ORIGINAL PAPER

Calibration-free concentration determination of chargedcolloidal nanoparticles and determination of effectivecharges by capillary isotachophoresis

Ute Pyell & Wendelin Bücking & Carolin Huhn & Barbara Herrmann &

Alexey Merkoulov & Joachim Mannhardt & Hartmut Jungclas & Thomas Nann

Received: 11 March 2009 /Revised: 29 May 2009 /Accepted: 3 June 2009 /Published online: 5 July 2009# Springer-Verlag 2009

Abstract Although colloidal nanoparticles show an elec-trophoretic heterogeneity under the conditions of capillaryelectrophoresis, which can be either due to the particle-sizedistribution and/or the particle shape distribution and/or thezeta-potential distribution, they can form correct isotacho-phoretic zones with sharp-moving boundaries. Therefore,the technique of isotachophoresis permits to generate plugsin which the co-ions and counter ions of the originalcolloidal solution are removed and replaced by thebuffering counter ions of the leading electrolyte. It isshown that analytical isotachophoresis can be used tomeasure directly, without calibration, the molar (particle)concentration of dispersed ionic colloids provided that thetransference number and the mean effective charge numberof the particles (within the isotachophoretic zone) can bedetermined with adequate accuracy. The method can also beused to measure directly the effective charge number of

biomacromolecules or colloidal particles, if solutions withknown molar (particle) concentration can be prepared. Thevalidity of the approach was confirmed for a model solutioncontaining a known molar concentration of bovine serumalbumin.

Keywords Effective charge number of colloidal particles .

Electrophoretic mobility . Isotachophoresis . Nanoparticles .

Zeta potential

Introduction

Analytical isotachophoresis (ITP) was developed as amethod for the determination of ionic constituents in acomplex sample [1, 2]. The method is based on thedevelopment of new moving boundaries if an electriccurrent passes an initially sharp junction between twoelectrolytes. It has been shown that moving boundarysystems are not only formed by strong electrolytes [3, 4]but also by weak electrolytes [5, 6]. In the latter case,moving reaction boundaries are formed. The conditions forthe formation of stable isotachophoretic zones have beeninvestigated in detail resulting in the concept of zone-existence diagrams [7].

In analytical ITP, the sample is injected as a zoneplaced between a leading electrolyte (containing a co-ionhaving a higher effective electrophoretic mobility thanthat of the analyte ion) and a terminating electrolyte(containing a co-ion having a lower effective electropho-retic mobility than that of the analyte ion). If stableisotachophoretic zones are formed the concentration ofanalyte ions in the zone behind the moving boundarywill be proportional to the concentration of the leadingco-ion. The proportionality constant will be given by the

Anal Bioanal Chem (2009) 395:1681–1691DOI 10.1007/s00216-009-2887-5

U. Pyell (*) : C. Huhn : B. Herrmann :H. JungclasDepartment of Chemistry, University of Marburg,Hans-Meerwein-Strasse,35032 Marburg, Germanye-mail: pyellu@staff.uni-marburg.de

U. Pyell :W. Bücking :A. Merkoulov : T. NannFreiburg Materials Research Centre (FMF),Albert-Ludwig University of Freiburg,Stefan-Meier-Strasse 21,79104 Freiburg, Germany

J. MannhardtJ&M Analytik AG,Willy-Messerschmitt-Strasse 8,73457 Essingen, Germany

T. NannSchool of Chemical Sciences and Pharmacy,University of East Anglia (UEA),Norwich NR4 7TJ, UK

ratio of the transference numbers of the analyte ion andthe leading co-ion.

For strong electrolytes, this relationship makes ITP themost precise method for the determination of transferencenumbers [8]. The migration of ions in stable isotachopho-retic zones was not only observed for atomic and molecularions but also for protonated/deprotonated proteins [9], ionicassociation colloids (micelles) [10, 11] and colloidal nano-particles [12, 13]. As isotachophoresis is primarily aseparation technique [14] it will (in the ideal case)transform the sample zone containing the ionic colloidtogether with ionic and non-ionic constituents coming fromthe synthesis procedure into a migrating analyte zonecontaining only the colloid and the (well-defined) counterion of the leading electrolyte. The concentration of thecolloid in this analyte zone will be readjusted to the valueneeded for fulfillment of the conservation law deducedfrom the Kohlrausch-regulating function (KRF) [15].

If the electric current strength passing the capillary filledwith electrolyte is kept constant, the electric charge trans-ported by the analyte zone moving at constant velocity canbe determined. These properties allow ITP to be used as anabsolute analytical method which permits the determinationof the amount of substance of analyte contained in theinjected sample volume without the requirement of calibra-tion [2] provided that the transference number and the meaneffective charge number of the analyte within the stableisotachophoretic zone (having sharp boundaries) can bedetermined with adequate accuracy. Bücking et al. [16]have therefore suggested to use ITP for the determination ofthe particle concentration of dispersed colloidal nano-particles. This method requires that correct isotachophoreticzones are formed which has to be confirmed because of theinherent electrophoretic heterogeneity of nanoparticles.

In the present paper, we investigate whether theserequirements are given so that ITP can be used withoutreference standards for the calibration-free determination ofthe molar (particle) concentration of dispersed colloidalnanoparticles. To this end, zeta potentials and effectivecharge numbers were calculated from measured electro-phoretic mobilities. The method was applied to solutionscontaining negatively charged colloidal nanoparticles:stabilized gold nanoparticles, CdSe/ZnSe nanocomposites,and CdTe nanoparticles (either coated with mercaptoethanol(ME) or bovine serum albumin (BSA)). The validity of theapproach was confirmed with a model solution containing aknown molar concentration of BSA. In this case ITP can beused to determine directly the effective charge number ofthe dissolved biomacromolecule (within the isotachopho-retic zone).

To date, the determination of the particle concentrationof nanoparticle dispersions is still a topic of intenseresearch. Many applications, e.g., technical applications of

semiconductor nanocrystals, also the regulation of thestoichiometry of nanoparticle–biomolecule conjugates andstudies of nucleation and growth mechanisms, require theknowledge of the particle concentration in colloidalsolutions of nanoparticles with adequate accuracy.

According to Yu et al. [17], the absorption spectrummethod is, in many cases, the most practical and convenientway to determine particle concentrations. However, theexperimental determination of the extinction coefficient fora given nanoparticle fraction is not a trivial task. The molarextinction coefficient for high-quality CdTe, CdSe, and CdSnanocrystals at the maximum of the first excitonicabsorption peak was found to be strongly dependent onthe particle radius. Kuçur et al. [18] deduced from theirexperimental studies that the molar extinction coefficientmight also depend on the synthesis procedure, on the usedcapping ligands, and, thus, on the crystallinity or quality ofthe nanoparticles investigated. These dependencies renderthe photometric concentration determination of colloidalnanoparticle solutions very inaccurate [18].

Although matrix-assisted laser desorption ionization-time of flight-mass spectrometry (MALDI-TOF-MS) hasbeen used very successfully to estimate the size distributionof nanoparticles (e.g., ZnS nanocrystals [19, 20]), accordingto the authors’ knowledge, quantitative applications of thisapproach have not been reported so far.

It is also possible to calculate the particle concentrationof colloidal particle solutions by the quantitative determi-nation of a nanoparticle component (e.g., Cd in CdSe) viaelemental analysis, e.g., after dissolving the particles innitric acid. If the mean number of atoms of the determinedelement per particle is known, particle concentrations areaccessible [18]. In general, the sketched approach requiresthe experimental determination of the concentration of thenanoparticle component, the experimental determination ofthe particle-size distribution and calculation of the meannumber of atoms of the quantified element per particleemploying specific crystal models. While the accuracy ofthe elemental analysis can easily be confirmed via theapplication of two independent methods (e.g., graphite-furnace atomic absorption spectrometry (GFAAS) and anodicstripping voltammetry (ASV)), the accuracy of the totalprocedure crucially depends on the accuracy of the determi-nation of the particle-size distribution and the accuracy of thecrystal model [18]. Due to error propagation, uncertainties ofthe calculations involved will result in large final measure-ment uncertainties.

This comparison shows that the development of simpleand accurate methods for the determination of the particleconcentration of colloidal particle solutions is highlydesirable in current nanoparticle research. It will beinvestigated whether these requirements can be met by theproposed procedure.

1682 U. Pyell et al.

Experimental section

Nanoparticles and chemicals

Stabilized gold nanoparticles (Goldpulver Typ 200-30) witha mean particle size of 2–3 nm and a maximum of thesurface plasmon resonance peak at 510 nm [21] wereprovided by Heraeus (Hanau, Germany). Bovine serumalbumin (>98%) was obtained from Roth (Karlsruhe,Germany).

CdSe/ZnS nanoparticles have been prepared accordingto a previously published procedure [22]. From thismaterial, ca. 10 mg were dispersed in approximately 1 mLtoluene. The nanoparticles were precipitated with ethanol(three times the volume of toluene), centrifuged and re-dispersed in toluene. The precipitation/re-dispersion cyclewas repeated three times to remove excess surface ligands.Finally, the nanoparticles were dispersed in 500 µL N,N-dimethylformamide (DMF). To the dispersion, 10 µL ofmercaptoethanol (ME) was added and the mixture wasstirred at 60 °C overnight. After stirring, 750 µL ofdiethylether was added to precipitate the ME-coatedCdSe/ZnS particles (Sample N1). Particles N1 werecentrifuged at 4,000 rpm and re-dispersed in an appropriateaqueous buffer.

For coating with bovine serum albumin (BSA), N1 wasdispersed in Tris–glycine buffer (pH 8.0). An excess ofBSA was added and incubated overnight. The resultingBSA-coated nanoparticles (Sample N2) were used withoutfurther purification.

CdTe nanoparticles were prepared by means of amicrowave-assisted synthesis procedure: 273.5 mg of acadmium stearate/stearic acid mixture (0.25 mmol cadmi-um), 1 mL of a 0.2 mol L−1 tellurium in trioctylphosphine(TOP) solution, 1.25 g dodecylamine (DDA) and 0.7 mLTOP were mixed in a reaction flask at 60 °C under gentleagitation. Subsequently, the flask was transferred into adomestic microwave oven and exposed to microwaveirradiation of 800 W for 90 s. The cold reaction mixturewas diluted with 3 mL of toluene and the resulting CdTenanoparticles were precipitated with ethanol (approximatelythree times the volume of toluene). Re-dispersion in tolueneand precipitation with ethanol was repeated three times.Finally, the CdTe nanoparticles were dispersed in 1 mLtoluene. ME coating of the CdTe nanoparticles (Sample N3)was achieved in the same way as described for the CdSe/ZnS nanoparticles.

CdTe nanoparticles were coated with BSA in the sameway as described for N2. Aqueous dispersions of the coatednanoparticles (Sample N4) were purified using a standarddialysis protocol: nanoparticle dispersions were loaded intoa cellulose dialysis tube with a molecular weight cut-off of25 kD (Sigma-Aldrich, Seelze, Germany). Then, the

dispersions were dialysed against the buffer used, whereasthe buffer was exchanged after 2 h for three times and thenleft stirring overnight. Finally, the purified product wasrecovered from the dialysis tube.

Benzoic acid (puriss. p.a.), 1,3-bis(tris(hydroxymethyl)-methylamino)propane (BTP, >99%), 4-(2-hydroxyethyl)piperazine–ethane–sulfonic acid (HEPES, MicroSelect),hydrochloric acid (37%, TraceSelect), and 2-morpholino-ethanesulfonic acid monohydrate (MES, Ultra) were fromFluka (Buchs, Switzerland) and 3-[4-(hydroxyethyl)-1-piperazinyl]-propanesulfonic acid (HEPPS, >99%) wasfrom Merck (Darmstadt, Germany). Tris-(hydroxymethyl)-aminomethane (Tris, ≥99.9%, p.a.) was fromRoth (Karlsruhe,Germany). 2-Mercaptoethanol (ME, for electrophoresis),N,N-dimethylformamide (DMF, 99.8%, anhydrous), diethy-lether (>99.7%, anhydrous), toluene (99.8%, anhydrous),ethanol (99.5%, anhydrous), trioctylphosphine (TOP, 90%),cadmium oxide (99.99%), stearic acid (>98.5%), dodecyl-amine (DDA, >99.5%, puriss.), and tellurium (purum p.a.)were purchased from Sigma-Aldrich (Seelze, Germany).

Determination of the hydrodynamic radii

Hydrodynamic radii were determined by dynamic lightscattering using a Nanosizer ZS (Malvern Instruments,Malvern, UK) in 0.1 mol L−1 Tris–HCl buffer pH 8.0 at25 °C.

Capillary electrophoresis

All measurements were carried out with a Beckman(Fullerton, CA, USA) Model P/ACE 2200 capillaryelectrophoresis system equipped with a UV absorbancedetector. Samples were injected by application of pressure.Unless otherwise specified, detection was performed at230 nm. Data were recorded with the Beckman SystemGold software. Further data treatment was done with Origin6.1G (Northampton, MA USA). Fused-silica capillaries(75 μm I.D. × 375 μm O.D.) were obtained from PolymicroTechnologies (Phoenix, AZ, USA). The repeatability ofreported data has been confirmed by repeated injections.Between runs, capillaries were equilibrated by rinsing withbackground buffer.

Capillary isotachophoresis

Initial measurements were carried out with the describedCE system equipped with an additional TraceDec contact-less conductivity detector (Innovative Sensor TechnologiesGmbH, Strasshof, Austria). Fused-silica capillaries(100 μm I.D. × 375 μm O.D.) were coated at the innerwall with polyacrylamide according to the method ofHjertén [23].

Charged colloidal nanoparticles and effective charges 1683

Concentration determinations were performed with theITP–ITP system ItaChrom II EA 202 M (JH-Analytik,Aalen, Germany) equipped with a 30-μL injection valve, anupper FEP capillary with 700 μm I.D. and a lower fused-silica capillary with 300 μm I.D. This apparatus permitsphotometric detection at selected wavelength and contact-less conductivity detection (inverted signal).

Theoretical considerations

Isotachophoresis is generally performed at constant electriccurrent strength I. As has been pointed out by Boček et al.[2], the characteristics of ITP allow the calibration-freedetermination of the injected amount of substance. Underthese conditions:

Q ¼ I Δt ð1Þwhere: Q = electric charge, Δt = time difference. Thecharge in each zone is transported by both the analyte andthe counter ion:

Q ¼ Q1 þ Q2 ¼ n1 F zeff ;1�� ��þ n2 F zeff ;2

�� �� ð2Þwhere: Q1 = charge transported by the analyte, Q2 = chargetransported by the counter ion, n = amount of substance,F = Faraday constant, zeff = effective charge number. Theindices 1 and 2 refer to the analyte ion and the counter ion,respectively. The charge transported by the anion or thecation in a homogeneous zone is dependent on thetransference number tN.

Q1 ¼ Q tN1 ¼ Qm1j j

m1j j þ m2j j ð3Þ

Q2 ¼ Q tN2 ¼ Qm2j j

m1j j þ m2j j ð4Þ

where: μ = effective electrophoretic mobility. Rearrange-ment yields:

n1 ¼ I Δt

F zeff ;1�� �� m1j j

m1j j þ m2j j ð5Þ

This equation allows to calculate the amount ofsubstance of analyte within an isotachophoretic zonedirectly from the zone length Δt provided that for thezone investigated, the effective electrophoretic mobilityof the analyte ion μ1, the effective electrophoreticmobility of the counter ion μ2 and the effective chargenumber of the analyte ion zeff,1 are known. It also allowsdetermining directly the effective charge number zeff,1 ofthe analyte ion, if the amount of substance injected isknown. While effective electrophoretic mobilities can be

determined directly by capillary electrophoresis or by ITP,the effective charge number zeff of a colloidal particle hasto be calculated from the electric charge density σ at theplane of shear (available via the electrokinetic potential ζ)and the hydrodynamic radius Rh. In these calculationsbased on the Gouy–Chapman theory, it is assumed that theparticle is a sphere and that the charge of the sphere isuniformly distributed on the surface [24].

For nanoparticles (arbitrary values of κRh and ζ, κ =Debye–Hückel parameter) no simple analytical formula isavailable for the calculation of the electric potential ζ atthe plane of shear from the electrophoretic mobility μ.Taking the electrostatic force exerted by the externalelectric field, the Stokes friction, the electrophoreticretardation, and the relaxation effect into consideration[25], a numerical calculation is required. This was firstdone by Wiersema et al. [24] and further extended tohigher values of ζ by O’Brien and White [26]. In 2001,Ohshima [27] presented the following approximate ana-lytical formula, which is valid for 1:1-electrolytes and ζ≤100 mV [28, 29]:

m ¼ 2"r"0z3h

f1 kRhð Þ � zezkT

� �2

f3 kRhð Þ � mþ þ m�2

zezkT

� �2

f4 kRhð Þ" #

ð6Þwhere εr = relative electric permittivity, ε0 = electricpermittivity of vacuum, η = viscosity of the electrolyte, κ =Debye–Hückel parameter, Rh = hydrodynamic particleradius, k = Boltzmann constant, T = absolute temperature,f1, f3, and f4 are a function of κRh and are given by

f1 kRhð Þ ¼ 1þ 1

2 1þ 2:5= kRh 1þ 2e�kRhð Þf g½ �3 ð7Þ

f3 kRhð Þ ¼ kRh kRh þ 1:3 e�0:18kRh þ 2:5ð Þ2 kRh þ 1:2 e�7:4kRh þ 4:8ð Þ3 ð8Þ

f4 kRhð Þ ¼ 9kRh kRh þ 5:2 e�3:9kRh þ 5:6ð Þ8 kRh þ 1:55 e�0:32kRh þ 6:02ð Þ3 ð9Þ

The dimensionless ionic drag coefficients (m+ and m−)are easily accessible from the limiting conductances of thecation Λ0

þ and the anion Λ0� in the electrolyte considered:

m� ¼ 2"r"0kTNA

3hzΛ0�

ð10Þ

where NA = Avogadro number and z = charge number ofelectrolyte ions (symmetrical electrolyte).

Assuming a spherical geometry of the particle and auniform distribution of the charge of the sphere on itssurface allows to calculate exactly the electric charge

1684 U. Pyell et al.

density, σ, at the plane of shear for a given ζ and a given Rh

by numerical treatment [30]. For a symmetrical electrolyte,the magnitude σ can also be calculated with an approximateempirical formula [30, 31]:

s ¼ Q

4pRh2 ¼ "r"kT

ezk 2 sinh

ez2kT

� �þ 4

kRhtanh

ez4kT

� �� �

ð11ÞFor a 1:1 electrolyte, Loeb et al. [30] compared the

results of the numerical treatment with those obtained withEq. 11. For 0.5<κRh<∞ the maximum deviation is only 5%(independent of ζ).

Results and discussion

Electrophoretic mobilities, electrokinetic potential,and effective charge number

Electrophoretic mobilities of the nanoparticles, of BSA, andof benzoic acid (reference substance) were determined inleading electrolyte (5 mmol L−1 HCl titrated to pH=8.60with BTP) by capillary electrophoresis with an uncoatedfused-silica capillary. The velocity of the electroosmoticflow was determined with DMSO. Figure 1 shows a typicalelectropherogram obtained for the gold nanoparticles (Au-NP). The mean effective electrophoretic mobility m wascalculated from:

m ¼ LD LTt1 U

� LD LTt2 U

ð12Þ

where: LD = length of capillary to detector, LT = total lengthof capillary, t1 = migration time corresponding to themaximum of the recorded peak, t2 = migration time of aneutral EOF-marker, U = applied voltage. The repeatabilityof the data was confirmed by repeated injections. Thecalculated values for m are given in Table 1 together withdata on repeatability. This table also contains the hydrody-namic radii determined from dynamic light-scatteringexperiments (see “Experimental” section).

In Fig. 2, these experimental data are compared withelectrophoretic mobilities calculated with Eqs. 6–10. Thecharge number of the counter ion BTP can be approxi-mated to be very close to one (pKS1=6.8, pKS2=9.0 [32]).The following parameters were taken into the calculations:η=0.000891 Ns m−2, εr=78.3, T=298.16 K, m+=0.221,m−=0.146. The electrolyte is modeled as 5 mmol L−1

NaCl (κ=0.2325 nm−1). The comparison of theoretical

0 10 12 1442 6 8

0

20

40

60

80

100

120

140

160A

bsor

banc

e(23

0 nm

)/10

-3 A

U

t/min

Fig. 1 Electropherogram of Au-nanoparticles. Electrolyte:5 mmol L−1 HCl titrated with BTP to pH=8.6, fused-silica capillary269 (201) mm × 75 μm, voltage 10 kV, pressure injection 2 s, ambienttemperature

Table 1 Hydrodynamic radius Rh, electrophoretic mobility m, electric charge density σ at the plane of shear and effective charge number zeff forinvestigated nanoparticles and reference substances, T=25 °C

Sample Rh/nm m=mm2 kV�1 s�1 Na srel/% ζ/mV σ/As m−2 zeff

Au-NP (3)b −32.2 2 0.2 −68 −0.0282 −19.9N1 6 −9.4 2 9.7 −17 −0.0047 −13.4N2 11 −29.0 2 1.0 −60 −0.0154 −146N3 (6)b −15.4 2 2.0 −29 −0.0082 −23.1N4 11 −21.2 3 1.5 −40 −0.0095 −90.3BSA (3.4)c −26.6 2 0.8 −52 −0.0197 −17.8BA – −34.4 11 3.8 – – −1.00

Au-NP stabilized gold nanoparticles, N1 CdSe/ZnS particles coated with mercaptoethanol, N2 CdSe/ZnS particles coated with BSA, N3 CdTeparticles coated with mercaptoethanol, N4 CdTe particles coated with BSA, BSA bovine serum albumin, BA benzoic acida Number of measurementsb Estimated valuec Data from [38]

Charged colloidal nanoparticles and effective charges 1685

and experimental data permitted the estimation of ζ (seeTable 1). If the hydrodynamic radius Rh is known withadequate accuracy, the parameters σ and mean effectivecharge number zeff are then accessible via Eq. 11 (seeTable 1). It should be noted that those results (data notshown) which can be obtained via the Einstein relation-ship (D = (μkT)/(ze)), which is valid for small ions, arestrongly different from the results obtained via theprocedure presented in this paper (diffusion coefficient Dcalculated from Rh via the Stokes–Einstein equationD ¼ kTð Þ= 6phRhð Þð Þ).The mean effective charge number determined for BSA

via determination of m (−17.8; see Table 1) is in accordancewith those results obtained for the titration of this protein[33, 34]. In these studies, a higher mean charge number of−20 was reported for an ionic strength of 0.02 at pH=8.5.The difference in the determined values is expected fromtheory, as the mean effective charge number obtainable viaEqs. 6–11 (the electrokinetic charge number), which isinfluenced not only by dissociation equilibria but also bycounter ion condensation, should be smaller than the mean

charge number obtained via titration [35–37]. If theelectrokinetic charge number zeff is taken as a measure forthe number of BSA units associated to one semiconduc-tor nanoparticle ligand number ¼ zeff N2ð Þ=zeff BSAð Þ orðzeff N4ð Þ=zeff BSAð Þ; respectivelyÞ, a particle/BSA ratio of1:8 would be expected for N2 and a particle/BSA ratio of1:5 for N4.

Au-nanoparticles

The validity of Eqs. 1–5 is only given if correctisotachophoretic zones are developed. Under these con-ditions the length of the zone is directly proportional to theamount of analyte in the sample and the concentration of

0 2 4 6 80

1

2

3

4

5

0 2 4 6 80

1

2

3

4

5

90 mV 80 mV 70 mV 60 mV 50 mV 40 mV 30 mV 25 mV 20 mV 15 mV 10 mV 5 mV

µ/(-

1)(1

0-8 m

2 s-1 V

-1)

κ Rh

Au-NP N1 N2 N3 N4 BSA

Fig. 2 Calculated curves for the dependence of the electrophoreticmobility on the product κr for different values of ζ together withmeasured data points obtained by capillary electrophoresis (forexperimental conditions refer to Fig. 2) 5 10 15

0

50

100

150

200

250a

Abs

orba

nce(

230

nm)/

10-3

AU

t/min

5 6 7 8 9 100

50

100

150

200

b

Con

duct

ivity

/a.u

.

t/min

Fig. 3 Isotachopherograms recorded for Au-nanoparticles with aphotometric detection (230 nm) or b conductometric detection. Leadingelectrolyte: 1 mmol L−1 HCl titrated with BTP to pH=7.88, terminatingelectrolyte: 10 mmol L−1 MES titrated with Tris to pH=6.00, fused-silica capillary coated with polyacrylamide 483 (415) mm × 100 μm,I = 1.0 μA, pressure injection 15 s, ambient temperature

1686 U. Pyell et al.

analyte in the migrating zone follows the conservation lawdeduced from the Kohlrausch-regulating function.

In a first study, the original sample (Goldpulver Typ 200-30) was diluted with water by factor 0.100, 0.075, 0.050,and 0.025. Measurements were done with a modified CEapparatus equipped with a contactless conductivity detectorand a fused-silica capillary coated with linear polyacryl-amide. As leading electrolyte, 1 or 2 mmol L−1 hydrogenchloride was titrated with BTP to pH=8.60. As terminatingelectrolyte, 10 mmol L−1 MES was titrated with Tris to

pH=6.00. Figure 3 shows the isotachopherograms recordedwith the photometric detector and the conductivity detector.There is a clear sharp isotachophoretic zone correspondingto the zone of gold nanoparticles with very high absorbanceand a relative step height similar to that of benzoic acid,which would be expected from the effective electrophoreticmobilities determined under CE conditions (Table 1).

Plotting the zone length (taken from the isotachophero-grams recorded with the conductivity detector) against theconcentration of the sample at fixed injection volume undervariation of the concentration of the chloride ion in theleader demonstrates that the conditions for the developmentof correct isotachophoretic zones are given. The zonelength is directly proportional to the concentration ofnanoparticles in the sample and inversely proportional tothe concentration of the leading ion (Fig. 4).

Further experiments were performed with a commercialITP unit equipped with a 30-μL injection valve. Sampleswere prepared by diluting the original sample with water byfactor 0.010, 0.015, 0.020, and 0.025. In these experiments,the leading electrolyte was 10 mmol L−1 hydrogen chloridetitrated with BTP to pH=8.60 and the terminating electro-lyte was 10 mmol L−1 HEPPS titrated with BTP to pH =8.00. Isotachopherograms recorded with a photometricdetector and a conductivity detector are superpositioned inFig. 5. The broad zone recorded with the conductivitydetector corresponds to hydrogen carbonate/carbonate. Thezone of Au-nanoparticles is selectively recorded with thephotometric detector. Within this zone, there is a constantincrease in the resistivity due to the electrophoreticheterogeneity of the investigated particles. The isotacho-phoretic zone which can be ascribed to migrating nano-particles tails into the preceding hydrogen carbonate/

absorbance (280 nm)

resistivity

Fig. 5 Isotachopherogram forAu-nanoparticles. Leading elec-trolyte: 10 mmol L−1 HCl titratedwith BTP to pH = 8.60, termi-nating electrolyte: 10 mmol L−1

HEPPS titrated with BTP topH = 8.00, fused-silica capillary200 mm × 300 μm, I=50 μA,injection 30 μL, ambienttemperature

0 42 6 8 100

2

4

6

8

10

12

14

16

18

20

22

24 c(leader) = 2 mmol L-1

c(leader) = 1 mmol L-1

Zon

e le

ngth

/mm

c/arbitrary units

Fig. 4 Zone length for Au-nanoparticles dependent on the particleconcentration for different concentrations of the leading electrolyte,data evaluated for repeated injections. Isotachopherograms recordedwith conductometric detection. Leading electrolyte: a 1 mmol L−1 HCltitrated with BTP to pH = 7.88, b 2 mmol L−1 HCl titrated with BTPto pH = 7.88, terminating electrolyte, 10 mmol L−1 MES titrated withTris to pH = 6.00, fused-silica capillary coated with polyacrylamide483 (415) mm × 100 μm, a I=1.0 μA, b I=2.0 μA, pressure injection20 s, ambient temperature

Charged colloidal nanoparticles and effective charges 1687

carbonate zone, while there is a clear sharp secondboundary. This observation might be due to the presenceof oligomers of the gold nanoparticles in low concentration.It is also possible that the fraction of nanoparticles withhighest electrophoretic mobility forms a stable mixed zonewith the hydrogen carbonate/carbonate zone. In spite of theobserved fronting, there is a linear increase in zone lengthwith increasing amount of substance injected (Fig. 6)supporting the conclusion that correct isotachophoreticzones are formed also for the migrating nanoparticles underthe conditions selected.

BSA-coated nanoparticles

Under those conditions which were successfully applied tothe stabilized gold nanoparticles, also the isotachophoreticbehavior of the other particles listed in Table 1 wasinvestigated. As an example, the isotachopherogram devel-oped for N2 is shown in Fig. 7. Isotachophoretic zones withsharp boundaries have been obtained. There is a repeatableincrease in the absorbance at the end of the analyte zone,which is followed by an increase in the electric resistivity(marked with circles). This increase can be attributed to thepresence of a small molar fraction of dimers, trimers, andhigher oligomers. Similar isotachopherograms wereobtained for N4, while isotachophoresis revealed thepresence of impurities contained in Sample N1. With avery slow terminating anion (10 mmol L−1 lysine, pH =10.3) isotachophoretic zones with sharp boundaries couldbe obtained also for N3.

For N2, the zone length dependent on the amount ofsubstance was studied by varying the volume of sampleinjected. To this end, injection (1–4 μL) was performedwith a 10-μL microsyringe. The data plotted in Fig. 8confirm the presence of correct isotachophoretic zones.There is a linear increase in the zone length with increasingamount of substance of the analyte.

The theory of isotachophoresis predicts that theelectric conductivity κ within a zone is proportional tothe effective electrophoretic mobility of the analyte [1, 2].In Fig. 9, the relative step heights (relative step heightRSHð Þ ¼ hA � hLð Þ= hT � hLð Þ; hA = step height of theanalyte zone; hL = step height of the leading zone; hT =step height of the terminating zone) determined from the

0 31 42 50

5

10

15

20

25

30

35

40

45

50

55

60

c(Au-NP)/arbitrary units

Cor

rect

ed z

one

leng

th/s

Fig. 6 Zone length for Au-nanoparticles dependent on the particleconcentration, data evaluated from isotachopherograms recorded withconductometric detection or photometric detection, mean values fortwo repeated injections (four data points), standard deviation given aserror bars. For experimental conditions refer to Fig. 5

absorbance (280 nm)

resistivity

Fig. 7 Isotachopherogram forCdSe/ZnSe nanoparticles coatedwith BSA (N2). Leading elec-trolyte: 10 mmol L−1 HCl titratedwith BTP to pH = 8.60, termi-nating electrolyte: 10 mmol L−1

HEPPS titrated with BTP topH = 8.00, fused-silica capillary200 mm × 300 μm, I=50 μA,injection 3 μL, ambienttemperature

1688 U. Pyell et al.

isotachopherograms recorded by conductivity detection(inverted signal) are plotted against the effective electro-phoretic mobilities determined by capillary electrophore-sis. As expected, there is an increase in RSH withdecreasing absolute effective electrophoretic mobility ofthe analyte, which further confirms the formation ofcorrect isotachophoretic zones.

Calculation of particle concentrations

If the mean effective charge number zeff is known, Eq. 5allows calculating the amount of substance of particlesfrom the length of the isotachophoretic zone. Thecalculation requires knowledge of the transference numbertN (tN ¼ mAj j= mAj j þ mCj jð Þ; mAj j = absolute effectiveelectrophoretic mobility of the analyte ion; mCj j = absoluteeffective electrophoretic mobility of the counter ion). Themagnitude mCj j was determined indirectly via Eq. 5 fromthe length of the isotachophoretic zone for benzoic acid(c=0.50, 1.00, 1.50, and 2.00 mmol L−1, V=30 μL,I=50 μA, mAj j see Table 1), which has, under theconditions selected (see Fig. 10), an effective chargenumber of −1. For HEPPS, this calculation yielded mCj j ¼22:81 mm2 kV�1 s�1. The effective electrophoretic mobi-lities of the nanoparticles investigated had been deter-mined by capillary electrophoresis (see Table 1).

In Table 2, the injected sample volumes, the determinedzone lengths and the calculated molar particle concentra-tions ccalc are listed for several solutions of nanoparticlesand reference substances. In the case of benzoic acid andBSA, solutions with known analyte concentrations wereprepared by weighing of the pure compound. The proce-

dure allows the calibration-free concentration determinationof colloidal solutions with good repeatability of data. Theresulting data proved to be independent of the volume ofsample injected or the dilution factor of the sample.

With the nanoparticles investigated, κRh is, in allinstances, larger than 0.5. Loeb et al. [30] have shown thatin this range (independent of ζ) the maximum error due tothe use of Eq. 11 (approximate empirical formula) is 5%.For 1<κRh<∞, the maximum error is 2%, and for 2<κRh

<∞, the maximum error is <1%. Further main sources ofuncertainty are the determination of m, the determination ofRh, and the calculation of ζ from m via the approximateanalytical formula developed by Ohshima [27], which isvalid in the range of ζ≤100 mV for a symmetricalelectrolyte with similar limiting conductances of the cationand the anion (see Eqs. 6–11). It should be also noted thatthe isotachophoretic zone (in the ideal case) contains onlythe charged analyte and the counter ion of the leadingelectrolyte. The concentration of the analyte and the counterion in the migrating zone is adapted to the value needed forfulfillment of the conservation law deduced from theKohlrausch-regulating function. This difference betweenthe composition of the electrolyte, in which m is deter-mined, and the composition of the isotachophoretic zone, inwhich the colloidal particles migrate, might induce someadditional uncertainty concerning the mean effective chargenumber of colloidal particles [34].

BSA as model colloid

The protein BSA has been selected as a model colloidbecause of its spheroidal shape, its high solubility in waterand the independence of Rh of the pH within pH=4–9 andof the temperature up to 50 °C [38]. It is known from size-

-35 -30 -25 -20 -150,6

0,7

0,8

0,9

1,0

N2

BSA

N4

N3

BA

RS

H

µ/(mm2 kV-1 s-1)

Fig. 9 Relative step heights determined from isotachopherograms (forexperimental conditions refer to Figs. 5 and 7) plotted against theelectrophoretic mobility calculated from electropherograms (forexperimental conditions refer to Fig. 1)

1,0 1,5 2,0 2,5 3,0 3,5 4,050

100

150

200

250

300

350

400 conductivity detection (Method 1) conductivity detection (Method 2) photometric detection

Zon

e le

ngth

/s

VP/µL

Fig. 8 Zone length for CdSe/ZnSe nanoparticles coated with BSA(N2) dependent on the injection volume, data evaluated fromisotachopherograms recorded with conductometric detection (Method1 = automated length determination, Method 2 = manual lengthdetermination) or photometric detection. For experimental conditionsrefer to Fig. 7

Charged colloidal nanoparticles and effective charges 1689

exclusion chromatography and dynamic light-scatteringexperiments that a very high molar fraction of the proteinis present in solutions as monomer, although the presenceof dimers, trimers, and higher oligomers was also observed[34, 38].

A model solution of BSA in water (c=0.100 mmol L−1)was analyzed by ITP. Figure 10 shows the recordedisotachopherogram. BSA migrates in an isotachophoreticzone with sharp boundaries. There is, however, no constantsignal height within the zone neither for the absorbance(280 nm) nor for the electric conductivity. While the firstboundary is very sharp with constant relative step height,the second boundary can be characterized to be morediffuse. The resistivity within the zone shows a constantincrease when approaching the second boundary. Thisbehavior is clearly different from that observed for amonomeric ionic compound in solution (e.g., benzoic acid).The distortion of the isotachophoretic zone in the case ofBSA can be explained by the presence of dimers and otheroligomers, which cause an observable electrophoreticheterogeneity [34]. Although the kinetics of the aggrega-tion/dissociation process can be regarded to be slow, noseparation of monomers from dimers and other oligomerswas possible with the ITP procedure developed.

From the zone length, Δt, the estimated mean effectivecharge number, zeff (see Table 1), and the effectiveelectrophoretic mobilities for BSA and the counter ion(determined by CE and ITP), the molar concentration of thesample solution was calculated to be 0.104 mmol L−1 (N=5, RSD=3.1%, Table 2) employing Eq. 5 assuming a purelymonomeric solution. This concentration is in excellentagreement with the theoretical value of 0.100 mmol L−1

confirming that ITP can be used for the calibration-free

concentration determination of colloidal solutions. Thisagreement of data, however, also confirms the validity ofthe method employed to determine mean effective chargenumbers. It should be emphasized that ITP generally offersthe possibility to determine effective charge numbers ofproteins and other biomacromolecules from the length of anundisturbed isotachophoretic zone, provided that solutionswith known molar concentration can be prepared and thatthe biomacromolecule is present in solution as monomer.

Table 2 Injected sample volume VS, calculated molar particleconcentrations ccalc and theoretical molar particle concentrations ctheorfor different analytes, for experimental conditions refer to Fig. 5

Analyte VS/μL Zone length/s ccalc/mmolL−1 ctheor/mmolL−1

BA 30 50.5 0.52 0.50

BA 30 103.6 1.08 1.00

BA 30 145 1.51 1.50

BA 30 199 2.06 2.00

BSA 30 (202)b (0.104)b 0.100

N2 1 79 0.156 –

N2 2 156 0.154 –

N2 3 224 0.148 –

N2 4 319 0.158 –

N3 (3)a 44 0.133 –

N4 20 38 0.003 –

Au-NP (3)a 23.2 0.118 –

Au-NP (4.5)a 33.7 0.114 –

Au-NP (6)a 45.5 0.116 –

Au-NP (7.5)a 56.2 0.114 –

aVS × dilution factorbN=5, RSD=3.1%

absorbance (280 nm)

resistivity

Fig. 10 Isotachopherogram forBSA (c=0.100 mmol L−1).Leading electrolyte:10 mmol L−1 HCl titrated withBTP to pH = 8.60, terminatingelectrolyte: 10 mmol L−1

HEPPS titrated with BTP topH = 8.00, fused-silica capillary200 mm × 300 μm, I=50 μA,injection 30 μL, ambienttemperature

1690 U. Pyell et al.

Conclusions

Correct isotachophoretic zones can be obtained for chargedcolloidal nanoparticles, which make it possible to determinewithout the need for calibration particle concentrations ofcolloidal solutions. In accordance with the principles of ITP,this concentration determination can be combined on-linewith a separation step (e.g., separation from matrix constitu-ents resulting from the synthesis procedure) due to differencesin the effective electrophoretic mobility between the nano-particles and further constituents, which offers the possibilityto integrate sample preparation steps into the determinationprocedure. Data for a BSA solution confirm the concept ofcalibration-free concentration determination of colloidal sol-utions [16]. From the data can be also concluded that theprinciple of ITP offers a very simple route to determine meaneffective charge numbers for proteins and other colloidalmacromolecules or nanoparticles.

The presented approach provides a method for the direct,accurate, and rapid determination of the particle concentra-tion of colloidal nanoparticle solutions provided that thetransference number and the mean effective charge numberof the particles (within the isotachophoretic zone) can bedetermined with adequate accuracy. Being different fromapproaches developed so far, the presented method will beespecially useful when combined on-line with a separationstep or when a second independent method is necessary toconfirm the accuracy of a measurement procedure.

Acknowledgments Financial support from the Bundesministeriumfür Bildung und Forschung, Germany (BMBF, FKZ: 13 N8645) isgratefully acknowledged. The authors thank Fuad Al-Rimawi (De-partment of Chemistry, University of Marburg, Marburg, Germany)for his assistance with the preparation of the coated fused-silicacapillary, Jürgen Hins (JH-Analytik, Aalen, Germany) for the loan ofthe ITP unit and the contactless conductivity detector, and W.C.Heraeus GmbH, Hanau, Germany for providing a sample of stabilizedgold nanoparticles.

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