Particle Charges in Nonaqueous Colloidal Dispersions

Inan Chen

Wilson Center for Research and Technology, Xerox Corporation, Webster, New York 14580

Received November 29, 1995. In Final Form: May 2, 1996X

Numerical solutions of the Poisson-Boltzmann equation for potentials in the electrical double layersurrounding a particle are used to derive a new relationship between the particle charge q and the surfacepotential ú. Unlike the linear ú-q relation for a particle in charge-free media, the new relation shows thatas the particle charge increases, the initial linear increase of the potential slows down and asymptotesto a finite value. The asymptotic values of the potential at high particle charges are dependent on theDebye length (or the ionic charge density) of the media and are of the order of a few hundred millivoltsfor particles in typical nonaqueous dispersions, e.g., those used as liquid developers for electrographicimages. Thus, with this relationship, the reported values of charge and the electrophoretic mobilitydetermined experimentally for these dispersions correspond to physically reasonable values ofúpotentials,which are smaller than that expected from the linear relation by more than an order of magnitude. Inaddition, the variations of particle charges with ionic charge densities and particle concentrations areexamined.

I. IntroductionThe charge q and the mobility µ of colloidal particles

are two important figures of merit in electrophoreticapplications of nonaqueous colloidal dispersions, anexample ofwhich is the liquiddeveloper for electrographicimages.1-4 The two quantities are related by the Stokes’law5

where R is the particle radius and η is the coefficient ofviscosity of the fluid. Denoting themassdensity ofparticleby Fm and the mass of particle by m, the above relationcan be rewritten in terms of the “charge-to-mass ratio”,q/m as

With Fm ) 1 g/cm3,R) 1 µm, and η ) 3 cP, a value of q/m) 200 µC/g corresponds to a mobility of µ ) 1.5 × 10-4

cm2/(V s).The mobility is also related to the surface potential, or

“zeta potential” ú, by the Hueckel equation6

where ε is the permittivity of the fluid. With ε) 2× 10-13

F/cm, a typical value for hydrocarbon liquids, the abovevalue of µ (or q/m) gives a zeta potential of ú ) 3.4 V! Apossible cause leading to this counterintuitively largevalue can be suggested as follows.Eliminating µ from eqs 1 and 3, or eqs 2 and 3, ú is

related to q, or q/m by

ú ) q/4πεR ) (q/m)FmR2/3ε (4)

This can be recognized as the expression for the surfacepotential of a spherical particle with charge q in a charge-

free medium.4 In colloidal dispersions, the chargedparticles are dispersed, not in a charge-free medium, butinamediumcontaining surfactants ions of bothpolarities.The charged particle attracts ions of opposite polarity(counterions), and repels those of the same polarity (co-ions), forming an electrical double layer near the surface.Thecounterionandco-iondistributionsaroundtheparticleare determined by the competition between the Coulombforce and the thermal motion of ions, resulting in aBoltzmann distribution at equilibrium (Gouy-Chapmanmodel).7 The ú-q relation for such a particle can beexpected to differ from eq 4.Theprimaryobjectiveof thiswork is toderive thesurface

potential-charge (ú-q) relation for particles surroundedby an electric double layer, from the solutions of Poisson-Boltzmann equations.8 It will be shown that with thenew relationship, the experimentally determined q/m(∼100 µC/g) or µ values (∼10-4 cm2/(V s))9 correspond tomore reasonable values of zeta potential (e400 mV). Themathematical procedure for the evaluation of particlecharge is described in the next section (section II). Theresultant ú vs q/m relations and the nature of particleand ion chargedistributions associatedwith the electricaldouble layer are presented and discussed in section III.The Appendix describes the features of the potentialdistributions which are used to calculate the charges.

II. Mathematical ProcedureThe Poisson-Boltzmann equation for the electrical

potential ψ in spherical symmetry, with the origin at thecenter of particle, is given by

with the net space charge density F given in terms of thepotential as

where k is the Boltzmann constant, T is the absoluteX Abstract published in Advance ACS Abstracts, July 1, 1996.(1) Gibson, G. A.; Luebbe, R. H. J. Imaging Technol. 1991, 17, 207.(2) Larson, J. R.; Lane, G. A.; Swanson, J. R.; Trout, T. J.; El-Sayed,

L. J. Imaging Technol. 1991, 17, 210.(3) Larson, J. R.; Morrison, I. D.; Robinson, T. S. IS&T’s Int Congr.

Adv. Non-Impact Print. Technol., Final Program Proc., 8th 1992, 193.(4) Morrison, I. D.; Tarnawskyj, C. J. Langmuir 1991, 7, 2358.(5) Wiersema, P.H.On theTheory of Electrophoresis; Pasmans: The

Hague, 1964.(6) Ross, S.; Morrison, I. D. Colloidal Systems and Interfaces; John

Wiley & Sons: New York, 1988; p 346.

(7) Gouy, G. J. Phys. Theor. Appl. 1910, 9, 457. Chapman, D. L.Philos. Mag. 1913, 25, 475. Reference 6, p 232.

(8) Loeb, A. L.; Overbeek, J. Th. G.; Wiersema, P. H. The electricaldouble layer arounda spherical colloidparticle;MITPress: Cambridge,MA, 1961.

(9) Chen, I.; Mort, J.; Machonkin, M. A.; Larson, J. R. Program &Abstract, 69th Colloid & Surface Science Symposium, Salt Lake City,Utah, 1995; p 80.

µ ) q/6πηR (1)

µ ) 2(q/m) FmR2/9η (2)

ú ) 3ηµ/2ε (3)

(1/r2)(d/dr)[r2dψ/dr] ) -F/ε (5)

F ) F+ + F- ) zen0[exp(-zeψ/kT) - exp(zeψ/kT)] (6)

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temperature, (ze is the charge on each positive/negativeion, and n0 is the number density of surfactant ions ofeachpolarity in the absence of (or at a large distance from)the particle. This “nominal ionic charge density”, zen0, isrelated to the conductivity of the surfactant dispersion byσ ) 2(zen0)µi, where µi is the ion mobility (assumed to beidentical for anions and cations). In terms of the nominalionic charge density, a “Debye length” D can be definedas

D can be chosen over zen0 as an input parameter for thePoisson-Boltzmann equation, because its dimensionenables direct comparison with the particle radius R.The potential value at the particle surface, commonly

denoted as the zeta potential is determined by thedifference in the electrochemical properties (such aschemical potentials or electron affinities) of the particlesand the surfactant ions. The zeta potential ú serves asone of the boundary conditions, at r)R, for the Poisson-Boltzmann equation

Another boundary condition at a large distance isspecified as follows. Let f be the fractional volume ofparticles in the dispersion, then the number of particlesin a unit volume of dispersion is

Defining the “particle space” as the average sphericalspaceperparticle of volume1/N)4πR3/3f, then the “radiusof particle space” S is given by

For example, with the particle concentration of 2%, i.e.,f ) 0.02, the radius of particle space S is 3.68 times theparticle radius R, and for f ) 0.2 (20%), S ) 1.71R.The boundary condition at a large distance can now be

stated as the vanishing of the potential gradient at r )S, due to the symmetry

From the numerical solutions, ψ(r), of the Poisson-Boltzmann equation, various charge quantities can becalculated as follows.The charge on a particle is equal and opposite in sign

to the total net space charge surrounding the particle.Therefore, the charge per particle q is given by

where the space charge density F is expressed in terms ofthe derivative of potentialψ using eq 5, and the boundarycondition, eq 11, is applied. Dividing this expression ofq by the particle mass m ) 4πR3Fm/3, the charge/massratio is given by

The gradient of ψ can be calculated from the solutionof the Poisson-Boltzmann equationwith a given value ofúas the boundsary condition, eq 8. Thus, eq 13 representsimplicitkly the q/m vs ú relation.

Furthermore, multiplying q of eq 12 by the number ofparticles per unit volume N, eq 9, the particle chargedensity, Q, is given by

Similarly, the positive and the negative ion charges (qpand qn) surrounding the particle can be expressed as

where I( denote the integrals that can be calculated fromthe potential ψ(r). Multiplying these quantities by thenumberofparticlesperunitvolumeN, the chargedensitiesof (positive) co-ions Qp and (negative) counterions Qn are

The variations of Q, Qp, and Qn with the nominal ioniccharge density, the zeta potential, and the particleconcentration are shown with numerical examples in thenext section. The accuracy of numerical results can beverifiedby thechargeneutrality conditionamongthe threecharge densities, Q, Qp, and Qn,

III. Results and DiscussionAllnumerical examples in this sectionand theAppendix

are given in normalized units defined below. Typicalvalues of practical interest for the units are also given:length (R) ≈1 µm; potential (kT/ze) ≈25 mV (with z ) 1at room temperature); charge density (εkT/zeR2)≈0.5 µC/cm3 (with ε) 2× 10-13 F/cm and R ) 1 µm); charge/massratio (εkT/zeR2Fm)≈0.5 µC/g (with Fm ) 1 g/cm3 and aboveε and R).III. 1. Potential-Charge Relation. The surface

potential vs the charge/mass ratio relation (ú vs q/m),calculated by the procedure described in the previoussection, in particular, eq 13, is shown in Figure 1 for thecase of 2% solid content (f) 0.02). The three solid curvesdiffer in the values of the Debye lengthsD, given in unitsof R, the particle radius. With the typical values ofparameters given above, the three cases with D ) 3R, R,and 0.3R correspond to the nominal ionic charge densityof zen0)0.05, 0.5, and5µC/cm3, respectively. Thedashed

D ) [ε(kT/ze)/(zen0)]1/2 (7)

ψ(R) ) ú (8)

N ) 3f/4πR3 (9)

S ) R/f1/3 (10)

dψ/dr ) 0 at r ) S (11)

q ) -4π∫RSFr2 dr ) 4πε∫RS(d/dr)[r2dψ/dr] dr )

-4πεR2[dψ/dr]R (12)

q/m ) -(3ε/RFm)[dψ/dr]R (13)

Figure 1. Calculated ú vs q/m relations for a 2% solid contentdispersion with three values of Debye length D (solid curves)and the linear relation in charge-free medium, eq 4 (dashed).The normalized unit for the potential is kT/ze (≈25 mV) andthat for q/m is εkT/zeR2Fm (≈0.5 µC/g). See section III for thedetails of the normalized units.

Q ) -(3fε/R)[dψ/dr]R (14)

qp ) 4π(zen0)∫RSexp(-zeψ/kT)r2 dr ≡ 4π(zen0)I+ (15)

qn ) -4π(zen0)∫RSexp(zeψ/kT)r2 dr ≡ -4π(zen0)I-(16)

Qp, Qn ) (4πN(zen0)I( ) (3(f/R3)(zen0)I( (17)

Q + Qp + Qn ) 0 (18)

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curve shows the linear ú vs q/m relation in charge-freemedium, of eq 4 (equivalent to D ) ∞ or zen0 ) 0).In the presence of electrical double layer, the surface

potential ú is seen to increase linearly at small values ofq/m (e a few tens of µC/g) as in the charge-free medium,but asymptotes to a finite value at higher values of q/m(g50 µC/g). The asymptotic value, although dependenton the Debye length D (or zen0), is only slightly largerthan 10(kT/ze) (≈250 mV). For example, with the abovecited parameter values of practical interest, a q/m ) 100µC/g corresponds to ú ≈ 240, 190, and 125 mV for Debyelengths of D ) 3, 1, and 0.3 µm (or zen0 ) 0.05, 0.5, and5 µC/cm3), respectively. This is a reduction by about anorder of magnitude from the values expected from thelinear relation, eq 4. The reduction is more significantfor a larger value of q/m.It should be noted that in this discussion the ú values

refer to the potentials at the physical surface r ) R of theparticles. The decrease of potential near the particlesurface is faster for the larger ú, as shown in Figure 5 oftheAppendix. Thus, if the particle size includes the innerlocalized portion of the double layer (Stern layer),10 theobserved ú potential in reality can be somewhat smallerthan that expected from the above calculations.III. 2. Particle and IonCharges. The variations of

the particle charge density Q, eq 14, and the co-ion andcounterion charge densities, Qp and Qn, eqs 17 and 18,with the úpotential are shown inFigure2withanexamplefor particle concentration f ) 0.02 and Debye length D )R. As the ú potential increases from 4 to 16 kT/ze (i.e.≈100 to 400 mV), both the particle charge density Q andthecounterionchargedensityQn increasebyabout4ordersofmagnitude,while theco-iondensityQpdecreasesslightlyfrom the nominal value zen0. An efficient charging ofparticles by the surfactants is represented by the particlecharge density Q exceeding the co-ion charge density Qp,which is seen to occur for larger ú values (>4kT/ze for thisexample). Similar features are observed for other valuesof D within the range of practical interest.Thedependenceofparticle chargeQon theDebye length

or equivalently, on thenominal ionic charge density, zen0,is shown inFigure3A, for the caseofparticle concentrationf ) 0.02 and various ú values. Q is seen to vary as thesquare-root of zen0 at smaller values of ú (<12kT/ze).Onlyfor larger values of ú, Q increases linearly with zen0(compare with the dashed lines of logarithmic slopes 1/2and 1, respectively).The corresponding relations for co-ion and counterion

charge densities, Qp and Qn, are shown in Figure 3B

(respectively, below and above the dotted line whichrepresents the values of zen0). Qp and Qn are seen todeviate from thenominal ionic chargedensity (the dashedline) in opposite directions and increasingly more as theú potential increases and the zen0 decreases. The loss ofco-ions is much less than the increase in counterions,especially at high values of ú potential.The conductivity σ of the dispersion consists of three

components

where µ,µp, andµn represent themobilities of the particle,the co-ions, and the counterion, respectively. Because ofthedominantsizesofQandQn overQp, as showninFigures2 and 3, the conductivity is determined by the particlecontribution if the particle mobility µ is larger than theion mobilities, µp, and µn. Also, because the particlemobility is proportional to the particle charge, as repre-sented by eq 1, the conductivity σ, then, varies as thesquare of Q. The fact that Q varies as the square root ofzen0 at (lower) values of ú < 12kT/ze, as shown in Figure3A, means that the dependence of conductivity σ on zen0is linear. This is consistentwith the experimental resultsthat, in the regimeof lowsurfactant concentrationswherethe ionic charge density, zen0, is proportional to theconcentration, the conductivity of the dispersion varieslinearlywith the surfactant concentration (i.e., the specificconductivities are constant).3,11Similar figures of Q, Qp, and Qn versus zen0 for 20%

particle concentration, f) 0.2, are shown in Figure 4. Theparticle charge density Q (Figure 4A) is seen to increaseby about a factor of 10 from those in Figure 3A, in

(10) Stern, O. Z. Electrochem., Angew. Phys. Chem. 1924, 30, 508.Reference 3, p 236.

(11) Morrison, I. D. Proceedings of the Engineering FoundationConferences on Dispersion and Aggregation: Fundamentals andApplications, Palm Coast, FL, March 15-20, 1992, p 55.

Figure 2. Particle charge densityQ, co-ion charge densityQp,and counterion chargedensityQn, as functions of zetapotential.The normalized units are εkT/zeR2 (≈0.5 µC/cm3) for chargedensities and kT/ze (≈25 mV) for zeta potential.

Figure 3. (A) Particle charge density Q and (B) co-ion chargedensity Qp and counterion charge density Qn, as functions ofnominal ionic charge density zen0, at 2%particle concentration(f ) 0.02). All charge densities are in units of εkT/zeR2 (≈0.5µC/cm3).

σ ) µQ + µpQp + µnQn (19)

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proportion to the concentration f. The square-root de-pendence of Q on zen0 at ú < 12kT/ze is similar to that inFigure 3A for f ) 0.02. From Figure 4B, it can be seenthat the counterionchargedensityQn also increasesnearlyby a factor of 10 as the particle concentration increasesfrom f ) 0.02 to f ) 0.2 (compare Figure 3B).

VI. Summary and Conclusion

A new relation between the surface potential ú andcharge q of particles surrounded by an electrical doublelayer in colloidal dispersions has been derived from thesolutions of the Poisson-Boltzmann equation. The newrelation indicates that the ú vs q/m relation is linear atsmall values of q/m as in the charge-free medium, but athigher values of q/m, it asymptotes to a value muchsmaller than that expected from the formula for charge-free medium. The asymptotic value of ú is dependent onthe nominal ionic charge density of the medium and is inthe range of (10-15)(kT/ze) for the range of ionic densityof practical interest. For example, for toner particles intypical liquid developers, with a charge-to-mass ratio ofa few hundred µC/g, and a mobility of the order of 10-4

cm2/(V s),9 the surface potential is in the low multiples of100 mV.

Acknowledgment. The author wishes to thank hiscolleagues, C. Chen, G. Gibson, J. Larson, I. Morrison, J.Mort, and T. Smith for stimulating discussion on thesubject of this investigation as well as on colloid sciencein general.

Appendix: Potential Distributions

Examples of potential distributions obtained fromnumerical solutions of the Poisson-Boltzmann equationdescribed in section II are presented and discussed in thisAppendix.

Figure 5 shows the potential distributions calculatedfor various surface value ú, a fixed Debye length D, andparticle concentration f. With the typical values ofparameters given in section III, the Debye length D ) Rcorresponds to the nominal ionic charge density of zen0≈ 0.5 µC/cm3. The radius of particle space correspondingto the concentration of f ) 0.02 is S ) 3.68R. It can beseen that the larger the ú value, the sharper the initialdecay is. Therefore, for larger ú values the potential inthe Stern layer, which includes an inner localized portionof the double layer, can be significantly smaller than thenominal ú value at the physical surface.Figure6shows thepotential distributionswithú )4(kT/

ze), f ) 0.02, and D ) 0.3 to 3R. The range of D coveredin the figure corresponds to the nominal ionic chargedensities of zen0)0.05 to5µC/cm3,with the typical valuesof parameters given in section III. It can be seen that theslope of potential dψ/dr at r ) R decreases (in absolutevalue) as theDebye lengthD increases. This is consistentwith the expectation that the particle charge, eq 12 or 13,decreases as zen0 decreases.

Figure 4. (A) Particle charge density Q and (B) co-ion chargedensity Qp and counterion charge density Qn, as functions ofnominal ionic chargedensity zen0, at 20%particle concentration(f ) 0.2). All charge densities are in units of εkT/zeR2 (≈0.5µC/cm3).

Figure 5. Variations of potential distribution, ψ(r), with thesurface value ú.

Figure6. Variations of potential distribution,ψ(r),withDebyelength D.

Figure 7. Variations of potential distribution, ψ(r), withparticle concentration f.

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ForagivenúpotentialandDebye length, thedependenceof potential distributions ψ(r) on the particle concentra-tion f or, equivalently, on the radius of particle spaceS, eq 10, is shown in Figure 7. It can be seen that thepotential near the edges of the particle space (dashedlines) varies significantly with the concentration f(or S). However, the slope [dψ/dr]R at the surface re-mains almost unchanged. This indicates, from eqs 12

and 13, that under the condition of constant ú and D,the particle charge per particle q, or q/m, is not muchaffected by the particle concentration f. Consequently,the particle charge per unit volume, Q of eq 14, shouldincrease approximately in proportion to the particleconcentration f.

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