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Deformation Kinetics of Potassium Bromide Crystals Predict Tablet Stress Relaxation
Deformation Kinetics of Potassium Bromide Crystals Predict Tablet Stress Relaxation W. C. DUNCAN-HEWITT’ AND E. A. PAPADIMITROPOULOS~ Received November 24, 1992. from the *Faculty of Pharmacy, Universw of Toronto, 19 Russell Street, Toronto, Ontario, Accepted M5S 1A 1, Canada, and #The R. W. Johnson Pharmaceutical Research Institute, Toronto, Ontario, M3C 1L9,Canada. for publication May 13, 1993’. Abstract 0 A model relating the interparticulate contact stress within a tablet matrix with the compaction stress was developed previously to permit the nonlinear deformation kinetic analysis of the viscoelastic behavior of pharmaceutical tablets with the known properties of the
tablet constituents. The present research was undertaken to determine whether the inverse operation (is., using tablet stress relaxation to determine singlecrystal properties)was possible. The stressrelaxation of potassium bromide (KBr) compacts was evaluated as a function of temperature and relative density,and an attempt was made to calculate the deformation kinetic parameters. The stress relaxation of KBr did not fit the model under ambient conditions for two reasons: ( 7) KBr has two slip systems with approximately the same shear stress at room temperature: and (2)KBr strain-hardens. When these com-
plications were taken into consideration,the stressrelaxation behavior could be explained. Therefore, whereas single crystal tests are capable of yielding parameters that can be used to predict compact behavior, the inverse process of quantifying fundamental material parameters from compact behavioris problematicdue to the difficulty of determining, a priori, all the processes that operate simultaneously. Tablet formulation would be greatlyfacilitiated if the response of the formulation during compaction and the strength of the resultant tablet could be predicted. Two approaches currently exist. Traditionally, attempts to elucidate the compaction process seek to discover relationships between compaction parameters and tablet failure, treating the die contents rather like a “black box”. This approach attempts to circumvent the difficultiesassociated with measuringthe mechanical properties of the very small, brittle, single particles that comprise many pharmaceutical compacts. Alternatively, predictions can be derived from knowledge of the properties of the tablet components if the relevant material properties can be identified, measured, and related modelistically to tablet behavior.1.2 The production of a coherent tablet from a particulate mass requires the formation of many strong interparticulate bonds. Practically, this is achieved by applying a compaction stress that is intended to decrease porosity (thus increasingthe number of particle-particle contacts) and, through plastic deformation, to produce relatively large areas of interparticle contact where adhesion can occur. Unfortunately, the applied stress also can give rise to elastic strain that, when released, can cause tablet failure. Thus, the ideal pharmaceutical tablet material should be ductile and exhibit little elasticity. This requirement is complicated by the fact that all materials respond to stresses in a time-dependent manner: they are viscoelastic. Whether a particular material is considered to be ductile or elastic (or hard or brittle) depends on the rate of load application and its relationship with the rate at which the various deformation mechanisms are able to respond. Quantitation of the “viscoelasticity”of a pharmaceutical substance is prerequisite to any modelistic approach to the understanding of tablet compaction. Linear viscoelasticmodeling of tablet responseto compaction stress and stress relaxation without reference to the properties Abstract published in Advance ACS Abstracts, November 15,1993.
0 1994, American Chemical Sock?@and American Pharmaceutical Association
of the constituent particles has had limited success: simple, satisfactory models have not been found.3 It is true that one can fit experimental data by using a large number of viscoelastic parameters, but their physical meaning and their statistical significanceare questionable. Furthermore, tablets of different porosities appear to possess different viscoelastic parameters, which is unreasonable if one can assume that the material being compacted does not change significantlyduring the compaction process itself. Deformationkinetic analysis,which postdata that irreversible deformation or plastic flow of a solid is controlled by thermally activated phenomena, is an alternate means to study viscoelasticity. Furthermore, deformation kinetic analysis can be incorporated into a modelistic approach to the understanding of tablet compaction from the properties of single particles and has been used successfully in many engineering and a number of pharmaceutical application^.^*^ In the latter case, the temperature- and porosity-dependent stress relaxation of sodium chloride (NaC1) tablets was correlated with the microscopic deformation mechanism (characterized by a defined activation energy) with two intervening models to link each with the crystal hardness, a parameter that can be measured easily with a microindentation tester.’ These models will be used to evaluate the experimental behavior of potassium bromide (KBr). One might hope that knowledge of a model that successfully predicts the behavior of a number of materials might provide the means whereby single particle characterization could be avoided. For example, it is possible that stress relaxation experimentsalone might provide deformationkineticparameters that could be used to predict the kinetics of tablet compaction. In essence,this means that the contents of the “black box”would be defined. The research described below explores this possibility. Because the application of deformation kinetic analysis had been validated with NaCl compacts, we extended this approach to evaluate the stress relaxation of KBr compacts. KBr was chosen as an alternate material because although its flow and compactionbehavior is as well characterized as that of NaC1, its flow behavior is unique in that it is markedly softer and compacts much more readily than NaC1.
Experimental Section Crystalline KBr (Fisher Scientific, P205, lot 895777) with a Vickers Hardness of 88 MPa (10-sindentation time at 25 “C)was used as received from the manufacturer. Compacts were made at temperatures rauging from -45 to 100 “C. Prior to each compression, the cylindrical die and flat-faced punches (13-mm diameter) were cleanedwithmethanol,dried, and then filled with 1.5g of KBr. An INSTRON (model 4201) stressstrain analyzer,which was equippedwith Series IX comprebion software, was electronically calibrated prior to each compression and used to manufacture compacts of varying relative density. Compacta of f i i relative densities ranging from 0.77 to 0.89 were compressed at a punch displacement rate of 1 mm/min. At maximum punch travel (at loads of 2.5,3.5, or 4.5 kN),the movement of the cross-head was arrested,and stress relaxation was monitored for 2 min. At low temperatures (i.e., from -45 to 0 “C), stress relaxation was studied with a differential
0022-3549/94/1200-9 1$04.50/0
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Jownal of Pharmaceutical Sciences / 91 Vol. 83, No. 1, &nmry 1994
Table 1-Summary sample
kn
F-,
KBr2 KBr3 KBr4 KBr5 KBr6 KBr7
ol Deformatlon Klnetlc Data at 298 K
2.5 2.5 3.5 3.5 4.5 4.5
Weight, g
Thlckness, mm
1.4957 1.461 1.497 1.4985 1.4935 1.4943
5.35 5.18 5.02
Density, ~ c m - ~Relative Denslty 2.105 2.124 2.246 2.235 2.324 2.436
5.05
4.84 4.62
r2
V.a, b9
0.988 0.983 0.989 0.987 0.986 0.983
58 59 59 63 66 61
Equatlon
0.766 0.772 0.817 0.813 0.845 0.886
y y y y y y
= 1.415~ - 25.331 = 1.447~- 25.844
= 1.452~- 26.085 = 1 . 5 3 1~ 27.363 = 1.614~- 28.806 = 1.489~- 26.72
4.73
A
16.5.
A
16.25.
m
3
16. 15.75-
+
;15.5.
2 15.25. 15 14.75. 14.5.
14.254
0
2L 0
20
40
60
80
100
120
time, s
Flguro 1-Punch
force versus time plots for the stress relaxation of KBr compacts compressed with maximurn loads (F,) of (0)2.5, (0)3.5,and (A)4.5 kN. The rate of stress relaxation varies with compaction force and tablet relative denslty. scanning calorimeter cooling unit. This unit employs dry nitrogen to avoid condensation problems. Cornpacts of KBr were compressed at room temperature a t varying loads, and, once the target load was reached, compaction was terminated by lifting of the cross-head. These compacts were then placed in the hot-air oven (200 "C) and were permitted to anneal for 2 days. Subsequently, the cornpacts were removed from the oven and placed in the environmental chamber at room temperture.8 The fan in the chamber was used to promote air circulation and more rapid and reproducible cooling. The stress relaxation of these compacts waa then monitored at 278 and 348 K. Hardness values were obtained from Frost and Aahby.9
Results Stress Relaxation of Compacts of As-Received Material-The data collected from the stress relaxation studies (Table I) were used to calculate the activation volume ( Vaa) with eq 1 (see Discussion; (*V,,, = 61 b3, SEM = 0.044,p = O.OOO1)). Literature values of V,, for this material are not available; however, the order of magnitude is indicative of a dislocation (Peierls-Nabarro) mechanism, which is appropriate for this material at these high s t r e ~ s e s . ~ Figure 1 illustrates the average force versus time plots and Figure 2 illustrates the same data normalized by the procedures described above for the tablet model plotted as a function of In (time) (Figure 2a) or according to eq 3 (Figure 2b). Had the simple model (1,rate-controlling deformation mechanism) been adequate, one would have expected the procedure to give rise to collinear plots. In fact, not only are the plots not collinear, the individual plots are curved which is also contrary to expectation. The activation energy (Eaa) of KBr was not 92 /Journal of phermaceutlcal Sciences Vol. 83, No. 1. January 1994
.
.
.5
0
1.5 2 2.5 In (time, s)
1
-I
3.5
4
4.5
5
0
-1.5
._I.
3
0
.
14.25 14.5 14175
I5
15125. 15.5 -15175
I 6 -16:25 16.5 .16:75
17
shear stress, MPa
Flgure 2 3 A , top)Shear stress versus In (tlme) plots for thestress relaxation of KBr compacts compressed with maximum loads (F,) of (0)2.5, (0) 3.5,and (A)4.5 kN. Shear stress Is calculated with eq 2 and the punch force and the hardness values for single crystals. The plots are derlved from the integral of eq 1 and are expected to be linear if one mechanlsm controls deformatlon. (B, bottom) Plots of in (shear stress rate) versus shear stress for the stress relaxation of KBr compacts compressed with maximum loads (F,) of (0)2.5, (0)3.5,and (A)4.5 kN. Derived from eq 1, the plots are expected to be linear If one mechanlsm controls deformation.
calculated from a plot of In i (stress rate) versus 1/K due to the curvilinearity of this plot under the experimental conditions. Stress Relaxation of Annealed Compacts-The data collected from the stress relaxation experiments performed at 298 and 348 K were plotted as In i versus mean shear stress ( 7 ) . The results (Table 2) indicate that the slope and intercepts of the curves have changed with respect to those of the unannealed materials (Table 1j. However, the annealing process caused the stress relaxation curves to become superimposable (Figure 3). Low Temperature Data-The Vaafor the deformation of KBr compacts a t subambient temperatures, calculated from the
of Deformallon Klnellc Data at 298 and 348 K for Annealed KBr
Table 2-Summary
F-,
Samples
kn
Relative Density
Equation
Temperature, K
r2
0.768 0.821 0.868 0.803 0.852 0.883
y = 5 . 5 7 5 ~- 93.317 y = 3 . 9 1 -~ 66.554 y = 4 . 3 4 8 ~- 73.507 y = 3 . 4 9 7 ~- 46.1 13 y = 4 . 8 2 4 ~- 62.499 y = 4.119X - 53.839
298 298 298 348 348 348
0.971 0.981 0.997 0.981 0.986 0.987
2.5 3.5 4.5 2.5 3.5 4.5
KBr33/348&39/40 KBr35/36&43/44 KBr37/38&45/46 KBr73/74&79/80 KBr75/76&81182 KBr77f78&83/84
Model 1: Halsey-Eyring Nonlinear Three-Element Model4-It is assumed that at high levels of stress, one deformation mechanism predominates. That deformationmechanism is controlled by a kinetic activation barrier that is characterized by a defined End (the height of the barrier) and Vact (the physical extent of the barrier). The rate of stress relaxation is related to the strain rate (9) and the parameters that characterize the activation mechanism as shown in eq 1: [- Endl 17 Vad1 i = -Ei. = -EA exp -exp -
-6.5 15.4
0
15.5
15.6
15.7
15.8 15.9 16 shear stress, MPa
16.1
16.2
16.3
16.4
Figure 3-The In (shear stress rate) versus shear stress plots for the stress relaxation of KBr compacts that were annealed after cornpactlon and prlor to stress relaxation testing. The annealing process causes the plotsto become linear and superimposable. Key to F, values: (0) 2.5; ( 0 )3.5; (A)4.5 kN.
B
8 - 4 4 . . 23.4
23.6
.
.
23.8
.
.
24
24.2
24.4
24.6
24.8
25
shear stress, MPa
Flgure 4-The In (shear stress rate) versus shear stress plots for the stress relaxationof KBr compactscompressedat -45 'C. One mechanism appears to control deformation at this temperature. Key to F, values: (0)2.5; ( 0 )3.5; (A)4.5 kN.
plot of In i versus average T, was 112 b3 (SEM = 0.042, p = 0.O001; Figure 4) and is of the order of magnitude indicative of a Peierls-Nabarro mechanism.' Lowering the temperature of compaction and stress relaxation had two effects: (I)the plots became linear, and (2) the plots became superimposed.
Discussion The tablet model is developed based on the assumption that the geometry and physics of the Vickers microindentation configuration and particleparticle contact are similar. The behavior of the single particle and compacts are related by the following two intervening models.
(1) kT kT In eq 1,E is the apparent elasticmodulus,A is a preexponential factor, T is the absolute temperature, and k is the Boltzmann constant. The activation parameters provide some information about the mechanisms that control the deformation behavior under the test conditions. For example, a Vad of the order of unity correspondswith a moleculardeformationmechanism such as diffusion or creep, whereas V,, of the order of 10 to 100 indicates that a dislocation mechanism may be controlling deformation. The nonlinear three-element model gives i as a function of i. and is required because activation theory defines i. values, whereas i (as stress relaxtion) is measured experimentally. Model 2 Tablet Modello-A tablet model is required to link single particle and compact behaviors. A tablet is porous and its stress relaxation is known to vary with porosity. The tablet model provides the means to normalize tablet stress relaxation curvesto make them independent of porosity. Normalizedstress relaxation curvesare expected to be superimposable. The tablet model used previously assumes that: (I)the particles contact each other only at asperities, (2) the interparticle contact area generated during compaction does not change during stress relaxation and (3)the material properties are independent of strain. Under these conditions, the hardness, characterized by microindentation testa, best characterizes the deformation behavior of the material. Because the hardness is measured from the contact area produced by a given load after 10 s, it can be used to calculate the total true interparticle contact area from the measured force on the punch after 10 s of stress relaxation with eq 2:
Fp= HA, In eq 2, Fpis the punch force, H i s the Vickers Hardness, and A , is the true cross-sectional interparticulate contact area. The Mises Yield Criterion (which relates the yield and shear stresses") combined with a hardness constraint factor (which relates the hardness with the yield stress') is used to relate the hardness and the shear stress in eq 3 r = H/3& (3) In eq 3, 7 is the shear stress and H is the Vickers Hardness. However, eq 3 is true only if no strain hardening occurs. Vickers microindentations are geometrically similar; that is, regardless of the load, the contact and deformation geometry remains constant. The result is that under most conditions, the Vickers hardness is independent of load, regardless of whether strain hardening occurs. The particle-particle contact zone is like the Journal of pharmaceutical Sciences / 83 VOL 83, NO. 1, JenUap' 1994
1 ~
~~
. .. .
.
+
* **
.
.
*
~
*
.
. .. . .
25
. .
.
12
*.
. . .. . . .. . *
*.
*.
.*
4
*.
* *
.*
4
*
.*
**
*. *
**. -0
c.
Figwe 5-The deformationof NaCl occurs within distinct dislocationbands (Figure 5a. 1 to 3);hence,the environment in which dislocations move is constant. KBr, however, does not form discrete dislocation bands during deformation (Figure 5B);therefore, the environment changes constantly and the crystals strain-harden.
intersection of spheres and so more closely resembles the Brinell hardnesstest configuration. In this test, the hardness of materials that strain-harden varies as a functionof load because the contact geometryvaries continuously. When no strain-hardening occurs, the Brinell and Vickers hardness values are similar, but deviate from one another when strain-hardeningoccurs. The observation that the normalized slope of the stress relaxation plot is a function of compact force may indicate that strain-hardening is occurring. Komnik et a1.12studied the deformation behavior of a number of halides, including NaCl and KBr. The series of pictures in Figure 5 show how the etched surfaces of these crystals appear after increasing amounts of strain. Each etch pit, represented in Figure 5 by a small diamond, shows the point of emergence of a dislocation line and illustrates the behavior of NaCl (Figure 5a) and KBr (Figure 5b) under uniaxial stress. Figure 5a shows that deformation of NaCl occurs within distinct dislocation bands. Upon application of stress, deformation is localized at the edges of these bands as they grow (Figure 5a to 3); hence, the environment in which dislocations move is constant. This uniformityof the environment is associatedwith minimal strainhardening. For such a material, the stress relaxation curves of contact stress versus In (time) plots should overlap once they are normalized, which was the case for NaC1. However, KBr, does not form discrete dislocationbands during deformation (Figure 5b).12 Upon application of stress, the dislocation motion is not localized but is occurring diffusely throughout the crystal in an environment that is constantly changing as the average concentration of dislocations increases. This results in strain-hardening due to dislocation interactions and could cause the poor overlap of the curves observed in Figure 2. Strain-hardening that occurs during compaction may be removed if the compactsare annealed subsequent to compaction but prior to stress relaxation testing. Keeping in mind that annealing could alter the behavior of the crystals substantially, the compacts were annealed as described above to determine whether this would cause the normalized curves to become 94 /Journal of Pharmaceutical Sciences Vol. 83, No. 1, January 1994
50
100
150
200 T (K)
250
300
350
400
450
500
Flgure 64ritical resolved shear stressfor two slip systems In KBr, plotted as a function of temperature. At room temperature, the stress required to activate both slip systems is similar. Adapted from Klmnik eta/.’*
superimposed. Examining Table 2 one sees that the slope and intercepts of the i versus y curves have changed with respect to those of the unannealed materials probably due to alterations of intrinsic crystal defect concentration during the annealing process (TableI). That the procedurecausedthestress relaxation curves to become superimposed supports the proposition that the original effect arose from strain-hardening. The data calculated for annealed compacts cannot be compared with that for “fresh” compacts because the annealing process changes the intrinsic structure of the crystals. The second observation,that the individualplots are distinctly curvilinear,may be due to the operation of two or more competing mechanisms with similar Eactvalues as shown by Skrotzki and Haasenla at room temperature (Figure 6). If two mechanisms possess similar parameters it would be difficultto extract reliable values from the plot. However, it is has also been shown that one of the slip systems becomes much more resistant to slip a t subambient temperatures. If this is so, then stress relaxation plots to KBr compacts should become linear as the temperature is decreased. For this reason, experiments were performed at temperatures ranging from -5 to -45 “C. In this temperature range, the activation plots did indeed become linear. An Arrhenius plot of the low temperature data yielded an experimental Eactof 2.7 X 10-19 J (SEM = 0.057, p = 0.0033, Figure 7). 55) and KBr, one By comparing the Esetfor NaCl(2.0 X may come to the somewhat paradoxical conclusion that KBr deformation must involve a larger activation barrier than that of NaC1. This is surprising because the latter material is harder. But, one must note that the homologous temperatures (temperatures measured relative to the meltingtemperature) at which these values are measured are different. The “softness”of KBr may simply arise from the fact that two mechanismsare operating at room temperature. The interaction of the two slip systems operatingconcurrentlymay have intensifiedthe strain-hardening effect as has been shown to be true in other cases.’
Conclusions The stress relaxation of a compact is a functionof the relaxation at the interparticulate contact regions,modified by the porosity. The effect of porosity can be accounted for with a tablet model, in much the same way that the elimination of drugs from the body can be accounted for by compartmental modeling.14 However, the use of present models requires prior knowledge of the single particle hardness to permit the elimination of the porosity variable. If measurements are made as a function of
;I "1 2
contact areas during relaxation) or to probe the underlying material structure and stress/strain response. The structural characteristics of interest include defects, impurities, and the strain-dependent dislocation distribution. To predict tablet response under these conditions, it is probable that the tablet model would become substantially more complex than the one presented in this paper. To predict the stress/strain/time response of a material under arbitrary conditions, for example during compaction or ejection from the die, one requires a complete characterization of the material being studied. The results of tablet stress relaxation experiments can be used for the purpose of such a prediction as long as one can ensure that the parameters derived from these tests are completeand correct.
\
k
9C
-
-8 -1
-12
\
-14
-1
,0032
\
,0034
,0036
,0038
.oOM
,0042
,0044
inverse absolute temperature, K-'
-
Figure7-Anhenius plot for the stress relaxationof KBr compactsbetween -45 and -5 O C . The fad is 156 kJ-mol-I (y = -18837~ 66.467, r2 = 0.976).
+
temperature, the hardness must be determined for all the temperatures employed in the stress relaxation tests. If normalizationof the force versus time stress relaxation data is accomplished, then it becomes possible to use deformation kinetic theory to interpret the stress relaxation behavior in terms of a finite number of deformation mechanisms, each of which is characterized by given E,, and Vae values. In practice, the total number of mechanisms that can be evaluated concurrently is less than or equal to three for statistical reasons. Normally, one mechanism controls the deformation a t very high stresses (i.e.,for the short time period at the beginningof stress relaxation) or very long times. The measurement of long-term relaxation is complicated by the need for stringent control of the environmental conditions. These concerns notwithstanding, the authors have not yet encountered a situation of interest in the field of practical tablet compaction in which dwell times are short,where more than two deformation mechanismsare required to explain deformation kinetic behavior. Interpretation of stress relaxation data becomes problematic when normalization by Vickers hardness data does not give rise to collinear force versus time curves. Under these conditions, it becomes necessary either to reevaluate the assumptions of the tablet model (in particular, constancy of the interparticulate
References and Notes 1. Duncan-Hewitt, W. C.; Weatherly, G. C. J. Pharm. Sci. 1990,79, 147-152. 2. Duncan-Hewitt, W. C.; Weatherly, G. C. J. Pharm. Sci. 1990, 79, 273-278. 3. Cole, E. T.;Rees, J. T.; Hersey, L. A. Pharm. Acta Helu. 1975.50. . . 28-32. 4. Krausz, A. S.;Eyring, H. Deformation Kinetics; Wiley: New York, 1975. 5. Papadimitropoulos, E. A.; Duncan-Hewitt, W. C. J. Pharm. Sci. 1992,81,701-704. 6. Duncan-Hewitt W. C.: Weatherlv, - . G. C. Pharm. Res. 1989.6.1060, . 1066. 7. The Science of Hardness Testing and i t s Research Applications, Westbrook, J. H.; Conrad, H.; Eds.; American Society for Metals: Metals Park, OH, 1973. 8. Cottrell, A. H.; Stokes, R. J. Proc. Roy. Sco. A. 1955,233,17-34. 9. Frost, H. J.; Ashby, M. F. Deformation Mechanism Maps; Pergammon: Oxford, 1982. 10. Papadimitropoulos, E. A., M. Sc. Pharm. Thesis; University of Toronto, Toronto, Ontario, Canada, 1988. 11. McClintwk, F. A.; Argon, A. S. MechanicaZBehaoiourofMateriaZs; Addison-Wesley: Reading, MA, 1966. 12. Komnik, S.;Bengus, V. Z.; Lyak, E. D. Phys. Stat. Sol. 1967,19, 533-541. 13. Skrotzki, W.;Haasen, P. In Deformation of Ceramic Materials ZI; Tressler, R. E.; Bradt, R. C., Eds.; Plenum: New York, NY, 1984, pp 429-444. 14. Gibaldi, M.; Perrier, D. Pharmacokinetics, 2nd ed.; M. Dekker: New York, NY, 1982.
Acknowledgments SupportfromtheMedicalResearchCouncilofCanadaandtheNatural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Journal of Pharmaceutical Sciences / 95 Vol. 83, No. 1, Januaty 1994
tablet constituents. The present research was undertaken to determine whether the inverse operation (is., using tablet stress relaxation to determine singlecrystal properties)was possible. The stressrelaxation of potassium bromide (KBr) compacts was evaluated as a function of temperature and relative density,and an attempt was made to calculate the deformation kinetic parameters. The stress relaxation of KBr did not fit the model under ambient conditions for two reasons: ( 7) KBr has two slip systems with approximately the same shear stress at room temperature: and (2)KBr strain-hardens. When these com-
plications were taken into consideration,the stressrelaxation behavior could be explained. Therefore, whereas single crystal tests are capable of yielding parameters that can be used to predict compact behavior, the inverse process of quantifying fundamental material parameters from compact behavioris problematicdue to the difficulty of determining, a priori, all the processes that operate simultaneously. Tablet formulation would be greatlyfacilitiated if the response of the formulation during compaction and the strength of the resultant tablet could be predicted. Two approaches currently exist. Traditionally, attempts to elucidate the compaction process seek to discover relationships between compaction parameters and tablet failure, treating the die contents rather like a “black box”. This approach attempts to circumvent the difficultiesassociated with measuringthe mechanical properties of the very small, brittle, single particles that comprise many pharmaceutical compacts. Alternatively, predictions can be derived from knowledge of the properties of the tablet components if the relevant material properties can be identified, measured, and related modelistically to tablet behavior.1.2 The production of a coherent tablet from a particulate mass requires the formation of many strong interparticulate bonds. Practically, this is achieved by applying a compaction stress that is intended to decrease porosity (thus increasingthe number of particle-particle contacts) and, through plastic deformation, to produce relatively large areas of interparticle contact where adhesion can occur. Unfortunately, the applied stress also can give rise to elastic strain that, when released, can cause tablet failure. Thus, the ideal pharmaceutical tablet material should be ductile and exhibit little elasticity. This requirement is complicated by the fact that all materials respond to stresses in a time-dependent manner: they are viscoelastic. Whether a particular material is considered to be ductile or elastic (or hard or brittle) depends on the rate of load application and its relationship with the rate at which the various deformation mechanisms are able to respond. Quantitation of the “viscoelasticity”of a pharmaceutical substance is prerequisite to any modelistic approach to the understanding of tablet compaction. Linear viscoelasticmodeling of tablet responseto compaction stress and stress relaxation without reference to the properties Abstract published in Advance ACS Abstracts, November 15,1993.
0 1994, American Chemical Sock?@and American Pharmaceutical Association
of the constituent particles has had limited success: simple, satisfactory models have not been found.3 It is true that one can fit experimental data by using a large number of viscoelastic parameters, but their physical meaning and their statistical significanceare questionable. Furthermore, tablets of different porosities appear to possess different viscoelastic parameters, which is unreasonable if one can assume that the material being compacted does not change significantlyduring the compaction process itself. Deformationkinetic analysis,which postdata that irreversible deformation or plastic flow of a solid is controlled by thermally activated phenomena, is an alternate means to study viscoelasticity. Furthermore, deformation kinetic analysis can be incorporated into a modelistic approach to the understanding of tablet compaction from the properties of single particles and has been used successfully in many engineering and a number of pharmaceutical application^.^*^ In the latter case, the temperature- and porosity-dependent stress relaxation of sodium chloride (NaC1) tablets was correlated with the microscopic deformation mechanism (characterized by a defined activation energy) with two intervening models to link each with the crystal hardness, a parameter that can be measured easily with a microindentation tester.’ These models will be used to evaluate the experimental behavior of potassium bromide (KBr). One might hope that knowledge of a model that successfully predicts the behavior of a number of materials might provide the means whereby single particle characterization could be avoided. For example, it is possible that stress relaxation experimentsalone might provide deformationkineticparameters that could be used to predict the kinetics of tablet compaction. In essence,this means that the contents of the “black box”would be defined. The research described below explores this possibility. Because the application of deformation kinetic analysis had been validated with NaCl compacts, we extended this approach to evaluate the stress relaxation of KBr compacts. KBr was chosen as an alternate material because although its flow and compactionbehavior is as well characterized as that of NaC1, its flow behavior is unique in that it is markedly softer and compacts much more readily than NaC1.
Experimental Section Crystalline KBr (Fisher Scientific, P205, lot 895777) with a Vickers Hardness of 88 MPa (10-sindentation time at 25 “C)was used as received from the manufacturer. Compacts were made at temperatures rauging from -45 to 100 “C. Prior to each compression, the cylindrical die and flat-faced punches (13-mm diameter) were cleanedwithmethanol,dried, and then filled with 1.5g of KBr. An INSTRON (model 4201) stressstrain analyzer,which was equippedwith Series IX comprebion software, was electronically calibrated prior to each compression and used to manufacture compacts of varying relative density. Compacta of f i i relative densities ranging from 0.77 to 0.89 were compressed at a punch displacement rate of 1 mm/min. At maximum punch travel (at loads of 2.5,3.5, or 4.5 kN),the movement of the cross-head was arrested,and stress relaxation was monitored for 2 min. At low temperatures (i.e., from -45 to 0 “C), stress relaxation was studied with a differential
0022-3549/94/1200-9 1$04.50/0
-
Jownal of Pharmaceutical Sciences / 91 Vol. 83, No. 1, &nmry 1994
Table 1-Summary sample
kn
F-,
KBr2 KBr3 KBr4 KBr5 KBr6 KBr7
ol Deformatlon Klnetlc Data at 298 K
2.5 2.5 3.5 3.5 4.5 4.5
Weight, g
Thlckness, mm
1.4957 1.461 1.497 1.4985 1.4935 1.4943
5.35 5.18 5.02
Density, ~ c m - ~Relative Denslty 2.105 2.124 2.246 2.235 2.324 2.436
5.05
4.84 4.62
r2
V.a, b9
0.988 0.983 0.989 0.987 0.986 0.983
58 59 59 63 66 61
Equatlon
0.766 0.772 0.817 0.813 0.845 0.886
y y y y y y
= 1.415~ - 25.331 = 1.447~- 25.844
= 1.452~- 26.085 = 1 . 5 3 1~ 27.363 = 1.614~- 28.806 = 1.489~- 26.72
4.73
A
16.5.
A
16.25.
m
3
16. 15.75-
+
;15.5.
2 15.25. 15 14.75. 14.5.
14.254
0
2L 0
20
40
60
80
100
120
time, s
Flguro 1-Punch
force versus time plots for the stress relaxation of KBr compacts compressed with maximurn loads (F,) of (0)2.5, (0)3.5,and (A)4.5 kN. The rate of stress relaxation varies with compaction force and tablet relative denslty. scanning calorimeter cooling unit. This unit employs dry nitrogen to avoid condensation problems. Cornpacts of KBr were compressed at room temperature a t varying loads, and, once the target load was reached, compaction was terminated by lifting of the cross-head. These compacts were then placed in the hot-air oven (200 "C) and were permitted to anneal for 2 days. Subsequently, the cornpacts were removed from the oven and placed in the environmental chamber at room temperture.8 The fan in the chamber was used to promote air circulation and more rapid and reproducible cooling. The stress relaxation of these compacts waa then monitored at 278 and 348 K. Hardness values were obtained from Frost and Aahby.9
Results Stress Relaxation of Compacts of As-Received Material-The data collected from the stress relaxation studies (Table I) were used to calculate the activation volume ( Vaa) with eq 1 (see Discussion; (*V,,, = 61 b3, SEM = 0.044,p = O.OOO1)). Literature values of V,, for this material are not available; however, the order of magnitude is indicative of a dislocation (Peierls-Nabarro) mechanism, which is appropriate for this material at these high s t r e ~ s e s . ~ Figure 1 illustrates the average force versus time plots and Figure 2 illustrates the same data normalized by the procedures described above for the tablet model plotted as a function of In (time) (Figure 2a) or according to eq 3 (Figure 2b). Had the simple model (1,rate-controlling deformation mechanism) been adequate, one would have expected the procedure to give rise to collinear plots. In fact, not only are the plots not collinear, the individual plots are curved which is also contrary to expectation. The activation energy (Eaa) of KBr was not 92 /Journal of phermaceutlcal Sciences Vol. 83, No. 1. January 1994
.
.
.5
0
1.5 2 2.5 In (time, s)
1
-I
3.5
4
4.5
5
0
-1.5
._I.
3
0
.
14.25 14.5 14175
I5
15125. 15.5 -15175
I 6 -16:25 16.5 .16:75
17
shear stress, MPa
Flgure 2 3 A , top)Shear stress versus In (tlme) plots for thestress relaxation of KBr compacts compressed with maximum loads (F,) of (0)2.5, (0) 3.5,and (A)4.5 kN. Shear stress Is calculated with eq 2 and the punch force and the hardness values for single crystals. The plots are derlved from the integral of eq 1 and are expected to be linear if one mechanlsm controls deformatlon. (B, bottom) Plots of in (shear stress rate) versus shear stress for the stress relaxation of KBr compacts compressed with maximum loads (F,) of (0)2.5, (0)3.5,and (A)4.5 kN. Derived from eq 1, the plots are expected to be linear If one mechanlsm controls deformation.
calculated from a plot of In i (stress rate) versus 1/K due to the curvilinearity of this plot under the experimental conditions. Stress Relaxation of Annealed Compacts-The data collected from the stress relaxation experiments performed at 298 and 348 K were plotted as In i versus mean shear stress ( 7 ) . The results (Table 2) indicate that the slope and intercepts of the curves have changed with respect to those of the unannealed materials (Table 1j. However, the annealing process caused the stress relaxation curves to become superimposable (Figure 3). Low Temperature Data-The Vaafor the deformation of KBr compacts a t subambient temperatures, calculated from the
of Deformallon Klnellc Data at 298 and 348 K for Annealed KBr
Table 2-Summary
F-,
Samples
kn
Relative Density
Equation
Temperature, K
r2
0.768 0.821 0.868 0.803 0.852 0.883
y = 5 . 5 7 5 ~- 93.317 y = 3 . 9 1 -~ 66.554 y = 4 . 3 4 8 ~- 73.507 y = 3 . 4 9 7 ~- 46.1 13 y = 4 . 8 2 4 ~- 62.499 y = 4.119X - 53.839
298 298 298 348 348 348
0.971 0.981 0.997 0.981 0.986 0.987
2.5 3.5 4.5 2.5 3.5 4.5
KBr33/348&39/40 KBr35/36&43/44 KBr37/38&45/46 KBr73/74&79/80 KBr75/76&81182 KBr77f78&83/84
Model 1: Halsey-Eyring Nonlinear Three-Element Model4-It is assumed that at high levels of stress, one deformation mechanism predominates. That deformationmechanism is controlled by a kinetic activation barrier that is characterized by a defined End (the height of the barrier) and Vact (the physical extent of the barrier). The rate of stress relaxation is related to the strain rate (9) and the parameters that characterize the activation mechanism as shown in eq 1: [- Endl 17 Vad1 i = -Ei. = -EA exp -exp -
-6.5 15.4
0
15.5
15.6
15.7
15.8 15.9 16 shear stress, MPa
16.1
16.2
16.3
16.4
Figure 3-The In (shear stress rate) versus shear stress plots for the stress relaxation of KBr compacts that were annealed after cornpactlon and prlor to stress relaxation testing. The annealing process causes the plotsto become linear and superimposable. Key to F, values: (0) 2.5; ( 0 )3.5; (A)4.5 kN.
B
8 - 4 4 . . 23.4
23.6
.
.
23.8
.
.
24
24.2
24.4
24.6
24.8
25
shear stress, MPa
Flgure 4-The In (shear stress rate) versus shear stress plots for the stress relaxationof KBr compactscompressedat -45 'C. One mechanism appears to control deformation at this temperature. Key to F, values: (0)2.5; ( 0 )3.5; (A)4.5 kN.
plot of In i versus average T, was 112 b3 (SEM = 0.042, p = 0.O001; Figure 4) and is of the order of magnitude indicative of a Peierls-Nabarro mechanism.' Lowering the temperature of compaction and stress relaxation had two effects: (I)the plots became linear, and (2) the plots became superimposed.
Discussion The tablet model is developed based on the assumption that the geometry and physics of the Vickers microindentation configuration and particleparticle contact are similar. The behavior of the single particle and compacts are related by the following two intervening models.
(1) kT kT In eq 1,E is the apparent elasticmodulus,A is a preexponential factor, T is the absolute temperature, and k is the Boltzmann constant. The activation parameters provide some information about the mechanisms that control the deformation behavior under the test conditions. For example, a Vad of the order of unity correspondswith a moleculardeformationmechanism such as diffusion or creep, whereas V,, of the order of 10 to 100 indicates that a dislocation mechanism may be controlling deformation. The nonlinear three-element model gives i as a function of i. and is required because activation theory defines i. values, whereas i (as stress relaxtion) is measured experimentally. Model 2 Tablet Modello-A tablet model is required to link single particle and compact behaviors. A tablet is porous and its stress relaxation is known to vary with porosity. The tablet model provides the means to normalize tablet stress relaxation curvesto make them independent of porosity. Normalizedstress relaxation curvesare expected to be superimposable. The tablet model used previously assumes that: (I)the particles contact each other only at asperities, (2) the interparticle contact area generated during compaction does not change during stress relaxation and (3)the material properties are independent of strain. Under these conditions, the hardness, characterized by microindentation testa, best characterizes the deformation behavior of the material. Because the hardness is measured from the contact area produced by a given load after 10 s, it can be used to calculate the total true interparticle contact area from the measured force on the punch after 10 s of stress relaxation with eq 2:
Fp= HA, In eq 2, Fpis the punch force, H i s the Vickers Hardness, and A , is the true cross-sectional interparticulate contact area. The Mises Yield Criterion (which relates the yield and shear stresses") combined with a hardness constraint factor (which relates the hardness with the yield stress') is used to relate the hardness and the shear stress in eq 3 r = H/3& (3) In eq 3, 7 is the shear stress and H is the Vickers Hardness. However, eq 3 is true only if no strain hardening occurs. Vickers microindentations are geometrically similar; that is, regardless of the load, the contact and deformation geometry remains constant. The result is that under most conditions, the Vickers hardness is independent of load, regardless of whether strain hardening occurs. The particle-particle contact zone is like the Journal of pharmaceutical Sciences / 83 VOL 83, NO. 1, JenUap' 1994
1 ~
~~
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.
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.
.
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.
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Figwe 5-The deformationof NaCl occurs within distinct dislocationbands (Figure 5a. 1 to 3);hence,the environment in which dislocations move is constant. KBr, however, does not form discrete dislocation bands during deformation (Figure 5B);therefore, the environment changes constantly and the crystals strain-harden.
intersection of spheres and so more closely resembles the Brinell hardnesstest configuration. In this test, the hardness of materials that strain-harden varies as a functionof load because the contact geometryvaries continuously. When no strain-hardening occurs, the Brinell and Vickers hardness values are similar, but deviate from one another when strain-hardeningoccurs. The observation that the normalized slope of the stress relaxation plot is a function of compact force may indicate that strain-hardening is occurring. Komnik et a1.12studied the deformation behavior of a number of halides, including NaCl and KBr. The series of pictures in Figure 5 show how the etched surfaces of these crystals appear after increasing amounts of strain. Each etch pit, represented in Figure 5 by a small diamond, shows the point of emergence of a dislocation line and illustrates the behavior of NaCl (Figure 5a) and KBr (Figure 5b) under uniaxial stress. Figure 5a shows that deformation of NaCl occurs within distinct dislocation bands. Upon application of stress, deformation is localized at the edges of these bands as they grow (Figure 5a to 3); hence, the environment in which dislocations move is constant. This uniformityof the environment is associatedwith minimal strainhardening. For such a material, the stress relaxation curves of contact stress versus In (time) plots should overlap once they are normalized, which was the case for NaC1. However, KBr, does not form discrete dislocationbands during deformation (Figure 5b).12 Upon application of stress, the dislocation motion is not localized but is occurring diffusely throughout the crystal in an environment that is constantly changing as the average concentration of dislocations increases. This results in strain-hardening due to dislocation interactions and could cause the poor overlap of the curves observed in Figure 2. Strain-hardening that occurs during compaction may be removed if the compactsare annealed subsequent to compaction but prior to stress relaxation testing. Keeping in mind that annealing could alter the behavior of the crystals substantially, the compacts were annealed as described above to determine whether this would cause the normalized curves to become 94 /Journal of Pharmaceutical Sciences Vol. 83, No. 1, January 1994
50
100
150
200 T (K)
250
300
350
400
450
500
Flgure 64ritical resolved shear stressfor two slip systems In KBr, plotted as a function of temperature. At room temperature, the stress required to activate both slip systems is similar. Adapted from Klmnik eta/.’*
superimposed. Examining Table 2 one sees that the slope and intercepts of the i versus y curves have changed with respect to those of the unannealed materials probably due to alterations of intrinsic crystal defect concentration during the annealing process (TableI). That the procedurecausedthestress relaxation curves to become superimposed supports the proposition that the original effect arose from strain-hardening. The data calculated for annealed compacts cannot be compared with that for “fresh” compacts because the annealing process changes the intrinsic structure of the crystals. The second observation,that the individualplots are distinctly curvilinear,may be due to the operation of two or more competing mechanisms with similar Eactvalues as shown by Skrotzki and Haasenla at room temperature (Figure 6). If two mechanisms possess similar parameters it would be difficultto extract reliable values from the plot. However, it is has also been shown that one of the slip systems becomes much more resistant to slip a t subambient temperatures. If this is so, then stress relaxation plots to KBr compacts should become linear as the temperature is decreased. For this reason, experiments were performed at temperatures ranging from -5 to -45 “C. In this temperature range, the activation plots did indeed become linear. An Arrhenius plot of the low temperature data yielded an experimental Eactof 2.7 X 10-19 J (SEM = 0.057, p = 0.0033, Figure 7). 55) and KBr, one By comparing the Esetfor NaCl(2.0 X may come to the somewhat paradoxical conclusion that KBr deformation must involve a larger activation barrier than that of NaC1. This is surprising because the latter material is harder. But, one must note that the homologous temperatures (temperatures measured relative to the meltingtemperature) at which these values are measured are different. The “softness”of KBr may simply arise from the fact that two mechanismsare operating at room temperature. The interaction of the two slip systems operatingconcurrentlymay have intensifiedthe strain-hardening effect as has been shown to be true in other cases.’
Conclusions The stress relaxation of a compact is a functionof the relaxation at the interparticulate contact regions,modified by the porosity. The effect of porosity can be accounted for with a tablet model, in much the same way that the elimination of drugs from the body can be accounted for by compartmental modeling.14 However, the use of present models requires prior knowledge of the single particle hardness to permit the elimination of the porosity variable. If measurements are made as a function of
;I "1 2
contact areas during relaxation) or to probe the underlying material structure and stress/strain response. The structural characteristics of interest include defects, impurities, and the strain-dependent dislocation distribution. To predict tablet response under these conditions, it is probable that the tablet model would become substantially more complex than the one presented in this paper. To predict the stress/strain/time response of a material under arbitrary conditions, for example during compaction or ejection from the die, one requires a complete characterization of the material being studied. The results of tablet stress relaxation experiments can be used for the purpose of such a prediction as long as one can ensure that the parameters derived from these tests are completeand correct.
\
k
9C
-
-8 -1
-12
\
-14
-1
,0032
\
,0034
,0036
,0038
.oOM
,0042
,0044
inverse absolute temperature, K-'
-
Figure7-Anhenius plot for the stress relaxationof KBr compactsbetween -45 and -5 O C . The fad is 156 kJ-mol-I (y = -18837~ 66.467, r2 = 0.976).
+
temperature, the hardness must be determined for all the temperatures employed in the stress relaxation tests. If normalizationof the force versus time stress relaxation data is accomplished, then it becomes possible to use deformation kinetic theory to interpret the stress relaxation behavior in terms of a finite number of deformation mechanisms, each of which is characterized by given E,, and Vae values. In practice, the total number of mechanisms that can be evaluated concurrently is less than or equal to three for statistical reasons. Normally, one mechanism controls the deformation a t very high stresses (i.e.,for the short time period at the beginningof stress relaxation) or very long times. The measurement of long-term relaxation is complicated by the need for stringent control of the environmental conditions. These concerns notwithstanding, the authors have not yet encountered a situation of interest in the field of practical tablet compaction in which dwell times are short,where more than two deformation mechanismsare required to explain deformation kinetic behavior. Interpretation of stress relaxation data becomes problematic when normalization by Vickers hardness data does not give rise to collinear force versus time curves. Under these conditions, it becomes necessary either to reevaluate the assumptions of the tablet model (in particular, constancy of the interparticulate
References and Notes 1. Duncan-Hewitt, W. C.; Weatherly, G. C. J. Pharm. Sci. 1990,79, 147-152. 2. Duncan-Hewitt, W. C.; Weatherly, G. C. J. Pharm. Sci. 1990, 79, 273-278. 3. Cole, E. T.;Rees, J. T.; Hersey, L. A. Pharm. Acta Helu. 1975.50. . . 28-32. 4. Krausz, A. S.;Eyring, H. Deformation Kinetics; Wiley: New York, 1975. 5. Papadimitropoulos, E. A.; Duncan-Hewitt, W. C. J. Pharm. Sci. 1992,81,701-704. 6. Duncan-Hewitt W. C.: Weatherlv, - . G. C. Pharm. Res. 1989.6.1060, . 1066. 7. The Science of Hardness Testing and i t s Research Applications, Westbrook, J. H.; Conrad, H.; Eds.; American Society for Metals: Metals Park, OH, 1973. 8. Cottrell, A. H.; Stokes, R. J. Proc. Roy. Sco. A. 1955,233,17-34. 9. Frost, H. J.; Ashby, M. F. Deformation Mechanism Maps; Pergammon: Oxford, 1982. 10. Papadimitropoulos, E. A., M. Sc. Pharm. Thesis; University of Toronto, Toronto, Ontario, Canada, 1988. 11. McClintwk, F. A.; Argon, A. S. MechanicaZBehaoiourofMateriaZs; Addison-Wesley: Reading, MA, 1966. 12. Komnik, S.;Bengus, V. Z.; Lyak, E. D. Phys. Stat. Sol. 1967,19, 533-541. 13. Skrotzki, W.;Haasen, P. In Deformation of Ceramic Materials ZI; Tressler, R. E.; Bradt, R. C., Eds.; Plenum: New York, NY, 1984, pp 429-444. 14. Gibaldi, M.; Perrier, D. Pharmacokinetics, 2nd ed.; M. Dekker: New York, NY, 1982.
Acknowledgments SupportfromtheMedicalResearchCouncilofCanadaandtheNatural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Journal of Pharmaceutical Sciences / 95 Vol. 83, No. 1, Januaty 1994