Colloid and interface science in pharmaceutical research and development

509 Pages·2014·27.572 MB·English

by  Hiroyuki Ohshima, Kimiko Makino

Colloid and
Interface Science
in Pharmaceutical
Research and
Development
Colloid and
Interface Science
in Pharmaceutical
Research and
Development
Hiroyuki Ohshima
Kimiko Makino
AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON (cid:129) NEW YORK (cid:129) OXFORD
PARIS (cid:129) SAN DIEGO (cid:129) SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO
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ISBN:978-0-444-62614-1
CHAPTER
1
Interaction of colloidal
particles
Hiroyuki Ohshima
FacultyofPharmaceuticalSciences,TokyoUniversityofScience,2641
Yamazaki,Noda,Chiba278-8510,Japan
CHAPTER CONTENTS
1.1 Introduction ........................................................................................................2
1.2 PotentialDistributionAroundaChargedSurface:thePoisson–Boltzmann
equation ............................................................................................................2
1.2.1 HardParticle .....................................................................................3
1.2.2 SoftParticles ....................................................................................6
1.3 ElectricalDoubleLayerInteractionBetweenTwoParticles ....................................8
1.3.1 LinearSuperpositionApproximation ....................................................9
1.3.2 Derjaguin’sApproximation ................................................................11
1.3.2.1 TwoSpheres ........................................................................12
1.3.2.2 TwoCylinders ......................................................................13
1.4 vanderWaalsInteractionBetweenTwoParticles ...............................................14
1.4.1 TwoMolecules.................................................................................15
1.4.2 AMoleculeandaPlate ....................................................................16
1.4.3 TwoParallelPlates ..........................................................................17
1.4.4 TwoSpheres ...................................................................................18
1.4.5 TwoCylinders ..................................................................................19
1.4.6 TwoParticlesImmersedinaMedium ................................................20
1.4.7 TwoParallelPlatesCoveredwithSurfaceLayers.................................21
1.5 DLVOTheoryofColloidStability .........................................................................23
1.5.1 TotalInteractionEnergyBetweenTwoSphericalParticles ...................23
1.5.2 PositionsofaPotentialMaximumandaSecondaryMinimum..............23
1.5.3 TheHeightofaPotentialMaximumandtheDepthofaSecondary
Minimum ........................................................................................26
1.5.4 StabilityMap...................................................................................26
1.6 Conclusion........................................................................................................27
References ..............................................................................................................27
1
ColloidandInterfaceScienceinPharmaceuticalResearchandDevelopment.http://dx.doi.org/10.1016/B978-0-444-62614-1.00001-6
©2014ElsevierB.V.Allrightsreserved.
2 CHAPTER 1 Interaction of colloidal particles
1.1 INTRODUCTION
Thestabilityofcolloidalsystemsconsistingofchargedparticlescanbeessentially
explained by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory [1–12].
Accordingtothistheory,thestabilityofasuspensionofcolloidalparticlesisdeter-
mined by the balance between the electrostatic interaction and the van der Waals
interactionbetweenparticles.Inthischapterwestartwiththeelectricaldoublelayer
around acharge particle inanelectrolyte solution (Figure1.1).We treatboth hard
particles and soft particles, i.e., polyelectrolyte-coated particles [8, 10, 13–15]
(Figure1.2).Wediscusstheelectrostaticinteractionbetweentwoapproachingpar-
ticles due to the overlapping of the electrical double layers around them. We then
considerthevanderWaalsinteractionbetweenparticles.Finallywediscussthesta-
bilityofacolloidalsuspensiononthebasisofthetotalinteractingenergy(theelec-
trostatic energy andthe vander Waals energy)between particles.
1.2 POTENTIAL DISTRIBUTION AROUND A CHARGED
SURFACE: THE POISSON–BOLTZMANN EQUATION
Aroundachargedcolloidalparticleimmersedinanelectrolytesolution,mobileelec-
trolyteionsformanioniccloudofthickness1/k(calledtheDebyelength),kbeing
the Debye–Hu¨ckel parameter (Figure 1.1). As a result of Coulomb interaction
FIGURE1.1
Electricaldoublelayerofthickness1/k(Debyelength)aroundasphericalchargedparticle.
1.2 Potential distribution around a charged surface 3
FIGURE1.2
Softparticle(polyelectrolyte-coatedparticle).
betweenelectrolyteionsandparticlesurfacecharges,intheioniccloudtheconcen-
trationofcounterions(electrolyteionswithchargesofthesignoppositetothatofthe
particlesurfacecharges)becomesmuchhigherthanthatofcoions(electrolyteions
with charges of the same sign as the particle surface charges). The ionic cloud
together with the particle surface charge forms an electrical double layer, which
isoftencalledanelectricaldiffusedoublelayer,sincethedistributionofelectrolyte
ionsintheioniccloudtakesadiffusivestructureduetothermalmotionofions.Elec-
trostaticinteractionsbetweencolloidalparticlesdependstronglyonthedistributions
ofelectrolyteionsandtheelectricpotentialacrosstheelectricaldoublelayeraround
the particle surface [1–12].
1.2.1 HARD PARTICLE
Firstweconsiderauniformlychargedplate-likehardparticleimmersedinaliquid
containingMionicspecieswithvalenceziandbulkconcentration(numberdensity)
ni1 (i¼1,2...M)(inunitsofm(cid:2)3).Wetakeanx-axisperpendiculartotheplatesur-
facewithitsorigin0sothattheregionx>0correspondstotheelectrolytesolution
(Figure 1.3(a)). From the electroneutrality condition, we have
XM
zin1i ¼0 (1.1)
i¼1
The electric potential c(x) at position x, measured relative to the bulk solution
phase,wherecissetequaltozero,isrelatedtothechargedensityr (x)atthesame
el
point by the Poisson equation, viz.,
d2c r ðxÞ
¼(cid:2) el (1.2)
dx2 EE
r 0
4 CHAPTER 1 Interaction of colloidal particles
+ + − +
− −
+ + + +
−
− + −
Particle + − − Solution Particle − + + Solution
surface + −−− + core + + − +
− − +
+ + + − + + −−
+ − + + + + + +
x − − − x
0 −d 0
Surface layer
Y(x) Y(x)
Y
DON
Y
0
Y
0
0 x 0 x
−d
1/k 1/k1/k
(a) (b)
FIGURE1.3
Ionandpotentialdistributionsaroundahardplate(a)andasoftplate(b).
whereE istherelativepermittivityoftheelectrolytesolution,andE isthepermit-
r 0
tivity of a vacuum. We assume that the distribution of the electrolyte ions obeys
Boltzmann’s low,viz.,
(cid:2) (cid:3)
niðxÞ¼n1i exp (cid:2)ziekcTðxÞ (1.3)
whereni(x)istheconcentration(numberdensity)oftheithionicspeciesatpositionx,
eistheelementaryelectriccharge,kisBoltzmann’sconstant,andTistheabsolute
temperature. Thecharge density r (x) at position xis thus given by
el
(cid:2) (cid:3)
relðxÞ¼XM zieniðxÞ¼XM zien1i exp (cid:2)ziekcTðxÞ (1.4)
i¼1 i¼1
Combining Eqs.(1.2) and (1.4) gives
(cid:2) (cid:3)
dd2xc2¼(cid:2)E1E XM zien1i exp (cid:2)ziekcTðxÞ (1.5)
r 0i¼1
ThisisthePoisson–Boltzmannequationforthepotentialdistributionc(x),which
is subject tothe following boundaryconditions:
cð0Þ¼c attheparticlesurface (1.6)
0
cðxÞ!0asx!1 (1.7)
1.2 Potential distribution around a charged surface 5
Iftheinternalelectricfieldsinsidetheparticlecanbeneglected,thenthesurface
charge density s of the particle is related to the potential derivative at the particle
surface as
(cid:4)
(cid:4)
dc(cid:4) s
(cid:4) ¼(cid:2) (1.8)
dx EE
x¼0þ r 0
Ifthe potential cislow, viz.,
(cid:4) (cid:4)
(cid:4)(cid:4)ziec(cid:4)(cid:4)
(cid:4) (cid:4)(cid:3)1 (1.9)
kT
then Eq. (1.5) reduces to the following linearised Poisson–Boltzmann equation
(Debye–Hu¨ckel equation):
d2c
¼k2c (1.10)
dx2
with
!
XM 1=2
1
k¼ EE kT z2ie2n1i (1.11)
r 0 i¼1
where k is called the Debye–Hu¨ckel parameter. The reciprocal of k (i.e., 1/k ),
whichiscalledtheDebyelength,correspondstothethicknessofthedoublelayer.
Notethatni1intheaboveequationsisgiveninunitsofm(cid:2)3.Ifoneusestheunitsof
M (mole/litre), then ni1 must be replaced by 1000NAni1, NA being Avogadro’s
number.
Linearised equation(1.10) can besolvedto give
cðxÞ¼c e(cid:2)kx (1.12)
0
wherethe surface potential c isrelatedto the surface chargedensity s as
0
s
c ¼ (1.13)
0 EE k
r 0
where Eq. (1.8) has been used. For arbitrary potentials c(x), we need to solve the
nonlinear Poisson–Boltzmann equation (1.5). This equation can easily be solved
for a planar surface in contact with a z–z symmetrical electrolyte solution of bulk
concentration n. Inthis case Eq. (1.5) with Eq.(1.11) becomes
d2y
¼k2sinhy (1.14)
dx2
with
(cid:2) (cid:3)
2z2e2n 1=2
k¼ (1.15)
EE kT
r 0
wherey(x)(cid:4)zec(x)/kTisthescaledpotential.Equation(1.14)canbesolvedtogive
(cid:2) (cid:3)
2kT 1þge(cid:2)kx
cðxÞ¼ ln (1.16)
ze 1(cid:2)ge(cid:2)kx
6 CHAPTER 1 Interaction of colloidal particles
with (cid:2) (cid:3)
zec
g¼tanh 0 (1.17)
4kT
Thesurface potential c isrelated tothe surface charge density s as
0
(cid:2) (cid:3)
2EE kkT zec
s¼ r 0 sinh 0 (1.18)
ze 2kT
Considertheasymptoticbehaviourofc(x)atlargex,whichwillbeusedlaterfor
calculating the electrostatic interaction between two particles. It follows from
Eq.(1.16) thatc(x) at large kxtakes the form
(cid:2) (cid:3)
4kT 4kT zec
cðxÞ¼ ge(cid:2)kx¼ tanh 0 e(cid:2)kx (1.19)
ze ze 4kT
ComparingEq.(1.19)withthelinearisedform(Eq.(1.12)),wefindthattheeffec-
tivesurface potential c of the plate isgiven by
eff
(cid:2) (cid:3)
4kT kT zec
c ¼ g¼ 4tanh 0 (1.20)
eff ze ze 4kT
Thepotentialdistributionina2–1electrolytesolutionandthatforamixedsolu-
tion of 1–1 and 2–1 electrolytes are given in Refs. [16, 17]. Also, the results for a
spherical particle ora cylindrical particle are given inRefs.[16–19].
1.2.2 SOFT PARTICLES
We next consider the case where the particle core is covered by an ion-penetrable
surfacelayerofpolyelectrolytes,whichwetermasurfacechargelayer(or,simply,
a surface layer). Polyelectrolyte-coated particles are called soft particles [8, 10,
13–15]. Soft particles serve as a model for biocolloids such as cells. Figure 1.3(b)
gives schematic representation of ion and potential distributions around a soft sur-
face,whichshowsthatthepotentialdeepinsidethesurfacelayerispracticablyequal
to the Donnan potential c , if the surface layer is much thicker than the Debye
DON
length1/k.Alsowetermc (cid:4)c(0)(whichisthepotentialattheboundarybetween
0
thesurfacelayerandthesurroundingelectrolytesolution)thesurfacepotentialofthe
polyelectrolyte layer.
Considerasurfacechargelayerofthicknessdcoatingaplanarhardsurfaceina
symmetricalelectrolytesolutionofbulkconcentrationnandvalencez.Wetreatthe
casewherefullyionisedgroupsofvalenceZaredistributedatauniformdensityofN
inthesurfacechargelayerandtheparticlecoreisuncharged.Wetakeanx-axisper-
pendiculartothesurfacechargelayerwithitsoriginx¼0attheboundarybetween
thesurfacechargelayerandthesurroundingelectrolytesolutionsothatthesurface
chargelayercorrespondstotheregion(cid:2)d<x<0andtheelectrolytesolutiontox>0
(Figure1.3(b)).ThePoisson–Boltzmannequationsfortheregionsinsideandoutside
the surface chargelayerare givenby
d2y
¼k2sinhy, 0<x<þ1 (1.21)
dx2
1.2 Potential distribution around a charged surface 7
(cid:5) (cid:6)
d2y ZeN ze
¼k2sinhy(cid:2) , (cid:2)d<x<0 (1.22)
dx2 EE kT
r 0
WehavehereassumedthattherelativepermittivityE takesthesamevalueinthe
r
regionsinsideandoutsidethesurfacechargelayer.NotethatEq.(1.21)istheusual
Poisson–Boltzmann equation (Eq. (1.14)) while the right-hand side of Eq. (1.22)
containsthecontributionofthefixed-chargesofdensityr ¼ZeNinthepolyelec-
fix
trolyte layer. Now we introduce the Donnan potential c , which is obtained by
DON
setting the right-hand side ofEq. (1.22) tozero,
(cid:2)kT(cid:3) (cid:2)ZN(cid:3) (cid:2)kT(cid:3) 2ZN ((cid:2)ZN(cid:3)2 )1=23
4 5
c ¼ arcsinh ¼ ln þ þ1 (1.23)
DON ze 2zn ze 2zn 2zn
then Eq.(1.22) can be rewritten as
d2y
¼k2ðsinhy(cid:2)sinhy Þ, (cid:2)d<x<0 (1.24)
dx2 DON
where y the scaled Donnan potential y (cid:4)zec /kT. The boundary condi-
DON DON DON
tions for Eqs. (1.21) and (1.24) are
(cid:4)
(cid:4)
dc(cid:4)
(cid:4) ¼0 (1.25)
dx
x¼(cid:2)dþ
(cid:4) (cid:4)
(cid:4) (cid:4)
cð(cid:2)0(cid:2)Þ¼cð(cid:2)0þÞanddc(cid:4)(cid:4) ¼dc(cid:4)(cid:4) (1.26)
dx dx
x¼(cid:2)0(cid:2) x¼(cid:2)0þ
dc
cðxÞ!0and !0asx!1 (1.27)
dx
Equation (1.25) corresponds to the situation in which the particle core is
uncharged. For the case where the surface layer d is much thicker than the Debye
length1/k,asimplerelationbetweenthesurfacepotentialc (cid:4)c(0)andthevolume
0
chargedensityr ¼ZeNcanbederivedasfollows.ByintegratingEqs.(1.21)and
fix
(1.24) once, we obtain
(cid:2) (cid:3)
dy 2
¼k2½coshy(cid:2)coshyð(cid:2)dÞ(cid:2)2sinhy fy(cid:2)yð(cid:2)dÞg(cid:5), (cid:2)d<x<0 (1.28)
dx DON
(cid:2) (cid:3)
dy 2
¼k2ðcoshy(cid:2)1Þ, 0<x<þ1 (1.29)
dx
Notethatifd(cid:6)1/k,theny((cid:2)d)ispracticallyequaltothescaledDonnanpoten-
tial.Byusingthisfact,evaluatingEqs.(1.28)and(1.29)atx¼0andequatingthem
with the help ofEq. (1.26),we finally obtain