Journal of Colloid and Interface Science 253, 70-76 (2002)
`doi:10.1006/jcis.2002.8476
`
`Stability of Nanodispersions: A Modelfor Kinetics
`of Aggregation of Nanoparticles
`
`Department ofChemistry, Faculty of Science, University ofZagreb, Maruliéev trg 19, P.O. Box 163, HR 10000 Zagreb, Croatia
`
`Nikola Kallay! and Suzana Zalac”
`
`Received October 29, 2001; accepted May 9, 2002
`
`In the course of aggregation of very small colloid particles
`(nanoparticles) the overlap of the diffuse layers is practically com-
`plete, so that one cannot apply the common DLVOtheory. Since
`nanopoarticles are small compared to the extentofthe diffuse layer,
`the process is considered in the samewayas for two interacting ions.
`Therefore, the Brgnsted concept based on the Transition State The-
`ory was applied. The charge of interacting nanoparticles was calcu-
`lated by means of the Surface Complexation Model and decrease of
`effective chargeofparticles was also taken into account. Numerical
`simulations were performed using the parameters for hematite and
`rutile colloid systems. The effect of pH and electrolyte concentra-
`tion on thestability coefficient ofnanosystems was found to be more
`pronounced but similar to that for regular colloidal systems. The
`effect markedly depends on the nature of the solid whichis char-
`acterized by equilibrium constants of surface reactions responsible
`for surface charge, i.e., by the point of zero charge, while the speci-
`ficity ofcounterions is described by their association affinity,i.e., by
`surface association equilibrium constants. The most pronounced is
`the particle size effect. It was shown that extremely small particles
`cannotbe stabilized by an electrostatic repulsion barrier. Addition-
`ally, at the same mass concentration, nanoparticles aggregate more
`rapidly than ordinary colloidal particles due to thier higher number
`concentration.
`© 2002 Elsevier Science (USA)
`Key Words:stability of nanodispersions.
`
`INTRODUCTION
`
`It is commonly accepted that the stability ofcolloidal systems
`is, inmostcases, the result ofan extremely slow aggregation pro-
`cess. The main reason for such a slow aggregation processis a
`high electrostatic energy barrier, and in some cases a protective
`layer of adsorbed chains. The theory of Colloid Stability consid-
`ers collision frequency and efficiency (1, 2). Collision frequency
`wastheoretically solved by Smoluchowski (3), while the basis
`for evaluation ofthe collision efficiency was given by Fuchs(4).
`In orderto use the Fuchs theory one should knowthe interaction
`energy as a function of the distance between interacting parti-
`
`'To whom correspondence should be addressed. Fax: +385-1-4829958.
`E-mail: nkallay @ prelog.chem.pmf.hr.
`2 Present address: PLIVA d.d., R&D—Research,Prilaz barunaFilipovi¢a 25,
`HR 10000 Zagreb, Croatia.
`
`0021-9797/02 $35.00
`© 2002 Elsevier Science (USA)
`All rights reserved.
`
`cles. The effect of dispersion forces was solved by Hamaker(5),
`Bradly (6), and de Boer (7), while electrostatic repulsion could
`be evaluated on the basis of the Derjaguin, Landau, Vervey,
`Overbeek (DLVO)theory (1, 8). Recently, more sophisticated
`models were elaborated (9-14). In most of the cases the theory
`of Colloid Stability explains the experimental data, especially
`if the correct values of the electrostatic surface potentials, as
`obtained from the Surface Complexation model (15-19), are
`used (20-22). However, small particles, with sizes below 10 nm
`(called nanoparticles), generally do not show electrostatic sta-
`bilization. According to de Gennes (23), the reason for the in-
`stability of nanocolloidal systems might be in their low charge
`(surface charge density times surface area). In somecasessta-
`ble systems of nanoparticles could be prepared (24, 25) but no
`kinetic measurements were published.
`In this paper we analyze the theoretical aspect of the kinetics
`of aggregation of nanoparticles based on the Brg@nsted theory
`(26, 27), which was developed for the salt effect on the kinetics
`of ionic reactions (primary salt effect). The reason for such a
`choice lies in the fact that the classical DLVO approach cannot
`be used for nanoparticles: nanoparticles are small with respect to
`the thicknessofthe electrical diffuse layer, so that in the course
`of the collision of two nanoparticles a complete overlap of two
`diffuse layers takes place. Let us consider extension ofthe diffuse
`layer. According to the Gouy-Chapman theory, depending on
`the ionic strength and surface potential, the latter is reduced to
`10% of its original value at a distance of 2 to 2.5 reciprocal x
`values. This meansthat at the ionic strength of 10~? mol dm~3
`the diffuse layer is extended up to 6 nm from the surface. As
`shownon Fig. 1, in such a case overlap of diffuse layers of two
`nanoparticles is practically complete. In the case of ordinary
`colloid particles the overlap is partial so that the DLVO theory
`is applicable.
`A nanoparticle surrounded by a diffuse layer is similar to an
`ion situated in the centerofan ionic cloud. In the course ofcolli-
`sion two nanoparticles in contact have a commondiffuse layer or
`“fonic cloud.” Therefore, interaction of nanoparticles could be
`considered in a mannersimilar to that for two interacting ions,
`and consequently described by the Brgnsted theory. This theory
`considers the “transition state” or “activated complex” which is
`a pair of two interacting ions with a commonionic cloud. The
`
`70
`
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`Actavis LLC v. Abraxis Bioscience, LLC
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`
`

`

`
`
`FIG. 1. Overlap ofelectrical interfacial layers for two ordinary colloid par-
`ticles (r = 30 nm) and for two nanoparticles (r = 3 nm).
`
`STABILITY OF NANODISPERSIONS
`
`the concentration of the transition state
`
`v= k'[ABZ4+28] |
`
`71
`
`[4]
`
`where k’is the rate constant (coefficient) of the second process.
`Equilibration of the first step is fast so that one calculates the
`concentration of the transition state [AB*4t*®] from the rele-
`vant equilibrium constant K* taking into accountthe activity
`coefficients y of reactants and of the transition state
`
`[AB*4t28)
`yAB*At28)
`# . Jeae
`y(A*)y(B8) [Az][B2]
`
`(5)
`
`equilibration of the transition state is fast, while the transforma-
`tion of the transition state into product(s) is slow, and thus the
`rate determiningstep.
`
`The equilibrium constant K* is definedin termsofactivities, and
`consequently its value does not dependonthe ionicstrength;i.e.,
`it correspondsto infinite dilution. Equations [4] and[5] result in
`
`THEORY
`
`Introduction of the Bronsted Concept to Kinetics
`of Aggregation of Nanoparticles
`
`(peOo eres
`v=k KraBesy ][B* ].
`
`[6]
`
`According to the above equation, the overall rate constant, as
`defined by Eq.[2], is given by
`
`The quantitative interpretation of kinetics of aggregation of
`nanoparticles will follow the Brgnsted concept (26, 27). It will
`1eeVOA*)yYB*)
`be based on the Transition State theory using the activity coef-
`ficients as given by the Debye-Hiickellimiting law.
`Aggregation of two charged nanoparticles A** and B*® could
`be represented by
`
`A* + B® —> ABS,
`
`[1]
`
`where z denotes the charge number. The rate of aggregation v
`is proportional to the product of concentrations of interacting
`particles [A*4][B**]
`
`v = k[A**][B*],
`
`[2]
`
`where k is the rate constant (coefficient) of aggregation.
`According to the Brgnsted concept, in the course of aggrega-
`tion two charged nanoparticles undergo reversible formation of
`the transition state with charge numberbeing equal to the sum of
`the charges of interacting species. The transition state AB*4*7#
`undergoesthe next step (binding) which is slow andis therefore
`the rate determining step
`
`Aza + Bs <> ABAtzZB —} AB*48,
`
`[3]
`
`Note that equilibration of the interface may result in a change
`of the total charge of the doublet. In such a case zq + zg # Zap.
`Since the equilibration ofthe first step is fast, and the second
`processis slow, the overall rate of reaction (v) is proportionalto
`
`k=k Sey
`
`[7]
`
`It is clear that the overall rate constant k depends on the ionic
`strength of the medium through activity coefficients. Activity
`coefficients could be obtained from the Debye-Hiickel equation
`derived for ionic solutions. The same equation is assumedto be
`applicable for extremely small particles i of charge number z;
`
`log y =—
`
`2?Apu ype
`os.
`1+abI2”
`
`[8]
`
`Theionic strength J, for 1 : 1 electrolytes is equal to their concen-
`tration. The Debye-Hiickel constant Apy dependsontheelectric
`permittivity of the medium ¢(=€0é;,)
`
`\3/2
`F2
`1/2
`Apy = =——— |}
`
`8xL1n10 (sr)
`
`PH
`
`0]
`
`where L is the Avogadro constant and R, T, and F have
`their usual meaning. (For aqueous solutions at 25°C: Apy =
`0.509 mol~!/? dm3/?.) Coefficient b in Eq. [8] is equal to
`
`ase
`b=(—>]
`ERT
`
`.
`
`[10]
`
`while parameter a is the distance of closest approach of the
`interacting charges, whichis in the case of nanoparticles related
`
`

`

`72
`
`KALLAY AND ZALAC
`
`to their radius. By introducing Eq. [8] into Eq.[7] one obtains
`
`electrical interfacial layer
`
`logk = logk' + log K* +
`6
`6
`
`2ZaZRpADH-2
`1+abr”
`
`[11]
`
`k=
`
`
`21.F?\\/
`
`( eRT )
`
`‘
`
`[16]
`
`Equation [11] suggests that the plot of the experimental log k
`value as the function of 1:/?/(1 + abiZ/") shouldbe linear with
`the slope of 2z4zpApu which is true if charges of interacting
`species do not dependonthe ionic strength. However,as it will
`be shownlater, the charge of a colloidal particle decreases with
`ionic strength due to association of counterions with surface
`charged groups.
`
`Estimation of the Equilibrium Constant of the Transition
`State Formation
`
`To analyze the effect of repulsion between two charged par-
`ticles on the equilibrium constant K* weshall split the Gibbs
`energy ofthe transition state formation A*G°into electrostatic
`term, A* G9, andtherest, which weshallcall the chemical term,
`A*G4,. Thelatter includes van der Waals dispersionattraction
`
`—RT in K* = —RT In(K2. K4) = A*G°
`= A*G%, + A*GS,
`
`[12]
`
`where A¥G°, = —RT In KZ, and A*G2,=—RT In KZ.
`Asnoted before, the equilibrium constant K* is based on the
`activities of the interacting species andits definition (Eq. [5])
`considers the corrections for the nonideality. It corresponds to
`the zero-ionic strength so that the value of Kz could be ob-
`tained considering simple Coluombic interactions between two
`nanoparticles. Accordingly, the (molar) electrostatic energy be-
`tween particles A** and B*8 ofradii r, and rg in the medium of
`the permittivity ¢ is
`
`At the zero-ionic strength (7, > 0)the surface potential y of
`a sphere of radius r and the charge numberz is
`
`?
`
`=
`
`
`ze
`4ner
`
`7
`
`[17]
`17
`
`(Note that g potential is in fact the electrostatic potential at the
`onset of diffuse layer.) Under such a condition the diffuse layer
`extendsto infinity (xe — 0), so that for zero separation (x > 0)
`Eq.[15] reads
`
`A*Giup =
`
`2,2
`
`2 F
`
`8reLr
`
`In2.
`
`[18]
`
`The comparison ofEq. [18] with Eq. [14] shows that HHF theory
`results in ~30% lower value of energy than the Coulomblaw.
`This discrepancy is not essential for the purposeof this study,
`so thatin further analysis we shall use the Coulomb expression.
`By introducing Eqs. [12] and [13] into Eq. [11] for the rate
`constant of aggregation of nanoparticles A** and B**, one ob-
`tains
`
`ZazpF?
`+1
`logk= logk'
`OFay
`OBE
`NORE TE Re 4reL(ra+rp)
`
`2zazpApule”
`1+abii”?
`
`[19]
`
`or in another form
`
` 2?
`logk= logkp — Zaz ( —2A—=a) [20]
`
`i
`PN ra tre
`PF + abhi?
`zazpF?
`°
`dare+8)
`"
`
`where ra + rg is the center to center distance between inter-
`acting particles in close contact. In the case of two identical
`particles (rg =rg =r and za = zp =z)
`
`where
`
`ata = 2=—.
`a”
`8reLr
`
`and
`
`14
`[14]
`
`F*1n10
`= 21
`4neLRT
`Py
`
`This approach, based on the Coulomb law, could be tested by
`the Hogg-Healy-Fuerstenau (HHF) theory (9). For two equal
`spheres of the same surface potential y, separated by surface to
`surface distance x, the electrostatic interaction energy, expressed
`on the molar scale, is equal to
`
`A* Gop = 20eLrg’ In{1 + exp(—«x)],
`
`[15]
`
`ko =k’ Ki.
`
`[22]
`
`At high ionic strength the counterion association is so pro-
`nounced that the effective charge number of nanoparticles ap-
`proachesto zero. In such a casethe electrostatic repulsion dimin-
`ishes and the aggregation is controlled by the diffusion (k = kaise),
`as described by the Smoluchowski theory. Accordingly,
`
`where « is the Debye-Hiickel
`
`reciprocal
`
`thickness of the
`
`ko = kaiss.
`
`[23]
`
`

`

`STABILITY OF NANODISPERSIONS
`
`73
`
`Thestability coefficient (reciprocal of the collision efficiency),
`commonly defined as W = kgige/k, is then equal to
`
`pi?
`ko
`log W = lo — = ZaZR| ——— — 2Apy ——————
`ee
`1+ abhi?
`ra +rp
`
`B
`
`(
`
`).
`
`[24]
`
`In the case of aggregation of identical nanoparticles the above
`equation is reduced to
`
`ko
`log W = log — =
`
`8 y
`
`08
`
`2z (¢
`
`a
`.3(B
`— — App —— |.
`
`PHT +abhi?
`
`25
`
`[25]
`
`Evaluation of the Charge Number
`
`Fora given electrolyte concentration, the stability coefficient
`of the nanodispersion could be obtained by Eq. [24] (or by
`Eq. [25], in the case of uniform particles), once the charge
`numberof particles is known. The surface potential (as used
`in the theory of Colloid Stability) and charge numberare deter-
`minedbythe ionic equilibrium atthe solid/liquid interface which
`will be considered here for metal oxide particles dispersed in
`aqueouselectrolyte solutions. The Surface Complexation model
`(2-pK concept) considers (15-22) amphotheric surface =MOH
`groups, developed by the hydration of metal oxide surfaces, that
`could be protonated (p) or deprotonated (d)
`
`From the d-plane (onset ofdiffuse layer, potential $4), ions
`are distributed according to the Gouy-Chapman theory.
`The total concentration of surface sites Tio; is equal to
`
`Tot = P(MOH) + (MOH?) + PCMO-)
`+T(MO™- . Ct) + P(MOH}- A-).
`
`Surface charge densities in the 0- and B-planes are
`
`oo = F(T(MOH}) + (MOH} - A~)
`—T(MO~) —- (Mor - C*))
`og = F(T(MO™ -Ct)—T(MOH} - A>).
`
`[30]
`
`(31)
`[32]
`
`The net surface charge density o, corresponding to the charge
`fixed to the surface is opposite in sign to that in the diffuse
`layer og
`
`05 = —04 = 09 +05 = F(T(MOHS)—T(MO7)).
`
`[33]
`
`The relations between surface potentials, within the fixed part
`of electrical interfacial layer (EIL), are based on the constant
`capacitance concept
`
`og
`
`Os
`
`dp — da
`= hoon = i
`
`Ba
`34
`
`Cc;
`
`[26]
`
`=MOH + Ht > =MOH?;
`_
`T(MOH})
`Ky = exp(Fo0/RT)Mon(MOH)
`where C, and C2 are capacities of the so-called inner and outer
`layer, respectively. The general modelofEIL could be simplified
`=MOH -> =MO- + H*;
`(19) by introducing ¢g =a, which corresponds to Cz — oo.
`
`The equilibrium in the diffuse layer is described by the Gouy-
`T'(MOH)
`Chapman theory.
`For planar surfaces(relatively large particles)
`
`Kg= exp(—F¢p/RT)MO2D) [27]
`
`=
`
`+
`
`K, and Kg are equilibrium constants of protonation and depro-
`tonation,respectively, oo is the potential of the 0-plane affecting
`the state of charged surface groups MOH} and MO,I is the
`surface concentration (amountper surface area), and a is activity
`in the bulk ofsolution.
`Charged surface groups bind counterions, anions A™ (surface
`equilibrium constant K,), and cations C* (surface equilibrium
`constant K,)
`
`=MOH} + A~ -> =MOH}- A™;
`T'(MOH} - A~)
`K.= exp(—Fp/RT)5
`=MO7 + Ct > =Mo- - ct:
`— ;
`Ke = exp(Fp/RT) =o [29]
`a(C+)r(Mo-)’
`
`(8
`
`oa
`
`RTex
`
`o,=—ay=—
`
`sinh(—Fg/2RT)
`
`[35]
`
`and for small spherical particles (nanoparticles)
`
`r
`Ed
`=—[(1—-——__}
`
`3 (
`
`r+k7!)
`
`rt
`
`.
`
`36
`
`[Bo]
`
`Once the system is characterized, the Surface Complexation
`mode! enables calculation of the colloid particle charge num-
`ber under given conditions. This means that one should know
`equilibrium constants of surface reactions, capacitances ofin-
`ner and outer layers, and total density of surface sites. By an
`iteration procedure one obtains the net surface charge density o,
`(defined by Eq. [33]) from which the particle charge numberis
`
`where ¢g is the potential of 6-plane affecting the state of asso-
`ciated counterions.
`
`z=4r’xo,/e.
`
`[37]
`
`

`

`74
`
`KALLAY AND ZALAC
`
`5
`
`0<
`
`=
`
`=
`
`Numerical Simulation and Discussion
`
`The above theory, developed for kinetics of aggregation of
`nanoparticles (nanocoagulation), will be demonstrated on a few
`examples. Two systems (hematite and rutile) underdifferent con-
`ditions will be examined. The values of equilibrium parameters,
`used in calculation of the particle charge number, were obtained
`by interpretation of adsorption and electrokinetic data for ordi-
`nary colloid particles (21, 22). It was assumedthat these param-
`eters approximately describe the properties of corresponding
`nanosystems.In the evaluation the Gouy-Chapmenequation for
`sphericalinterfacial layer, Eq. [36], was used. Once the charge
`numberwas obtained, the stability coefficient was calculated via
`Eq.[25].
`Figure 2 demonstrates the effect of electrolyte concentration
`on the stability of hematite nanodispersions containing parti-
`cles of r = 3 nm.It is obvious that the stability of the system
`decreases rapidly with electrolyte addition. At pH 4, particles
`are positively charged so that association of anions with the
`surface charged groups takes place. Nitrate ions were found to
`aggregate the system more effectively with respect to the chlo-
`ride ions, which is due to lower values of the surface associa-
`tion equilibrium constant of the latter counterions. The effect
`of electrolyte concentration is explicitly included in Eq. [25]
`through ionic strength. However, particle charge number also
`dependson the electrolyte concentration due to counterion as-
`sociation so that both effects result in a decrease ofstability at
`
`HEMATITE
`pH =4
`r=3nm
`
`IgW
`
`-3
`
`-2
`
`“1
`
`0
`
`ig(/,/mol dm")
`
`FIG. 2. Effect of electrolytes on the stability of hematite aqueous
`nanodispersion (r-=3 nm) at T=298 K and pH 4, as obtained by
`Eq.
`[25]. The charge number was calculated by the Surface Com-
`plexation model
`(Eqs.
`[26]-[39]) using parameters obtained (21) with
`hematite colloid dispersion (r =60 nm): Tio =1.5 x 1075 mol m7?; K,=
`5x 104; Kg=15x10""; pHp.=7.6; K(NOS)=1410; K(CI-)=525;
`Cy(NOJ) = 1.88 Fm~?; Cy(CI~) = 1.81 F m™?; Cy = 00;¢, = €/€9 = 78.54.
`
`HEMATITE
`
`KNO,
`r=3nm
`
` 10
`
`3
`
`-2
`
`-1
`
`0
`
`ig(/,/mol dm™)
`
`FIG. 3. Effect of pH on the stability of hematite aqueous nanodispersion
`(r =3 nm)in the presence of potassium nitrate at T = 298 K. The parameters
`used in calculations are the sameas in Fig,2.
`
`higherelectrolyte concentrations. Figure 3 demonstrates the ef-
`fect of the activity of potential determining H* ions. At lower
`pH values particles are more positively charged, the system is
`more stable, and higher electrolyte concentration is necessary
`for aggregation. Theeffect of particle size on the stability of the
`system is dramatic. As shownin Fig. 4, systems with smaller
`
`HEMATITE
`KNO,
`pH =4
`
`IgW
`
`-3
`
`-2
`
`-1
`
`0
`
`ig(/,/mol dm“)
`
`FIG.4. Effect particle size on the stability of hematite aqueous nanodisper-
`sions in the presenceof potassium nitrate at pH 4 and T = 298 K. The parameters
`used in calculations are the sameas in Fig.2.
`
`

`

`STABILITY OF NANODISPERSIONS
`
`75
`
`HEMATITE
`
`RUTILE
`pH=9.5
`
`LiNO3
`
`The absolute values of surface potentials are approximately the
`same. However, due to the different values of the equilibrium
`parameters, rutile was found to be significantly less stable than
`hematite.
`It may be concluded that application of the Brgnsted concept
`to the stability of nanodispersions shows that the presence of
`electrolytes may completely reduce the stability, and that the
`electrolytes with counterions exhibiting higheraffinity are more
`effective for association at the interface. Lower surface poten-
`tial (pH closer to the point of zero charge) slightly reduces the
`stability. In the above analysis the parameters used for calcula-
`tions may differ from the reality since they were obtained from
`measurements with relatively large colloid particles; however,
`the general behavior of nanosystems maybe still explained. The
`comparison between hematite and rutile showed howsensitive
`the stability of nanosystems is on the inherent characteristics
`of the solid. Different equilibrium parameters result in a very
`pronounced difference in the stability. However, the major char-
`acteristics ofthe system are the particle size; smaller particles are
`significantly less stable. The interpretation based on the Bronsted
`theory is more closeto the reality if the particles are very small.
`In this paper we have introduced the Brgnsted concept for
`interpretation of the kinetics of aggregation of nanoparticles.
`Both DLVOand the Brénsted approach result in reduced sta-
`bility at high electrolyte concentrations, but according to the
`latter approachthe stability coefficient does not directly depend
`on the Hamaker constant governing dispersion attraction. The
`presented concept suggests that dispersion forces are cause for
`binding ofparticles but do notaffect the kinetics of aggregation
`of nanoparticles. In considering the stability of nanosystems one
`should also take into accountthe effect ofparticle concentration.
`Let us compare two systems with the same mass concentration
`but different in particle size. One system is an ordinary col-
`loidal system (r = 30 nm), while the second oneis a nanosystem
`(r =3 nm). Thedifference in particle size by a factor 10 results
`in 1000 times higher concentration of nanoparticles. Since the
`aggregationrate is proportionalto the squareof the particle con-
`centration the collision frequency in the nanosystem will be a
`million times higher. Thus, the aggregation rate will be a million
`times higher while the aggregation half-time will be reduced by
`a factor of 1000. Accordingly, one may conclude that nanopar-
`ticles can hardly form a stable dispersion without additives,
`(e.g., surface-active agents) because their aggregation is fast
`due to both low stability coefficient and high particle number
`concentration,
`
`REFERENCES
`
`1. Vervey, E. J. W., and Overbeek,J. Th., “Theory of the Stability ofLyophobic
`Colloids.” Elsevier, Amsterdam, 1948.
`2. Kallay, N., and Zalac, S., Croat, Chem. Acta 74, 479 (2001).
`3. von Smoluchowski, M., Z. Physik. Chem. (Leipzig) 17, 129 (1916).
`4. Fuchs, N., Z Physik. 89, 736 (1934).
`5. Hamaker, H. C., Physica 4, 1058 (1937).
`6. Bradly, R. S., Trans. Faraday Soc, 32, 1988 (1936).
`7. de Boer, J. H., Trans. Faraday Soc. 32, 21 (1936).
`
`-3
`
`~2
`
`-1
`
`0
`
`Ig(/_/mol dm")
`
`FIG. 5. Effect of kind of material on the stability of aqueous nanodisper-
`sion (r =3 nm) at T = 298 K. For hematite particles, the parameters used in
`calculationsare the sameas in Fig. 2. Forrutile rods (length is 100-240 nm and
`width is 45 nm) the parameters are (22): Pitot = 10-5 mol m~?; Kp =6x 10’;
`Kg=6x 107; PHpze = 6.0; K(Lit) = 380; C)(Lit) = 1.58 F m7*; C2 =00,
`
`particles are markedly less stable. In fact, such a system is un-
`stable even at low electrolyte concentrations.It is interesting to
`analyze the particle size effect on the basis of Eq. [25], assum-
`ing that surface charge density does not dependsignificantly on
`the particle size. In such a case the particle charge numberis
`proportional to r? so that first term associated with B is pro-
`portional to r?. Accordingly, one would expect an increase of
`stability coefficient with particle size. The second term,associ-
`ated with ionic strength, acts in the opposite direction. However,
`this term is not significant at low ionic strength so that the first
`one prevails. At high ionic strength the second term becomes
`influential, but also the charge density and charge numberare
`reduced so much that log W approaches zero for any particle
`size. A similar conclusion would be obtained if one assumes
`that surface potential does not depend significantly on parti-
`cle size. However, according to Eq. [36], the size effect on the
`stability will be less pronounced. The reality is between the
`above-noted extremes, but the use of the Surface Complexa-
`tion model enabled us to avoid such speculations. Generaly, one
`may conclude that smaller particles are less stable. Indeed, as
`shown for zirconia (24) the nanodispersion coud be stable at
`very high surface potentials which were achieved 6 pH units
`below the point of zero charge. The specificity of the material
`comprising nanoparticles could be seen through the values ofthe
`equilibrium parameters. Figure 5 showsthe difference between
`the predicted behavior of hematite and rutile nanosystems. The
`pH values were chosen to be approximately equally far from the
`point of zero charge, but in the opposite direction. At pH 9.5
`rutile is negatively while at pH 4 hematite is positively charged.
`
`

`

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