DLVO theory applied to TiO pigments and other materials in ...download.xuebalib.com › xuebalib.com.36612.pdfDLVO theory that incorporates a less rigorous method of in-tegrating the

Progress in Organic Coatings 44 (2002) 131–146

DLVO theory applied to TiO2 pigments and other materials in latex paints

Stuart Croll∗Department of Polymers and Coatings, North Dakota State University, Dunbar 64, Fargo, ND 58105, USA

Received 18 May 2001; received in revised form 25 May 2001; accepted 13 August 2001

Abstract

Understanding how a paint formulation translates into comparative numbers of particles, how the spacing between particles compares totheir size and what controls their stabilization mechanisms improves efficient formulation design. The application of Derjaguin, Landau,Verwey and Overbeek (DLVO) theory of the electrostatic stabilization of colloids is reviewed by calculating the inter-particle potentialsfor typical titanium dioxide pigment particles in a conventional aqueous paint formulation. The calculations show that composition andstructural details of the particles need to be input, in order for DLVO theory to model the stability of the pigment particles. It is necessaryto include the extent of the surface treatment on the pigment particles, the ionic strength of the continuous phase and the thickness of anyadsorbed layer of dispersant polymer as well as knowing the zeta-potential of the particles under the prevailing conditions. Ionic strengthdetermines the range of DLVO forces, so the conductivity of various salt solutions was determined in order that ionic strength could bemeasured from simple determinations of paint conductivity. Other calculations compare the likely stability of an extender and a typical latexto that of the pigment particles in both DLVO calculations and in calculations of settling rates. The spacing between particles in a randomdispersion is estimated from their particle sizes, their concentrations and the characteristic packing fraction assigned to the dispersion.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:DLVO; Titanium dioxide; Adsorbed layer; Ionic strength; Particle separation

1. Introduction

Charge stabilization of colloidal particles is the most eas-ily used and cost effective method of maintaining the dis-persion quality in aqueous paints. The Derjaguin, Landau,Verwey and Overbeek (DLVO) theory [1] is a very usefulformalism for describing the balance between the attractiveVan der Waals forces between particles and the stabilizingrepulsion due to their electrostatic charge. Although theunderlying concept is simple, the theory manifests itself inmore or less complicated equations, so it is usually assumedto describe the stability of dispersions in paint withoutany quantitative understanding being attempted. Usually, awell-designed latex paint is stable and none of the compo-nents are flocculated or settle quickly. This paper reviewsthe environment within a liquid paint so that DLVO theorymay be applied in a realistic way and also shows what otherinformation has to be included in order to have the theoryto correlate with the dispersion stability of TiO2 pigment, atypical latex and an extender.

The simple representation of the DLVO theory commonlyfound in descriptions of paint stability does not include

∗ Tel.: +1-701-231-9415; fax:+1-701-231-8439.E-mail address:[email protected] (S. Croll).

a detailed description or appreciation of the system as awhole. In order to apply these ideas usefully or successfully,we need to have a clearer understanding of the structure ofthe particles and their environment. The continuous phase(water) of commercial paint has a much higher level ofdissolved salts than most studies in the colloid literatureconsider. Thus, there is a much higher ionic strength andconductivity, which means that the water cannot supporthigh levels of electrostatic stabilization. Paint is formulatedat much higher volume concentrations than many literaturestudies of colloidal systems; the particles are comparativelyclose together, less than their own diameter apart. Underthese circumstances, conventional descriptions of charge sta-bilization have been questioned and other interactions havebeen invoked to account for stability particularly at shortrange, e.g. hydrophobicity and acid–base interactions [2,3].

The primary pigment in paint is usually a grade of titaniumdioxide. The light scattering properties of the pigment aredue to the core titanium dioxide of the particle. However,these particles are coated with one or more layers of othermaterials, so their surface characteristics are not of TiO2.Similarly, it is well known that latex particles have a layeredstructure, as well as possibly other features. In particular,latex particles have a more or less “fuzzy” surface of acidic(charged) polymer and steric stabilizing entities external to

0300-9440/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0300-9440(01)00261-2

132 S. Croll / Progress in Organic Coatings 44 (2002) 131–146

the core polymer binder. There is no understanding of stericstability similar in status to DLVO, so this paper will focusmore on titanium dioxide pigment particles because they aredesigned and used with electrostatic charge stabilization inmind. It will show the value of the pigment structure in thedispersion stability and include the effect of “dispersant”polymer.

2. Background

Molecules are attracted to each other by London disper-sion forces, Keesom orientation forces and Debye inductiveforces. It is assumed that the molecules in colloidal, andother particles, are attracted to each other in a pairwisefashion similar to the way they would be in a gas. Twolevels of approximation are available for calculating theoverall attractions between solid or liquid materials.

The more rigorous approximation is to use “Lifschitz”calculations [4,5] to calculate the strength of the attractionbetween material bodies. This is a level of approxima-tion that is derived from continuum electrodynamics andrequires knowledge of the frequency dependent dielec-tric response of all the materials in the system, even theintervening medium. Unfortunately, sufficiently detaileddielectric properties of materials are not been readily avail-able by experimental or theoretical means. Depending onparticle geometry and material homogeneity, any of thesecalculations may require up to six numerical integrationsnone of which converge very quickly. The calculations areponderous and do not give useful, explicit algebraic results.

The second, less ponderous, approximation is to use theDLVO theory that incorporates a less rigorous method of in-tegrating the dispersion forces and makes some geometricalapproximations. One of the results is a “Hamaker” constantthat is characteristic of the attractive forces between materi-als. Useful algebraic expressions can be obtained for somecommon geometries [6], including flat planes and spheresthat need not have equal diameters. The more complicatedequations can be programmed into a computer then usedquickly and conveniently to explore a variety of situations.DLVO theory is often used with Hamaker constants that arecalculated using Lifschitz calculations or, indeed, it is usedto analyze experimental data to measure Hamaker constants.One must remember that this approach employs geometricaland numerical approximations and should be used carefully[1,7–10].

The simplest forms of the DLVO theory can be foundin many textbooks on colloidal and interface science [1,6]and this is the form that is most often assumed by painttechnologists. As mentioned before, it describes the inter-action potential between any two particles as the sum ofthe stabilizing repulsive electrostatic potential,Vrep, and theattractive potential,Vatt. The attractive potential is given by

Vatt = − AR12h

(1)

for R � h, whereA is the Hamaker constant which is aproperty of the material that describes the strength of theattraction. The force of attraction is higher in materials withhigh values of this characteristic,h the separation betweenthe particles, andR the particle radius. We need, therefore,to know how the separation between particles compares totheir size before we know whether this approximation issuitable.

The repulsive potential is given by

Vrep = 2πεε0Rς2 ln[1 + exp(−κh)] (2)The overall, total potential is given by the sum ofVatt andVrep. Here,ε0 is the permittivity of free space,ε the relativedielectric constant of the continuous phase (∼79 for waterat room temperature). The zeta-potential,ζ , is the potentialon the pigments measured by determining their mobilitythrough the continuous phase. Although it is due to thecharge on the surface of the particles it is measured at someradius typical of the hydrodynamics of the particle and thecharges surrounding it and thus is not exactly the potentialat the surface, but at some “shear plane”.

Eq. (2) is the simple equation usually quoted forκR >10, which we shall see is the situation in most aqueouspaint because the ionic strength of the medium is high.κis the inverse Debye length (the exponential decay distancefor the screened electrostatic potential) and as the ionicstrength increases, this length becomes shorter and shorter.That is, as the medium becomes more conductive it is lessable to support the electrostatic field:

κ =[∑

j (zj e)2nj

εε0kT

]1/2(3)

Here nj is the concentration, number per unit volume, ofthe jth type of ions, of chargezje, wheree is the elementarycharge. One has to be careful with the units ofn to ensurethat they are consistent with the calculation if convertingfrom molar units. At the pH of aqueous housepaint, 8–9.5,many other multivalent ions are hydrated or hydrolyzed andcarry a net single charge, so a reasonable approximationis that z = 1. One also has to remember to include theconcentration of both the anion and cation from a dissolvedsalt.

In fact, when one includes values of the physical constantsin Eq. (3), it becomes much simpler for a single type ofelectrolyte where the anions and cations both have the samemagnitude of charge:

1

κ= 0.304c−1/2[z]−1 (4)

Herec is the concentration in mol/l. All the numerical con-stants have been incorporated so that the screening length(1/κ) is expressed in nanometers and the temperature is takenas 25◦C, or 298 K.

The expressions so far have been for one type of particlein terms of size and material composition and for particles

S. Croll / Progress in Organic Coatings 44 (2002) 131–146 133

that are very close each other. This paper will deal withtypical paint ingredients. Many paint ingredient particlescontain layers of different materials, e.g. the alumina andsilica surface treatments on titanium dioxide pigments andthe non-ionic surfactants and “protective colloids” on latexparticles. We need to allow for particles in paint that are notsolely one material; we need to know how far apart they are;the effect of any polymers adsorbed on their surface and weneed to know the ionic strength of the intervening medium,i.e. its composition (or conductivity). Unless these inputs aremade, the inaccuracies in applying the DLVO theory willovershadow its use.

DLVO equations are available for application to thecomplicated situation in aqueous paint [11]. They providefor a spherical particle to be covered with two concentric,continuous layers of different materials, as in Fig. 1. AllDLVO expressions incorporate approximations. The formreproduced here allows investigation into the effect of lay-ered particles, but other, simpler forms may be more usefulin simpler circumstances:

−12VA = (A1/2o − A1/2m )2Ho + (A1/2i − A1/2o )2Hi+ (A1/2p − A1/2i )2Hp + 2(A1/2o − A1/2m )× (A1/2i − A1/2o )Hio + 2(A1/2i − A1/2o )× (A1/2p − A1/2i )Hip + 2(A1/2o − A1/2m )× (A1/2p − A1/2i )Hop (5)

where theH functions are defined in the equations displayedbelow. For the various situations:

Fig. 1. Schematic diagram of single titanium dioxide pigment particle, showing the layer structure.

Ho : ∆ = h, R1 = R2 = R + δt,Hi : ∆ = h + 2δo, R1 = R2 = R = δi,Hp : ∆ = h + 2δt, R1 = R2 = R,Hio : ∆ = h + δo, R1 = R + δt, R2 = R + δt,Hip : ∆ = h + δo + δt, R1 = R, R2 = R + δi,Hop : ∆ = h + δt, R1 = R, R2 = R + δt (6)

whereh is the separation between the particles, i.e. betweenthe two outer layers,R the radius of the central core ofthe particle (material p),δi the thickness of the inner layer,δo the thickness of the outer layer andδt the combinedthickness of the inner and outer layers.

Subscript ‘p’ refers to the central part of the particle, ‘i’refers to the inner layer, ‘o’ refers to the outer part of theparticle’s layers and ‘m’ refers to the intervening continu-ous medium between these structured particles. The othersubscripts are for algebraic use as shown later.

The separation between the particles in a slurry is∼30 nm and in a typical paint∼100 nm so the transit timefor the electromagnetic field between the particles becomescomparable with the period of the dipole vibrations in thematerials. In this case, the so-called “retarded” potential hasto be used that allows for the non-instantaneous characterof the interaction [10].

The expression for theH functions is given by

HS = a[y

u+ y

u + y]

+ 2 ln(

u

u + y)

+ 8bR21

C

[2y + (2u + y) ln

(u

u + y)]

(7)


or

H L = a′

10C

[y(1 + y)2

u2+ y(1 − y)

2

(u + y)2 −2(y2 + y + 1)

u

+ 2(y2 − y + 1)U + y + 4 ln

(u + yu

)]

+ b′

60R21

[2

u + y −2

u+ y

2 + y + 1u2

− y2 − y + 1(u + y)2 −

y(1 + y)2u3

− y(1 − y)2

(u + y)3]

(8)

whereC = R1+R2, x = ∆/2R1, y = R2/R1, andu = x2+xy+ x anda = 1.01, b = 0.14(2π/λ), a′ = 2.45(λ/2π),b′ = 2.04(λ/2π). λ is a numerical constant that has thedimensions of length= 100 nm.

HS is the approximation to be used when∆ is less than afitted parameter,∆∗; HL the appropriate approximation forlonger ranges and should be used when∆ is greater than∆∗. ∆∗ was determined empirically by Vincent for use inthis version of the DLVO equations [10]:

∆∗ = 109− 107 log10R + 37.5(log10R)2 − 4.5(log10R)3(9)

whenR and∆∗ are expressed in nanometers.These expressions for retarded attractive potentials were

given in a paper that only considered one surface treatmentlayer, but allowed the two particles that to have differentcore materials and have that surface treatment layer materialto be different in each case. Here, these equations are usedin conjunction with expressions from the same author in arelated paper that incorporates two surface layers [11].

VDW forces are also transmitted between dipoles viaelectromagnetic fields so if the electrostatic repulsion isscreened by the conductivity of the medium then so mustthe VDW forces. This has been recognized before; thescreening factor used here,(1 + 2κh)exp(−2κh) reducesthe VDW potential in highly conducting media [1,8]. Thisis the usual approximate approach. A more rigorous calcu-lation would apply the screening reduction to the ‘zeroth’term in a Lifschitz calculation. Fortunately, this ‘zeroth’term is usually a major contribution to the overall result sothe approximation is useful [9].

The electrostatic repulsive potential,VR, used here is arecent and rigorous derivation [12] that enables the potentialto be calculated for particles of different sizes and surfacepotential:

VR = ε4

(kT

e

)2R1R2

R1 + R2 + h [(y1 + y2)2 ln(1 + e−κh)

+ (y1 − y2)2 ln(1 − e−κh)]Here R1 and R2 are the overall particle radii of possiblydifferent particles, including the core and layer thicknesses,εthe dielectric constant of the medium,T the absolute temper-ature (K),k Boltzmann’s constant,e the charge of a proton,

andy1 andy2 are the reduced surface potentials of the twotypes of particle (typically identified with the zeta-potential,and divided bykT/e).

In order to apply the theory, it is evident that somequantitative understanding of the system is required. Thatknowledge includes the composition and structure of theparticles, some idea of the relative separation between par-ticles, the ionic strength of the continuous phase (so theDebye screening can be estimated), the zeta-potential on theparticles, the size of the adsorbed layer of dispersant andthe size of the particles themselves. Also, it is useful to un-derstand the system well enough to reassure ourselves thatwhen we investigate instability of paint, we are consideringthe correct ingredients in some appropriate priority.

3. Paint systems

3.1. Settling rates

Particles settle under the influence of gravity whichdepends on their density and size. The forces that act inopposition are buoyancy (depends on the density of themedium relative to the particle and on its size) and viscousdrag which is greater for large particles and for viscousmedia. The well-known “Stokes” equation describes the sed-imentation rate of an isolated particle in a viscous medium:

v = 2g(ρparticle− ρmedium)R2

9η

whereg is the gravity of acceleration constant, 9.81 m s−2,ρ density,η viscosity,R the particle radius, as before, andv the terminal settling velocity.

This is an approximation for a crowded system like paintbut it provides illustrative data. If we consider, as exam-ples, latex particles that have a diameter of 300 nm and adensity of 1190 kg/m3 (like polymethylmethacrylate), silicaextender particles of radius 7.5�m and density 2650 kg/m3

(quartz) and a titanium dioxide pigment that has a densityof 4030 kg/m3 and a diameter of 319 nm [13]. Water has adensity of 1000 kg/m3 and a viscosity of 1 mPa s (0.01 P),but paint viscosity at the low shear rates typical of settlingis increased due to the volume of solids in suspension andwith the help of thickeners by several orders of magnitudeand can be greater∼100 Pa s (1000 P). Results of calcula-tions with these parameters are given in Table 1.

The right-hand column has been translated into moreobvious units and demonstrates the usefulness of thicken-ing paint to hinder settling. Naturally, the denser inorganicparticles are more likely to settle than the almost neutrallybuoyant latex particles. A well-dispersed suspension of anyof these materials should not settle quickly under normalcircumstances in a viscous paint medium. The onus onthe paint formulator is to ensure that the particles do notflocculate into large enough agglomerations that they settlequickly or affect the other properties of the paint film.


Table 1Settling rates of paint ingredients

Particle In paint (viscosity, 100 Pa s) (m/s) In paint (distance settled in a day) In pure water (1 mPa s)

Titanium dioxide pigment 1.68× 10−12 0.145�m ∼1.5 cm/daySilica extender 5.06× 10−10 0.044 mm ∼2 cm/hLatex 9.32× 10−14 8.0 nm ∼2.4 mm/month

3.2. Relative numbers of particles in a paint formulation

In a practical sense, paint formulations will be moresensitive to problems with the majority ingredients thanthose that are present in the minority. Provided that the sizeand density of the ingredient particles in a formulation areknown then it is straightforward to calculate the relativenumbers of each type of particle present in the paint. How-ever, the author does not know of any published examplesof this. The numbers presented in Table 2 were calculatedfor a simple, typical formulation. Paint formulations arecommonly expressed as pounds per hundred gallon, butmost scientific calculations are in metric units, so the tablecontains a mixture of physical units.

For every one of the larger extender particles there arenearly 20,000 TiO2 or latex particles. The exact result of thistype of calculation depends on the formulation and exactparticle size, but it is clear that the dominant species, asfar as number of particles are concerned are the latex andtitanium dioxide particles.

Clearly, the titanium dioxide pigment and the latex arethe important species to examine first if a formulation issuspected of being flocculated.

It is possible to do a similar comparison by surface area,with similar conclusions, although less dramatic results.Titanium dioxide pigment and latex have comparativelyhigh specific surface areas and represent about 90% ofthe surface area available in the mixture, compared to theextenders which together only provide about 10% here.

3.3. Ionic strength

In order to estimate the inverse Debye layer thicknessin calculating the colloidal stability of the dispersions, it isnecessary to determine a realistic value for the ionic strengthof the continuous medium. Typical anions found in paintare given in Table 3 [14]. One can see that in this case,an approximate description of the supernatant compositionwould be a sodium sulfate solution.

Table 2Relative number of particles with respect to the extender

Material Amount, dry (lb/100 gallons) Particle diameter (nm) Material density (kg/m3) Relative number of particles

TiO2 pigment 225 319 4030 18300Silica extender 160 7500 2610 1Calcined clay 50 1500 2630 39Latex 180 300 1190 17700

Table 3Partial (inorganic) composition of paint serum

Ion Concentration (�g/ml)

Aluminum 100Sodium 2030Silicon (silica fines) 250Phosphorus 78Titanium


Fig. 2. Solution conductivity as a function of concentration for three sodium salts (chloride, sulfate and orthophosphate) that may simulate the ioniccharacter of a typical aqueous paint serum.

Eq. (4) can be used to calculate an appropriate value of theinverse Debye thickness. For the 1:1 electrolyte, NaCl, theinverse Debye thicknesses is∼1.6 nm, a very short rangedpotential. The value would be slightly less if we used valuesof concentration from the sodium sulfate conductivity curve.For comparison, this value is less than 1% of the size ofthe diameter of the titanium dioxide pigments or the latexparticles.

3.4. Particle separation

The other useful conceptual comparison is with theinter-particle separation. In addition, many DLVO expres-sions, including those given here are for situations whereinthe particles are large compared to the separation betweenthem. Hence the ratio of particle size to separation needsto be estimated. Appendix A contains the background tothe calculation. It is an approximate geometrical packingcalculation that assumes all the particles in the paint are, onaverage, equally separated regardless of their size, charge,or composition.

The composition used as an example in Table 1 wasused to calculate the particle–particle separation in a paintof 34.5% volume solids. It has been found before thatwell-dispersed titanium dioxide suspensions sediment toa packing fraction of 0.57 [13] which is close to “looserandom packing” [15] which is usually assigned a value of0.59 (this volume fraction has also been associated with anorder–disorder transition in colloidal suspensions [16]). Of-ten when rheological studies are conducted on particulatesuspensions the viscosity diverges at a packing fraction of

0.63 [17] which is often called the “dense random” packingfraction. For illustration, the separation calculation has beendone at packing geometries that correspond to both looserandom packing and dense random packing.

For a geometry corresponding to loose random packing,0.59, the calculation for this mixture of ingredients givesa spacing of 76 nm. For a geometry that corresponds todense random packing the spacing calculation gives 88 nm(a denser packing arrangement actually permits the samenumber of particles to be further apart). Particle separationsare much less than the particle sizes which complies withthe assumptions made in deriving the particular DLVO equa-tions used here. It also implies, even in a well dispersedand stabilized paint that particles tend to maintain their spa-tial juxtaposition because motion involving several particlesis necessary before one could change its local relationshipwith others. It is easy to see how the motion of particlesbecomes restricted by their neighbors, leading to increasedviscosity and the possibility of their getting jammed in anorder–disorder transition [16].

The exponential decay length of the electrostatic potentialis a small fraction of the separation between the particles andso for many purposes, the stabilized particles in an aqueoushousepaint can be visualized as “hard” spheres.

3.5. Structure of titanium dioxide pigment particles

All titanium dioxide pigments supplied to the coatingsindustry have a layered structure. They are surface treatedwith other, hydrated oxides. Many general purpose pig-ments and pigments for exterior use have a layer of silica


Fig. 3. (a) Transmission electron micrograph of a general purpose titanium dioxide grade, DuPont TiPure® R706. The dark core materials are crystallitesof titanium dioxide. The denser components of the surface treatments can be seen although with less contrast. The amorphous component of the surfacetreatment will have very little contrast. (b) Atomic force microscope phase image of a particle of a general purpose titanium dioxide grade, DuPontTiPure® R706. The background is a glass microscope slide. This image shows that the topography of the surface treatment is smoother than might havebeen expected from conventional transmission electron microscopy.


deposited over the base titania pigment in order to slowdown the production of free radicals at the pigment surface.This layer also helps in providing the surface of the pigmentwith a negative charge at the alkaline pH of typical aqueoushousepaint. All pigments for use in coatings have an outerlayer of hydrated alumina as a processing aid in the pigmentmanufacture, to render the surface more hydrophilic andto provide a good substrate for stabilizer adsorption. Thehydrated alumina also helps determine the surface potentialof the pigment [18].

One can calculate the layer thicknesses from the amountsof these surface treatments given by the manufacturers.Using a density of 2190 kg/m3 for amorphous silica [19]and a silica content of 3%, a silica layer proves to be∼3 nmthick. The effect of a silica layer on the Hamaker constantof titanium dioxide pigments in water has been exploredelsewhere and shows that there is comparatively little dif-ference in effect between fully dense silica and a very dilutesilica gel [20]. Thus, the Hamaker constant of unhydratedsilica is used here in the calculations.

If one uses a density of 2420 kg/m3 for hydrated alumina(Gibbsite) [19] and remembers that the surface treatmentis presented as alumina by the manufacturers so an adjust-ment has to be made, e.g. 3% alumina represents∼5% asGibbsite. This amount of hydrated alumina corresponds to athickness of about 5 nm. These values assume that the pig-ments are spherical and that each of the layers is perfectlycompact, smooth and even. It also assumes that the hydratedalumina is only Gibbsite which is not the case [18], but it isa reasonable assumption for the purposes of this paper. Il-lustrative transmission electron micrographs or atomic forcemicroscope images in Fig. 3(a) and (b) (DuPont TiPure®

Fig. 4. Influence of a 1% solution of sodium polyacrylate,Mw 8500, at different ionic strengths on the zeta-potential of a TiO2 pigment.

R706) show that the surface layers are not perfectly compactor even but the above values will be used for illustration inthe quantitative work presented here. They also demonstratethat the particles are not spherical, nor are they all the samesize, however the images do convey a realistic microscopicview of pigment particles and, perhaps for other purposes,one can gain a better idea of pigment structure and thedetailed structure of the surface treatment.

3.6. Adsorbed polyelectrolyte layer

In order to apply DLVO theory, the action of pigmentdispersants needs to be included since the main reason forhaving these materials is to increase the stabilizing charge(negative at the high pH of paint) and thus the potential onthe surface of the pigment.

Fig. 4 shows the effect of adding a sodium polyacrylatepolymer as a “dispersant” to titanium dioxide at a pH of 9.0for a range of ionic strength that is expressed as conduc-tivity. Sodium polyacrylate adsorption was determined tobe between 0.3 and 0.4% by weight on the plateau regionof the adsorption isotherms, see Fig. 5. The isotherms weredetermined by separating the pigment with a centrifuge andusing carbon analysis on dried pigment [14]. It is apparentthat the polyacrylate polymer adds a significant amount tothe magnitude of the stabilizing zeta-potential,−39 mVin contrast to−15 mV at a conductivity of 0.5 S/m. Bothcurves show that the stabilizing potential is considerablyreduced as the conductivity of the medium increases andthus increasingly screens the effect of the surface charge onthe particles. It is clear that decreasing the ionic strength ofthe paint helps dispersion stability in the sense of achieving


Fig. 5. Adsorption isotherms of sodium polyacrylate of various molecularweights on TiO2 pigment at pH 9.0.

a higher zeta-potential, particularly in the case when thereis no added charge due to dispersants. At the conductivityrepresentative of the housepaints examined here, 0.31 S/m(0.034 M equivalent electrolyte concentration), the magni-tude of the zeta-potentials can be obtained from the curvesas 41 mV with the dispersant and 17.5 mV when there is nodispersant.

The other factor that will be incorporated into thedescription is the effective thickness of the polyelectrolytelayer that is adsorbed onto the surface of the pigment.This was determined by sedimentation experiments [13]to be approximately 5 nm for the polyelectrolyte stabilizerused here. The polymer diameter including the neutral-ized side-groups was determined to be approximately 1 nmwhich might be representative of the layer thickness if allthe polymer adsorbs flat to the surface of the pigment.

4. Application of DLVO theory

Before embarking upon this discussion, it is well toremember that all versions of DLVO theory containapproximations and compromises. Different versions in theliterature use different forms of the equations and may em-ploy different values of Hamaker constant, etc. It would beunrealistic to assume that these calculations would exactly

match calculations with other equations given elsewhere.The objective was to examine the effect of known structurein the particles and to see whether the description of theireffect was echoed in practical experience. This work wasdone in order to understand the effect of various aspects ofthe particles at large in paint and not to endorse a particularexpression for the DLVO forces.

Hamaker constants for many materials are available inthe literature. They tend to be updated periodically as bettermeasurements or calculations become available. Valuesof 42.1 kT for rutile, 16.05 kT for silica, 35.3 kT for alu-mina [21] are used, assuming that the temperature is 25◦C(298 K). Similarly, the value for water is taken as 13.9 kT[22].

A Hamaker constant for the outer surface treatment layerof hydrated alumina [18] was calculated as 21.8 kT. Thiswas done using an approximation for composite materials[10] and assuming that this region could be approximatedby a 1:1 molar mixture of alumina and water. This is anoversimplification but there does not appear to be a suitablevalue in the literature.

Unless otherwise specified, the ionic strength of the paintmedium will be assumed to be equivalent to 0.034 M of a1:1 electrolyte, as explained before. This corresponds to aninverse Debye thickness of 1.6 nm. Since the calculationswill be for the potential between like particles they can bepresented here as a function of the distance from a particlesurface since this is a more intuitively useful here than theusual method in the literature of representing it as a functionof the separation between particles (twice the distance fromone particle). This section will attempt to show the contri-bution that each level of detail makes to understanding thestability of TiO2 pigment and other suspensions.

4.1. The simplest case

If we apply the simple DLVO theory, assuming that thepigment is comprised only of titanium dioxide we get thecurves in Fig. 6. The potential does not much exceed 1 kT,even with the added charge imparted by the stabilizing dis-persant. Generally, a barrier height of 10 kT [23] is thoughtto be necessary to provide a barrier against flocculation.Thus, this calculation suggests that such a pigment dis-persion will flocculate, but this is not true. Evidently, thispicture is incomplete.

4.2. Pigment surface treatment structure

One can repeat the calculation above, but now includingthe effect of the inner layer of silica and the outer layer ofhydrated alumina. This graph was calculated for the case ofthe pigment without adding polyelectrolyte stabilizer so thatthe effect of the surface treatment layers was clear. Fig. 7shows that there is a higher potential barrier in the case of thesurface treatment structure. The calculations were repeatedby assuming that there was only one layer of the combined


Fig. 6. The application of simple DLVO theory, assuming that the pigment is comprised only of titanium dioxide. The potential does not exceed muchto 1 kT, even with the added charge imparted by the stabilizing dispersant.

thickness of either silica or hydrated alumina. There wasvery little difference. Assuming that both surface layers werealumina was very close to results allowing there to be acomposite layer structure. Clearly, the material of the innerlayer does not exert much (separate) influence and the moreimportant structure is the outer layer.

The increase in potential barrier is because the attractionbetween the particles is reduced by now allowing an outerfew nanometers on each particle to be of a material witha lower Hamaker constant (less attractive force). However,this calculation result for potential barrier is still much toolow to predict stability and the situation remains incomplete.

The TEM image, Fig. 3(a), shows that in places the surfacetreatment layers, particularly hydrated alumina, may extendmuch more than the calculated value suggests. Thus, therewill be a greater stabilizing effect. Conversely, there mustalso be places where the layers are very thin so their effectis reduced. This implies that the surface charge density willfluctuate from place to place on the surface of the particlesand may lead to flocculation [24,25]. The hydroxyl contentof the hydrated alumina ensures that the pigments are much

more hydrophilic than they would be if they were coatedonly with silica (for UV durability control). This hydroxylfunctional surface provides a basic surface that is a goodsubstrate for the polyacid salt stabilizing molecules. These“dispersants” add charge to the surface and thus may accountfor particle stability.

4.3. Allowing for the adsorbed polyelectrolyte

The lowest curve in Fig. 8 is the potential for the particleswith surface treatment structure and adsorbed dispersant,but where no allowance has been made for the adsorbedpolymer to have any physical thickness. The increasedzeta-potential (−41 mV) has increased the peak potentialbarrier to almost 5 kT in contrast to the value of∼0.2 kTwithout the polymer. Clearly, the added surface charge fromthe adsorbed “dispersant” is very valuable for dispersionstability. Nevertheless, it is not enough to meet the criterionof being more than∼10 kT.

There is a considerable effect, as soon as we allow forthe effect on the repulsive electrostatic potential of a finite


Fig. 7. DLVO potentials showing the effect of including a typical inner layer of silica and outer layer of hydrated alumina. This graph was calculatedfor the case of the pigment without added polyelectrolyte stabilizer so that the effect of the surface treatment layers was clear.

adsorbed layer of polymer. This can be done in a simpleway by shifting the origin of the potential into the externalphase by an amount equal to the thickness of the adsorbedlayer [26]. One could argue that the polymer would con-tribute to the attractive force, but polymers have a lowHamaker constant (∼17 kT) compared to titanium dioxide[21], and one would expect the polymer to be very hydrated(leading to an even lower Hamaker constant) and present adiffuse interface to the surrounding continuous phase so itis a very reasonable approximation to neglect the attractivecontribution of the very thin polymer layer.

Fig. 8 shows that the potential barrier rises to about 15 kTat a distance that corresponds approximately to the adsorbedlayer thickness. The adsorbed polymer layer thickness al-lows the repulsive potential to rise to a value that wouldprovide dispersion stability. The layer thickness of 5 nm wasthat measured for the dispersant used here [13]. Another,narrower layer thickness of 1 nm approximates to the poly-mer diameter and a potential curve for an intermediate layerthickness of 2.5 nm is provided as an illustration.

With some exceptions, most dispersant polyelectrolytesused in paint formulations have molecular weights closer to

∼3000 g/mol and so might form tighter layers on a pigmentsurface since they would be less inclined to form loops andtails [27] that of MW 8500. One might then imagine thata value of 1 nm might be more typical. However, polyelec-trolytes are themselves charged so their effective size willbe increased somewhat due to their intrinsic repulsive po-tential. In addition, results from a different colloid systemshow how effective only a few molecular tails are in provid-ing steric stability [28]. Thus, the range of layer thicknessesin Fig. 8 probably encompasses most situations met withtypical dispersants in aqueous housepaints.

The curves that achieve a potential barrier of 10 kT, implydispersion stability which is indeed the case of titanium diox-ide suspensions made with this amount of polyelectrolyte.Using input (ionic strength, sizes, etc.) that is representativeof typical paints, DLVO theory appears only to describe sta-bility successfully when enough detail about particle struc-ture is included. In doing so, the function of pigment surfacetreatments and “dispersant” are made more apparent.

Adsorbed polyelectrolyte not only adds surface chargeand thus increases the stabilizing zeta-potential but it alsodisplaces the location of stabilizing potential away from the


Fig. 8. Effect of adsorbed layer thickness on DLVO predictions of inter-particle potential. All curves use zeta-potential (−41 mV) that includes theadsorbed sodium polyacrylate and allowing for pigment surface treatment layers; ionic strength 0.034 M (1:1 electrolyte).

locus of the attractive potential and so has a major effect onthe particle stability. Its layer thickness is not large comparedto the particle–particle separation but a major part of itsinfluence is because of its spatial extent and the combinedaction has been labeled an ‘electrosteric’ mechanism [29].Although they do not attempt to describe the nature of asteric barrier, the example calculations here reinforce thecommon formulating practice of including components thatprovide some steric stabilization.

One must still remember that titanium dioxide and otherinorganic materials are substantially denser than the contin-uous phase in paints. They do eventually settle and needre-agitation prior to use.

4.4. Variations in ionic strength

Prior calculations have been done at an ionic strengthequivalent to that of 0.034 M of a 1:1 electrolyte. This waschosen as representative of the paints evaluated by conduc-tivity measurement. However, the highest value measuredcorresponded to an ionic strength of 0.062 M. Fig. 9 showsthe effect of varying the ionic strength. An ionic strength of0.01 M was chosen as a plausible lower figure for typicalaqueous paint systems, although even lower values may beachieved.

Data on the pigment showed that it had a higherzeta-potential at lower ionic strength. This has the effectof increasing the maximum height of the potential bar-rier. Otherwise, if the zeta-potential was independent ofionic strength the only effect would be that the stabilizing

potential would be appreciable over a longer range sincethe screening length (inverse Debye length) would be in-creased. Both effects lead to greater dispersion stabilitywhich is the advantage of designing systems with low ionicstrength if longer term stability is required.

4.5. Other paint suspensions

For comparison, as in the settling calculations givenearlier, potential–distance curves have been generated fora silica extender and a latex, Fig. 10. Values were takenfrom the literature for use in these calculations. A Hamakerconstant for polymethylmethacrylate of 17 kT at 25◦C [6]was used as typical of most acrylic latexes used in paintalthough actual values determined in the literature seem tobe uncertain [30] and this value is higher than many. Forcomparison, the zeta-potential on the silica and latex waschosen to be the same as on titanium dioxide surface withadsorbed polyelectrolyte at 0.034 M, i.e.−41 mV; the par-ticle sizes were the same as in Table 1. Silica particles andtitanium dioxide particles were both assumed to have a 2 nmlayer of polyelectrolyte adsorbed to them since that wouldbe more typical of common 3000 g/mol dispersants used inpaints.

Commercially available latex usually has a hydrated outerlayer originating from carboxylic acid co-monomers used inthe synthesis particles together with some non-ionic stericstabilizing polymer present at the surface. The curve herewas generated by allowing for a 5 nm displacement of theeffective position of the charge from the main material body


Fig. 9. DLVO calculations showing that reducing ionic strength increases the range of the potential barrier and allows a greater barrier size due toincreased zeta-potential.

of the particle. Often the zeta-potential on latex has a muchhigher magnitude than 41 mV and one could justify alsoa much deeper “fuzzy” layer [31] so the curve for the la-tex almost certainly underestimates the stabilizing poten-tial. However, even the notional latex here has a potentialcurve that indicates stability, even at the fairly high ionicstrength.

Both the latex and silica have Hamaker constants that aresignificantly smaller than that of titanium dioxide and so theparticles are intrinsically less attractive to one another. Thesilica extender also benefits from its size since the net effectof size is to dilute the Van der Waals forces compared tothe electrostatic potential that originates at the surface of theparticles.

4.6. Other possibilities of suspension instability

The latex and pigment particles here have potentials thatextend only a few nanometers into the surrounding medium.Thus, if they are all well stabilized and dispersed they havelittle influence on each other and so the viscosity of the sys-tem will be quite low. As an example, the Krieger–Doherty[32] relationship can be used to estimate the relative

viscosity,ηrel, of a suspension (with respect to the medium):

ηrel =[1 −

(φ

p

)]−2.5pwherep is the maximum packing possible in the suspension,assumed to be 0.64 here.

If we allow the particles in the paint formulation (origi-nally at 34.5% volume solids, see earlier) to have a addedlayer of 5 nm around them due to an adsorbed dispersantthen the effective volume solids becomes 35.9%. The rela-tionship above predicts that the suspension will only be 3.7times as viscous as the medium, i.e. water. Clearly, paintformulators must increase the viscosity of the continuousphase.

The separation between particles in the paint formulationexamined here is approximately 80 nm. This is the sameorder of magnitude as the size of some thickener molecules.For example, if we use a very common approximation tocalculate the radius of gyration of a polymer in solution[33] (〈R2g〉1/2 (nm); 0.06 M1/2), we find that associativethickeners of 70,000 g/mol molecular weight have a radiusof gyration of 15 nm. The radius of gyration is an estimateof the radius that the polymer sweeps out in solution, so


Fig. 10. Possible DLVO inter-particle potential curves for paint components. The zeta-potential on the silica and latex was chosen to be the same as ontitanium dioxide surface (with adsorbed polyelectrolyte at 0.034 M, i.e.−41 mV); the particle sizes are as in the table.

these molecules may extend even further. Elsewhere, theradius of gyration of hydroxyethylcellulose thickeners hasbeen measured at∼60 nm for the high molecular weight,1,000,000 g/mol, types that are often used [34]. The highmolecular weight (∼1,000,000 g/mol) polyelectrolyte thick-eners will have a radius of gyration of tens of nanometers insolution [35]. Depending on their concentration and adsorp-tion onto particle surfaces there may be significant depletionforces in play because the depletion volume [36,37] occu-pies much of the interstitial volume between the particlesand thus increases the effective forces between them. Thismay lead to depletion flocculation [38–40] as much be-cause there are so many small particles in the paint, whichforces a small separation between them, as the polymersize issues that are usually presented as the source of theflocculation.

Another possible interaction that may be increased is thatof bridging flocculation. This occurs when the same poly-mer molecule adsorbs on more than one particle and thuscauses an agglomeration. Although this tends to occur at

low polymer concentrations when there is a greater chanceof unoccupied adsorption sites on neighboring particles, onecan easily see that this tendency might increase in formula-tions wherein the separation between particles is not biggerthan the size of the polymer molecules in solution.

4.7. Latex film drying and film formation

The high ionic strength of typical aqueous paints is suchthat the inverse Debye length is∼2 nm so that it representsa small fraction of the particle–particle separation and theparticle size. Thus the particles, although crowded, do notsense each other unless they approach closely and the par-ticles and their surrounding DLVO potential can be thoughtof as hard spheres that have essentially the same size as thematerial particles.

However, this hard sphere nature has an impact on theway that latex particles come together to form a film. If therewas a long range repulsion between the particles, as therewould be if the ionic strength of the continuous aqueous


phase were very low, then the position of one particle wouldbe influenced by the repulsion from its neighbors. For manyyears, conventional colloidal stability studies were carriedout in continuous phases that were very pure which allowedthe stabilizing potential to extend comparatively far from theparticle surface. In this situation, one can appreciate how, asthe paint dried, the particles would avoid each other and en-sure that their concentration was maintained evenly through-out the suspension as it increased during drying. This wouldnaturally lead to a more gradual first phase of latex filmdrying as originally proposed [41], wherein the concentra-tion of latex particles increases but remains homogenous inthe film.

If the particles do not sense others until they are veryclose then this allows most particles to move freely. Theywould only pack together when they are very close to theirneighbors. Thus, the later ideas of latex drying that involvea more sudden particle interaction during drying and amoving particle consolidation front [42] with the formationof a packed layer of latex [42,43] are better supported inpaints that have the high ionic strength characteristic of acommercial aqueous paint formulation. Thus the two viewson latex paint drying result from a different appreciation ofthe ionic strength typical in paint.

5. Conclusions

DLVO theory has been reviewed. It fails if ionic strengthvalues are high enough to be representative of commerciallysupplied paint are put into the common, simple expressionsof the theory because they do not predict inter-particle po-tentials large enough to provide stability. It is successfulonly when details on particle composition and polymeradsorption can be incorporated. Although the theory workshere for particles that are at least roughly spherical and thuscorrespond to geometrical approximations used to derivethe equations, there are instances when the theory cannotbe applied to particles that do not meet these constraints,e.g. clay particles [44].

The main source of the effectiveness of the detaileddescription is the effect of polyelectrolyte dispersant poly-mers that not only add charge to the particles but they alsohave the effect of displacing the locus of that charge (andthus the origin of the zeta-potential) a few nanometers awayfrom the particle surface into the aqueous phase. The lowerHamaker constant of the inorganic surface treatment layerson the TiO2 pigment contribute much less to the reductionin attraction between particles.

The scheme presented here to account for the stabilizationof pigment particles could be invoked to describe the stabil-ity of other structured particles, e.g. latexes. This approachwould also be valuable in studying latex particles that have“fuzzy” stabilizing layers.

A simple packing calculation has been presented that al-lows a calculation of the particle–particle separation in a

randomly arranged suspension of particles. This shows thatthe separation between particles is very small compared totheir size. However, even at these close packed situations therange of the repulsive electrostatic forces is a small fractionof the separation. This short range interaction is consistentwith ideas of latex paint film drying that involve localizedand comparatively sudden consolidation of particles in anouter layer rather than the more conventional, more gradual,three stage process.

The separation calculated here shows that it is compara-ble with the size of the polymeric thickener molecules thatare typically in solution to viscosify paints. Thus, depletionor bridging flocculation may become more possible de-pending on the concentration of these materials. Furtherresearch should be done to determine the relative impor-tance of depletion forces and DLVO forces in determiningthe stability behavior of aqueous paints.

Appendix A. Separation in composite particulate system

The approximate separation calculation here is for dense,mixed systems that do not adopt a regular, crystalline geo-metry. The strategy is to calculate the packing fraction ofspherical particles where, for the purposes of the calcula-tion, each particle includes the separating layer around it.Each particle is assumed to be equally separated, on aver-age, from its neighbors regardless of size or composition.This should be a reasonable approximation in a randomlydistributed, well-mixed system where the particles arebombarded continually in Brownian motion:

Volume occupied by all the individual particles

× (including their separating layers) =∑i

nivi

whereni is the number of particles/unit volume belongingto the ith type of material,vi the volume occupied by oneof the ith type of particles.

The volume,vi , here is the effective volume that includesthe particle surrounded by a shell of thickness12δ, whereδexpresses the separation between the particles:

vi = 43π

(ri + δ

2

)3(A.1)

whereri is the radius of anith particle.If VT is the overall volume of the suspension, including

the interstices between particles in the packing arrangement:

Occupied volume fraction= 1VT

∑nivi (A.2)

Now the calculation that remains is to find an expressionfor the number of each type of particle (expressed in usefulquantities). Typically paint formulations, or any other com-posite recipe, are expressed by weight so it is convenient touse the mass fractions,βi . The βi are defined here as the


mass fraction of theith material with respect to all the otheringredients, including the water or solvent:

Mass of theith type of material= MTβiwhereMT is the mass of the complete system:

The volume of each of the particles of theith material

= 43πr3i

Thus, if the density of theith material isρi , the number ofparticles can be calculated as

ni = MTβi 1ρi(4/3)πr3i

(A.3)

Using Eqs. (A.1) and (A.3) and substituting into Eq. (A.2),we obtain

Occupied volume fraction= ρT∑i

βi

ρi

(1 + δ

2ri

)3(A.4)

HereρT = MT/V T, the overall density of the system (oftenexpressed by paint formulators in lb/g) which can be easilymeasured or calculated from the recipe.

If the separation between particles is known, then theoccupied volume fraction can be calculated. However, ourinterest is in the inverse calculation; Eq. (4) also allows theseparation,δ, in a system to be calculated if the occupiedvolume fraction is specified. If the layer thickness is zero,then Eq. (4) reduces to an equation for calculating the overallvolume fraction from the mass fractions of the ingredients.

Particle–particle arrangements can be chosen from knownvalues of packing fractions, e.g. 0.64 (dense random pack-ing) 0.59 (loose random packing, or the colloidal glasstransition). There are other packing fractions that one coulduse, based on periodic structures, e.g. 0.745 (hexagonalclose packing), etc.

Although a suspension concentration may be much lessthan these packing fractions, the assumption is that the par-ticles, on average, have the same type of spatial relationshipregardless of their concentration. This is reasonable becauseseparations calculated in this paper for a formulation of34.5% volume solids are still substantially less than anyparticle diameters so several particles must cooperate if anyone is to change juxtaposition with its neighbors.

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DLVO theory applied to TiO2 pigments and other materials in latex paintsIntroductionBackgroundPaint systemsSettling ratesRelative numbers of particles in a paint formulationIonic strengthParticle separationStructure of titanium dioxide pigment particlesAdsorbed polyelectrolyte layer

Application of DLVO theoryThe simplest casePigment surface treatment structureAllowing for the adsorbed polyelectrolyteVariations in ionic strengthOther paint suspensionsOther possibilities of suspension instabilityLatex film drying and film formation

ConclusionsSeparation in composite particulate system

References

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DLVO theory applied to TiO pigments and other materials in ...download.xuebalib.com › xuebalib.com.36612.pdfDLVO theory that incorporates a less rigorous method of in-tegrating the