Mechanical Properties of Compacts and Particles that Control TabletingSuccess

EVERETT N. HIESTANDX

Received March 12, 1997, from the residence at 11378 East G Avenue, Galesburg, MI 49053. Accepted forpublication June 30, 1997X.

Abstract 0 The use and limitations of linear-elasticity equations fordescribing mechanical properties of compacts is discussed; the limitationsoccur because compacts are porous, viscoelastic, nonhomogeneous, Mohrbodies. Awareness of these properties permits meaningful comparisonsto be made. Ignoring limitations may result in unjustified conclusions.Special care during measuring mechanical properties of compacts isrequired. The mechanical criteria for a successful formulation are goodflowability for powders and adequate strength without fracture forcompacts. Interparticle attraction is spontaneous but particle contactnumbers and size are limited by mechanical preclusion. Plasticdeformation and fracture mechanics are controlling mechanisms with themagnitude of elastic constants having little effect on the successfulprocessing. General compaction characteristics can be appraised by usingcombinations of properties, e.g., specific tableting indices.

1. IntroductionBecause mechanical properties determine the success or

failure of powder-to-tablet processing,1 a concise discussionmay be useful, especially since some pharmaceutical studiesprovide only limited discussions of these properties. Herein,solid processing refers only to physical operations. For adefinition of basic terms or detailed descriptions of funda-mental principles, the reader should consult standard texts.2For this discussion, it is assumed that capillary forces fromliquids are negligible; however, modified mechanical proper-ties from changing water content are inseparable, e.g., phen-acetin,3 microcrystalline cellulose,4 and susceptible materialslike polymers and indomethacin.5 The assumed criteria forsuccess are adequate tensile strength (bonding) and nonfrac-ture (nonbrittle) for compaction3 and flowability4 for powders.In practice, physical measurements are used because me-chanical properties are not predictable a priori. Manytechniques6-8 and compaction concepts described by others9are not included herein. While some views expressed in thesereferences differ from the author’s, the papers provide worthyinformation, perspective, and advanced concepts. A fewspecific, questionable statements are cited here and in ref 1.Airing differences is the first step toward resolution of meritand to meaningful progress.

2. BackgroundMechanical properties pervade and control solid processing.

Powder bed flow is plastic deformation. Compaction producesincreased strength by particle fracture and packing rear-rangement; the bed is sheared and the particles are deformed.Pharmaceutical compacts and powder beds are complexstructures that present difficult challenges when measuringtheir mechanical properties. Primarily these originate fromthe interrupted continuity from pores and changes of consoli-

dation. Furthermore, pharmaceutical compacts are made byuniaxial compression; thus, they are not isotropic. Prodigiousevidence exists that successful compaction requires plasticdeformation. The increased radii of curvature of contacting,plastically deformed surfaces1,10,11 increases the number ofclose atoms. Quantitatively, properties are described usingstresses; strain is seldom used because porosity changescomplicate the stress-strain relationship. Since quantitativeequations provide guidance not available for most geometries,the contacting spherical segment model (CSM) and variationsof it10-15 are used for guidance. While powder particles arenot spheres, qualitatively the principles apply to contactingsurfaces. Because plastic deformation “erases some memory”of original shape, spherical similarity of the region aroundthe contact is improved. With nonspherical surfaces,14 e.g.,hard crystals deforming softer particles, and with rotation andshearing stresses9 present, the spherical similarity diminishes.Including the ever present viscoelastic effects11,15 add furthercomplications.Interparticle contact lowers the free energy; there is de-

creased surface energy. This is attraction. Only mechanicalbarriers inhibit contacts from forming spontaneously. Bothinternal attraction and external compaction stresses arecounterbalanced by Born repulsion within the contact; attain-ing balance determines contact size. Rarely do individualbonds (as with chemical bonds) contribute significantly to theattraction. Tablet bond indicates total attraction, but will bereferred to as bonding to avoid the unintended interpretationas one or more “single” bond(s). Contrary to some views,16bonding is an exothermic process.17 Repulsion requires work,not vice versa. Some18 correctly clarify this. At distancesgreater than for Born repulsion, the indigenous interactionenergy vs separation plot has no inflections, no minimum.Thus at contacts, only a few atoms near the contact perimeter,would have the spacing and juxtaposition required for opti-mumH-bonds and/or polar interactions. The dispersion forceswould dominate the interaction between solid, organic par-ticles. Even with organic liquids, which have unrestrictedmolecular juxtapositioning, the ratio of dispersion to polarforces acting at interfaces always exceed unity.1 Furthermore,empirical equations relating surface tension and solubilityparameter data19,20 confirm that interactions at interfaceshave lesser polar fraction than for solutions. Thus, directlyapplying to interfaces the data for energy ratios from extendedsolubility parameters is problematical.While these forces produce co/adhesion that may induce

deformation,13 they are very small compared to compactionloads. Contacts are inherently a kind of solid “bridge”;1 thechasm between particles has been eliminated. Short rangedispersion interactions dominate. Note that the crystalstructure21 of many organic compounds are grossly influencedby molecular shape conformity that maximizes the molecule-to-molecule dispersion force interaction. Also, solubility19depends on intermolecular interaction to which dispersionforces contribute significantly. Originally solubility parameterconcepts were derived by assuming dominance of dispersioninteractions. Some exclude the universal close range interac-

x Retired from the Upjohn Co., Kalamazoo, MI.X Abstract published in Advance ACS Abstracts, August, 15, 1997.

© 1997, American Chemical Society and S0022-3549(97)00106-8 CCC: $14.00 Journal of Pharmaceutical Sciences / 985American Pharmaceutical Association Vol. 86, No. 9, September 1997

tions in tablets by using an unrealistic model which considersonly molecular interactions for surfaces separated by somedistance.22 What stabilizes this separation, i.e., preventscontact? Since the dispersion interactions are inverselyproportional to separation to some power (the exponentmagnitude depends on the geometry), the interaction withimmediate neighbors, those in contact, dominate. However,summing the attraction among all atoms results in significantcontributions from more remote atoms.

3. Tractableness sPowders and Compacts

Mechanics is response to stress; it includes (1) elasticdeformation, (2) plastic deformation, and (3) fracture. Theusual linear elastic equations apply to an ideal continuum,one that neglects only atomic size volumes. For powders andcompacts, continuumc is used because the neglected sizebecomes a pore/particle volume. This distinction is needed;a mean value has meaning only for areas/volumes much largerthan the neglected size. Furthermore, the ideal continuumis a homogeneous, isotropic solid with no volume changeduring shear deformation. The continuumc is not isotropic,contains solid fraction variations, has strain rate dependent(SRD) properties, and may change consolidation during plasticdeformation. Consequently, applying equations rigorous fora continuum to a continuumc has limitations; but whenproperly used, valuable results are obtained.23,24

For characterizing the SRD properties of the continuumc,fitted “spectra” for viscoelasticity have had some success,25but the parameters may not always9 be identified with realmaterial properties. Nevertheless, by using sequential seg-ments of the physical process, and linear viscoelastic analysisof each segment, significant insight of events during theprocess23 may be postulated. When only relative values for acontinuumc are sought, the use of simpler linear elasticequations applied at the operative strain rate or dwell timeprovide acceptable stress estimates.24 The discussion hereinuses the latter approach. Forgetting and/or ignoring thecompromises made can lead to serious misinterpretations.Undeniably, the continuumc properties must reflect those

of the constituent particles.9 Particle coordination number(contacts/particle) as a function of solid fraction (Fr), inter-particle forces acting and contact mechanics have beencombined in attempts to explain strength.9,11 While themicroregion of particles are in the continuum domain, formi-dable problems exist in conversion of particle properties tocontinuumc properties: (1) Measurements on very smallparticles are difficult or impossible to obtain. (2) Often singlecrystals are not isotropic; measurements on one surface maynot represent the particle behavior in the compact. (3) A widerange of mechanical values occurs among particles of the samematerial (those in the same compact). (4) Some values, e.g.,yield strength and brittleness, may change during processingbecause work hardening was induced. (5) The yield criterionfor the particle (von Mises is usually assumed) may beunknown. (6) Commercial products have many unlike com-ponents. And (7) methods used for conversion are equivocal.This fundamental approach is essential, but currently, theassumptions made when starting with particle properties arenumerous.9,11 Efforts to avoid some of the above problems relyon effective “particle” properties, i.e., properties for thenonporous continuumc, which have been estimated by exten-sion of continuumc properties to Fr ) 1. At commonly usedsolid fractions, semilogarithmic plots of many mechanicalparameters vs Fr appear to be linear; however, extrapolationmay yield unrealistic values for the nonporous solid. Use ofa nonlinear method26 or of one segment based on percolationthresholds27 should be considered for these estimates.

4. Elastic DeformationElastic deformation is a reversible energy storage and force

balancing mechanism that is a part of any stressed state.Young’s modulus, E, and Poisson’s ratio, ν, characterize it.The combined two constants are designated, E′, where

The subscripts designate different interacting materials. Fora continuum in the elastic range, often, the dilatational valuefor ν is assumed to be approximately 0.3. Making thisassumption28 for organic materials is risky, e.g., for polymers,nylon, and rigid PVC, values of 0.4 are reported.29 Further-more, when plastic deformation occurs in a continuum, ν )0.5 (required for constant volume, see section 5). Conversely,plastic deformation of the continuumc nearly always is ac-companied by a volume change and/or a consolidation change;add to this the SRD effects, and the interpretation of ν forcompacts becomes obfuscated. Clearly, one cannot confidentlyassume, as has been done,6 ν ) 0.3 for a continuumc. Otherpublished, undocumented statements may be misleading, e.g.,“....the fact that the compressive modulus should always begreater than the tensile modulus, ....”.6 One may ask, by howmuch? For either measurement, the particles are in contact.Thus, any measurable (tensile or compression) modulus isdominated by the Born repulsion gradient. The transitionfrom compression to tension would be a continuous functionwithout an inflection; a large difference for the two regions isunexpected.During compaction, contacts are at all stages of develop-

ment. New contacts are forming while older ones continueplastic deformation. Similarly during unloading from com-pression and tensile loading, last formed contacts may un-dergo either ductile extension or brittle fracture10-14 whileothers are being reduced in area. Attraction continues untilseparation, i.e., compact fracture. Even with limited strainsome existing contacts may undergo plastic deformation.However, once significant “at rest” relaxation of stress hasoccurred, very short duration, micro-strain should be elasticonly. Thus, dynamic methods are necessary to measureinstantaneous elastic moduli.With rough, hard materials, E′/H is an important adhesion

parameter. H is indentation harness, the mean pressureunder an indenter. Low values of E′/H and high roughnessare associated with elastic, low ad/cohesion, interactions;30 i.e.,very high hardness may reduced bonding. For the limitedcompression of powder beds, low values of E′/H shouldpromote flowability. For pharmaceutical compacts, Ho/E′ isthe strain index3 obtained during impact hardness, Ho,measurements. Its magnitude does not correlate with tablet-ing performance, probably because the magnitude for organicmaterials is in the ad/cohesion, plastic deformation range; itis not in a transition range.With contacts and rounded indenters, the stress pattern

within the contact region builds from zero at the edge to highcompression at the center. Regardless of the original stressdistribution, viscoelastic decay reduces the stress gradientwithin the contact area.31 Relaxed stress distributions areexpected to increase unloaded radii of curvature; contactstrengthening is expected. However in practice, weakeningmay occur; an expansion of the compact may accompany thisinternal stress decay; even fracture may occur.23 Clearly,details of how the multifarious stress distributions in thecontact region influences compact strength are elusive.Fracture prevents theoretical, potential strengths from

being observed. For fracture (crack growth), elastic energyfrom applied stress is used for both forming new surfaces andproducing plastic deformation in the fracture zone, the crack

1/E′ ) (1 - ν12)/E1 + (1 - ν2

2)/E2

986 / Journal of Pharmaceutical SciencesVol. 86, No. 9, September 1997

tip. Materials are brittle if they use little energy for plasticdeformation, including the SRD part. Increased strength fromSRD properties have been included in the CSM by using SRDvalues of E′,15 viz (1) E′o, the instantaneous value (insignificantstress relaxation), and (2) E′∞, the relaxed modulus. The forcenecessary to slowly separate two surfaces is increased by afactor equal to E′o/E′∞.15 (A variation of this was includedwhen using the CSM for tablet bonding.10,11) Some studiesshow that “...there is no relationship between the Young′smodulus and the fracture strength of the materials”.32 Sup-porting this: (1) With the CSM the elastic modulus15 occursin the strength equations only as a ratio, E′o/E′∞. And (2) thestrain index, Ho/E′o, values3,33 show that fracture and elasticstrain of decompression following plastic deformation do notcorrelate. E′/H may be an ad/cohesion parameter for veryhard materials; E′ alone is not.

5. Plastic Deformation and Yield StressTo describe a stress state, stresses are categorized as either

normal or shear stresses. Beyond a limited shear deforma-tion, elastic deformation loses reversibility; the shear stresseshave exceeded the shear strength. With unloading, the solidhas changed shape; elastic, partial shape recovery occurs. Theunrecovered energy was used for plastic deformation. Anormal only stress is a principal stress, a shear component isnot present in that direction. Nonprincipal stresses have bothnormal and shear components in those directions. For a givenstress state, the magnitude of both the shear, τij, and normal,σi, stresses (subscripts indicate specific orthogonal directions)vary with the choice of direction considered, and only onespecific direction choice would be the orthogonal principalstress directions. (This is describing stress mathematicallyusing chosen axes directions.) Any isothermal volume changeof a continuum is elastic only. Importantly the sum of theorthogonal, normal stresses is constant, regardless of thedirections considered. The mean normal stress is the hydro-static stress component, σ; (σx + σy + σz)/3 ) σ. Also, this iscalled the dilatational stress. σ induces only elastic, notplastic, deformation of a continuum. Uniquely, when σx ) σy) σz, there are no shear stresses within the continuum.Conversely in the continuumc, a large applied hydrostaticstress, even when σx ) σy ) σz, is expected to induce plasticdeformation of particles, i.e., there are shear stresses withinthe continuumc. Clearly, the discontinuity, the pores, changesthe practical results.The shear stress component is called the distortional or

deviatoric stress and produces elastic and, when the shearstrength is exceeded, plastic deformation. In a continuum,the deviatoric stress does not produce volume change. Dis-tortional strain energy, Ud, can be obtained from principalstress differences:

This defines the equivalent stress, σj:2

σj g Y is the von Mises yield criterion, and defines a “shear”stress state that induces plastic deformation in a continuum.The magnitude of Y is obtained from a continuum underuniaxial stress, σi, in tension where the maximum shearstress, τmax. ) σimax/2, induces plastic yielding, this σimax ) Y.Tension is considered as positive and compression as negativestress. (For powders this sign custommay be reversed.) Withtwo variable orthogonally applied stresses, the shear stressthat would produce yield, i.e., equal Y/2, can be shown on a

plot of τ vs σn (σn is normal stress). For the continuum, Y/2 isa straight line parallel to the σn axis extending into both thetensile and compression regions, i.e., independent of hydro-static stress. This is not applicable to the continuumc, whoseyield criterion is a function of both the distortional andhydrostatic stress. This defines the continuumc as a Mohrbody. A reasonable explanation is that the area of contactbetween particles changes, shear strength changes, with theapplied hydrostatic stress. Furthermore, compacts, powderbeds, and most individual particles34 of organic compoundsexhibit SRD properties; thus for a continuumc, the magnitudeof Y is SRD.All compacts are Mohr bodies; the tensile strength is much

less than the strength under compression. Again using twovariable applied stresses, the yield strengths on a τ vs σn plotmay be either linear or curved, but they would slope andextrapolate to a tensile intersect of the σn axis. Shear celldata provide these plots for powders,4,8 but have rarely beenused for compacts. Specific yield criteria for a continuumc

35,36

that incorporate (1) both dilatational and deviatoric stresses,(2) SRD properties, and (3) changing consolidation have notgained acceptance. Nevertheless, much can be learned fromcomparison of values when the stress conditions and stressstate are known. The commonplace comparisons, which usethe same compaction pressure for different materials, do notmeet these conditions, and conclusions may be flawed becauseobserved results may be from different porosities and SRDproperties. Models for tablet strength use solid fraction asthe reference condition; the number of interparticle contacts(a fundamental strength parameter) are estimated from solidfraction.Plots of die-wall pressure vs punch pressure confirm that

extensive plastic deformation occurs throughout the compac-tion-unloading cycle. If plastic deformation were absentduring compaction, complete unloading would yield essentiallyzero die wall pressure. With plastic deformation duringcompaction and only elastic unloading, a straight line to a highdie wall pressure intercept would result. This very highintercept is unrealized because plastic deformation duringunloading decreases die wall pressure faster than elasticunloading. At zero punch pressure, the die wall stress isdetermined by the Mohr body (compact) shear strength. Diewall and punch pressures are only the hydrostatic component(usually a mean value); the deviatoric stress, wall friction, isunknown. Excellent lubrication reduces the friction to aminimum level. (As with all continuumc values, the particlestresses, the critical point of action, are unknown.) Undeni-ably, these plots demonstrate that there is extensive, universalplastic deformation associated with tableting. Some investi-gators22 downgrade the importance of plastic deformation andstate that “...materials often possess inadequate plasticity forthe development of large zones that could take part in theinterparticulate attraction by intermolecular forces.” Sincedie wall-punch pressure plots suggest the opposite, thatjudgement needs documentation and clarification. Smallamounts of plastic deformation can change the radius ofcurvature and have substantial effects on total dispersioninteractions, i.e., on bonding.The ever present, plastic deformation provides an explana-

tion of why lot-to-lot problems are common. The yield valueof the particles is dependent on defects in the crystals, andchanges may occur in nearly all processing. Usual productionspecifications do not include criteria for mechanical properties.Observed punch pressures and associated porosities were

used by Heckel37 and plotted as ln[1/(1 - Fr)] vs compactionpressure. This is an idealized interpretation of compactiondata to infer the events happening in various regions of theprocess. These plots are said, when properly done, to providea mean yield stress of the powder. Using this, a strain rate

Ud ) {(1 + ν)/6E}{(σx - σy)2 + (σy - σz)

2 + (σz - σx)2}

Ud ) {(1 + ν)/3E}σj

Journal of Pharmaceutical Sciences / 987Vol. 86, No. 9, September 1997

sensitivity term has been defined6,28 to provide insight for timedependent effects. It is reported that the yield stresses, σh(assuming σh ) H/3), from indentation hardness measure-ments are comparable to Heckel values.6 σh ) H/3 is anapproximation for a specific continuum that obeys the vonMises criterion; the application to Mohr bodies is equivocal.The theoretical link to the Heckel theory is very weak. Aninteresting perspective on the Heckel method is quoted:38“...in...compaction studies for pharmaceutical powders, devia-tions from Heckel’s theories have been utilized in obtainingadditional information. So, most often the Heckel equationis strictly speaking invalid on most stages of compaction ofpharmaceutical powders. It might be advantageous to utilizeother porosity-pressure functions and methods of materialscience contemporaneously with the Heckel equation.” Datacollection using a sophisticated tableting device does notremove these inherent problems. Nevertheless, the Heckelmethod has proponents for its use to characterize tabletformulations.

6. Fracture

Because of the particulate structure, continuumc fracturehas brittle characteristics. Crack growth (brittle) occurs whentensile loading supplies locally enough elastic energy for bothnew surfaces and the associated local plastic deformation. Theweakening crack originates from the defect because it con-centrates stress; it is the first region to exceed strength. Thecrack tip readily becomes a moving stress concentrator andcrack propagation across the stressed solid continues. How-ever, when plastic deformation, including the SRD part,consumes energy and prevents the accumulation of adequateelastic stress for crack growth, the material is not brittle. Thismechanism describes both the continuum crack growth andbrittle separation of particles. The matrix of particles incontact provide islands of ad/cohesion, with stress concentra-tors, the pores, dispersed among them. Thus, fracture in thecontinuumc is “zipper-like”, hopping from contact to contact;ductile fracture of a continuumc would be unexpected evenwhile some individual particles undergo ductile extension.The ideal CSM defines conditions for particle separations

by either ductile extension or brittle fracture.10-15 Using thesimpler “no creep” equations for viscoelastic materials whileusing an uncontrolled strain rate, as with the centrifuge,39cannot provide definitive data. Neighboring contacts in thecontinuumc may separate by different mechanisms. Further-more, the multiple histories and orientations of contactscauses the applied stresses at particles to range from tensileto shear at various contacts. The failure mechanism for acontact under shear is less well-understood.In a continuumc, large shear strain without fracture must

involve contact “maintenance”, i.e., changing of particle andcontact shape or possibly the breaking of existing and formingreplacement contacts. When confined in the die by a largehydrostatic stress, fracture is impeded and plastic deformationmay increase strength; nevertheless, brittle fracture withinthe die during unloading is commonplace.40,41 Decreasing thebrittleness (by reformulation) eliminates these fractures. Toooften, air entrapment is blamed for capping. This is claimedwithout supporting data and without indicating causative,physical differences between capping and noncapping formu-lations. While fine powders may entrap air in pores, they alsoimpede flowability and seldom are dominant in a ready-for-production formulation. Usually, brittleness causes capping.A measure of the degree of compact brittleness is valuable;

e.g., the brittle fracture index3,33 (BFI), the critical stressintensity factor6 (KIC), and the stress intensity factor (oftencalled the fracture toughness) divided by hardness9 all provide

a brittleness indicator. The latter have acceptance in othertechnologies but are more difficult to use for routine screening;also, the test specimen may be less similar to a productiontablet. Deceptive effects from density gradients, work hard-ening, etc. are not predictable. Thus, a test specimen similarto a regular tablet lowers the risk of obtaining unrepresenta-tive results for tableting use. The square tablets used withthe tableting indices are made by uniaxial compaction; thus,internal density gradients and work hardening should besimilar to the usual round tablet case. However, the decom-pression is triaxial and the reduced shear deformation andminimized “microfracture” during decompression are unique.The purpose is to produce a flawless compact, one requiredfor tensile strength testing, as for the BFI to indicate brittle-ness. With unconfined compacts, few test procedures producesignificant plastic deformation without brittle fracture. Usu-ally, indentation hardness is an exception; it provides enoughhydrostatic stress to avoid fracture. Even for indentation, the“pile-up” just outside the perimeter of the indentation mayproduce localized tension and a dent circumscribing, “surface”crack; the compressed region seldom is disturbed by it. Thehardness data are valid. An unconfined, column-shapedcompressed compact, in uniaxial compression (platens cover-ing the entire end of the column compact), exhibited cata-strophic fracture41 at a typical Mohr body failure angle, ∼60°.Apparently, the failure occurred at the angle where shearstrength equaled shear stress; i.e., it was shear-stress-inducedbrittle fracture. With platens that apply stress to a partialcross section, e.g., diametral compression of a disk or trans-verse compression of square compacts, a tensile stress occursbetween the platens normal to the compression stress. Atfracture, it gives a tensile strength, σT (the observed σT is forthe specific stress state imposed by the procedure.). When alarge defect (defect . particles/pores) occurs in the tensileregion, the stress concentration factor is essentially thatpredicted by linear elastic equations. The BFI success, whichuses this principle, attests3,33 to this.

7. Practical Use sA Practical System

An important part of tablet formulation is selecting andcombining materials to obtain needed mechanical properties,viz., flowability for powders and desired bonding and lowbrittleness for tablets. Definitive measurements should beuseful for both individual components and mixtures. Thefollowing focuses primarily on measurements preferred by theauthor. For flowability, the compressibility index42 providesa simple, general, useful criterion. Shear cell data providemore information4,8 because a defined stress state is used. Forcompacts, tableting indices have been defined. Qualitativestatements of what each tableting index measures3,33 (two forbonding, the brittle fracture, and the strain index) that weremade when introducing the concepts are useful, generaldescriptions but are oversimplified. Test methods and condi-tions were chosen for simplicity while retaining valid, mean-ingful43 comparisons. The indices3,33 use defined stress statesand systematic measurement of selected, comparable me-chanical properties of test specimens formed much as intableting. This avoids making measurements during process-ing induced changes and with uncontrolled stress states.Square tablets were chosen for test specimens because (1)

they accommodate triaxial decompression to minimize un-wanted defects in the test compact; (2) for σT measurements(transverse compression44), they provide success with morematerials than any other known to the author; and (3) thetensile strength of the central region of the compact isobtained. Thus, the same portion of the compact is used forboth σT and H measurements for the bonding index,3,33 σT/H.

988 / Journal of Pharmaceutical SciencesVol. 86, No. 9, September 1997

The equipment and procedure choices have been explainedin detail.33 Using more conventional shapes and varying theprocedures such as compaction rates45 may not provideequivalent data; furthermore, they often fail with problemmaterials, their raison d’etre.The mean pressure, regardless of stress distribution, under

the indenter is the indentation hardness. For a continuum,the stress distribution under spherical indenters is well-understood.46,47 The short dwell time of the impact hard-ness,48 Ho, minimizes the SRD effect and a similar stressdistribution is assumed. (Ho is calculated from energy terms,not stress per se.) With a slow strain rate or a longer dwelltime, the decay of stress produces an unknown, multifari-ous15,31 stress distribution. The selected dwell time (b) hard-ness is designated Hb. Comparison of Hb with Ho indicatesstrain rate sensitivity and is used with the best case, σT/Hb,and worst case, σT/Ho, bonding indices.43,49 The same valueof tensile strength is used for both bonding indices.Meaningful comparison of σT among very different SRD

materials results from imposing common times to fracture;i.e., a selected, constant time lapse from the stress (1/e)σT,0.368σT, to σT is used. The effect is real; e.g., the tensilestrength of sorbitol was reduced by 42% by increasing the timeconstant by a factor of 54.33 Others7 have reported an oppositeeffect for loading rate, perhaps due to changing failure modewith their tensile test method. Square compacts avoid this.Often instrumented tablet presses are used to detect

compaction variations. One of these, a sophisticated singlepunch tablet press referred to as a simulator, uses smallquantities of powder. Too often, this is cast as a competitorto measuring the tableting indices. When used, they arecomplementary approaches. The indices provide more anddifferent information and can measure mixtures or individualmaterial. The dimensionless indices provide criteria valuesapplicable to all materials. Also, experienced operators finduseful information in the tensile strength, compaction pres-sures, and hardness values used for the indices. Measuringindices requires more powder than many like to commit.However, the formulation ruggedness43,50 becomes clear; i.e.,can the formulation accommodate reasonable lot-to-lot varia-tions? Furthermore, lot-to-lot variation of one ingredient,possibly from alternate suppliers, can be determined withoutpreparing a mixture. Of the indices, the BFI is the mostimportant; values above 0.2 should be avoided. Also, a verylow bonding index, below 0.01, indicates a lack of ruggednessand suggests concern. While the tolerable lack of ruggednessvaries with the formulation and manufacturing choices,50 acaution signal is clear.Perhaps the biggest problem with the indices is the failure

of potential users to understand the concepts and the proce-dure restrictions.33 The objective, a single system applicableto all materials, is indeed difficult to meet. For example, rarematerials have been identified where the BFI results appearto be anomalous, e.g., the tensile strength of the compact witha hole is greater than the strength of the compact withoutthe hole. Fortunately, the aberration is obvious; the explana-tion is unknown. However, the data suggest it is not aproblem, brittle material; practically, it need not be pursued.Resorting to using mixtures to obtain useful data is a commonpractice with other systems; it becomes a last resort here asa way to confirm that the substance that produced theanomalous result is not a problem material.

8. Conclusions

Mechanical properties determine the success of powder-to-tablet processing. The use of linear elastic equations todescribe stress in compacts is useful but requires great

caution. Strain rate sensitivity must be accommodated.Compacts are Mohr bodies and have very different yieldcriteria from a continuum. Despite the complexity of themechanics of pharmaceutical systems, mechanical propertymeasurements provide the most systematic, rational approachto formulation design.

9. List of Acronyms and SymbolsCSM contacting spheres modelSRD strain rate dependentBFI brittle fracture indexFr solid fractionE Young’s modulusν Poisson’s ratioE′ 1/E′ ) (1 - ν12)/E1 + (1 - ν22)/E2, subscripts identify

particlesE′o, E′∞ instantaneous and relaxed value of E′, respectivelyH indentation hardnessHo, Hb instantaneous and dwell time value of hardness,

respectivelyτ shear stress; subscripts i and j indicate orthogonal

directionsτmax maximum elastic shear stress, yield stressσ hydrostatic stress ) (σx + σy + σz)/3); with subscript

x, y, and z indicating orthogonal directions ofnormal stress; n or i for general, normal stressand σimax is yield value

σj equivalent stressσh a possible yield stress estimated from indentation

hardnessσT tensile strengthUd distortional strain energyY von Mises yield criterion; σj g YKIC critical stress intensity factor

References and Notes1. Hiestand, E. N. Principles, Tenets and Notions of Tablet Bonding

and Measurements of Strength. Eur. J. Pharm. Biopharm. Inpress.

2. Polakowski, N. H.; Ripling, E. J. Strength and Structure ofEngineering Materials; Prentice-Hall: Englewood Cliffs, NJ,1966.

3. Hiestand, E. N.; Smith, D. P. Indices of Tableting Performance.Powder Technol. 1984, 38, 145-159.

4. Amidon, G. E.; Houghton, M. Powder Flow Testing in Prefor-mulation and Formulation Development. Pharm. Res. 1995, 12,923-929.

5. Hancock, B. C.; Shamblin, S. L.; Zografi, G. Molecular Mobilityof Amorphous Pharmaceutical Solids Below Their Glass Transi-tion Temperatures. Pharm. Res. 1994, 11, 471-477.

6. Rowe, R. C.; Roberts, R. J. Mechanical Properties. In Pharma-ceutical Powder Compaction Technology; Alderborn, G., Nystrom,C., Eds.; Marcel Dekker: Inc: New York, 1996; pp 283-322.

7. Davies, P. N.; Newton, J. M. Mechanical Strength. In Pharma-ceutical Powder Compaction Technology; Alderborn, G., Nystrom,C., Eds.; Marcel Dekker, Inc: New York, 1996; pp 165-191.

8. Jenike, A. W. Gravity Flow of Solids. Bullitin. 108, UtahEngineering Experimental Station, University of Utah, SaltLake City, 1961.

9. Duncan-Hewitt, W. Uniaxial Compaction Modelled Using theProperties of Single Crystals. Drug Dev. Ind. Pharm. 1993, 19,2197-2240.

10. Hiestand, E. N.; Smith, D. P. Tablet Bond. II. ExperimentalCheck of Model. Int. J. Pharm. 1991, 67, 231-246.

11. Hiestand, E. N. Tablet Bond. I. A theoretical model. Int. J.Pharm. 1991, 67, 217-229.

Journal of Pharmaceutical Sciences / 989Vol. 86, No. 9, September 1997

12. Johnson, K. L.; Kendall, K.; Roberts, A. D. Surface energy andthe contact of elastic solids. Proc. R. Soc. London 1971, A324,301-313.

13. Maugis, D.; Pollock, H. M. Surface Forces, Deformation andAdherence at Metal Microcontacts. Acta Metall. 1984, 32, 1323-1334.

14. Maugis, D.; Barquins, M. Adhesive Contact of SectionallySmooth-Ended Punches on Elastic Half-spaces: Theory andExperiment. J. Phys. D: Appl. Phys 1983, 16, 1843-1874.

15. Greenwood, J. A.; Johnson, K. L. The Mechanics of Adhesion ofViscoelastic Solids. Phil. Mag. 1981, A 43, 697-711.

16. Mollan Jr., M. J.; Celik, M. The Effects of Lubrication on theCompaction and Post-compaction Properties of Directly Com-pressible Maltodextrins. Int. J. Pharm. 1996, 144, 1-9.

17. Coffin-Beach, D.; Hollenbeck, R. G. Determination of the Energyof Tablet Formation During Compression of Selected Pharma-ceutical Powders. Int. J. Pharm. 1983, 17, 313-324.

18. Ragnarsson, G. Force-Displacement and Network Measure-ments. In Pharmaceutical Powder Compaction Technology;Alderborn, G.; Nystrom, C., Eds.; Marcel Dekker, Inc. New York,1996, pp. 77-97.

19. Beerbower, A. Surface Free Energy: A New Relationship to BulkEnergies. J. Colloid Interface Sci. 1971, 35, 126-132.

20. Barton, A. F. M. CRC Handbook of Solubility Parameters andOther Cohesion Parameters. CRC Press, Inc.: Boca Raton, FL,1983.

21. Kitaigorodskii, A. I.Molecular Crystals andMolecules. AcademicPress, New York, 1973.

22. Nystrom, C., Alderborn, G., Duberg, M. and Karehill, P. BondingSurface Area and Bonding MechanismsTwo Important Factorsfor the Understanding of Powders and Compactability. DrugDev. Ind. Pharm. 1993, 19, 2143-2196.

23. Rippie, E. G.; Morehead, W. T. Structure Evolution of Tabletsduring Compression Unloading. J. Pharm. Sci. 1994, 83, 708-715.

24. Williams, J. G. Stress Analysis of Polymers, 2nd ed.; EllisHorwood Ltd.: Chichester, UK, 1980; p 124.

25. Danielson, D. W.; Morehead, W. T.; Rippie, E. G. Unloading andPost-compression Viscoelastic Stress versus Strain Behavior ofPharmaceutical Solids. J. Pharm. Sci. 1983, 72, 342-348.

26. Leuenberger, H. The Compressibility and Compactibility ofPowder Systems. Int. J. Pharm. 1982, 12, 41-47.

27. Holman, L.; Leuenberger, H. The Relationship between SolidFraction and Mechanical Properties of CompactssThe Percola-tion Theory Model Approach. Int. J. Pharm. 1988, 46, 35-44.

28. Roberts, R. J.; Rowe, R. C. The Effect of Punch Velocity on theCompaction of a Variety of Materials. J. Pharm. Pharmacol.1985, 37, 377-384 .

29. Park, J. B. Biomaterials Science and Engineering; PlenumPress: New York, 1984.

30. Greenwood, J. A.; Williamson, J. B. Contact of Nominally FlatSurfaces. Proc. Roy. Soc. London 1966, A 295, 300-319.

31. Lee, E. H.; Radok, J. R. M. The Contact Problem for ViscoelasticBodies. J. Appl. Mech, 27, Trans. ASME, Series E, 1960, 82,Sept 438-444.

32. Church, M; Kennerley, J. A Comparison of the MechanicalProperties of Pharmaceutical Materials Obtained by the FlexureTesting of Compacted Rectangular Beams. J. Pharm. Pharma-col. 1983, 35S, 43P.

33. Hiestand, E. Rationale for and the Measurement of TabletingIndices. In Pharmaceutical Powder Compaction Technology;Alderborn, G., Nystrom, C., Eds.; Marcel Dekker, Inc: NewYork, 1996; pp 219-244.

34. Lin, Meng-Chih; Duncan-Hewitt, W. Deformation Kinetics ofAcetaminophen Crystals. Int. J. Pharm. 1994, 106, 187-200.

35. Shima, S.; Oyane, M. Plasticity Theory for Porous Metals. Int.J. Mech. Sci. 1976, 18, 285-291.

36. Fukumori, Y.; Okada, J. Analysis of the Stress and StrainDistributions in the Compressed Powder in the Limiting Equi-librium States Specified by the Mohr Criterion. Theory andMeasurement of Mechanical Properties. Chem. Pharm. Bull,1977, 25, 1610-1635.

37. Heckel, R. W. Density-Pressure Relationships in Powder Com-paction. Trans. Metall. Soc. A.I.M.E. 1961, 221, 671-675; AnAnalysis of Powder Compaction Phenomena. Ibid. 1001-1008.

38. Paronen, P.; Ilkka, J. Porosity-Pressure Functions. In Pharma-ceutical Powder Compaction Technology; Alderborn, G., Nystrom,C., Eds.; Marcel Dekker, Inc.: New York, 1996, pp. 55-75.

39. Podczeck, F. Assessment of the Mode of Adherence and theDeformation Characteristics of Micronized Particles Adheringto Various Surfaces. Int. J. Pharm. 1996, 145, 65-76.

40. Rue, P.; Barkworth, P.; Ridgeway-Watt, P.; Rough, P.; Sharland,D.; Seager, H.; Fisher, H. Analysis of Tablet Fracture DuringTabletting by Acoustic Emission Techniques. Int. J. Pharm.Tech. Prod. Mfr. 1979, 1 (1), 2-5 .

41. Hiestand, E. N.; Wells, J. E.; Peot, C. B.; Ochs, J. F. PhysicalProcesses of Tableting. J. Pharm. Sci. 1977, 66, 510-519.

42. Carr, R. L. Evaluating Flow Properties of Solids. Chem. Eng.1965, 72, 163-168.

43. Hiestand, E. N. The Basis for Practical Applications of theTableting Indices. Pharm. Tech. 1989, 8 (9), 54-66.

44. Hiestand E. N.; Peot, C. B. Tensile Strength of CompressedPowders and an Example of Incompatibility as End-Point onShear Yield Locus. J. Pharm. Sci. 1974, 63, 605-612.

45. Watt, P. R. Interpretation of Data. In Tablet Machine Instru-mentation Pharmaceutics, Principles and Practice; HalstedPress: New York, 1988; Chapter 20, pp 397-428.

46. Tabor, D. The Hardness of Metals; Oxford University Press:Amen House, London, 1951..

47. Tabor, D. The Hardness of Solids. Rev. Phy. Technol. 1970, 1,145-179.

48. Hiestand, E. N.; Bane, J. M., Jr., Strzelinski, E. P. Impact Testfor Hardness of Compressed Powder Compacts. J. Pharm. Sci.1971, 60, 758-763.

49. In the Pharmacia & Upjohn laboratories, σT/Ho has been foundfor a given material to be an inverse flowability index: Amidon,G. E. Physical and Mechanical Property Characterization ofPowders. In Physical Characterization of Pharmaceutical Solids,Brittian, H., Ed.; Marcel Dekker, Inc.: New York, 1995; Chapter10, pp 281-319. The potential role of the strain index wasmentioned earlier in the text.

50. Formulations lacking ruggedness may fail or succeed in produc-tion because of machine choices, machine setup, tooling choices,storage humidity, etc. While the indices cannot sort amongthese, they predict that production problems would not be asurprise.

JS9701061

990 / Journal of Pharmaceutical SciencesVol. 86, No. 9, September 1997