Journal of Colloid and Interface Science 253, 70—76 (2002)
`doi:10.1006ljcis.2002.8476
`
`Stability of Nanodispersions: A Model for Kinetics
`of Aggregation of Nanoparticles
`
`Department ofChemistry, Faculty ofSct'ence, University ofZagreb, Moralicev rrg 19. BO. Bax I63, HR 1W0 Zagreb. Croatia
`
`Nikola Kallay‘ and Suzana Zalacz
`
`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 DLVO theory. Since
`nanopoarticles are small compared to the extent of the diffuse layer,
`the process is considered in the same way as for two interacting ions.
`Therefore, the Brnnsted 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 charge of particles 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 the stability coeflicient of nanosystems was found to be more
`pronounced but similar to that for regular colloidal systems. The
`effect markedly depends on the nature of the solid which is char-
`acterizcd 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 afinity, i.e., by
`surface association equilibrium constants. The most pronounced is
`the particle size effect. It was shown that extremely small particles
`cannot be 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.
`e :um Elwin-Science [USA]
`Key Words: stability of nanodispersions.
`
`INTRODUCTION
`
`It is commonly accepted that the stability of colloidal systems
`is, in mostcases, the result of an extremely slow aggregation pro-
`cess. The main reason for such a slow aggregation process is 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
`was theoretically solved by Smoluchowski (3), while the basis
`for evaluation of the collision efficiency was given by Fuchs (4).
`In order to use the Fuchs theory one should know the interaction
`energy as a function of the distance between interacting parti—
`
`“to whom conespondence should be addressed. Fax: +335-i-4329953.
`E—mail: nkallay®prelog.chem.pmf.hr.
`2 Present address: PLIVA d.d., R&D—Research, Prilaz bamna Filipoviéa 25,
`HR 10000 Zagreb. Croatia
`
`0021—9797100 $35.00
`0 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 Deriaguin, Landau, Vervey,
`Overbeek (DINO) 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 conect 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 am
`(called nanoparticles). generally do not show electrostatic sta—
`bilization. According to do Gennes (23), the reason for the in-
`stability of nanocolloidal systems might be in their low charge
`(surface charge density times surface area). In some cases sta~
`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 nanoparticlcs based on the Bronstcd theory
`(26, 2?), 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: nanoparticlcs are small with respect to
`the thickness of the electrical diffuse layer. so that in the course
`of the collision of two nanoparticles a complete overlap of two
`difiuse layers takes place. Let us consider extension of the 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 means that at the ionic strength of 10'2 mol din—3
`the diffuse layer is extended up to 6 rim from the surface. As
`shown on Fig. l, 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 center of an ionic cloud. In the course of colli—
`
`sion two nanoparticlcs in contact have a common diffuse layer or
`“ionic cloud.” Therefore, interaction of nanoparticles could be
`considered in a manner similar to that for two interacting ions,
`and consequently described by the Breasted theory. This theory
`considers the “transition state” or “activated complex” which is
`a pair of two interacting ions with a common ionic cloud. The
`
`Abraxis EX2020
`
`Actavis LLC v. Abraxis Bioscience, LLC
`|PR2017-O11OO
`
`

`

`STABILITY OF NANODISPERSIONS
`
`71
`
`
`
`FIG. I. Overlap of electrical interfacial layers for two ordinary colloid par-
`ticles (r = 30 nrn) and for two nanoparticles (r =3 urn).
`
`the concentration of the transition state
`
`U = kr[ABz.s+zn]’
`
`where k’ is the rate constant (coefficient) of the second procass.
`Equilibration of the first step is fast so that one calculates the
`concentration of the transition state {AB“+“] from the rele-
`vant equilibrium constant K 5‘ taking into account the activity
`coefficients y of reactants and of the transition state
`
`[Aanzn]
`y(ABZA+ZB)
`Ki'E _——.
`y(A“)y(B‘B) [Az*][B“]
`
`[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 determining step.
`
`The equilibrium constant K9‘ is defined in terms ofactivities, and
`consequently its value does not depend on the ionic strength; i.e.,
`it corresponds to infinite dilution. Equations [4] and [5] result in
`
`TI-EORY
`
`v : k
`
`{Kg y(Az‘)y(B“’)
`y(ABz,-i+ze) [Az‘HB‘n].
`
`[6]
`
`Introduction of the Breasted Concept to Kinetics
`of Aggregation of Nanoparticles
`
`The quantitative interpretation of kinetics of aggregation of
`nanoparticles will follow the Brensted concept (26, 27). It will
`be based on the Transition State theory using the activity coef-
`ficients as given by the Debye-Hiickel limiting law.
`Aggregation of two charged nanoparticles A“ and B1“ could
`be represented by
`
`A“ + B“ —> ABM,
`
`[1]
`
`where 2 denotes the charge number. The rate of aggregation u
`is proportional to the product of concentrations of interacting
`particles [Az*][B“]
`
`v = k[A‘“][Bz“],
`
`[2]
`
`where k is the rate constant (coefficient) of aggregation.
`According to the Bronsted concept, in the course of aggrega-
`tion two charged nanoparticles undergo reversible formation of
`the transition state with charge number being equal to the sum of
`the charges of interacting species. The transition state ABZAHB
`undergoes the next step (binding) which is slow and is therefore
`the rate determining step
`
`Azn +1315 9 ABZA'i'ZB _> ABZAB.
`
`Note that equilibration of the interface may result in a change
`of the total charge of the doublet. In such a case zit + z]; 95 mg.
`Since the equilibration of the first step is fast, and the second
`process is slow, the overall rate of reaction (v) is proportional to
`
`According to the above equation, the overall rate constant. as
`defined by Eq. [2], is given by
`
`;
`
`(NA) (3“)
`
`k = k Kaw.
`
`[7]
`
`It is clear that the overall rate constant It 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 assumed to be
`applicable for extremely small particles 1' of charge number 2:,-
`
`103 M = —
`
`ZEADH [cl/2
`—.
`1+ablcm
`
`[8]
`
`The ionic strength It for l : l electrolytes is equal to their concen-
`tration. The Debye-Hiickel constant ADH depends on the electric
`permittivity of the medium s(=ansr)
`
`Ann =
`
`
` 21/2 F2
`
`(SnLlnIO eRT)
`
`3,!2
`
`‘
`
`[9]
`
`where L is .the Avogadro constant and R, T, and F have
`their usual meaning. (For aqueous solutions at 25°C: An“ =
`0.509 mol“”2 dmm.) Coefficient b in Eq. [8] is equal to
`
`“’2
`2F2
`b = — ,
`eRT
`
`[10]
`
`while parameter a is the distance of closest approach of the
`interacting charges, which is 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
`
`lo kzlo k’+lo K7é+
`g
`g
`g
`
`22i«AKBADHIc'’2
`Ham,U2
`
`[11]
`
`k =
`
`
`21,1?2
`
`( sRT )
`
`"2
`
`.
`
`[16]
`
`Equation [ll] suggests that the plot of the experimental log k
`value as the function of r,‘ ’2/(1 + ab 12”) should be linear with
`the slope of 2zAzBADH» which is true if charges of interacting
`species do not depend on the ionic strength. However, as it will
`be shown later, the charge of a colloidal particle decreases with
`ionic strength due to association of counter-ions 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# we shall split the Gibbs
`energy of die transition state formation Aft? into electrostatic
`term, me 6" , and the rest, which we shall call the chemical term,
`N‘Ggh. The latter includes van dcr Waals dispersion attraction
`
`—RTln Kr = —RT 1n(K;K:f)=an°
`= M63, + are“at!
`
`[12]
`
`where Angh = —RT 1n K2; and AngI = —RT ln Kg.
`As noted before, the equilibrium constant K5‘ is based on the
`activities of the interacting species and its definition (Eq. [5])
`considers the corrections for the nonideality. It corresponds to
`the zero-ionic strength so that the value of K: could be ob-
`tained considering simple Coluombic interactions between two
`nanopam'cles. Accordingly, the (molar) electrostatic energy be-
`tween particles A“ and B“ of radii m and n; in the medium of
`the permittivity s is
`
`At the zero-ionic strength (I, —> 0} the surface potential 99 of
`a sphere of radius r and the charge number z is
`
`
`ze
`4ner
`
`.
`
`=
`
`‘0
`
`17
`
`]
`
`[
`
`(Note that 99 potential is in fact the electrostatic potential at the
`onset of diffuse layer.) Under such a condition the diffuse layer
`extends to infinity (1: —> 0), so that for zero separation (x —> 0)
`Eq. [15] reads
`
`are?”
`
`=
`
`
`ZZ F2
`l 2.
`87rer n
`
`[18]
`
`The comparison of Eq. [18] with Eq. [14] shows that I-II-IF theory
`results in ~30% lower value of energy than the Coulomb law.
`This discrepancy is not essential for the purpose of this study,
`so that in further analysis we shall use the Coulomb expression.
`By introducing Eqs. [12] and [13] into Eq. [1 l] for the rate
`constant of aggregation of nanoparticles A“ and B“, one ob-
`tains
`
`,
`
`uz
`2ZAZBADH c
`ZAZBFZ
`qr
`+iOchh—m W
`[19]
`
`or in another form
`
`1 k=l
`
`—
`
`
`
`1,132
`— 2A ‘— .
`
`20
`
`zaza F2
`Mo" =—
`°' 4mm, + m)’
`
`[13]
`
`where
`
`where n; + r]; is the center to center distance between inter-
`acting particles in close contact. In the case of two identical
`particles (rA =rB : r and Zn = Zn = Z)
`
`2
`2
`
`z F
`
`SfieLr
`
`.
`
`Ara“ =
`e1
`
`and
`
`[14]
`
`This approach, based on the Coulomb law, could be tested by
`the Hogg-Healy-Fuerstenau (HI-[F] theory (9). For two equal
`spheres of the same surface potential go, separated by surface to
`surface distance x, the electrostatic interaction energy, expressed
`on the molar scale, is equal to
`
`N‘G‘fim: =23r£Lrg02 ln[l + exp(—xx)],
`
`[15]
`
`F2111 10
`=—
`4nsLRT
`
`k, = as;
`
`21
`
`[
`
`]
`
`[22]
`
`At high ionic strength the counterion association is so pro-
`nounced that the effective charge number of nanoparticles ap-
`proaches to zero. In such a case the electrostatic repulsion dimin-
`ishes and the aggregation is controlled by the diffusion (k = kdjff),
`as described by the Smoluchowski theory. Accordingly,
`
`where x is the Debye-Htickel
`
`reciprocal
`
`thickness of the
`
`k0 = kdifi-
`
`[23]
`
`

`

`STABILITY OF NANODISPERSIONS
`
`73
`
`The stability coefficient (reciprocal of the collision efficiency),
`commonly defined as W = ltd-[ff] k, is then equal to
`
`
` to B
`1c”2
`log W = log ? =zAzB m + m — Zdnflm -
`
`[24]
`
`In the case of aggregation of identical nanoparticles the above
`equation is reduced to
`
`From the d—plane (onset of diffuse layer, potential dad), ions
`are distributed according to the Gouy-Chapman theory.
`The total concentration of surface sites I‘m is equal to
`
`PM = I‘(MOH) + l"(MOH3L) + l"(MO_)
`
`+ I‘(MO‘ -C+) + rat/Ion"; - A').
`
`[30]
`
`to
`1 W=l —:
`
`08k
`
`0g
`
`1,!”
`2 B
`——A —-——— .
`
`DH1+abIcU2
`
`23 (r
`
`Surface charge densities in the 0- and ,6-planes are
`
`25
`
`[
`
`l
`
`on = F(F[MOH;') + Ft'MOHg' -A_)
`
`— 1"(MO‘) — TIMO— - C+))
`
`Evaluation of the Charge Number
`
`as = F(l"(MO_ -C+)— “MOH; -A")).
`
`[31]
`
`[32]
`
`For a 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
`number of particles is known. The surface potential (as used
`in the theory of Colloid Stability) and charge number are deter-
`mined by the ionic equilibrium at the solidfliquid interface which
`will be considered here for metal oxide particles dispersed in
`aqueous electrolyte solutions. The Surface Complexation model
`(2-pK concept) considers (15—22) amphotheric surface EMOH
`groups, developed by the hydration of metal oxide surfaces, that
`could be protonated (p) or deprotonated (d)
`
`EMOH + H+ _+ among;
`
`rat/tong)
`_
`K, _ exp(F¢o/RT)a—-a_l+)r(MOH)
`
`EMOH —> EMO— + H+;
`
`—
`+
`K, = womanhw.
`I‘(MOH)
`
`The net surface charge density a, corresponding to the charge
`fixed to the surface is opposite in sign to that in the diffuse
`layer ad
`
`a, 2 —od = on + 05 = armour) — FWD—D.
`
`[33]
`
`The relations between surface potentials, within the fixed part
`of electrical interfacial layer (ElL), are based on the constant
`capacitance concept
`
`
`0'0
`0‘5
`‘. C2=
`
`.
`
`1:
`
`¢oh¢a
`
`ibfl—Qi'd
`
`[34]
`
`[26]
`
`[27]
`
`where C l and C2 are capacities of the so-called inner and outer
`layer, respectively. The general model ofBIL could be simplified
`
`(19) by introducing qbfi =¢d, which corresponds to C2 —+ 00.
`The equilibrium in the diffuse layer is described by the Gouy-
`Chapman theory.
`For planar surfaces (relatively large particles)
`
`KP and Kd are equilibrium constants of protonation and depro-
`tonation, respectively, a, is the potential of the O—plane affecting
`the state of charged surface groups MOB; and MO’, 1" is the
`surface concentration (amount per surface area), and a is activity
`in the bulk of solution.
`
`Charged surface groups bind counterions, anions A" (surface
`equilibrium constant K9), and cations C+ (surface equilibrium
`constant Kc)
`
`ZRTsx
`
`a, = —cd = —
`
`sinh(—F¢d/2RT)
`
`[35]
`
`and for small spherical particles (nanoparticles)
`
`as
`
`.9qu
`= — 1 —
`
`r (
`
`
`r
`
`r +KHI)
`
`_l
`
`36
`
`[
`
`1
`
`EMOH; +A‘ —> EMOH; . A‘;
`
`moon; . A")
`
`Kn I
`EMO‘ + C+ —> EMO— -C+;
`_ . +
`K. = exptFaa/RT) HMO C )
`a(C+)I‘(MO‘)'
`
`Once the system is characterized, the Surface Complexation
`model enables calculation of the colloid particle charge num—
`her under given conditions. This means that one should know
`equilibrium constants of surface reactions, capacitances of in-
`ner and outer layers, and total density of surface sites. By an
`iteration procedure one obtains the net surface charge density a,
`(defined by Eq- [33]) from which the particle charge number is
`
`[29]
`
`where an is the potential of ,B-plane affecting the state of asso—
`ciated counterions.
`
`z = 4r27t org/e.
`
`[37]
`
`

`

`74
`
`KALLAY AND ZALAC
`
`10
`
`a
`
`KN03
`
`HEMATITE
`
`r: 3 nm
`
`. 3
`
`-2
`
`-1
`
`0
`
`lchlmol dm'a)
`
`FIG. 3. Efiect 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 same as in Fig. 2.
`
`higher electrolyte 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. The effect of particle size on the stability of the
`system is dramatic. As shown in Fig. 4, systems with smaller
`
`HEMATITE
`KN03
`pH = 4
`
`t
`
`2a
`
`- 3
`
`-2
`
`-‘l
`
`0
`
`Iguclmol dm'a)
`
`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) under difl'erent conw
`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 assumed that these param~
`eters approximately describe the properties of corresponding
`nanosystems. In the evaluation the Gouy-Chapmen equation for
`spherical interfacial layer, Eq. [36], was used. Once the charge
`number was 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 chloH
`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
`depends on the electrolyte concentration due to counterion as-
`sociation so that both effects result in a decrease of stability at
`
`HEMATITE
`pH : 4
`
`r= 3 nm
`
`-a
`
`-2
`
`- 1
`
`o
`
`IgUJmol dm‘a)
`
`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 (2]) with
`
`hematite colloid dispersion (r =60 nm): Fm;=].5 x10"S mol m"2; KP =
`5x10“; Kd=l.5x10_”;
`lem=7.6; K(N0;}=1410; K(c1-)=525;
`clmo;)=1.ss Fm‘1; c.(c:1-) = 1.31 F m—1;c2 = on: 3.: em = 73.54.
`
`FIG. 4. Effect particle size on the stability of hematite aqueous nanodisper—
`sions in the presence of potassium nitrate at pH4 and T = 298 K. The parameters
`used in calculations are the same as in Fig. 2.
`
`

`

`STABILITY OF NANODISPERSIONS
`
`75
`
`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 Breasted concept
`to the stability of nanodispersions shOWS that the presence of
`electrolytes may completely reduce the stability, and that the
`electrolytes with counterions exhibiting higher affinity 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 may be still explained. The
`comparison between hematite and rutile showed how sensitive
`the stability of nanosystems is on the inherent characteristics
`of the solid. Different equilibrium parameters result in a very
`pronounced diiference in the stability. However, the major char-
`acteristics of the system are the particle size; smaller particles are
`significantly less stable. The interpretation based on the Brpnsted
`theory is more close to the reality if the particles are very small.
`In this paper we have introduced the Brensted concept for
`interpretation of the kinetics of aggregation of nanOparticles.
`Both DLVO and the Bronsted approach result in reduced sta-
`bility at high electrolyte concentrations, but according to the
`latter approach the stability coefficient does not directly depend
`on the Hamaker constant governing dispersion attraction. The
`presented concept suggests that dispersion forces are cause for
`binding of particles but do not affect the kinetics of aggregation
`of nanoparticles. In considering the stability of nanosystems one
`Should also take into account the effect of particle 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 run), while the second one is a nanosystem
`(r = 3 nm). The difference in particle size by a factor 10 results
`in 1000 times higher concentration of nanoparticles. Since the
`aggregation rate is proportional to the square of 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
`
`l. Vervey, E. J. W., and Overbeek, J. 11)., “Theory of the Stability ofLyophobic
`Colloids." Elsevier, Amsterdam, 1948.
`Kallay, N., and Zalac, S., Croat. Chem. Actn 74, 479 (2001}.
`von Smoluchowski. M., Z. Physilc Chem (bipzig) 17, 129 (19l6).
`Fuchs, N.. Z. Physik. 89, 736 (I934).
`Hamaker, H. C., Physica 4. 1058 (1937).
`Bradly. R. 3.. Trans. Faraday Soc. 32, 1938 (1936).
`de Boer, .l. H., Trans. Faraday Soc. 32, 21 (1936).
`
`HQV'PP.”
`
`HEMATITE
`
`pH = 4
`KNO3
`
`
`
`Rm
`
`-3
`
`- 2
`
`-1
`
`O
`
`lchImol dm'3)
`
`FIG. 5. Effect of kind of material on the stability of aqueous nanodisper-
`sion (1* :3 nm) at T = 293 K. For hematite particles. the parameters used in
`calculations are the same as in Fig. 2. For rutile rods (length is 100—240 nm and
`width is 45 nm) the parameters are (22}: Pm = 10—5 mol m"2; K = 6 x 101;
`Kd = 6 x 10—5; p119“ = 6.0; Kflsi+) = 330;C1(Li+}=1.58 F 111'
`;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 depend significantly on
`the particle size. In such a case the particle charge number is
`proportional to r2 so that first term associated with B is pro-
`portional to r3. 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 number are
`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 of the
`equilibrium parameters. Figure 5 shows the 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.
`
`

`

`76
`
`10.
`
`11.
`
`12.
`13.
`14.
`15.
`
`16.
`
`17.
`
`KALLAY AND ZALAC
`
`Derjaguiu. B. V., and 1.3111131), L, Acto Physicochim, USSR 14, 552
`(1941).
`flag. 11., Healy, T. w.. and Fuerstenau. D. W.. Trans. Faraday Soc. 62,
`1638 (1966).
`Bhaflachmjee, 3., Elimelech. M., and Borkovec. M.. Croat. Chem. Acm 71,
`883 (1998).
`Semrlfler. M., Ricka. 1., and Borkovec, M.. Cofbids Surf. 165, 79
`(2000).
`Hsu, J.—‘P., and Liu. B.—T.. J. Colloid Interface Sci. 217, 219 (I999).
`Ohshima, H., J. Colloid Interface Sci. 225, 204 (2000).
`Sun, N., and W312, Y.. J'. Colloid Inted'ace Sci. 234, 90 (2001).
`Yates, D. R, Levine. 5., and Healy. T. W..J. Chem. Soc, Faraday Trans. I
`70, 1807 (1974).
`Davis. J. A.. James, R. 0., and Leckic, J. 0.. J. Colloid Interface Sci. 63,
`430 (1978).
`Westall,
`J., and H011], H., Adv. Coiloid Interface Sci. 12, 265
`(1930).
`
`18.
`
`19.
`
`20.
`21.
`
`22.
`
`23.
`24.
`
`25.
`
`26.
`27.
`
`Dzombak, D. A., and Morel, F. M. M.. “Surface Complcxarion Modelling.”
`Wiley Interscicncc. New York, 1990.
`in “Interfanial Dynamics"
`Kallay. H., Kovaéevié. D., and Cop, A.,
`(N. Kallay, 56.}, Surfactant Science Series, Vol. 8. Dekker, New York.
`1999.
`
`Kallay, N., and Zalac, 5.. J. Coflotd Interface Sci. 230, l (2W0).
`Colié. M., Fuerstenau. D. W.. Kallay. N.. and Matijevié, E., Coflot'd Surf
`59, 169(1991).
`Kallay, N.. Colié. M., Fuerstenau. D. W.. Jang. H. M.. and Matijcvié. F...
`Colloid Polym. Sci. 272, 554 (1994).
`de Genncs, P.—G.. Croat. Chem. Acto 71, 833 (1998).
`Fayre, Vi. Spalla. 0., Belloni, 1.... and Nahavi, M.,J'. Colloid Interface Sci.
`187, 184 {1997}.
`van Hyning, D. L. Klemperer, W. G., and Zukoski, C. F., Langmuir 17,
`3128 (2001}.
`ansted. J. N.. Z. Physik. Chem. 102, 169 (1922).
`Chrisfiansen, J. A., Z. Physik Chem. 113, 35 (I924).
`
`