Chapter Seven (2).pdfhhhhgggggfdfgjkkjff

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About This Presentation

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Slide Content

Chapter seven
Rheology
1

Contents
•Introduction
•Newtonian and non-Newtonian systems
•Thixotropy
•Determination of viscosity
•Factors affecting rheological properties
•Pharmaceutical applications of rheology
2

Objectives
At the end of this lesson you will be able to:
Definerheology
DifferentiateNewtonianandnon-newtoniansystems
Determinethixotropy
Differentiatefactorsaffectingrheology
Determineviscosity
Applyrheologyinpharmaceuticalarena
3

•Rheology(Greekwords),rheo‘toflow’andlogos‘science’.
•Definedasthescienceconcernedwiththeflowanddeformation
propertiesofmatterundertheinfluenceofshearingstress.
•Theflowoffluids(liquidsandgases)andthedeformationofsolids
undertheinfluenceofshearingstress.
•Tensilestress:whenthestressisappliedperpendiculartothe
surfaceofthebody.
•Shearingstress:whenthestressisappliedtangentiallytothe
surfaceofabody.
•Sheardeformation:isameasureofthesamplesdeformation
(strain)
4
Introduction

•Thedeformationofanypharmaceuticalsystemcanbearbitrarily
dividedintotwotypes:
•ElasticDeformation:itisaspontaneousandreversible
deformation.
•Plasticdeformation:itisapermanentorirreversible
deformation.
•plasticdeformation:promotesflowandexhibitedbyviscousbodies.
Greatimportanceinanyliquiddosageformslike
suspensions,solutions,emulsions,etc.
5
Introduction….

Introduction…
•Idealsolidsdeformelastically
Theenergyrequiredforthedeformationisfullyrecoveredwhenthe
stressesareremoved.
•Idealfluidssuchasliquidsandgasesdeformirreversibly,thenthey
flow.
Theenergyrequiredforthedeformationisdissipatedwithinthefluid
intheformofheatandcannotberecoveredsimplybyremovingthe
stresses.
6

The deformation is recoverableif
the material returns to its initial
shape when the stress is removed.
Deformation is permanentif
the material remains deformed
when the stress is removed.
Whileelasticdeformationisrecoverable,plasticdeformationsarenot.
7
Recoverable vs permanent deformations

Types of materials based on flow
8
Materialsareclassifiedintotwogeneraltypesdepending
upontheirflowproperties:
Newtonian
Non-Newtonian
Theflowofliquidsorsemisolidsisdescribedbyviscosity.

Measure of the resistance of
liquid to deformationunder
shear stress.
9
Afluid’sinternalresistancetoflow
andmaybethoughtofasa
measureoffluidfriction.
The resistance offered when one
part of the liquid flows past
another.
Viscosity
Newtonian fluids: Viscosity

Newtonian fluids: Viscosity…
•Thephenomenonofviscosityisbestunderstoodbya
considerationofahypotheticalcubeoffluidmadeupof
infinitelythinlayers(laminae)whichareabletoslide
overoneanotherlikeapackofplayingcards.
10

Newtonian fluids: Viscosity…
•Thebottomlayerisconsideredtobefixedinplace.
•Theadhesiveforcebetweenthewallandtheflowinglayers.
•Inter-molecularcohesiveforces
Thisinter-molecularforce(viscousforces):viscosity
•Viscosityistheresistancetoflow,itischaracteristicofthe
medium.
•Inotherwords,viscositydescribestheinternalfrictionofa
movingfluid.
11

Newtonian fluids: Viscosity…
12

Newtonian fluids: Viscosity…
•Whenatangentialforceisappliedtotheuppermostlayer
Eachsubsequentlayerwillmoveatprogressively
decreasingvelocitythatthebottomlayerwillbe
stationary.
•Avelocitygradientwillthereforeexist,
•equalstothevelocity(ms
-1
)oftheupperlayer/theh
(m)ofthecube.
13

Newtonian fluids: Viscosity…
Thedifferenceinvelocity(dv)b/ntwoplanesofa
liquidseparatedbyaninfinitesimaldistance(dx)isthe
velocitygradientorarateofshear(dv/dxorG).
14
Shear rate (G) = dv/ dx

Newtonian fluids: Viscosity…
•Theforcerequiredforonelayerofaliquidtoslippastoveranother
layerwithagivenvelocitydepends
Directlyontheviscosityoftheliquidandontheareasoflayers
exposedtoeachotherand
Inverselyonthedistanceseparatingthetwosurfaces.
•Afluidwithhighviscosityresistsmotionbecauseitsmolecular
makeupgivesitalotofinternalfrictionwhenitisinmotion.
15

Newtonian fluids: Viscosity…
Theforceperunitarea(F/A)neededtobringaboutthis
flowisknownastheshearingstress,S.
Shear stress (S) = F/ A
16

Newtonian fluids: Viscosity…
•Newtonrecognizedthat:thehighertheviscosityofaliquid,the
greatertheforceperunitarearequiredtoproduceacertainrate
ofshear.
•Thus, the rate of shear is to the shearing stress
Newtonian law
17dr
dv
A
F

' dr
dv
A
F

' ShearofRate
StressShearing
G
S
dr
dv
A
F

'

Where(eta)isaconstantknownascoefficientofviscosity,
usuallyreferredtosimplyasviscosityorabsoluteviscosity.
Absolute viscosity

Newtonian fluids: Viscosity…
•Where,
•Sis the shearing stress (dyne.cm
−2
)
•G is the rate of shear (sec
−1
)
•Units: dynes-sec/cm
2
or poise
•SI unit: N-sec/m
2
•1 N-sec/m
2
= 10 poise
18
•1poiseisdefinedastheFrequiredtoproduceavelocity
differenceof1cm/secbetweentwoparallellayersofliquidsof
1cm
2
areaeachandseparatedby1cmdistance.

•Itistheabsoluteviscositydividedbythedensityofliquid
ataspecifiedtemperature.
•Thepreferredunitwhentheshearstressandshearrate
ofthefluidareinfluencedbythedensityis:
Stock(s)orcentistokes
19ρ
η
KV
Newtonian fluids: Kinematic Viscosity

•LiquidsthatobeyNewton’slawofflow:Newtonianliquids
•Newton’sequationfortheflowofaliquidis:
•PlotofshearstressVsshearrate:theslopegivestheviscosity.
•Thecurvealwayspassesthroughtheorigin
•Thereciprocalofviscosityiscalledfluidity:
1/η=Slope
20
S = η * G
Newtonian fluids

•Thepassagethroughtheoriginindicatesthatevenamild
forcecaninduceflowinthesesystems.
•Thelinearnatureofthecurveshowsthattheviscosity(η)
ofaNewtonianliquidisaconstantunaffectedbythevalue
oftherateofshear.
21
Characteristics of Newtonian flow

Newtonian fluids…
•Thusasingledeterminationofviscosityfromtheshearstressatany
givenshearrateissufficienttocharacterizetheflowpropertiesofa
Newtonianliquid.
•ExamplesofNewtonianfluidsinclude:
•Water
•Air
•Glycerinandmostmineraloil
•Truesolutions
•Verydilutesuspensionandemulsions
•Liquidparaffin
22

Non-Newtonian fluids
Anon-Newtonianfluidisdefinedasoneforwhichthe
relationshipbetweenF’andSisnotaconstant.
•Inotherwords,SubstancesthatfailtofollowtheNewtonian
equationofflow.
•Theviscositiesofnon-Newtonianfluidsvarywithshearrate.
•Statingasingleviscosityismisleading:apparentviscosity(I’e
viscositykeepsonchanging)
23

•Itcanbeseeninliquidandsolidheterogeneousdispersions
suchascolloids,emulsions,liquidsuspensionsandointments.
•Threeclasses:
Plasticflow
Pseudo-plasticflow
Dilatantflow
24
Non-Newtonian fluids…

•Aplasticmaterialdoesnotflowuntilacertainminimumshearing
stress,theyieldvalue,isapplied.
•Theydoesnotbegintoflowuntilashearingstresscorrespondingtothe
yieldvalueisexceeded.
•Thesubstanceinitiallybehaveslikeanelasticbodyandfailstoflow
whenlessamountofstressisapplied(atstressesbelowtheyield
value).
•Furtherincreaseinthestressleadstoanonlinearincreaseinthe
shearratewhichthenturnstolinearity.
25
Plastic flow (Bingham bodies)

•Theslopeoftherheogram
istermedasmobility,
analogoustofluidityin
Newtoniansystemsandits
reciprocalisknownasthe
Plasticviscosity,U
26G
fF
U


Plastic flow…

Plastic flow…
•Extrapolationsofthelinearplotgives‘x’intersect:yield
value
•Thiscurvedoesnotpassthroughtheorigin
•Intersectstheshearstressaxisataparticularpoint,yield
value
•Asthecurveaboveyieldvaluetendstobestraight,the
plasticflowissimilartotheNewtonianflowaboveyield
value
27

28
Shear
Flocculatedparticlesinaconcentratedsuspensionsusually
showplasticflow.
Flocculated Suspension: Mechanism of Plastic Flow

Plastic flow…
•Theyieldvaluerepresentsthestressrequiredtobreaktheinter-
particularcontactssothatparticlesbehaveindividually.
•Thusitisindicativeoftheforcesofflocculation
increasedbytheincreasedconcentrationofthe
dispersedphase.
•The more flocculated the suspension the higher will be the yield value.
29

Plastic flow…
30

Plastic flow…
•Otherexamplesofplasticflow:
Solidpowdermaterials
Topicalointments
Pastes
•Powdermaterialsthatarecompressedeasilyhavelowyieldpoint(highly
compressibleatminimumshearstress).
•Materialswithlowyieldvaluecompressedeasilysothismaterialscanbe
usedasabinder.
•Granulationofpowdermaterialscausesincreasedcompressibilitydueto
porosityandloweringofyieldvalue.
31

•Therelationshipbetweenshearstressandtheshearrateisnot
linearandthecurvestartsfromorigin.
•Theviscosityoftheseliquidscannotbeexpressedbyasinglevalue.
•Pseudoplasticflowisexhibitedbypolymerdispersionslike:
•Tragacanthinwater
•Sodiumalginateinwater
•Methylcelluloseinwater
•Sodiumcarboxymethylcelluloseinwater
•Most of Cream (o/w or w/o), gelliesand semisolids lotion
32
Pseudo-plastic Flow (shear-thinning)

Pseudo-plastic Flow…
•The viscosity of a pseudo-
plastic substance decreases
with increasing rate of shear
(shear-thinning systems)
33

•Atrest,thelinearpolymersdispersedatrandominthe
dispersionmedium.
•Asshearingstressisapplied,themacromoleculesbecome
alignedwiththelongaxisparalleltothedirectionofflow.
•Withthisorderedalignmentthemoleculespassoneanother
withlessfrictionalresistance,andtheviscosityisdecreased.
34
Pseudo-plastic flow behavior; Structural reasons

Pseudo-plastic Flow…
35

36
Pseudo-plastic Flow…

•IftheSisdecreased,theorientationofthemacromolecules
becomesmorerandom.
•Greaterfrictionalresistancetotheflowisreflectedinan
increasedviscosity.
•Sinceonlyamolecularalignmentisinvolved:nolagtime.
•E.g.Emulsionsanddispersionsofmanytypes:shear-thinning.`
37
Pseudo-plastic Flow…

•Resistancetoflow(viscosity)increaseswithincreaseinshearstress.
WhenSisappliedtheirvolumeincreasesandhencetheyarecalled
Dilatant.
Thispropertyisalsoknownasshearthickening.
•Dilatantflowisobservedinsuspensionscontaining
morethan50%W/Vofsolids,deflocculatedparticles.
•Deflocculatedwithlessthanorequalto5%or10%showspsudo-
plasticflow.
•Suspendingagentsuchashydrophilliccolloidslikebentonite,
magnisiumalmuniumsilicatesowsdilatantflow.
38
Dilatant Flow

Dilatant Flow…
39

Dilatant Flow…
•Atrest,theparticlesarecloselypackedwithaminimuminter-
particlevolume,orvoids.
•Thevehicleissufficienttofillthisvolume.
•Astheshearstressisincreased,thebulkofthesystemexpandsor
dilates.
Increaseintheinter-particlevoidvolume
Vehiclebecomeinsufficienttofilltheincreasedvoids.
Theresistancetoflowincreases:Thisprocessisreversible.
40

41
Dilatant Flow…

42
Summary of time independent flows

Non-Newtonian,TimeDependentbehavior
•Definition:Thixotropyisthedecreaseinviscosityasafunction
oftimeuponshearing,thenrecoveryoforiginalviscosityasa
functionoftimewithoutshearing.
•Itischaracteristicsfeatureofpseudo-plasticflow.
•Definedastheisothermalslowreversibleconversionofgelto
solution.
43
Thixotropy

•Thixotropicsubstancesonapplyingshearstressconvert
tosol(fluid)andonstandingtheyslowlyturntogel
(semisolid).
44
Thixotropy…

Thixotropy…
•Thixotropic systems are usually composed of asymmetric
particles or macromolecules which are capable of interacting
by numerous secondary bonds to produce a loose three-
dimensional structure, so that the material is gel-like when
unsheared.
•The energy imparted during shearing disrupts these bonds, so
that the flowing elements become aligned and the viscosity
falls, as a gel-sol transformation has occurred.
45

Thixotropy…
•When the shear stress is eventually removed the structure will
tend to reform, although the process is not immediate and will
increase with time as the molecules return to the original state
under the influence of Brownian motion.
•Furthermore, the time taken for recovery, which can vary from
minutes to days depending upon the system, will be directly
related to the length of time the material was subjected to the
shear stress, as this will affect the degree of breakdown.
46

•Thixotropicsystemscontainasymmetricparticleswhichset
upaloosethreedimensionalstructurewhichisrigidand
resemblesagelwhichisbrokendownonapplyingshear:
Solution
•Onremovingtheappliedstress,thematerialreformits
originalstructureofstate:Gel
47
Gel
Solution
Reason for thixotropic property

Reason for thixotropic…
48

•Rheogram:showahysteresiswhere,astheSisincreased
anup-curveisobtained,onreducingtheSgradually,a
downcurveshiftedtotheleftsideisobtained.
49
Thixotropy…

Thixotropy…
•With shear thinning system the down ward curve is frequently
displaced to the left of the up curve, showing that the material
has a lower consistency at any one rate of shear on the down
curve than it had on the up curve.
•This indicates the break down of structure that does not reform
immediately when the stress is removed.
•The presence of the hysteresis loop indicates that a breakdown in
structure has occurred, and the area within the loop may be used
as an index of the degree of breakdown
50

•Rheogram of thixotropic
materialsdependson:
Appliedshearrateorshearing
force.
Thelengthofthetimea
sampleissubjectedto
shearingorkinematichistory.
51
Theprevioushistoryofthesamplehassignificanteffectonthe
rheologicpropertiesofathixotropicsystem.
Thixotropy…

Thixotropy…
•The area of the hysteresis loop tells us that the time taken by the
material to regain its initial position.
•Large area taking more time to come into its initial position.
•It stands to reason that, the larger the enclosed area, more severe
is the time-dependent behavior of the material under discussion.
•Evidently, the enclosed area would be zero for a purely viscous fluid,
i.e., no hysteresis effect is expected for time-independent fluids.
52

•Thixotropicsubstancesarenowaday’smoreusedinsuspensions
togivestablesuspensions.
•Onstorageturntogel:viscosityincreasesinfinitely,donot
allowthedispersedparticlestosettledown:stable
suspension.
•Whenshearstressisappliedtheyturntosolutionandthusare
easytopourandmeasurefordosing.
•Solvetheproblemsofstabilityandpourability.
53
Thixotropy…

•Examplesofgelsthatshowthixotropicflowincludealuminum
hydroxidegelandbentonitemagma.
•Paintsarealsoanotherexampleofthixotropicfluids.
•Whenmodernpaintsareapplied,theshearcreatedbythebrushor
rollerwillallowthemtothinandwetoutthesurfaceevenly.
•Onceappliedthepaintsregaintheirhigherviscositywhichavoids
dripsandruns.
54
Thixotropy…

•ApplicationofFcausesbreakdownofstructurewithinthesystembut
•ThestructuredoesnotreformonremovaloftheF,or
•Thetimelagissolongthatfromapracticalpointofviewtheeffect
isirreversible.
Example:gelsproducedfromhigherM.Wtpolysaccharides
•Onapplicationofhighshear,
•The3Dstructureofthepolysaccharidesisreducedtoa2Done
and
•Theoriginalgel-likestructureisneverrecovered
55
Irreversible Thixotropy

•Antithixotropyisbelievedtoresultfromanincreasedcollisionfrequency
ofthedispersedparticlesinsuspensionswhichresultsinanincreased
inter-particlebondingwithtime.
•Thischangestheoriginalstateofthesystemfromanetworkofalarge
numberofindividualparticlesandsmallflocculestoanequilibriumstate
consistingofasmallnumberofrelativelylargefloccules:solution
Whichhelpsthedispersedparticlestoacquirearandomorientation
andthenetworkisestablished:Gel
Antithixotropicsystemshavelowsolidcontent(1-10%)andare
flocculated
56
Negative thixotropic

Negative thixotropic…
•Atrestthelargeflocculesbreakupandgraduallyreturn
totheoriginalstateofsmallflocculesandindividual
particles
viscosity decreases: Solution
57

•Rheopexyistheconversionofsolidstogeluponapplicationof
shearingstress.
•Substanceswithlowmeltingpointshowsrheopecticproperty.
•Example:polymerswithlowmeltingpoint
•Rheopecticfluidsarerarelyencountered
•Aretime-dependentdilatantbehavior.
58
Rheopexy

Rheopexy…
•The process of reformation of the gel structure after it has been deformed
can be accelerated by applying gentle and regular movement (rolling and
rocking motion).
•The equilibrium stateis gel.Just as the opposite behaviourof becoming
thinner with time is thixotropism(time dependent pseudoplastic
behaviour), rheopectic behaviourmay be described as time-dependent
dilatant behaviour.
•In a rheopectic system , the gel is the equilibrium form, whereas in
antithixothropy, the equilibrium state is solution
59

•Differentequipmentscalledviscometersareusedtomeasureviscosityof
differentfluidsandsemisolids.
•Successfuldeterminationofviscositydependsonthechoiceofthe
correctinstrument.
•ForaNewtoniansystem;
•Instrumentsthatoperateatasinglerateofshearsuchascapillary
viscometerscanbeused.
•Fornon-Newtoniansystems
•Multi-pointinstrumentsthatcanoperateatdifferentratesofshear
arerequired. 60
Determination of Viscosity

•It is a type of capillary viscometer
•This is ‘U’ shape tube with two bulbs and two marks
61
Ostwald Viscometer

Ostwald Viscometer…
•It is used to determine the viscosity of Newtonian liquids
Principle:
•Whenaliquidflowsbygravity,thetimerequiredfortheliquid
topassbetweentwomarks,uppermarkandlowermark,
throughaverticalcapillarytubeisdetermined.
•Thetimeofflowoftheliquidundertestiscomparedwiththe
timerequiredforaliquidofknownviscosity(usuallywater).
62

Ostwald Viscometer…
63
The viscosity of unknown liquid η
1 can be determined using the
equation
ρ
1= Density of unknown liquid
ρ
2= Density of known liquid
t
1= Time of the unknown liquid
t
2= Time of the known liquid
η
2= Viscosity of known liquid 2
22
11
1




t
t

•Consistsofcylindricaltransparenttubehavinggraduated
sectionnearthemiddleofitslengthandgenerallyasteel
ballthatisallowedtofallthroughthetube.
64
Falling Sphere Viscometer

Falling Sphere Viscometer…
•Thetubeisfilledwiththeliquidwhoseviscosityistobedeterminedand
theballisallowedtofall.
•Thevelocityofthefallingballismeasuredandviscosityiscalculated
usingstoke’slaw.
where d= Diameter of the falling ball; ρ
s=Density of the sphere;
ρ
l=Density of liquid; g= Gravitational acceleration; v = Terminal settling
velocity
65V
gd
ls
18
)(
2



•Asd
2
g/18isconstant,canbereplacedbyanotherconstant‘K'
Therefore,theequationwillbe:
•Stoke’slawstatesthatwhenabodyfallsthroughaviscous
medium,itexperiencesaresistancewhichopposesthemotionof
thebody
66V
K
ls)(



Falling Sphere Viscometer….

Chemicalfactors
ExtentofPolymerHydration
•Inhydrophilicpolymersolution,themoleculesareregardedas
completelysurroundedbyimmobilizedwatermoleculesforminga
solventlayer.
•Suchhydrationofhydrophilicpolymersgivesrisetoanincreased
viscosity.
67
Factors affecting rheological properties

Factors affecting rheological…
Impurities,TraceIonsandElectrolytes
•Chemicalimpuritiesarethemajorfactorsinchangingtheviscosity
ofnaturalpolymers.
•Athighconc. competeforadsorbedwatermolecules
surroundingthehydratedpolymer thepolymerbecome
dehydrated theviscosityofthedispersiondecreased
finallyprecipitationoccurs.
68

Factors affecting rheological…
Physicalfactor
Temperature
•Atemperatureincreaseusuallyproducesarapidviscosity
decrease.
•Prolongedheatingmayproducedrasticdecreaseinviscositydueto
decompositionofthepolymer.
e.g.,gelatin
AsTincreasestheviscosityofgasesincreaseswhereasasTincreasesthe
viscosityofliquidsdecreases.AsTincreaseskinematicviscosityof
liquidsdecrease.
69

Concentration
•Inconcentratedsuspensions:adecreaseinparticlesizeoran
increaseinthesurfaceareaofthesolidphaseproduceanincrease
intheviscosityofthesystem.
•Thisisduetoimmobilizationofthevehiclewithanincreasein
thefractionofthesuspensionvolumeoccupiedbythesolid.
70
Factors affecting rheological….

Pharmaceutical applications of rheology
1.Fluids
•Mixing
•Particlesizereductionofdispersedsystemswithshear.
•Passagethroughorifices,includingpouring,packagein
bottles,andpassagethroughhypodermicneedles.
•Fluidtransfer,includingpumpingandflowthrough
pumps.
•Physicalstabilityofdispersesystems.
71

Pharmaceutical applications…
2.Quasisolids(semisolids)
-Spreadingandadherenceontheskin
-Removalsfromjarsorextrusionfromtubes
-Releaseofdrugfrombase
3.Solids
-Flowofpowdersfromhoppersandintodiecavities
intabletingorintocapsulesduringencapsulation.
-Packagabilityofpowderedorgranularsolids
4.Processing
-Productioncapacityoftheequipment
-Processingefficiency
72

Pharmaceutical applications…
1-ProlongationofDrugAction:
•Therateofabsorptionofanordinarysuspensiondiffersfrom
thixotropicsuspension.
•Example:procainepenicillinG,aformofpenicillin,ofrelatively
lowwatersolubility.
•Aqueoussuspensionscontainingbetween40and70%w/vofmilled
ormicronizedprocainepenicillinG+smallamountofsodium
citrate&polysorbate80arethixotropicpastes&areofdepot
effectwheninjectedintramuscularly.
73

Ordinary suspension of
pencillinG
Thixotropysuspension of
pencillinG
I.M injection
Forms no depot, fast dispersion
&
absorption so maintain therapeutic
Level for short time
Forms spherical deposits at site of
injection which resists disintegration
by tissue fluids& Small surface area
( absorption) so maintain
Therapeutic Level for longer time
The formation of depot depends on:
a-high yield value
b-fast thixotropicrecovery after injection.
74

(2 )Effect on Drug Absorption:
•Theviscosityofcreamsandlotionsmayaffecttherateof
absorptionoftheproductsbytheskin.
•Agreaterreleaseofactiveingredientsisgenerallypossiblefrom
thesofter,lessviscousbases.
•Theviscosityofsemi-solidproductsmayaffectabsorptionofthese
topicalproductsduetotheeffectofviscosityontherateof
diffusionoftheactiveingredients.
75

(3) Thixotropyin Suspension and Emulsion
Formulation:
•Thixotropyisusefulintheformulationofpharmaceuticalsuspensionsand
emulsions.Theymustbepouredeasilyfromcontainers(lowviscosity)
•DisadvantagesofLowviscosity:
•Rapidsettlingofsolidparticlesinsuspensionsandrapidcreamingof
emulsions.
•Solidparticles,whichhavesettledoutsticktogether,producingsediment
difficulttoredisperse("cakingorclaying").
•Creaminginemulsionsisafirststeptowardscoalescence.(breakdownof
emulsion)
76

•Athixotropicagentsuchassodiumbentonitemagma,colloidal
silicondioxide,isincorporatedintothesuspensionsor
emulsionstoconferahighapparentviscosityorevenayield
value.
77
•Theagitationbreaksdownthethixotropic
structuresoreducingtheyieldvaluetozero&
loweringtheapparentviscosity.Thisfacilitates
pouring.

Thank you
78