Extracting Information from the Ionic Strength Dependence of

Article pubs.acs.org/ac

Extracting Information from the Ionic Strength Dependence of Electrophoretic Mobility by Use of the Slope Plot Amal Ibrahim,† Stuart A. Allison,‡ and Hervé Cottet*,† †

Institut des Biomolécules Max Mousseron, UMR 5247 CNRS, Université de Montpellier 1 and 2, place Eugène Bataillon CC 1706, 34095 Montpellier Cedex 5, France ‡ Department of Chemistry, Georgia State University, Atlanta, Georgia 30302-4098, United States S Supporting Information *

ABSTRACT: The effective mobility (μep) is the main parameter characterizing the electrophoretic behavior of a given solute. It is well-known that μep is a decreasing function of the ionic strength for all solutes. Nevertheless, the decrease depends strongly on the nature of the solute (small ions, polyelectrolyte, nanoparticles). Different electrophoretic models from the literature can describe this ionic strength dependence. However, the complexity of the ionic strength dependence with the solute characteristics and the variety of analytical expressions of the different existing models make the phenomenological ionic strength dependence difficult to comprehend. In this work, the ionic strength dependence of the effective mobility was systematically investigated on a set of different solutes [small mono- and multicharged ions, polyelectrolytes, and organic/inorganic (nano)particles]. The phenomenological decrease of electrophoretic mobility with ionic strength was experimentally described by calculating the relative electrophoretic mobility decrease per ionic strength decade (S) in the range of 0.005−0.1 M ionic strength. Interestingly, the “slope plot” displaying S as a function of the solute electrophoretic mobility at 5 mM ionic strength allows for defining different zones that are characteristic of the solute nature. This new representative approach should greatly help experimentalists to better understand the ionic strength dependence of analyte and may contribute to the characterization of unknown analytes via their ionic strength dependence of electrophoretic mobility. nonaqueous25−27 media or on peptide μep.28,29 For SI and nanoparticles (NP) that are globular in shape, modified Yoon and Kim modeling30,31 takes into account the ionic strength dependence including relaxation effects. The electrophoretic behavior of nano- or microparticles modeled as nonconducting spheres with a centrosymmetric charge distribution has been a subject of study for almost 100 years.32−36 It should be mentioned that Hückel32 and Henry33 accounted for the electrophoretic effect in their early works and that Overbeek34 deserves credit for first taking account to lowest order for the relaxation effect. More recent developments were introduced by O’Brien and White35 and then by Ohshima,36 taking relaxation and electrophoretic effects into account.35,36 For polyelectrolytes (PE), different sophisticated theories were developed by Manning,37 Stigter,38 Cleland,39,40 Muthukumar,41 and recently by Ohshima.42 Unfortunately, a simple analytical expression, that is, a full Poisson−Boltzmann solution, for a flexible, highly charged PE in an electric field does not exist yet. From a phenomenological point of view, a logarithmic dependence of the electrophoretic mobility on ionic strength was generally observed with some physical meanings.43

I

t is well-known that the electrophoretic mobility (μep) of solutes decreases with the ionic strength (I) of the electrolyte. The characteristic decrease is highly dependent on the nature of the solute (mono- or multicharged small ions, polyelectrolyte, nanoparticles, etc.).1,2 This dependence on ionic strength is often used to change the electrophoretic mobility of solutes contained in a mixture and, thus, to increase the separation selectivity between analytes, as exemplified in the literature for small ions,3−5 pharmaceuticals,6−8 oligomers and peptides,9,10 proteins,11,12 polyelectrolytes,13−15 and nanoparticles.16−18 However, the optimization of separation via the change in ionic strength is mainly empirical, based on a trial and error approach. The absence of a more rational approach is mainly due to the fact that the ionic strength dependence is rather complex. Ionic strength behavior depends on the solute nature, size, shape, charge number, and distribution and on the nature and concentration of the background electrolyte. Under certain limiting conditions, such as particles much larger than the Debye length, the electrophoretic mobility is known to scale as ∼I −1/2 according to the Smoluchowski law,19−21 where I is the ionic strength. For small ions (SI), a more complex dependence described by the Pitts equation22 or by Friedl et al.1 is generally observed. Several studies were devoted to systematic investigation of the effect of electrolyte concentration on small ion mobility in aqueous1,23,24 and © 2012 American Chemical Society

Received: August 5, 2012 Accepted: September 23, 2012 Published: October 10, 2012 9422

dx.doi.org/10.1021/ac302033z | Anal. Chem. 2012, 84, 9422−9430

Analytical Chemistry

Article

Beyond these existing models, it would be worthwhile for experimentalists to get a better insight on the phenomenological ionic strength dependence, according to solute physicochemical properties. This would help to improve and optimize separations and to better understand the ionic strength dependence according to the different classes of solutes. The goal of this work is to study the phenomenological ionic strength dependence of different kinds of solutes (small ions, polyelectrolytes, and spherical particles) to determine some general trends that are specific to each solute type. The objective was also to try to give simple and characteristic figures of merit relative to the ionic strength dependence for the three aforementioned classes of solutes.

μep =

⎤ ⎛ z eζ ⎞ 2 ⎜ ⎟ f4 (κR h)⎥ ⎥⎦ ⎝ kBT ⎠

THEORETICAL BACKGROUND Electrophoretic Mobility Dependence of Small Ions on Ionic Strength. The extended Onsager model or Debye− Hückel−Onsager equation22,24,28 set the dependence on ionic strength of the effective mobility μep of monocharged small ions as I 1 + Bri I

(1)

σ=

where μep∞ is the limiting mobility at infinite dilution, ri is the ionic radius, and I is the ionic strength. A1, A2, and B are the Pitts parameters, defined for fully dissociated 1:1 electrolytes (all numerical values are in the international units system) as

A1 =

A2 =

4.125 × 10−4 Fη ε rT

(1a)

8.201 × 105 (ε rT )3/2

(1b)

5.03 × 10 ε rT

8 ln[cosh(z eζ /4kBT )] ⎤ ⎥ sinh2(z eζ /2kBT ) ⎦

1/2

ζ=

⎛ ⎞ 2kBT eσ sinh−1⎜ ⎟ e ⎝ 2ε0εrκkBT ⎠

(3b)

Polyelectrolyte Electrophoretic Mobility Modeling. For polyelectrolytes, a logarithmic dependence of electrophoretic mobility on ionic strength is generally observed, in either random coil or long rod conformation:43

(1c)

zI

(3a)

An exact analytical expression giving the reciprocal of the last equation (ζ as a function of σ) is not available, but a good approximation can be obtained by use of46

where F is the Faraday constant, T is the temperature in kelvins, η is the viscosity of the electrolyte (pascal·seconds), and εr is the relative electric permittivity or relative dielectric constant. These parameters have the following values for aqueous solutions at 25 °C: A1 = 3.14 × 10−8 m2·V−1·s−1·mol−1/2, A2 = 0.229 mol−1/2, and B = 0.3288 Å·mol−1/2. By use of eq 1, the values of μep∞ and ri can be determined by nonlinear curvefitting of experimental effective mobilities. In the case of multicharged small ions, Friedl et al.1 proposed an empirical expression to predict the mobility of small ions of a charge number z = 2−6, and the relative difference between measured and calculated mobilities with that expression is about 5%: μep = μep∞e−0.77

⎛ z eζ ⎞ 2ε0εrκkBT sinh⎜ ⎟ ze ⎝ 2kBT ⎠ ⎡ 1 2 1 ⎢1 + + 2 κR h cosh (z eζ /4kBT ) (κR h)2 ⎣

11

B=

(3)

where εr is the relative electric permittivity, ε0 is the electric permittivity of vacuum, η is the viscosity of the electrolyte (pascal·seconds), z is the charge number of the electrolyte ions, e is the elementary electric charge, kB is the Boltzmann constant, κ is the Debye−Hückel parameter, and Rh is the particle hydrodynamic radius. f1, f 3, and f4 are functions of κRh and are given by eq 28 in ref 36. m+ and m− are dimensionless ionic drag coefficients given by eq 18 in ref 36. It is worth noting that the ionic strength dependence of the mobility of particles expressed in eq 3 is quite complex since both ζ and κ are dependent on the ionic strength. Assumption of a spherical geometry of the particle and a uniform distribution of the charge on the surface allows for the calculation of the surface charge density (σ) at the plane of shear for a given ζ and a given Rh by use of the following empirical equation derived by Makino and Ohshima:45

■

μep = μep∞ − (A1 + A 2 μep∞)

⎡ ⎛ z eζ ⎞ 2 m + m+ 2ε0εrζ ⎢ f1 (κR h) − ⎜ ⎟ f (κR h ) − − 3η ⎢⎣ 2 ⎝ kBT ⎠ 3

μep =

Q D ln(κ −1/b) 3πηκ −1

∼ ln(κ −1/b) ∼ ln(I )

(4)

where QD is the Manning effective charge of a rodlike polyelectrolyte subsection equal to the Debye length (κ−1), b is the characteristic dimension of a monomer, and η is the viscosity. In eq 4, note that QD is proportional to κ−1, so that QD/κ−1 is independent of the ionic strength. Equation 4 is valid only for polyelectrolytes having a typical size larger than the Debye length. Therefore, it is only valid on a limited range of ionic strength and for sufficiently large polyelectrolytes (not for oligomers). Phenomenological Ionic Strength Dependence of Mobility: Introduction to the Slope Plot. Whatever the solute considered, it can be very informative to have an idea of the order of magnitude of the decrease in mobility with ionic strength. As previously described, this decrease strongly depends on the nature of the solute (small ion, polyelectrolyte, or particle), and there is no general and simple analytical

(2)

Nanoparticle Electrophoretic Mobility Modeling. In 2001, Ohshima36 derived the following approximate analytical formula, which is valid for z:z electrolytes and zeta potential ζ ≤ 100 mV (eζ /kBT ≤ 4):36,44 9423



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Table 1. Sample Effective Mobilities at 5 and 50 mM and P and S Slopes Obtained by Logarithmic Fits

expression of this dependence. To roughly describe and evaluate the phenomenological electrophoretic mobility decrease with ionic strength, we propose to use the logarithmic dependence as described for polyelectrolytes in the previous section: μep = −P log(I ) + μ1M

samplea

(5)

where P is the slope of the μep = f(log I) plot and I is the ionic strength expressed in moles per liter. P value represents the absolute decrease of μep by ionic strength decade (P < 0). Equation 5 can be rewritten, in terms of relative electrophoretic mobility decrease compared to typical low ionic strength mobility investigated experimentally in CE (I = 5 mM), as μep μep5mM

= −S log(I ) + R (6)

where S represents the relative decrease of μep by ionic strength decade compared to the value obtained at I = 5 mM. R is a constant for a given solute. The μep/μ5mM = f(I) function has been studied in this work in a range of I between 5 and 100 mM. S values give valuable information on the phenomenological decrease of μep with ionic strength and can be considered as a characteristic of the solute since they depend on its charge density, size and nature. Interestingly, we noticed that plotting S = f(μep5mM) was even more informative on the solute characteristics, as demonstrated in the Results and Discussion section.

■

EXPERIMENTAL SECTION Chemicals. Sodium tetraborate (Na2B4O7·10H2O) was purchased from Aldrich (Steinheim, Germany). Poly(ethylene glycol) dodecyl ether (Brij 35) of 362.55 g/mol molar mass was purchased from Alfa Aesar (Karlsruhe, Germany). N,NDimethylformamide (DMF) and glacial acetic acid were purchased from Carlo Erba (Paris, France). Sodium chloride and sodium hydroxide were from VWR (Leuven, Belgium). Deionized water was further purified with a Milli-Q system from Millipore (Molsheim, France). Samples. 2-Acrylamido-2-methylpropanesulfonic acid (AMPS) and itaconic acid (IA), were purchased from Aldrich (Steinheim, Germany). Bovine serum albumin (BSA) was purchased from Acros Organics (Geel, Belgium). Polystyrenesulfonate polymers (PSS) of 148 000 and 962 000 MW were purchased from Polymer Standards Service (Mainz, Germany). Poly(glutamic acid) sodium salt (pGlu) of average degree of polymerization (DP) 220 was supplied by Flamel Technologies (Venissieux, France). Statistical copolymers (PAMAMPS) of acrylamide (AM) and 2-acrylamido-2-methylpropanesulfonic acid (AMPS) of different molar fraction (or chemical charge density) of 3%, 10%, 30%, 55%, 85%, and 100% were synthesized, at room temperature, by radical polymerization initiated by potassium persulfate and N,N,N′,N′-tetramethylethylenediamine (TMEDA) according to the procedure described by McCormick and Chen47 and Cottet and Biron.48 Polystyrene latexes of 10 and 40 nm hydrodynamic radii and carboxylated magnetic beads of 100 nm hydrodynamic radius were from Thermo Scientific and Ademtech, respectively. All samples studied in this work are introduced in Table 1 together with the experimental electrophoretic mobility data at 5 and 50 mM ionic strengths. Other electrophoretic mobility data taken from literature and considered in this work

μ5mM, 10−9 m2·V−1·s−1

AMPSb IAc

24.10 53.30

Lat 10 Lat 40 Mag 100

15.8 18.9 18.3

EOF (pH 6.2) EOF (pH 7.4) EOF (pH 9.2)

52.72 59.61 87.7

PAMAMPS 3% PAMAMPS 10% PAMAMPS 30% PAMAMPS 55% PAMAMPS 85% PAMPS 100% pGlu PSS 148 PSS 962

11.1

46.9 45.10 43.7 45

BSA

25.02

μ50mM, 10−9 m2·V−1·s−1

Small Ions (SI) 21.20 45.35 Nanoparticles (NP) 8.9 8.6 8.18 Microparticles 24.36 32.67 50.3 Polyelectrolytes (PE) 5.41

P, 10−9 m2·V−1·s−1·M−1

S

3.64 10.30

0.16 0.18

7.58 10.6 11.2

0.48 0.55 0.61

27.8 28.9 38

0.53 0.48 0.44

4.72

0.46

25.8

14.7

10

0.41

38.2

27.5

11.2

0.30

43.7

33.1

10.4

0.23

47.7

35.6

10.8

0.23

9.74 10.30 7.05 8.57

0.21 0.23 0.16 0.19

7.64

0.31

36.1 34.50 36.2 36.4 Protein 16.62

a

AMPS, 2-acrylamido-2-methylpropanesulfonic acid; IA, itaconic acid; Lat, latex; Mag, magnetic beads; EOF, electroosmotic flow; PAMAMPS, poly(acrylamido−stat-2-acrylamido-2-méthylpropanesulfonate) of different chemical charge densities f; pGlu, poly(glutamic acid); PSS, polystyrene sulfonate; BSA, bovine serum albumin. bz = 1. c z = 2.

are gathered in Table S1, available in the Supporting Information. Capillary Electrophoresis. CE was carried out on an Agilent 3D-CE instrument (Waldbronn, Germany) equipped with a diode array detector. Uncoated fused silica capillaries (Composite Metal Services, Worcester, U.K.) of 33.5 cm total length (25 cm from inlet to UV detector) × 50 μm i.d. were used for determination of μep for all solutes. New fused silica capillaries were conditioned by performing the following washes: 1 M NaOH for 30 min, 0.1 M NaOH for 15 min, and water for 10 min. Between runs, capillary was flushed at 1 bar with 0.1 M NaOH for 3 min and then 5 min with the background electrolyte (BGE). Sodium borate buffer at pH 9.2 was used as BGE at different ionic strength (I) ranging from 0.005 to 0.1 M for all solutes. The applied voltage was +10 kV at I = 0.005, 0.01, and 0.025 M; +5 kV at I = 0.05 M; +4 kV at I = 0.075 M; and +3 kV at I = 0.1 M. The temperature of the capillary cartridge was set at 25 °C. For latexes, anionic surfactants inherently present from the latex synthesis by emulsion polymerization may change the surface charge density. Therefore, for latexes, 2 mM Brij35 (neutral surfactant) was added to the BGE to replace residual anionic surfactants and to keep constant σ with ionic strength. For EOF measurements (model of large particles), BGE at different pH were usedsodium borate (pH 9.2), Tris-HCl (pH 7.4), 9424



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and BisTris/HCl (pH 6.4)to vary σ. Samples were prepared in deionized water (+2 mM Brij35 for latexes) and were injected hydrodynamically (17 mbar, 3 s). Before analysis, magnetic bead samples were rinsed three times with deionized water by use of a magnet and following the fabricant protocol. Data were collected at 200 nm. For each run, electroosmotic mobility was calculated from the migration time of DMF (neutral marker). Electropherograms were plotted in effective mobility scale (Figure 1) by use of the following equation:

⎛ 1 1 ⎞⎟ lL μep = ⎜⎜ − teo ⎟⎠ V ⎝ tapp

where P is the dissipated power and the factor 0.07 corresponds to the slope of the relative variation of the conductivity, κ(P)/κ (P=0), as a function of the dissipated power per unit of length (P/L) determined at 25 °C. In all experiments, the applied voltage was chosen so as to keep P/L values lower than 0.7 W·m−1 (i.e., corresponding to an increase of temperature of less than 2 °C).

■

RESULTS AND DISCUSSION Electrophoretic Behavior of Small Ions. μep measurements were performed for monocharged (AMPS) and doubly charged (IA) ions in borate buffer (pH 9.2) in the range of 0.005−0.1 M ionic strength. Figure 1 displays the electropherograms obtained for these samples at four different ionic strengths. For AMPS, tailing peaks (in mobility scale) were obtained since μep of AMPS is slightly lower than that of the borate co-ion, while fronting peaks were observed for IA. Figure 2A displays μep data as a function of I for AMPS and IA, together with other SI electrophoretic mobility data taken from the literature [monocharged oligoglycines28 (Gly2 and Gly10) and tricharged naphthalene-1,3,6-trisulfonic acid (NTSA)1]. The bars in Figure 2A represent ±1 standard deviation (SD) calculated for three repetitions. μep = f(I) data were fitted by use of eq 1 (for monocharged SI) or eq 2 (for di- and tricharged SI) and the fits are shown as solid lines. The same data can also be fitted by the phenomenological logarithmic model (eq 5) as presented by the dashed lines. Relative differences between the logarithm modeling and experimental data vary from 0.3% to 2.6% depending on the solutes. It can be noticed from Figure 2A that the higher the charge of SI, the higher the electrophoretic mobility decreases with I. This can be quantitatively determined by the P values (slopes in Figure 2A). Table 1 gathers the numerical P values from this work, while Table S1 (see Supporting Information) displays P values extracted from literature data.1,9,28 P values increase with the solute charge: P ∼ (2.3−4.5) × 10−9 m2·V−1·s−1·M−1 for z = 1; P ∼ (8.2−10.5) × 10−9 m2·V−1·s−1·M−1 for z = 2; P ∼ (10.6− 13.6) × 10−9 m2·V−1·s−1·M−1 for z = 3; P ∼ (14.6−16.8) × 10−9 m2·V−1·s−1·M−1 for z = 4; P ∼ 18 × 10−9 m2·V−1·s−1·M−1 for z = 5; and P ∼ 20 × 10−9 m2·V−1·s−1·M−1 for z = 6. Another way to estimate the influence of ionic strength on the electrophoretic mobility is to consider the relative decrease in mobility, taking the effective mobility at 5 mM ionic strength (μ5mM) as a reference, as displayed in Figure 3 for different solutes. Nonlinear fittings according to eq 6 (solid lines in Figure 3) give access to S values reported in Table 1 for different solutes. Phenomenologically, S represents the relative decrease in mobility per ionic strength decade. For SI, S values increases with the solute charge, ranging from ∼0.1−0.16 for z = 1, 0.18 for z = 2; 0.21 for z = 3; 0.24 for z = 4; 0.26 for z = 5; up to 0.29 for z = 6. Interestingly, while P values tends to increase with μ5mM within a class of SI of a given charge number (see open symbols in Figure S2 of the Supporting Information), S values were found to be almost independent of the SI electrophoretic mobility and therefore are more characteristic of the solute charge (see open symbols in Figure 4). A specific behavior was observed for mono-end-charged oligomers (oligoglycines), where S values were found to increase with solute size from 0.12 for Gly2 up to 0.23 for Gly11. The mobility of end-charged oligomers is therefore more sensitive to ionic strength (in relative variation) than the mobility of smaller monocharged ions. However, since the mobility of these oligomers at 5 mM ionic strength is clearly

(7)

Figure 1. Electropherograms in effective mobility scale obtained at different ionic strengths for SI, NP, and PE. Experimental conditions: fused silica capillary 33.5 cm (25 cm to UV detector) × 50 μm i.d.; BGE sodium borate buffer (pH 9.2) at 5, 10, 50, and 100 mM ionic strengths as indicated; voltage +10 kV at 5 and 10 mM, +5 kV at 50 mM, and 3 kV at 100 mM; hydrodynamic injection 17 mbar, 3 s; temperature 25 °C. For better clarity, only six solutes and four ionic strengths are displayed: 5, 10, 50, and 100 mM, as indicated on the graph. Peak identification: IA, itaconic acid; AMPS, 2-acrylamido-2methylpropanesulfonic acid; Mag 100, magnetic beads with Rh = 100 nm; Lat 40, nanolatex with Rh = 40 nm; PAMAMPS, statistical copolymers of acrylamide and AMPS having different chemical charge densities as indicated by the molar content (%) of AMPS.

where μep is the effective mobility, l is the effective capillary length to the detection point, L is the total capillary length, V is the applied voltage, teo is the detection time of the neutral marker, and tapp is the detection time of solute. tapp was taken at the apex (peak maximum) for symmetrical peaks. For PE and NP samples, which are intrinsically polydisperse, tapp was obtained by integration of the whole peak by use of tapp = [∫ h(t)t dt]/[∫ h(t) dt], where h(t) is the detector response (UV absorbance in the present work). For SI presenting triangular shape due to electromigration dispersion (EMD), tapp was taken at the beginning/end of the fronting/tailing peak (and not at the peak apex). Corrections of the electrophoretic mobility from joule heating have been performed by a previously described procedure28,49 according to the following equation: μ(P) μ(P = 0) = 1 + 0.07(P /L) (8) 9425



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Figure 3. Electrophoretic mobility decrease relative to mobility at 5 mM as a function of ionic strength (logarithmic scale) for SI, NP, and PE. Solid lines correspond to logarithmic fits (see eq 6). Experimental conditions were as described for Figure 1. Solute identification was as described for Figures 1 and 2.

Figure 4. Slope plot: S values as a function of μ5mM for different solutes. Open symbols represent SI: (□) z = 1 (AMPS, Gly2−11, ASA, NSA, and BA); (○) z = 2 (IA, ADSA, NDSA, BDSA, and PA); (Δ) z = 3 (ITSA, NTSA, ATSA, and TMA); (∇) z = 4 (ITeSA, ATESA, and PMA), (◇) z = 5 (APSA), (☆) z = 6 (AHSA). Particles: (×) Lat 10 and 40, (+) Mag 100, (I) Au 40 and 50, (*) EOF. Solid symbols represent PE: (■) PAMAMPS ( f = 3−100%), (●) pGlu, (▲) PSS 148, (▼) PSS 962, (★) BSA. See Table 1 and Table S1 (Supporting Information) for solute abbreviations and provenance.

different from multicharged SI, the position of these oligomers in the “slope plot” displayed in Figure 4 is clearly different from SI with z = 2, 3, or 4. When Figure S2 (Supporting Information) and Figure 4 are compared, the S = f(μ5mM) plot seems much more relevant to classify the different SI solutes compared to the P = f(μ5mM) plot. This trend will also be confirmed for the other kinds of solutes. Electrophoretic Behavior of Nano- and Microparticles. μep = f(I) data for organic (Lat 10 and 40, Mag 100) and inorganic50 (Au 40 and 50) NP as well as EOF values (considered as a model of microparticles) are given in Figure 2B (for clarity, not all solutes are shown). Experimental μep points were fitted with eq 3 (solid lines) by use of a constant σ value as adjusting parameter. The adjusted σ value was used to calculate ζ, which was injected in eq 3 to calculate μep.

Figure 2. Ionic strength dependence of electrophoretic mobility for (A) SI, (B) NP or EOF, and (C) PE. Solid lines are the fits by Pitts equation22 (eq 1) for monocharged SI (AMPS, Gly2, and Gly10); by Friedl modeling1 (eq 2) for multicharged SI [IA (z = 2) and NTSA (z = 3)]; and by OʼBrien−White−Ohshima calculations35,36 (eq 3) for NP and EOF. Dashed lines are the logarithmic fits according to eq 5. Error bars show SD for three repetitions of μep measurement inder the same conditions (for SI) and SD calculated by integration of the electrophoretic profile (for polydisperse NP and PE samples). Electrophoretic conditions were as described for Figure 1. Peak identification: Gly2, diglycine; Gly10, decaglycine; NTSA, naphthalenetrisulfonic acid; Lat 10, nanolatex with Rh = 10 nm; Au 40, gold nanoparticle with Rh = 40 nm; EOF, electroosmotic mobility in a fused silica capillary (model of large particle); others are as described for Figure 1. 9426



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Experimental data were fitted with the phenomenological logarithmic dependence (eq 5, dashed lines). As for SI, the logarithm modeling gives a good approximation of the ionic strength dependence (maximum relative difference between measured and calculated mobilities by use of the logarithm model 0.7%). Organic NP with low charge density σ ∼ 0.005 C·m−2 have a weak absolute μep decrease with I (low P values, 7.58−11.2 × 10−9 m2·V−1·s−1·M−1). Inorganic NP with higher σ (0.05 C·m−2) and the EOF (σ ∼ 0.02−0.04 C·m−2) have a much more important absolute μep decrease with I [P values of (18.1−20.4) and (27.8−38.0) × 10 −9 m2 ·V −1 ·s −1·M −1 , respectively]. When the relative decrease in mobility is considered, the trends are reversed with the organic NP with low charge density having the highest S values (0.48−0.61), followed by the EOF (0.44−0.53), and finally the inorganic (Au) NP (0.30−0.31), as displayed in Figure 4. From Figure S2 of the Supporting Information and Figure 4, it clearly appears that the slope plot with the S values (Figure 4) leads to a better discrimination of the different solutes behaviors compared to Figure S2, Supporting Information (with P values). Interestingly, S values were found to decrease with charge density in the case of NP, while the reverse trend was observed for SI. Size effect on ionic strength dependence can be studied with Lat 10, Lat 40, and Mag 100 which have approximately the same σ and increasing size. Both P and S values increase with size. This trend was also observed with SI for which S and P were found to increase with the hydrodynamic radius at a given charge number (oligoglycines). Electrophoretic Behavior of Polyelectrolytes. Figure 2C displays μep = f(I) data for PAMAMPS PE of different chemical charge densities (or molar content in AMPS, f) fitted with eq 5 (logarithmic dependence). The slopes of the lines in Figure 2C (P values) increases rapidly from 4.7 × 10−9·m2·V−1·s−1·M−1 for f = 3% up to about ∼(10−11) × 10−9 m2·V−1·s−1·M−1 for f higher than 10%. Above 10%, P values are similar. Considering the relative decrease in ionic strength, the S values tend to decrease strongly from 0.46 for f = 3% down to 0.21−0.23 for PE with f higher or equal to the Manning counterion condensation threshold ( f ∼ 36% for vinylic copolymer). The slope plot in Figure 4 is a good way to estimate the charge density of a PE (at least in the noncondensed PE range of f) with an almost linear regression of the S value with μ5mM. In this plot, condensed PE are centered around 0.2 (S) and 45 × 10−9 m2·V−1·s−1 (μ5mM) values as exemplified by the results on polyglutamates (0.23; 45.1 × 10−9 m2·V−1·s−1), PSS (0.18; 44.4 × 10−9 m2·V−1·s−1) and PAMPS (0.21; 46.9 × 10−9 m2·V−1·s−1). For PE, the electrophoretic mobility at finite ionic strength is known to be independent of the molar mass (so-called free draining behavior). Therefore, for the special case of PE, S values are almost independent of the molar mass while they were found to increase with size for SI and NP. This is clearly observed for the two PSS samples (148 and 962 × 103 g/mol), for which S values and mobilities are similar. It is worth noting that PE behavior with free draining behavior is generally observed for molar mass higher than ∼20 000 g/mol.13 Comparative Study of Small Ions, Nanoparticles, and Polyelectrolytes and Discussion about the Slope Plot Approach. From the results previously discussed for each solute type, it can be concluded that the slope plot presented in Figure 4 giving S (relative electrophoretic mobility variation) as a function of μ5mM is a very convenient way to compare the effect of ionic strength on electrophoretic behavior. The 2D

space occupied by the different solutes reported in this work is much more revealing in the slope plot (Figure 4) than when representing the absolute variation of the mobility [P = f(μ5mM); Figure S2, Supporting Information]. Interestingly, different zones can be distinguished in the slope plot according to the nature and characteristics of the solutes: (1) The z = 1 monocharged SI studied in this work clearly settle in the lower left corner of the plot with typical relative decrease S of 10−20% and mobilities at 5 mM lower than 35 × 10−9 m2·V−1·s−1. (2) The z = 2−6 SI display a steplike increase with z. Horizontal lines of given S for each z were separated (18%, 21%, 24%, 26%. and 29% for z = 2−6 respectively) with μ5mM values of (45−75) × 10−9 m2·V−1·s−1. (3) On the upper part, particles and NP display S values higher than 30% and up to 60%, which is generally much higher that what was observed for SI and PE. (4) PE section covers a linear zone starting from low charge density PE with S ∼ 45% to highly charged condensed PE with S ∼ 20%, with μ5mM varying concomitantly, (10−45) × 10−9 m2·V−1·s−1. Interestingly, BSA behaves as a moderately charged PE. Crossover regions can be observed between z = 6 SI and highly charged nanoparticles (Au NP). Other crossover also exists between low charge density PE (3% PAMAMPS) and small weakly charged NP (Lat 10 nm) or between the highly condensed PE and the z = 2 or z = 3 SI of low mobility. In most of these cases, the ambiguity of belonging to the solute type can be discarded if we consider the solute size. For instance, the distinction between a z = 2 SI of low mobility and the highly charged PE can be easily obtained if one consider the difference in size in addition to the slope plot information. To get a better insight in the relevance of the slope-plot approach and to confirm the general trends observed so far, a theoretical overview of the electrophoretic behavior was investigated by use of electrophoretic modeling based on a nonconducting sphere of radius a containing a centrosymmetric charge distribution of net valence charge Q. Although ions/ macroions of interest are not spherical nor do they necessarily contain a centrosymmetric charge distribution, the simple spherical model should be a fairly realistic model for nonspherical but globular solutes with more complex charge distributions.51,52 Henry33 investigated the problem of a nonconducting sphere but did not include the relaxation effect (distortion of the ion atmosphere around the particle due to the imposition of an externally applied electric or flow field34). Also, εi and εr denote the relative electric permittivity of the sphere interior and BGE, respectively. In modern notation, the “unrelaxed” mobility, μnr, can be written:53 μnr =

eQ + 6πηa

∫a

∞

3 ⎡ ⎛ a ⎞ 5 ⎞⎤ c ⎛⎛ a ⎞ r drρ0 (r )⎢1 − ⎜⎜ ⎟ − ⎜ ⎟ ⎟⎥ ⎝ r ⎠ ⎠⎥⎦ ⎢⎣ 2 ⎝⎝ r ⎠

(9)

Above, ρ0(r) is the local equilibrium charge distribution at a distance r from the center of the sphere, and c = (εr − εi)/(εi + 2εr). The physical significance of the first term on the righthand side of eq 9 is the mobility in the limit of zero ion concentration, and the second term represents the retarding influence of the ion atmosphere. The equilibrium charge distribution, which is derivable from the Poisson−Boltzmann equation, can be written: 9427



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ρ0 (r ) = e ∑ cjzje−ezj Λ 0(r)/ kBT j

≅−

e 2Λ 0(r ) kBT

∑ cjzj 2 + O[Λ 02(r)] j

(10)

Above, the sum over j extends over all ions of the BGE, cj and zj are the concentration and valence charge of species j, kB is the Boltzmann constant, and Λ0 is the local equilibrium electrostatic potential. The first term on the right-hand side of eq 10 is true regardless of the net charge, Q. In going from the second to third equality in eq 10 above, the exponential has been expanded and only the leading terms have been retained. The O term on the far right-hand side of eq 10 indicates terms of order Λ02 and higher. Neglecting these nonlinear terms is a good approximation for large and weakly charged particles. If only the linear term in Λ0 is retained, then Λ0(r) and consequently ρ0(r) are directly proportional to Q. If this is substituted into eq 9, then μnr is proportional to Q while S (see eq 6) is independent of Q. Under these conditions, eq 9 can be written: μnr =

eQg ′(x) 6πηa(1 + x)

Figure 5. S versus μ5mM for a solid sphere. Equation 9 was used where the relaxation effect is left out (solid symbols), and the method of O’Brien and White35 was used to account for the relaxation effect (open symbols). The BGE is aqueous NaCl at a temperature of 298 K. Three particle sizes are considered: a = 0.3 nm (diamonds), 1 nm (squares), and 5.0 nm (triangles). For each particle size, Q = jQ0, where j ranges from 1 to 10 and Q0 = 0.4, 1.5, and 12.0 for a = 0.3, 1.0, and 5.0 nm, respectively. Increasing charge corresponds to increasing values of μ5mM. Solid, dashed, and dotted lines represent S values for a = 0.3, 1.0, and 5.0 nm, respectively, when electrostatics and the charge density are treated at the level of the linear Poisson−Boltzmann equation.

(11)

In eq 11 at 25 °C, x = κa = 3.286√I, where a is in nanometers and I is the ionic strength in moles per liter. Also, g′(x) is the Henry function that is sigmoidal in shape and goes to 1 in the limit x → 0 and goes to 3/2 in the limit x → ∞. For weakly charged particles, it is straightforward to calculate S from eq 11. For more highly charged particles where the relaxation effect is ignored but the nonlinear terms in the charge density are included in eq 10, it is still possible to evaluate Λ0, μnr (using eq 9), and S. However, this must be done numerically. By varying the ionic strengths and then plotting μnr/μnr5mM versus log(I) and fitting using eq 6 (with “nr” subscript replacing “ep”), we can examine the influence of nonlinear electrostatic effects on S as a function of particle size. Plotted in Figure 5 is S versus μ5mM for a solid sphere at 298 K in an aqueous NaCl BGE where the relaxation effects are left out (solid symbols). The ionic strength range considered varies from 5 to 100 mM. Three particle sizes are considered: a = 0.3 nm to represent a small ion (diamonds), 1.0 nm to represent an intermediatesized ion (squares), and 5.0 nm to represent a nanoparticle (triangles). For each particle size, 10 charges are considered [Q = jQ0, where j = 1−10 and Q0 = 0.4 (a = 0.3 nm), 1.5 (a = 1.0 nm), and 12.0 (a = 5.0 nm)]. Successive data points (increasing μnr5mM for a given particle size) correspond to increasing net particle charge. Also shown in this figure are the “linear PB” results indicated by solid, dashed, and dotted lines for a = 0.3, 1.0, and 5.0 nm, respectively. As can be seen from this figure, S correlates strongly with particle size. The larger the particle, the larger S will be. Nonlinear electrostatic effects cause S to vary with increasing Q. For small particles, S increases with increasing Q, and for large particles, the opposite is true. For particles in the a = 1 nm range, S shows little variation with Q. For highly charged particles, the relaxation effect should be included in the calculation and this can be done via the O’Brien and White numerical procedure.35 Similar to the previous “unrelaxed” cases, the same ionic strength, particle size, and charge ranges are considered. By varying the ionic strengths and then plotting μ/μ5mM versus log(I) and fitting by use of eq 6, we can examine the combined influences of ion relaxation and

nonlinear electrostatic effects on S as a function of particle size. Plotted in Figure 5 is S versus μ5mM with the relaxation effects considered (open symbols). The inclusion of ion relaxation appears to amplify the nonlinear electrostatic effects observed for the solid symbols. The rule of thumb appears to be if a < 1 nm, S increases with increasing Q, and if a > 1 nm, S decreases with increasing Q. The dependence of S on Q is particularly pronounced for large particles.

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CONCLUSION In this work, we propose a new graphical representation, called the slope plot, displaying S, the relative electrophoretic mobility decrease with ionic strength (in lin-log scale), as a function of μ5mM. Several typical trends were observed according to the nature of the solutes: increasing S value with the charge number for SI, decreasing S value with charge density for NP and PE, and increasing S values with solute size for both SI and NP. The change in the charge dependence of S between SI and NP appears at ∼1 nm. Combining S values with the electrophoretic mobility (slope plot) allows for defining different zones according to the solute nature. In short, small ions occupy the lower part of the slope plot, and nano- and microparticles the upper part. Polyelectrolytes occupy a relatively localized zone form the upper left to the middle part of the graph. This new graphical representation offers a convenient and easy way to distinguish solutes according to their chemical nature (small ions, nanoparticles, or polyelectrolytes). It also gives phenomenological S values, which represent the percentage of relative mobility decrease per ionic strength decade, and that can be directly used by experimentalists. Such approach should be highly valuable to (i) identify or confirm the nature of a given solute or (ii) help in optimizing separation of mixtures containing different kinds of solutes by modification of the BGE ionic strength. Given the importance of nanoparticles in recent industrial applications and their relative risks and regulations, this new approach may also contribute in defining 9428



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simple experimental protocols for the identification of nanoparticles in industrial products.

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ASSOCIATED CONTENT

* Supporting Information S

One table, listing samples and their μep data taken from literature, and one figure, showing P values as a function of μ5mM for different solutes. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS H.C. gratefully acknowledges the supports from the Région Languedoc-Roussillon for the fellowship “Chercheurs d’Avenir” and from the Institut Universitaire de France.

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ABBREVIATIONS AND SYMBOLS μep electrophoretic mobility μep∞ electrophoretic mobility at infinite dilution called also limiting mobility or absolute mobility) μep5mM electrophoretic mobility at 5 mM ionic strength I ionic strength F Faraday constant T temperature kB Boltzmann constant η viscosity ε0 electric permittivity of vacuum εr relative electric permittivity or relative dielectric constant of the electrolyte z charge number e elementary electric charge κ Debye−Hückel parameter or reciprocal of Debye length Rh hydrodynamic radius m+ and m− dimensionless ionic drag coefficients ζ zeta potential σ surface charge density QD Manning effective charge κ−1 Debye or screening length b characteristic dimension of a monomer P slope of μep = f(log I) plot S slope of μep/μep5mM= f(log I) plot a sphere radius Q net valence charge of the sphere ρ0(r) local equilibrium charge distribution at a distance r from the center of the sphere εi relative dielectric constants of the sphere interior Λ0 local equilibrium electrostatic potential μnr unrelaxed mobility f molar fraction of AMPS or chemical charge density SI small ions PE polyelectrolytes NP nanoparticles

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REFERENCES

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Extracting Information from the Ionic Strength Dependence of

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