the hardness changes. 152 I Journal of Pharmaceutical Sciences Vol. 79, No. 2, February 7990
properties of their constituent crystals. This model must account for the ability of the material to densify (asthe present model does) since the size of the interparticulate contact areas is a maor determinant of the compact strength. However, many other factors affect strength as well. For example, (1)the particle brittleness will determine the ability of cracks and porosity to concentrate stress which can give rise to premature fracture; (2) contaminants can' alter the interparticulate bond strength; and (3) residual stresses will decrease the apparent fracture strength. It is hoped that the simple densification model described in this manuscript will provide the impetus for the development of more comprehensive models in the future.
References and Notes 1. Kawakita, K.; Ludde, K.-H. Powder Technol. 1970,4, 61. 2. Mathia, T. G.; Lamy, B. Wear 1986, 108,385. 3. Duncan-Hewitt, W. C.; Weatherly, G. J . Mat. Sci. Letters, in press.
4. Fischmeister, H. F.; Arzt, E. Powder Metall. 1983, 26, 82. 5. Artz, E. Acta Metall. 1982, 30, 1883. 6. Swinkels, F. B.; Wilkinson, D. S.; Ant, E.; Ashby, M. F. Actu Metall. 1983,31, 1829. 7. Rdigway, K.; Tarbuckle, K. J. Brit. Chem. Eng. 1967, 12, 384. 8. Scott, G. D.; Kilgour, D. M. Br. J . Appl. Phys. 1969,2,863. 9. Shaw, M. C. In The Science of Hardness Testin and Its Research A p lications; Westbrook, J. H., Conrad, H., fids.; A.S.M. Publisling: Metals Parks, OH, 1973; p 1. 10. Beeuwkes. R. Jr.: Chait. R.: Lin. H. H. In The Science ofHardmss Testmg and Its Research Applications; Westbrook, J. k; Conrad, H., Eds.; A.S.M. Publishing: Metals Parks, OH; 1973; p 75. 11. Atkins, A. G. In The Science of Hardness Testing and Its Research Applications; Westbrook, J. H.; Conrad, H., Eds.; A.S.M. Publishing: Metals Parks, OH, 1973; p 223. 12. O'Neill, H. Hardness Measurements ofMetals and Alloys, 2nd ed.; Chapman and Hall: London, 1967; p 17. 13. Schwartz,E. G.; Holland,A. R. Int. J . Powder Metall. 1969,5,79. 14. Hardman, J. S.; Lilley, B. A. In Proc. 1st Intl. C o n . on the Compaction and Consolidation of Particulate Matter, h r ' hton, 3 6 October, 1972, Powder Technol. Publ. Series, #4, GoBberg. A. S., Ed., 1972, p 115. 15. Green, A. P. J . Mech. Phys. Solids, 1954,2, 197. 16. Johnson, K. L. Contact Mechanics, Cambridge University Press: Cambridge, 1985; p 171. 17. Leigh, S.; Carless,J. E.; Burt, B. W. J . Pharm. Sci. 1967,56,888. 18. Ridgway, K.; Rupp, R. J . Pharm. Pharmacol. 1969,21,30S. 19. Obiorah, B. A. Int. J . Pharm. 1978, I , 249. 20. Summers, M. P.; Enever, R. P.; Carless, J. E. J . Pharm. Pharmacol. 1976,28, 89. 21. Duncan-Hewitt, W. C. Ph.D. Thesis; University of Toronto, Toronto, Canada, 1988. 22. McClintock, F. A.; Argon, A. S. Mechanical Behuviour of Materials, Addison-Wesley: Reading, MA, 1965, p 276. 23. Voronoi, G . F. J . Reine Angew. Math. 1908, 134, 198. 24. Venkatachari, K. R.; Raj, R. J . A m . Ceram. SOC.1986, 69,499. 25. Molerus, 0. Powder Technol. 1975, 12, 259. 26. Duncan-Hewitt, W. C.; Grant, D. J. W. Powder Technol. 1987, 52, 17. 27. Hersey,J. A.; Cole, E. T.; Rees, J. E. In Proc. 1st Zntl. Conf. on the Corn action and Consolidation of Particulate Matter, Brighton, 3 4 &t., 1972, Powder Technol. Publ. Series, #4, Goldberg, A. S., Ed., 1972; p 165. 28. Train, D. J . Pharm. Pharmacol. 1956,8, 745. 29. Verrall, R. A,; Fields, R. J.; Ashby, M. F. J . A m . Ceram. Soc. 1977, 60, 211. 30. Fell, J. T.; Newton, J. M. J . Pharm. Sci. 1971, 60, 1866. 31. York, P. J . Pharm. Pharmacol. 1979,31, 244. 32. Rue, P. J.; Rees, J. E. J . Pharm. Pharmacol. 1978,30, 642. 33. Huffine, C. L.; Bonilla, A. I. Chem. Eng. J . 1962, 8, 490. 34. Rippie, E. G.; Danielson, D. W. J . Pharm. Sci. 1981, 70, 476. 35. Hiestand, E. N.; Smith, D. Powder Technol. 1984,38, 145. 36. Lammens, R. F.; Liem, T. B.; Polderman, J.; de Blaey, C. J. Powder Technol. 1980,26, 169. 37. Subramanian, B.; Bansigir, K. G. J . Muter. Sci. 1980, 15, 2889. 38. Gilman,J. J. In TheScience o Hardness Testin andlts Research Applications; Westbrook, J. .; Conrad, H., Efs., ASM Publishing: Metals Parks, OH, 1973, p 51. 39. Brace, W. F. J . Geophy. Res. 1960, 65, 1773.
6
c. DUNCAN-HEWITT*’AND GEORGEc. WEATHERLY’
Received February 6, 1989, from the ‘Faculty of Pharmacy, University of Toronto, 19 Russell Street, Toronto, Ontario, M5S 1Al Canada and the ‘Department of Metallurgy and Materials Science, University of Toronto, Toronto, Ontario, M5S 1A4 Canada. Accepted for publication May 31, 1989. - -__~-
Abstract 0A model is presented which uses the Vickers microinden-
tation hardness of ductile crystals such as sodium chloride to predict the uniaxial compaction behavior of compacts. A general approach first developed in the materials science field to predict the densification of particulate matter under hydrostatic loading was followed. However, modifications to account for the effects of particle geometry and the closed-die loading conditions were considered. Using the standard microindentation hardness value of sodium chloride,the model predicted the densification behavior of this material at a punch displacement rate of 1 mmlmin. Densification at higher compaction rates was predicted by considering the effect of deformation kinetics on the hardness. Secondary factors which affect compaction, such as particle size effects and die-wall friction, are also briefly discussed.
1
313
05! 0.0
.
I
0.1
.
, 02
.
,
.
0.3
nomalizcd mean compaction rtreu, ;I
It is difficult to make tablets which are sufficiently strong to survive, intact, until they are ingested by a patient. Tablet strength is a strong function of the interparticulate contact area and the primary function of tablet compaction is to increase this area of contact. Therefore, an understanding of the compaction process will provide considerable insight into the origins of tablet strength, an insight which should facilitate the subsequent production of tablets having controlled strength characteristics. Many investigations of tablet compaction have attempted to elucidate putative correlations between measurable tableting parameters such as compaction rate, punch and die wall stresses, stress relaxation, and tablet porosity to evaluate tablet “compactability” (the relative ease with which a given material consolidates). The simplest set of these parameters consists of the maximum force applied to the punch and the relative density of the resultant tablet. Although many empirical equations have been developed to describe the relationship between these compaction parameters,’ the most popular of these is the Heckel relationship (eq l),the form of which can yield useful information about the nature of the materials being compacted:
ln(l/(l - p)) = KO + k
~
q
(1)
where pis the relative density of the compact, KOis a constant, K, is the slope of Heckel plot (“inverse mean yield pressure”), and a, is the compaction stress (punch stress). The primary factors which are believed to influence the compaction behavior of a given material are its yield stress (or hardness) and its fracture toughness. The magnitude of the “mean yield pressure” in the Heckel relationship indicates the relative ease with which the material in question may be compacted and must be a function of the hardness of single particles. The effect of fracture toughness is reflected by the shape of the Heckel plot: if it is linear (Figure la) then oO22-3~9/90/02OO-0 147$0 1 .OO/O 0 1990, American Pharmaceutical Association
, 0.4
.
I
05
.
4
0.6
Y
Flgure 1-Heckel plots for sodium chloride, a model ductile matela1 (a) and sucrose, a brittle material (b). The mean compaction stress (u), the average of the punch and die-wall stresses, is normalized by dividing by the plastic yield stress characteristic of the material. In many instances, the yield stress is approximately one-third of the hardness.
essentially ductile behavior is indicated, while a curvilinear plot (Figure lb) is believed to indicate brittle behavior. An alternate method, adopted in the present study, models the process of tablet compaction and attempts to provide an explanation for the relationship between measurable variables in terms of the microscopic properties of the system. In the case of tablet compaction, the compaction stress and relative density relationship is modeled using the mechanical properties of the constituent crystals. Two models of tablet compaction, one for ductile materials and one for brittle materials, are required. However, it is necessary to find a criterion a priori to establish which mode of compaction behavior a given material will exhibit. “Brittleness” factors (e.g., the ratio of fracture toughness to hardness) have been used extensively in the wear literature: and appear to be reasonable criteria to differentiate compaction behavior.3 The ratio of the fracture toughness and hardness for sodium chloride is an order of magnitude larger than those of brittle materials such as sucrose and acetaminophen.3 This effectively divides the materials into two groups and correctly reflects the Heckel plot and microscopic behavior of these materials during compaction. Models of Compaction Behavior-Models of densification of ductile metal powders under hydrostatic loading have been developed by Fischmeister and Arzt4.6 and Swinkels et al.6 We will follow their approach but will consider, where appropriate, modifications to account for particle geometry and the uniaxial closed-die loading conditions of our experiments. Four factors must be considered in any model of powder compaction: (1) the change in the number of particle contacta (coordination number = Z) as densification proceeds; (2) the Journal of Pharmaceutical Sciences I 147 Vol. 79, No.2, February 1990
condition for local yielding at individual contacts; (3) the average area of each contact as a function of the particle geometry and relative density of the compact; and (4) the relationship between the far-field stresses and the local stresses a t each contact. The role that each of these factors plays in the models of tablet compaction is described below for ductile materials and in a companion paper for brittle materials. Expression for Z-Several relationships between relative density and coordination number have been described in the literature, ranging from lists of relative densities associated with various perfect packing of spheres to descriptions of random packings of spheres.6.7.8The relationship proposed by Ant6 and which was used throughout the present study is expressed by the following set of equations:
p 2 0.85
(2b)
where p is the relative density and Z is the coordination number. These equations were derived from the packing geometry associated with a random packing of spheres; since the packing of particles in a powder bed is random, the A n t 6 relationship is preferred. While the coordination number relationships given by A n t are strictly correct only for relative densities h 0.64, Figure 2 indicates that the actual values which result from back-extrapolation using eq 2 are similar to the values proposed by Ridgway and Tarbuckle.7 Therefore, for the sake of simplicity, the relationship given in eq 2a was used for relative densities less than 0.64. Although many pharmaceutical compacts are made from angular particles, the particles chosen for this study possessed shape factors which differed from that of a sphere by two or less. (Shape factors are used to indicate the degree to which the shape of a particle departs from sphericity, and are defined in terms of factors such as relative surface area and relative length.) Furthermore, after a little compaction, any angular asperities will become rounded by plastic deformation and the particle shape will approximate a sphere. The distribution function was derived for a bed of equalsized spheres. Large deviations from this condition will give rise to other relationships. To avoid this possibility it is necessary experimentally to limit the range of the particle size distribution (e.g., by sieving).
7
12 L
$
11:
E,
10.
C
c
0 .c m .-c E 0 0 0
/
I
’
9.
8.
7. 6. 5.
4.
31 .2
. , . , . , . . . , . . . . . . .3
.4
.5
.6
.7
.a
.9
uc=
H
=
FJA
(3)
where a, is the interparticulate contact stress, H is the hardness of the crystal, F, is the contact force, and A is the contact area. For angular particles, this relationship should be modified by a geometric factor that accounts for the particular geometry of the contact. However, numerous studies”12 have shown that provided the angle of the cone of the indenter is ?goo, the correction factor is less than 8%and eq 3 should give an adequate representation of the limiting area of a given contact under hydrostatic loading. However, the non-hydrostatic loading conditions in a closed die lead to substantial shear stresses a t the contact points, enhancing the densification. Schwartz and Holland13 found that the slope of the Heckel plot for iron (l/mean yield value) derived from isostatic compaction was 4 . 9 ~ 1 0 -MPa-l, ~ while those quoted by Heckel for iron powders undergoing uniaxial compaction, ranged from l.l.10-7 to 1.4-10-7 MPa-’. While they attributed this effect to the presence of fines, it is more likely that shear, while not causing densification, facilitates it. Using a die which permitted the direct application of shear stresses, Hardman and Lilley14 found that shear facilitated the densification of sodium chloride. While the contact area in the absence of shear stresses will be governed by the hardness, any tangential movement will enlarge the area of contact if the material is ductile.16 Johnson16 explored the effect of combined shear and normal pressure when a plastic wedge is deformed by a flat, rigid die. This geometry may be used to represent asperity contact during compaction,and provides a simple means of comparing the modes of deformation in the two limiting conditions. Johnson16 found that the contact area increased from a value of FJH if no shear stress is present to a value of 5.2FJH when the shear stress reaches the same value as the normal stress. In the latter case the area of contact is determined by the shear strength of the material. The problem now is to relate these values to the behavior of an average contact in closed-die pressing. The following approach was used to calculate the proportionality between the mean contact stress in the die and the hardness of the material. For closed-die pressing, the punch to die-wall stress ratio is typically observed to be 2.5:l for pharmaceutical materia1~.1~-~0 We assume that the three principal stresses are then crx, 0.4 u.,,and 0.4 a, and that they are aligned parallel and perpendicular to the loading direction. For contacts aligned parallel to principal stresses the shear stress is zero and the contact area would be given by FJH. For orientations where the shear stress is greater than or equal to the normal stress, the contact area is given by 5.2FJH. Between these two limiting cases, the proportionality factor (linking the contact area and the hardness) was assumed to vary linearly from 1 to 5.2, as discussed by Johnson. When these factors are averaged over the range of possible orientations of interparticulate contacts, we find that the proportionality factor is 3.421 and eq 3 should be replaced by:
1
Relative Density Flgure 2-Relationships between coordination number and relative density. That proposed by Ridgway (1) closely approximates the relationship describing a progression of perfect packing configurations of spheres while that proposed by Tarbuckle and Arzt (2) arises from the packing distributionfunction of a random close-packedarray of spheres. 148 I Journal of Pharmaceutical Sciences Vol. 79, No. 2, Februaty 1990
Local Yielding-The problem of local yielding a t particleparticle contacts is closely related to the hardness indentation test or the mutual indentation of two crystals. For hydrostatic loading, it is usually assumed that the limiting contact area is given by an expression of the type:
a, = W3.4 = FJA
(4)
This equation applies for uniaxial compaction with die wall stresses equal to 0.4 times the punch stress. The following argument suggests that a proportionality factor of 3.4 is about right. Using the ratio between the
hydrostatic (normal) stress and deviatoric (shear) stress within the die, the yield criterion of Green15 predicts that yielding should commence when the ratio between the mean normal stress and the shear strength of the material is approximately 3.2. If it is assumed that the hydrostatic part of the total stress causes consolidation, these values correspond closely to the ratio which may be calculated from the experimental results reported by Schwartz and Holland13 for ion (-3.9),as discussed above. Two additional factors make the hypothesis that shear strain facilitates consolidation attractive. It provides a means of explaining the effect of lubricants and glidants on the densification process. These materials facilitate densification by decreasing the interparticulate shear stress, but weaken the resultant compact. Since very thin layers of lubricants are present on the surfaces of the crystals, they would play a minor role if only normal stresses caused densification. It also provides a mechanism of consolidation which should be more rate dependent than indentation, since shear at the interface is inhibited by the roughness of the contacting surfaces. Friction is especially rate dependent in the presence of surface contaminants.22 Average Area of ParticlelParticle Contact as a Function of Relative Density-The crystals used in this study possessed relatively compact habits, and it is assumed that their behavior resembles that of spheres. As noted earlier, while the asperities of the particles are initially angular, small amounts of deformation will tend to round them so that on average, the contacts will deform as sections of spheres. A n t 6 developed an elegant model to describe the densification of spherical powders based on the radial distribution function described above. A spherical particle with an initial radius of unity (R;Figure 3) is assumed to grow concentrically within a polyhedral cell (Voronoi ce1l29, the dimensions of which are defined by the packing. The cell is constructed from the bisectors of the lines which join the particle being considered and its nearest neighbors. If the particle contacts a side of its cell, this indicates that the particle is contacting its neighbor in that direction. When the particle contacts and Figure 3), the volume of overlaps a face of the polyhedron (R’; the material removed by the contact (Vex) is assumed to be
distributed evenly across the remaining free surface (S,) of the particle which increases the particle radius further (R). Using a geometrical argument based on the radial distribution function, A n t derived the following relationships which provide an expression for the average contact area in terms of the relative density:
+ (15.5.~/12).(R‘ - 1)3.(3.R’+ 1)
(5c)
+ 15.5 + R2.15.5.(2-R-3)]
(5e)
where pis the present relative density, p I is the initial relative density, Sf is the free surface remaining on sphere, R is the present radius of sphere with deposited material, R’ is the enlarged radius of sphere before material deposition, Vex is the volume of material cut off by face of Voronoi cell, Zi is the initial coordination number, Z is the present coordination number, and A is the average contact area. The constant 15.5 is derived from the radial distribution function. Local Stresses at Particle Contacts as a Function of Far-Field Stresses-Relationship between Uniaxial Compaction Stress and the Mean Stress Which Causes Densification-Die compaction differs from isostatic pressing since the load is applied from one direction only. It is assumed that densification is caused by the hydrostatic component of the stress within the die, which is smaller than the punch stress, while the shear component of the stress causes changes in shape and helps to enlarge the contact areas.24 The hydrostatic component is the mean of the principal stresses within the die. This calculation is not trivial, since the effects of friction and the discontinuous nature of the powder compact will cause rotation of the direction of the principal stresses with increasing depth and radius within the compact, and in fact will probably change from particle to particle. Therefore, to simplify the problem, it was assumed that the first principal direction is parallel to the axis of the die; the others are perpendicular to this, but otherwise are arbitrarily directed (uy= uz). The ratio’of the die-wall stress (a to the punch stress (ux)has been measured for sodium ch?oride17-19 and sucrose,~7~~9~20 and appears to be essentially constant (-0.4) for uwvalues between 0 and 197 MPa, which is greater than the range of stre_ssesemployed in the present experiments. The mean stress (u), assumed to be equal to the hydrostatic component of stress, is then:
-
u = (ux+ cry i- uz)/3
Flgure &View of the densification model. The central particle is assumed to possess the initial radius (R) and is located within a polygonal cell (Voronoi cell). During densification the patticle is assumed to grow concentrically within its cell and impinge upon other nearby particles. Material which overlaps the boundaries of the Voronoi cell is redistributed across the remaining free surface of the particle.
(6)
where ax is the punch stress and uy and uzare the die wall stresses. Rezationshipbetween the Mean Stress and the Averwe Force at One Cont~t-Equation 7, Proposed by M o l e r ~ s expressed ,~~ the relationship between the loads at the contacts between spherical particles and the relative density of a powder mass: Journal of Pharmaceutical Sciences I 149 Vol. 79, No. 2, February 1990
(7) where o is the hydrostatic external stress (-3, F, is the contact force, Z is the coordination number, and r is the particle radius. A n t 5 and Swinkels et a1.6 used this relationship to complete their models of densification. This relationship assumes that the particles are spherical and that the load is transmitted through point contacts. This assumption is not generally true during compaction. Therefore, the following is proposed to replace eq 7. Since the hydrostatic stress gives rise to densification, it is assumed that the mean compaction stress is distributed over the surface of a sphere. The average relative density of the contents of the sphere is equal to the present relative density of the compact. It is assumed that the size of the Voronoi cell (or equivalently, the reference sphere over which the external stress is distributed) is constant, while the particle within the reference sphere grows. Then the surface area of the reference sphere also remains constant and may be calculated from the initial relative density as follows: fi =
SA
=
v,/v, 4.rr.r:
where pi is the initial relative density, V, is the particle volume, V, is the volume of reference cell, rr is the radius of reference cell, and SA is the surface area of the external reference. The local contact stress is then related to the far-field stresses as follows:
F,
=
r,*A
where is given by eq 6, Z is defined by eq 2, and A, the area of contact, can be found from the relationship presented in eq 5. Application of the Model-The model is applied as follows: (1)the initial relative density is that of the powder before consolidation begins; (2)a set of increasing relative densities is prescribed; (3) the coordination number for each relative density is calculated using eq 2; (4) the contact area per asperity is calculated as a function of the relative density using eq 5; (5) the total force per particle is calculated using eq 4; (6) the hydrostatic or mean stress is calculated using eq 9; and (7) the punch stress is calculated from eq 6. In summary, this procedure calculates the punch stress a t each prescribed relative density during uniaxial compaction from the mechanical properties of the crystals being considered.
Experimental Section Sodium chloride was chosen as the model “ductile” material for study. The sodium chloride employed (Fisher, lot 740662A, density = 2.165 glmL, Mellor, 19643was recrystallized from water according to the method described by Duncan-Hewitt and Grant.26 The cubic crystals thus obtained were sieved, and the fractions ranging from 500-710 pm were collected for testing. Twenty crystals were prepared and indented (1 or 2 indentations per crystal) at room temperature with a load of 147 mN using a Leitz-Wetzlar Miniload hardness tester which was equipped with a Vickers dlamond indenter as described by Duncan-Hewitt and Weatherly.3 The indentation time was 10 s. For the compaction studies a 1.3cm diameter die was brushed with a slurry of 10% magnesium stearate in ethanol and allowed to dry for 150 I Journal of Pharmaceutical Sciences Vol. 79, No. 2, February 7990
5 min. Approximately 3.0 g of the crystals was weigned accuretelv using a Mettler balance (model 1-911) and poured into the die. A Euny dense tablet would then possess a thickness of 1am. The die was then clamped onto the table of a n electronically controlled shaker (Janke and Kunkel IKA-Vibrax-VXR) and vibrated a t 35 Hz for 1min. The top flat faced punch was inserted into the die and the assembly was shaken for an additional 1 min to minimize the variability of the density distribution in the green compact. The punch was guided by a face plate which was equipped with a rubber ring. Complete insertion of the upper punch required pushing the face plate to meet the top of the die. This resulted in precompression loads ranging from 50-200 N, depending upon the amount of lubricant which penetrated under the rubber ring. As a result, the initial relative density of the bed of crystals ranged from 0.55 to 0.58. Compacts were prepared using a n Instron Universal Testing Apparatus (Model 4206). The assembled die was placed on the lower platform of the Instron, and the crosshead was lowered until a compressive force of 0.01 N was measured. The initial position of the crosshead relative to the bottom of the die cavity was then recorded and the initial relative density of the bed of crystals within the die was calculated. The crosshead was lowered a t controlled rates of 1,10, and 50 m d m i n and the maximum compressive force (up to 25 kN) was recorded. The force required to overcome the friction exerted by the rubber ring, estimated as the apparent yield value at the beginning of the test, was subtracted from the value of the punch force. After compaction, the tablets were removed from the die, and their dimensions were measured using a micrometer. These values were employed to calculate the relative density of the compacts. This calculation is based on the assumption that the compacts possess a uniform density distribution; the validity of the assumption will be discussed below. Some compacts were cut into halves either vertically or horizontally. The surfaces of these pieces were ground and polished to produce a relatively smooth finish and examined microscopically (Leitz Microscope).
Results and Discussion Hardness of Sodium Chloride Crystals-The sodium chloride crystals were difficult to test, as the crystals sometimes adhered to the indenter, invalidating the results of that experiment. Chipping was not observed. The Vickers Hardness Number (H) was calculated from the mean diagonal length as follows:3
H
=
loadlarea of contact = 1.854.P/d2
(10)
where P is the load and d is the mean length of the indentation diagonal. The hardness of the sodium chloride crystals used in this study was found to be 213 2 50 MPa (95% confidence interval, n = 15). Appearance of Compacts-When the sodium chloride tablets were examined microscopically,the crystals exhibited very little fracture. This observation has also been reported in the literature.27 A t high relative densities, the crystals were usually oriented in columns. These columns tended to follow the contours of stress which are interpolated from the density distribution in a tablet produced by uniaxial compression.= The crystals retained their essentially cubical shape, but the contact areas were rounded. Compaction Stress versus Relative Density Relations h i p T h e effect of strain rate on the compaction of the sodium chloride compacts is shown in Figure 4. A t a compaction rate of 1 m d m i n , the experimental data appear to agree with those reported by Hersey et al.27 for crystals in the 40-80 mesh size range compressed a t the same rate in an unlubricated die (diameter = 33 mm) using an Instron Universal testing apparatus. A t any given value of compaction stress, the compacts were less dense a t a higher rate of compaction (Figure 4). Fit of Model to Experimental Results-The average ratio of punch stress to die wall stress was taken to be 0.4, the
-. P
1.70
,
1.50
I
m o
v-
D'
1.30
-E
1.10
g
0.90
$ 1 I
=
m o
c
i
'
m o
9
0.70) 0
a1 mm/min 0 1 0 mmlmin
m50 mmlmin '
' 10
'
' 20
.
'
"
.
40
30
'
'
'
60
50
Punch Stress, MPa
Flgure +Heckel plots for sodium chloride tablets compacted at punch displacement rates of 1, 10, and 50 mm/min.
average value found in the literature.17-19 Using this value, and the hardness value of 213 MPa, the model predicted the densification of sodium chloride at a compaction rate of 1 m d m i n (Figure 5). This conclusion is justified on the basis that the predicted and experimental curves coincide within the range of the experimental variability associated with the hardness values and the compaction data. Figure 5 also indicates that accuracy of the hardness is crucial to the agreement between theory and experiment. Effect of Rate of Compaction-The plastic yield stress is a function of the strain-rate; therefore the use of hardness values, or values calculated from them, is strictly valid only if the contact times during hardness testing and compaction are comparable. As stated above, a hardness value of 213 MPa could predict densification only for crystals compacted at a rate of 1m d m i n . At higher compaction rates, larger stresses were required to produce tablets of a given relative density. Assuming that the increased strain-rate causes the apparent interparticulate contact stress to increase concurrently, the following approach was employed to estimate the contact stress at higher compaction rates. Verrall et 81.29 constructed a deformation mechanism map for sodium chloride, which shows that the hardness will increase with strain rate. Assuming that increasing strainrate has a similar effect on both the indentation hardness and the resistance to compaction, the contact stresses at 10 and 50 m d m i n were interpolated from the deformation mechanism map, by finding strain rates 10 and 50 times greater than that 2.0 1
corresponding to a hardness value of 213 MPa. The resultant hardnesses were found to be 278 MPa and 335 MPa, respectively. The ability of these values to predict compaction at 10 and 50 m d m i n is shown in Figure 6. It may be concluded that the predictive capability of the model is improved if the effect of strain-rate is considered in the manner described above. This suggests that indentation creep experiments are wellsuited to the characterization of materials which are to be compacted. It must be emphasized that the foregoing analysis provides only a rough approximation: the correspondence between a compaction rate of 1m d m i n used in the compaction tests and a standard Vickers Hardness Test (i.e., 10 s indentation time) was found empirically. Furthermore, the strain-rate is not constant during compaction: both strain-hardening, and undetermined changes in the contact geometry may cause additional complications which have not been addressed in this study. Other Factors Affecting Compaction-Other variables which are believed to have a predictable effect on the densification behavior of pharmaceutical materials include elasticity, particle size, initial packing of the powder in the die, and the range of stresses employed.SG33 Elastic effects may in part be related to the rate dependence of compaction through the concept of viscoelasticity:34 as the rate of compaction increases, the amount of reversible (elastic) deformation increases relative to the amount of irreversible (plastic) deformation. In this study, the complications that the consideration of elastic deformation entails were avoided by relating the compaction stress to the density of the compacts once they have been removed from the die (i.e,,after elastic recovery has occurred). The consequence of this approximation is important: tablets of some materials are known to fail by capping and laminating, either during decompression or as the tablet is ejected from the die. Neglecting elasticity effects will compromise the ability of the models to predict the strength of such compacts. In future studies it will be essential to address the elastic and plastic properties of individual crystals in much more detail since the amount of stored elastic strain determines the stress available to break the bonds which are formed while the compact is still under pressure.35 There appears to be considerable disagreement in the literature about whether an increase in particle size facilitates densification or whether it renders densification more difficult, and opposite trends have been reported.3" The effect of particle size is obviously complex, and may arise from size-dependent differences in particle shape, size distribution,
1.8
I
10 mm/mln, pledictcd using H = 278 MPa
130 125
(expt'l. if H = 150 MPa) 120
0.8
0a
experimental data, H = 213 MPa 0.1
02
03
0.4
-
1.15 05
0.6
1.10
normalized mean compaction stress, 0 I Y
Flgure %Fit of the densification model to the experimental Heckel plot for sodium chloride compacted at a rate of 1 mmimin. The effect of
varying the hardnesgvaluesused in the model is also shown. The mean compaction stress (u), the average of the punch and die-wall stresses, is normalized by dividing by the plastic yield stress characteristic of the material. Because the punch stress value is normalized, the position of the experimental curve shifts as the hardness changes.
/
.a
1
.21
.25
M mm/rnin, predicted using H = 33.5 MPa I
.29
.
,
33
.
,
.
normalized mean compaction stress,
,
41
37
.
,
.45
IY
Flgure &Predicting the densification curves for sodium chloride cornpacted at punch displacement rates of 10 and 50 mm/min. The hardness values are derived by considering the effect of strain rate on plastic
yielding.
Journal of Pharmaceutical Sciences I 151 Vol. 79, No. 2,February 7990
or mechanical properties or the effect may arise from differences in particle packing in the die. It was felt that the effects of particle size could be ignored in this study because the hardness tests were performed on the same crystals which were employed in the compaction studies. Extrapolation of the results of this study to much smaller particle such as occur in pharmaceutical powders is not a priorz justified, and requires careful theoretical consideration. Fedion between the contents of the die and the die wall causes the effective compaction stress to vary through the powder bed, and as a result, the density of the compact is not homogeneous. The relative effects ofdie-wall friction are greater the smaller the tablet; therefore, the pressure required to produce a tablet of a given relative density increases concurrently.36 Application to Other Materials+ would be useful to determine whether this mddel predicts the behavior of materials with mechanical properties similar to those of sodium chloride. Heckel plots for potassium chloride are available in the literature (particle size fraction 60-80 mesh).*7 The literature provided hardness values ranging from 92 MPa37 to 220 MPa.38 A hardness of 125 MPa, which is close to the median value, fits the data best (Figure 7). It is remarkable that the range of hardness values reported in the literature is quite large for both sodium chlonde (150-314 MPa) and potassium chloride (92-220 MPa). Brace39 observed that the hardness value of sodium chloride may be doubled if work-hardening has occurred. Rapid crystallization may also give rise to a large number of defects in the crystal, with a concurrent increase in hardness. Crystals which are employed to form compacts are not annealed, and handling procedures probably introduce some surface defects. It appears reasonable, therefore, that in the absepce of experimental data, the median hardness ought to predict the densification behavior reasonably well, as observed for potassium chloride.
Conclusions A model has been developed which predicts the densification behavior of sodium chloride during uniaxial compaction to within experimental error. The effects of compaction rate were considered explicitly and other factors which affect compaction were discussed briefly. Further development of the model should consider the effects of elastic strain, which may cause tablet failure due to capping or lamination. Ultimately, a model is required which can predict the strength of pharmaceutical compacts from the mechanical (expt'l. if H = 220 MPa) \
0.0
01
0.2
/
03
0.4
normalized mean compaction stress, / Y
05
0.6
0.7
(T
Flgure 7-Fit of the densification model to the experimental Heckel plot for potassium chloride compacted at a rate of 1 mmlmin. The effect of varying the hardnessvaluesused in the model IS also shown. The mean compaction stress (o),the average of the punch and die-wall stresses, is normalized by dividing by the plastic yield stress characteristic of the material. Because the punch stress value is normalized, the position of the experimental curve shifts as the hardness changes. 152 I Journal of Pharmaceutical Sciences Vol. 79, No. 2, February 7990
properties of their constituent crystals. This model must account for the ability of the material to densify (asthe present model does) since the size of the interparticulate contact areas is a maor determinant of the compact strength. However, many other factors affect strength as well. For example, (1)the particle brittleness will determine the ability of cracks and porosity to concentrate stress which can give rise to premature fracture; (2) contaminants can' alter the interparticulate bond strength; and (3) residual stresses will decrease the apparent fracture strength. It is hoped that the simple densification model described in this manuscript will provide the impetus for the development of more comprehensive models in the future.
References and Notes 1. Kawakita, K.; Ludde, K.-H. Powder Technol. 1970,4, 61. 2. Mathia, T. G.; Lamy, B. Wear 1986, 108,385. 3. Duncan-Hewitt, W. C.; Weatherly, G. J . Mat. Sci. Letters, in press.
4. Fischmeister, H. F.; Arzt, E. Powder Metall. 1983, 26, 82. 5. Artz, E. Acta Metall. 1982, 30, 1883. 6. Swinkels, F. B.; Wilkinson, D. S.; Ant, E.; Ashby, M. F. Actu Metall. 1983,31, 1829. 7. Rdigway, K.; Tarbuckle, K. J. Brit. Chem. Eng. 1967, 12, 384. 8. Scott, G. D.; Kilgour, D. M. Br. J . Appl. Phys. 1969,2,863. 9. Shaw, M. C. In The Science of Hardness Testin and Its Research A p lications; Westbrook, J. H., Conrad, H., fids.; A.S.M. Publisling: Metals Parks, OH, 1973; p 1. 10. Beeuwkes. R. Jr.: Chait. R.: Lin. H. H. In The Science ofHardmss Testmg and Its Research Applications; Westbrook, J. k; Conrad, H., Eds.; A.S.M. Publishing: Metals Parks, OH; 1973; p 75. 11. Atkins, A. G. In The Science of Hardness Testing and Its Research Applications; Westbrook, J. H.; Conrad, H., Eds.; A.S.M. Publishing: Metals Parks, OH, 1973; p 223. 12. O'Neill, H. Hardness Measurements ofMetals and Alloys, 2nd ed.; Chapman and Hall: London, 1967; p 17. 13. Schwartz,E. G.; Holland,A. R. Int. J . Powder Metall. 1969,5,79. 14. Hardman, J. S.; Lilley, B. A. In Proc. 1st Intl. C o n . on the Compaction and Consolidation of Particulate Matter, h r ' hton, 3 6 October, 1972, Powder Technol. Publ. Series, #4, GoBberg. A. S., Ed., 1972, p 115. 15. Green, A. P. J . Mech. Phys. Solids, 1954,2, 197. 16. Johnson, K. L. Contact Mechanics, Cambridge University Press: Cambridge, 1985; p 171. 17. Leigh, S.; Carless,J. E.; Burt, B. W. J . Pharm. Sci. 1967,56,888. 18. Ridgway, K.; Rupp, R. J . Pharm. Pharmacol. 1969,21,30S. 19. Obiorah, B. A. Int. J . Pharm. 1978, I , 249. 20. Summers, M. P.; Enever, R. P.; Carless, J. E. J . Pharm. Pharmacol. 1976,28, 89. 21. Duncan-Hewitt, W. C. Ph.D. Thesis; University of Toronto, Toronto, Canada, 1988. 22. McClintock, F. A.; Argon, A. S. Mechanical Behuviour of Materials, Addison-Wesley: Reading, MA, 1965, p 276. 23. Voronoi, G . F. J . Reine Angew. Math. 1908, 134, 198. 24. Venkatachari, K. R.; Raj, R. J . A m . Ceram. SOC.1986, 69,499. 25. Molerus, 0. Powder Technol. 1975, 12, 259. 26. Duncan-Hewitt, W. C.; Grant, D. J. W. Powder Technol. 1987, 52, 17. 27. Hersey,J. A.; Cole, E. T.; Rees, J. E. In Proc. 1st Zntl. Conf. on the Corn action and Consolidation of Particulate Matter, Brighton, 3 4 &t., 1972, Powder Technol. Publ. Series, #4, Goldberg, A. S., Ed., 1972; p 165. 28. Train, D. J . Pharm. Pharmacol. 1956,8, 745. 29. Verrall, R. A,; Fields, R. J.; Ashby, M. F. J . A m . Ceram. Soc. 1977, 60, 211. 30. Fell, J. T.; Newton, J. M. J . Pharm. Sci. 1971, 60, 1866. 31. York, P. J . Pharm. Pharmacol. 1979,31, 244. 32. Rue, P. J.; Rees, J. E. J . Pharm. Pharmacol. 1978,30, 642. 33. Huffine, C. L.; Bonilla, A. I. Chem. Eng. J . 1962, 8, 490. 34. Rippie, E. G.; Danielson, D. W. J . Pharm. Sci. 1981, 70, 476. 35. Hiestand, E. N.; Smith, D. Powder Technol. 1984,38, 145. 36. Lammens, R. F.; Liem, T. B.; Polderman, J.; de Blaey, C. J. Powder Technol. 1980,26, 169. 37. Subramanian, B.; Bansigir, K. G. J . Muter. Sci. 1980, 15, 2889. 38. Gilman,J. J. In TheScience o Hardness Testin andlts Research Applications; Westbrook, J. .; Conrad, H., Efs., ASM Publishing: Metals Parks, OH, 1973, p 51. 39. Brace, W. F. J . Geophy. Res. 1960, 65, 1773.
6