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Electrophoretic interpretation of PEGylated NP structure with and without peripheral charge Reghan J. Hill, Fei Li, Tennyson L. Doane, and Clemens Burda Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b02809 • Publication Date (Web): 02 Sep 2015 Downloaded from http://pubs.acs.org on September 7, 2015

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Electrophoretic interpretation of PEGylated NP structure with and without peripheral charge Reghan J. Hill,∗,† Fei Li,† Tennyson L. Doane,‡ and Clemens Burda‡ Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, QC, H3A 0C5, Canada, and Department of Chemistry, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, United States E-mail: [email protected]

Abstract Anchoring poly(ethylene glycol) (PEG) to inorganic nanoparticles (NPs) permits control over NP properties for a variety of technological applications. However, the core-shell structure tremendously complicates the interpretation of the ubiquitous ζ -potential, as furnished by electrophoretic light scattering, capillary electrophoresis or gel electrophoresis. To advance the ζ -potential—and the more fundamental electrophoretic mobility—as a quantitative diagnostic for PEGylated NPs, we synthesized and characterized Au NPs bearing terminally anchored ∗ To

whom correspondence should be addressed University ‡ Case Western Reserve University † McGill

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5 kDa PEG ligands with univalent carboxymethyl (CM) end groups. Using the electrophoretic mobilities, acquired over a wide range of ionic strengths, we developed a theoretical model for the distributions of polymer segments, charge, electrostatic potential, and osmotic pressure that envelop the core: knowledge that will help to improve the performance of soft NPs in fundamental research and technological applications.

Introduction Electrophoresis is a ubiquitous research tool for characterizing colloidal particulates. When combined with knowledge of the particle size, as typically furnished by light scattering or electron microscopy, the electrophoretic mobility furnishes an electrokinetic surface charge, which plays a decisive role in controlling dispersion stability 1 . In bio-medical applications, for example, inorganic NPs are often grafted with poly(ethylene glycol) (PEG) to improve their biocompatibility and reduce cellular uptake, making them promising candidates for drug delivery 2 . Interpreting the electrophoretic mobility, often reported as a ζ -potential, requires knowledge of other surface properties, many of which are difficult to obtain 3 . This is especially true for decorated inorganic nanoparticles (NPs), the electronic and optical properties of which depend exquisitely on size and shape 4–7 . In this study, we synthesized a model NP dispersion and interpreted the mobilities furnished by a commercial light-scattering electrophoresis instrument over a wide range of ionic strengths. The novelty of this study is that it directly unites NP synthesis and an electrokinetic theory addressing the inhomogeneous structure of a soft-corona with peripheral charge, also accounting for the socalled polarization and relaxation effects. We show that charged terminal groups occupy positions at the corona periphery, with a dependence upon ionic strength that has important ramifications for conjugation reactions, cell targeting and electrostatic assembly. Our results elucidate fluidic behaviour within the polymer corona, which is directly relevant to drug delivery and catalytic applications. It is our hope that this study will motivate future efforts to report more complete NP characterization data, and to interpret these using soft-sphere models instead of the ubiquitous 2 ACS Paragon Plus Environment

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bare-sphere models that often furnish poorly defined and inappropriate ζ -potentials for functional NPs. A detailed understanding of the NPs with polymeric coronas is critical for understanding protein resistance, ligand efficacy, and small-molecule diffusion to and from NP surfaces 8 . However, the interpretation of their electrophoretic mobilities is challenging. Firstly, the small particle size a places the κ a parameter (particle radius divided by the Debye length κ −1 ) outside the well-known Smoluchowski regime (κ a ≫ 1). Moreover, particle size is difficult to define because of the soft, penetrable characteristics of polymeric coronas. Next, the high surface curvature tends to accentuate the corona inhomogeneity, also increasing the electrostatic potential so that the Debye-Hückel approximation may break down 9 . Finally, the distribution of charge is difficult to assign, because it may be adsorbed to the core and integrated into the architecture of the functional ligands. Thus, few studies have directly tackled the quantitative ability of electrophoretic mobility measurements to characterize functionalized NPs. These require the particle synthesis to be accompanied by a thorough particle characterization with a systematic variation of key experimental parameters. The requisite experimental data for a robust theoretical interpretation is often either unavailable or dispersed among a large body of research literature. Hanauer et al. 5 systematically synthesized several series of functionalized gold and silver NPs, measuring electrophoretic mobilities in agarose gels to furnish the grafting densities of terminally anchored, univalent PEG chains. However, they adopted the well-known Henry mobility formula to convert measured mobilities to ζ -potentials, which they converted to surface charge densities using the Guoy-Chapman (GC) equation. The effective particle radii were specified to be those of the metal core plus the hydrodynamic thickness of the PEG layer, which led to grafting densities ≈ 0.08 nm−2 and ≈ 0.23 nm−2 at the surfaces of silver and Au NPs, respectively. These are significantly lower than the ≈ 2.4 nm−2 grafting densities for uncharged PEG with the same molecular weight ascertained by others using TGA 10 . This brings into question the Henry-GC methodology, since it requires the core-shell NPs to behave as bare impenetrable spheres, which they are not. Measuring the hydrodynamic size and electrophoretic mobility of functionalized NPs is clearly


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challenging. Uncharged polymer coatings, such as PEG, can mask charge on the NP core, reducing electrophoretic mobilities to the level of instrumental noise. Similar challenges are faced when attempting to measure hydrodynamic size using light scattering. For example, size distributions can be very broad, multimodal, and fluctuating, so that the mean and standard deviation have limited value. In the present study, the peripheral charge on the PEG-ligands produced mobilities that were robustly measured using a commercial light-scattering electrophoresis instrument (Nano ZetaSizer ZS series, Malvern Instruments). However, the hydrodynamic sizes furnished by light scattering (using three commercial light-scattering methods) were inconsistent, even with extensive efforts to systematically disperse (by ultra-sonication) and purify (by centrifugation and filtration) the dispersions. These challenges highlight the benefits of being able to indirectly ascertain NP size from electrophoretic mobility data, as explored in this study. Interestingly, Pyell et al. 9 have recently demonstrated a compelling application of capillary electrophoresis and Taylor dispersion to ascertain NP mobility and size. Note that it is customary in the literature to convert electrophoretic mobilities to ζ -potentials, e.g., using the Henry formula, of which the Hückel and Smoluchowski formulas are special cases. As highlighted by Doane et al. 3 , a meaningful physical interpretation is possible by adopting a much more sophisticated electrokinetic model than is presently available in commercial electrophoresis instrument software. Unfortunately, NP ζ -potentials are often reported without reference to the solution ionic strength or the theory/formula by which the measured mobility was converted to a ζ -potential 11 . This obviously limits the degree to which the data can be theoretically interpreted and compared among closely related studies. While we convert measured and calculated mobilities to ζ -potentials using the Smoluchowski formula, this is solely to expedite a convenient conversion of the mobility units to millivolts. For the functional NPs in this study, the electrostatic potential is a non-monotonic function of radial distance, with a shear plane that is well displaced from the core. These unusual features, which deviate substantially from the models integrated into electrophoresis instrument software, may prevail for other classes of NP-based therapeutics, including quantum dots 11 . The present work


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provides a foundation for future attempts to rigorously interpret the electrophoretic mobilities of protein- and DNA-conjugated NPs.

Experimental section Following Doane et al. 10 , dodecylamine coated gold nanoparticles (DDA-Au NPs) were synthesized via the Brust-Schiffrin two-phase method 12,13 (see SI for details). Briefly, the surface ligands were then exchanged for CM-PEG-HS, verified, in part, by characteristic peaks for gold and PEG in the EDAX and FTIR spectra. The SAED (selected area electron diffraction) image indicated crystalline Au with a primary lattice spacing ≈ 0.2 nm. TEM micrographs revealed spherical NP cores with an average diameter ≈ 5.4 nm (standard deviation ≈ 2.3 nm), as furnished by digital image analysis using ImageJ 14 . From DLS, the NP hydrodynamic radius was ∼ 30 nm. Note, however, that this comes from a broad volume-weighted distribution of hydrodynamic diameters spanning the range ∼ 25–200 nm. From TGA, the mass loss ≈ 46.1% in the range T ∼ 50– 500◦ C furnishes ≈ 146 PEG chains per particle when the average core diameter is 5.4 nm. This is equivalent to a grafting density ≈ 1.6 nm−2 , which is higher than that of Au NPs synthesized using the method of Manson et al. 15 , and lower than the value ≈ 2.4 nm−2 for 2 kDa PEG 10 and ≈ 7 nm−2 for tetraethylene glycol (TEG) (< 0.2 kDa) 16 . Electrophoretic mobilities M are plotted in figure 1 as a ζ -potential using the well-known Smoluchowski formula M = ζ εs ε0 /η , where

ε0 εs is the solvent dielectric constant and η is the shear viscosity. Note that we have identified a range for the ionic strength I of reverse osmosis (RO) water: the lower limit is from the measured pH ≈ 6.1 and the upper limit is set by the ionic strength ascertained from the measured conductivity ≈ 6.6 µ S cm−1 .1 Electrophoretic mobilities were interpreted by advancing the standard electrokinetic model for soft NPs 19,20 to account for peripheral charge on the corona. The solution is by an efficient numerical solution of the coupled Poisson equation, Nernst-Plank ion conservation equations, and solvent 1 According

to Ponnamperuma et al. 17 and Griffin and Jurinak 18 , the ionic strength (M) is approximately 0.013 times the conductivity (mS cm−1 ) at 25◦ C.


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mass and momentum conservation equations. There are no practical limits on the Debye length or electrostatic potential, and the so-called polarization and relaxation terms—necessary for highly charged particles, and oftentimes neglected in more approximate electrokinetic theories—are accounted for. The model requires a prescribed surface charge density σ on the spherical particle core, and a prescribed radial distribution of polymer segments and charge for the corona. The bathing electrolyte has a prescribed bulk ionic strength I, which (for the calculations undertaken here) was an aqueous 1-1 electrolyte (NaCl) with the same value of I as the electrolytes used in the experiments. In addition to the electrophoretic mobility, the model furnishes the hydrodynamic radius Rh , which is related to the Brownian diffusion coefficient Ds by the Stokes-Einstein relationship, Ds = kB T /(6πη Rh ). Note that Rh is necessarily smaller than the physical extent of the polymer layer from the particle centre because the layer has a finite hydrodynamic permeability, which depends on the size and density of the peripheral polymer segments.

Results and discussion For PEGylated NPs, we prescribe the radial distribution (radial distance r from the centre of the core with radius a) of polymer segments and fixed charge densities, respectively, with Gaussian functions: 2 /L2

ns = ns,0 e−(r−a)

2 /δ 2 e

and nc = n f ,0 e−(r−a−Le )

.

Here, Na =

Z ∞ a

ns 4π r2 dr = 4π a3 ns,0 f (L/a, 0)

and Nc =

Z ∞ a

nc 4π r2 dr = 4π a3 n f ,0 f (δe /a, Le /a),

with √ 2 f (x, y) = x2 e−(y/x) (y + 2)/2 + x(1/2) π (y2 + 2y + x2 /2 + 1)[1 + erf(y/x)],


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are, respectively, the numbers of segments (chains, statistical segments or monomers) and terminal charges, which are directly related to the grafting density, and the PEG-chain molecular weight and valence. Moreover, geometrical parameters Le and δe are prescribed to mimic the distribution of chain ends from self-consistent mean-field computations, which also connect Le to L. To account for the influence of ionic strength I on the PEG coating thickness, we prescribe the interpolating formula L/L0 = (L∞ /L0 − 1)/(I/I0 + 1) + 1,

(1)

where L∞ is the maximum coating thickness at vanishing ionic strength, L0 is the minimum coating thickness at high ionic strength, and I0 is a transition ionic strength. These are all taken to be fitting parameters where (i) L0 must be comparable to the thickness of an uncharged layer, (ii) L∞ must be less than the polymer chain contour length, and (iii) I0 is an ionic strength where κ −1 ∼ L0 . With L0 = 10 nm, for example, the Debye length κ −1 = L0 when I ≈ 1 mM. According to the scaling/blob theory of Biver et al. 21 , the layer thickness L′ (for uncharged polymer) principally depends on the grafting density γ = Na /(4π a2 ) and substrate curvature, as measured by the core radius a: L′ = a{[1+(5/3)Lmax /a)]3/5 −1}, where Lmax = (3/5)kLc (γ v/l)1/3 with k an order-one dimensionless constant, Lc = Nl the chain contour length, v . l 3 the excluded volume of a statistical segment, and l the length of a statistical segment. Specific values for PEG 1 are available as SI (table 1) with k = 1. For polyelectrolyte chains, l and v will increase with the linear charge density and decrease with the ionic strength. For chains bearing a single terminal charge, such as the CM-PEG ligands used in this work, l and v should be the same as for PEG ligands, but the distribution of chain ends may deviate from the blob model. The electrostatic energy among the chains is minimized by the ends uniformly occupying a volume enclosed by concentric spheres having radii a and a + Lc . However, this state must be attained at the cost of decreasing the chain conformational entropy. Moreover, when the chain ends interact electrostatically with equally signed charge on the substrate, there is an additional penalty for the ends to occupy positions close to the corona. Thus, we expect that, at low ionic strength, the layer will expand by an amount that depends, in part, on the core charge. 7 ACS Paragon Plus Environment

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We performed self-consistent mean-field computations for terminally anchored PEG ligands. These furnish the total segment density and the density of end-segments 22,23 , available as SI (figure 7). The calculations were undertaken using self-consistent potential U = kB T vn with v the excluded volume of a segment and n the segment number density. Computations were unstable with the excluded volume greater than v ≈ 0.25l 3 . Therefore, to assess the slightly larger value v = 0.35l 3 , we plotted (see SI, figure 8) a measure of the layer thickness 22

h=

sR

∞ − a)2 4π r2 dr a n(r)(r R∞ 2 a n(r)4π r dr

versus v/l 3 for 5 kDa chains (N = 83, l = 0.6 nm) grafted to a NP core with radius a = 2.7 nm and to a flat plate. Extrapolation to v = 0.35l 3 furnished h ≈ 4.5 nm for the NP core, which is about twice the value for a layer of ideal chains; on a flat substrate, h ≈ 5.2 nm. The minimal set of independent parameters required to quantitatively interpret our electrophoresis data for CM-5kPEG-Au NPs is summarized in table 1. Note that the hydrodynamic segment size as influences the mobility and hydrodynamic radius. Its numerical value depends on an arbitrary definition of a segment, which, for convenience, we have taken to be an entire polymer chain with each chain bearing one fixed charge. The hydrodynamic radius of a statistical segment (length l), as /N ≈ 0.020 nm, is close to the value 0.0175 nm ascertained for terminally anchored PEG on liposomes 23 , albeit at lower grafting densities and on substrates with much weaker curvature. An important characteristic of the corona is its nominal permeability (square of the Brinkman screening length ℓ) ℓ2 =

1 2a3 [(1 + L/a)3 − 1] = , ns 6π as Fs (φs ) Na 9as Fs (φs )

where the segment drag coefficient Fs (φ ) ≈ 1 when the segment volume fraction φs = ns 4π a3s /3 ≪ 1. Note that ℓ depends on Na and as only through their product Na as when Fs = 1. With the parameters in table 1, we find φs ≈ 0.24 and ℓ ≈ 0.87 nm when L = 5 nm (with Fs = 1) and

φs ≈ 0.0030 and ℓ ≈ 7.6 nm when L = 30 nm. These demonstrate the significant extent to which the corona thickness affects the segment density and permeability. 8 ACS Paragon Plus Environment

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Table 1: Principal model parameters to interpret measured mobilities of CM-5kPEG-Au NPs with an aqueous NaCl electrolyte. parameter core radius corona segment distribution parameter corona charge distribution parameter corona charge distribution parameter low-ionic strength value of L high-ionic strength value of L transition ionic strength parameter 5 kDa PEG-ligand friction radius statistical segment friction radius monomer friction radius number of PEG ligands number of corona charges core surface charge density

symbol a L Le δe L∞ L0 I0 as as /N as /Nm Na Nc σ

value 2.7 nm Eqn. (1) 3L/2 L/3 18 nm 4 nm 1 mM 1.7 nm 0.020 nm 0.015 nm 146 Na −15 mC m−2

Another important characteristic, which depends on the charge, is the electrophoretic mobility of a segment. This is obtained by equating the electrical force on a segment Ezs e to the hydrodynamic drag force on a segment 6πη as Fs (φs )Ueo , giving Ueo zs e/η = , E 6π as Fs (φ ) where η ≈ 0.001 Pa s is the solvent viscosity and zs e is the product of the segment valence zs and the fundamental charge e. With zs = −1 and as = 0.9 nm, we find Ueo /E ≈ −1.1 × 10−8 m2 s−1 V−1 , which, when expressed as a Smoluchowski ζ -potential, furnishes ≈ −14 mV. By comparison to the NP mobilities (reported as ζ -potentials) in figure 1, Ueo /E correctly establishes their order of magnitude. Figure 1 (top panel) compares measured and calculated CM-5kPEG-Au NP electrophoretic mobilities reported as ζ -potentials) versus electrolyte ionic strength. Changing the electrolyte composition, using TAE and PBS buffers with NaCl, did not significantly affect the mobilities, indicating that the CM groups are highly dissociated at the prevailing pH values (in the range


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6.1 for RO water to 8.2 for 1×TAE buffer). This is consistent with the Henderson-Hasselbalch equation, which predicts > 98% COOH dissociation (pKa = 4.5) when pH > 6.1 24 . Plotting the mobility with a logarithmically scaled ionic strength axis reveals a characteristic ‘knee’ in the curve at 5 mM. The dashed line is the mobility of the bare core (radius a = 2.7 nm) bearing a fixed surface charge density σ = −15 mC m−2 according to the standard electrokinetic model. Coincidently, this has almost the same magnitude as for the CM-5kPEG functionalized Au NPs at high ionic strength. Note that grafting an uncharged PEG corona with a structure that is independent of the ionic-strength substantially reduces the mobility to the values indicated by the red/dash-dotted line. This theoretical extrapolation of the model for CM-5kPEG functionalized Au NPs passes through the locus of data for 5kDa PEG grafted to Au cores (average 5.4 ± 2.1 mV registered at 0.1 and 10 mM) 10 , and is consistent with the electrophoretic mobilities reported for these particles in agarose gels at higher ionic strengths (squares). Note that no corrections for possible frictional coupling to the gel or differences in the PEG grafting density or core charge have been undertaken. The solid/blue line passing through the data is the fit of the electrokinetic model for CM5kPEG-Au NPs. To reproduce the ‘knee’ at 5 mM, it was necessary for the corona thickness to increase monotonically with decreasing ionic strength, according to Eqn. (1). Electrokinetic models for charged polyelectrolytes have been developed in the literature to address unusual electrokinetic behaviour arising from charge-regulated polyelectrolytes when varying the pH and ionic strength 25,26 . Recall, we have assumed that the bulk pH of the dispersions is sufficiently high (relative to the pKa ≈ 4.5 for CHOOH) at all ionic strengths that practically all the CM groups bear a negative charge 16,24 . Note, however, that the local pH (within the corona) may be somewhat lower than in the bulk, because positive electrolyte ions are electrostatically attracted to the negatively charged core and polymer corona (i.e., concentrated in the two diffuse layers, as illustrated in the bottom panel in figure 2). This will decrease the corona charge in regions where the local pH is low (see Kusters et al. for a recent application of this principle to NP encapsulation 27 ). Such processes demand non-trivial modifications to the presently available computational model, so their


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0

ζ (mV)

−10 −20 −30 −40 −50 10

−2

0

10 I (mM)

10

2

30 25 R h (nm)

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20 15 10 5 0 10

−2

0

10 I (mM)

10

2

Figure 1: (top) Electrophoretic mobility reported as a Smoluchowski ζ -potential (mV) and (bottom) hydrodynamic radius Rh (nm) versus ionic strength I (mM) for CM-5kPEG-Au NPs. Symbols are experimental data [PBS buffer (blue circles), TAE buffer (red circles), RO water (lower rectangle)] with solid lines the theoretical calculations for charged (passing through the experimental data) and uncharged (curve with the smallest mobility magnitude) coronas (both with the same segment density distributions, and thus the same Rh ). The dashed line is for the bare core (with a fixed surface charge density), the dash-dotted line is for an uncharged corona with a fixed segment density distribution (constant Rh , all other parameters the same), and the upper rectangle identifies the mobility-ionic strength locus for 5kPEG-Au NPs 10 . Squares are the mobilities of 5kPEG-Au NPs in agarose gels 10 (corrected for the gel electroosmotic flow and converted to Smoluchowski ζ -potentials at 25◦ C).


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influences (particularly in lower pH electrolytes), will be addressed elsewhere. The accompanying hydrodynamic radius (figure 1, bottom panel) decreases from ≈ 27 nm at vanishing ionic strength to ≈ 10 nm at high ionic strength. Note that the pair of solid/blue lines in both panels are for coronas with exactly the same distribution of Stokes resistance centers, but with and without the CM peripheral charge. Thus, the slightly larger hydrodynamic radius for the charged corona at low ionic strength can be attributed to the so-called electroviscous effects. Because the Au core has a radius 2.7 nm (dashed line), the 5kDa PEG corona contributes between ≈ 7.3 and 24 nm to the hydrodynamic radius, significantly less than the 5kDa PEG chain contour length (≈ 50 nm). The 7.3 nm layer thickness at high ionic strength, where the electrostatic forces within the layer a completely screened by the electrolyte salt is slightly smaller than the scaling theory prediction L ≈ 9.3 nm, perhaps due to the higher hydrodynamic permeability of the peripheral region. The ≈ 24 nm layer thickness at low ionic strength is comparable to the scaling-theory prediction of the thickness that would be achieved for an uncharged brush on a flat substrate, Lmax ≈ 18 nm. Interestingly, the ionic strength I0 ≈ 1 mM that denotes the transition from the compressed/unperturbed to expanded states occurs when the Debye length κ −1 ≈ 9.6 nm is slightly thicker than the unperturbed hydrodynamic thickness. This suggests that electrostatic repulsion between the core surface charge and the CM charges might be responsible for extending the reach of the CM-5kPEG chains. Note that the core surface charge density σ = −15 mC m−2 corresponds to only ≈ −π a2 σ /e ≈ 2 charges per core, much smaller than the number of grafted chains. However, if the CM peripheral charges were all placed on a spherical surface with radius ≈ 10 nm, then σ ≈ −19 mC m−2 is only ≈ 24% higher than on the Au surface. Note that extrapolating the model to ionic strengths corresponding to RO water (0.01–0.1 mM) predicts mobilities that fall within the measured range of ζ -potentials (∼ 40 mV). Finally, removing the CM charges, but maintaining the same ionic-strength dependence of the layer thickness, produces the upper solid blue line in figure 1. These mobilities are ostensibly lower than with a fixed layer thickness, illustrating the significant role that hydrodynamic screening by PEG has on the electrophoretic mobility.


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Radial profiles at a physiological ionic strength I = 100 mM are shown in figure 2. Here, the electrostatic potential ψ (top) decreases from a maximum value ψ ≈ −15 mV at the core surface to a local maximum at the corona periphery ≈ −6 mV. There are commensurate changes in the concentrations of the co- and counter-ions (bottom). The excess concentration of counter-ions in the periphery are important, because these may ultimately contribute to a repulsive electrostatic interaction between NPs and substrates to which the particles might attach in practical applications. Because the peripheral charge is fixed, the electrostatic potential increases with decreasing ionic strength. Below, we will see that much higher electrostatic potentials prevail at the lower ionic strengths at which gel electrophoresis has been undertaken. Streamlines of the accompanying flows at I = 100 mM are available as SI (figure 9), also comparing the flows for charged and uncharged PEG layers. Without charge, electroosmotic flow is driven solely by the counter charge of the Au core. This must overcome significant hydrodynamic resistance in the core, where the polymer segment density is very high, thereby producing slow flow with an intricate internal circulation. With a charged corona, electroosmotic flow is pumped through the peripheral regions of the corona. This produces a much stronger flow, and thus an higher electrophoretic mobility, with flow within the corona translating with the core as a rigid body. These observations suggest that this class of NPs may be effective at trapping and transporting small molecules. Next, we examine how the mobility varies with the fraction χ of charged to uncharged PEG ligands. Such an approach was demonstrated experimentally by Hanauer et al. 5 to precisely control the mobility of gold and silver NPs in agarose gels. Here, we vary the fixed charge density by changing χ with all other parameters fixed at an ionic strength I = 20 mM. This ionic strength is the value prevailing in gel electrophoresis 5 . The principal question we seek to address here is the extent to which the mobility departs from the Henry-GC prediction, which, recall, is for bare, impenetrable spheres bearing a low charge. With NPs having a larger core radius a ≈ 10 nm, the gel mobilities of Hanauer et al. 5 increase approximately logarithmically with χ in the low and intermediate regimes. Note, however, that Hanauer et al. interpreted the results as being linear in χ at intermediate values of 0.05 . χ .


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(a)

(b)

0

1

−0.1 0.8 n 0j /(2I )

ψe/(k B T )

−0.2 −0.3 −0.4 −0.5

0.6

0.4

−0.6 −0.7 0

5

10

0.2 0

15

5

κ(r − a)

10

15

10

15

κ(r − a)

(c)

(d) 0.5

200

0.4 n f /(2I )

150 n s (mM)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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100

0.3 0.2

50

0.1 0 0

5

10

15

0 0

5 κ(r − a)

κ(r − a)

Figure 2: Radial profiles of (a) scaled electrostatic potential ψ e/(kB T ), and (b) scaled equilibrium mobile ion concentrations n0j (co- and counter-ions), (c) segment density ns , and (d) scaled immobile charge density n f /(2I) for CM-5kPEG-Au NPs: I = 100 mM (charged corona).


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0.75, speculating that the deviation as χ → 1 might be due to steric interaction with the gel, since electrostatic repulsion between chain ends could increase the particle size. Based on our analysis of the light scattering electrophoresis data in figure 1, this seems unlikely when I = 20 mM. Note that Hanauer et al. did not address the notably rapid variation of the mobility with respect to χ as

χ → 0; this is incongruent with the Henry-GC interpretation, which furnishes a linear increase in the mobility with χ . Daniel et al. 16 reported gel electrophoretic mobilities of Au NPs with 5.3 and 12.3 nm diameter cores when systematically varying the ratio of TEG-CM to TEG-OH ligands. These data furnish an approximately linear increase in mobility with respect to the percent of TEG-CM ligands for the smaller NPs, with a weaker increase for the larger NPs. While this may be because of steric interaction of the larger particles with the gel, it might also be explained by polarization of the diffuse double layer, since the weaker curvature of the larger cores could impart a higher peripheral corona charge density. Note that this hinges on the grafting density being weakly dependent on the surface curvature/core size, which is suggested by TGA data for these particles. We assume that the brush structure is independent of the charge density, thereby specifying the structural parameters according to uncharged PEG brushes. Again, based on our data for CM-5kPEG-Au NPs, this seems reasonable at the prevailing ionic strength (I = 20 mM). Next, to extrapolate our model (fitted to CM-5kPEG-Au NPs over a wide range of ionic strengths) to the PEGylated NPs of Hanauer et al. 5 , we accounted for the larger core radius by adjusting L, δe and Le according to the scaling theory of Biver et al. 21 . Thus, for example, the fitted value of L at I = 20 mM furnished by Eqn. (1) with a = 2.7 nm was multiplied by an appropriate factor to account for the larger core radius a = 10 nm and the possibility of a different grafting density. A similar correction was applied to the width of the charged region δe to account for the smaller peripheral blob size that accompanies a more compact layer on a less curved substrate. The calculations presented in figure 3 have the same grafting density that we measured using TGA for much smaller Au NPs, whereas the grafting density on the larger NPs of Hanauer et al. 5 may be lower because of the increased steric barrier for ligand exchange. We therefore performed


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calculations (not shown) with higher and lower grafting densities, again with commensurate adjustments to L, δe and Le , as described above. Interestingly, these furnished poorer comparisons to the data, suggesting that the grafting densities for the 10 nm cores are within ≈ 25% of the value we ascertained for the smaller CM-5kPEG-Au NPs. The model prediction is shown in figure 3 as the red/dashed curve. For convenient reference, also shown are mobilities according to the HenryGC methodology (blue/solid lines), as described by Hanauer et al. 5 . These predict perfectly linear increases in the mobility according to the particle size and charge density, as detailed in the figure caption. Similarly to the data, the soft-sphere model predicts a decrease in the rate at which the mobility increases with increasing charge. Rather than the steric influence suggested by Hanauer et al. 5 , this effect can be attributed to electroosmotic flow through and around the soft corona. Whereas our mobilities (see figure 1) were obtained using a conventional light-scattering instrument (Nano ZetaSizer ZS series, Malvern Instruments), the mobilities of Hanauer et al. were measured using gel electrophoresis. Here, the large pore size 5 circumvented Ferguson analysis, whereby the mobilities from experiments with varying gel concentrations are extrapolated to zero gel concentration 28–30 . Doane et al. 10 reported electrophoretic mobilities of 5kPEG-Au NPs in agarose gels from −0.043 to −0.124 µ m cm s−1 V −1 at buffer ionic strengths in the range 10.5–42 mM; these are consistent with the data of Hanauer et al. 5 shown in figure 3 with χ = 0 . When Doane et al. 10 systematically varied the agarose gel concentrations (in the range 0.5– 2.0%), the impact on mobility was weak compared to the much larger variations that come from varying the PEG-ligand molecular weight. Thus, the large pore sizes that accompany the low gel concentrations in these studies seem to justify neither study performing a Ferguson extrapolation; nevertheless, Ferguson analysis may still prove helpful for explaining deviations between the model and the gel-mobilities in figure 3 when 0 < χ . 0.3. Note that part of the discrepancy may be accentuated by our shifting of all the mobilities by a constant value 0.315 µ m cm s−1 V −1 to account for electroosmotic flow in the gel, whereas a slightly smaller value may be more appropriate 5 . Other sources of uncertainty might be attributed to the inherent challenge of accurately


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measuring low mobilities (with uncharged PEG-ligands), Joule heating, and the possibility that the particles with low peripheral charge may interact with the gel differently than when highly charged; although we might generally anticipate that a stronger non-hydrodynamic interaction would hinder the mobility rather than enhance it 10 . Comparing radial profiles for the NPs with a = 10 nm (and χ = 1) to those in figure 2 with a = 2.7 nm at I = 100 mM reveals an approximately 50% increase in the polymer segment density, a narrower spatial distribution of peripheral charge, and a substantial peripheral electrostatic potential, ψ ≈ −60 mV. Flow streamlines during electrophoresis (figure 3, right) reveal a strong peripheral electroosmotic flow (c.f., SI figure 9), with fluid within the corona translating with the same electrophoretic velocity as the core and corona. Again, this reflects the high peripheral charge and high concentration of polymer close to the core.

Conclusions A theoretical interpretation of the electrophoresis of PEGylated Au NPs furnishes the internal structure, namely the radial distributions of charge, polymer segments, and ions in the corona. These are important for designing and assessing the performance of NPs for drug delivery, imaging, and optics applications. Measuring electrophoretic mobilities over a wide range of electrolyte ionic strengths is a potentially simple and powerful means of probing the NP structure. Uniting the experimentally measured ζ -potential with a rigorous electrokinetic model, guided by a selfconsistent-field description of the polymer layer, avoided having to rely on dynamic light scattering to determine the hydrodynamic size. This may be beneficial when the absolute hydrodynamic sizes of core-shell NPs are prohibitively difficult to measure. Our model suggests that charged end-groups on the corona may interact with charge on the core, possibly increasing the size at low ionic strengths (e.g., less than 1 mM) where the Debye length is comparable to or larger than the unperturbed corona thickness. Theory is required to (i) assess the coupling of electrostatics to the structure of terminally anchored PEG bearing charged ends, and (ii) account for the influences of


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2

−M (µm cm s − 1 V

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1.5

1

0.5

0 0

0.2

0.4

χ

0.6

0.8

1

Figure 3: (top) Electrophoretic mobility (µ m cm s−1 V −1 ) of PEGylated gold NPs versus the fraction of univalent 5 kDa PEG chains at ionic strength I = 20 mM with a fixed coating structure (furnishing an hydrodynamic radius Rh ≈ 21 nm). Symbols are the mobilities of SH-PEGCOOH/SH-PEG-OCH3 functionalized NPs in agarose gel as reported by Hanauer et al. 5 (taken from their figure S3). A mobility 0.315 µ m cm s−1 V −1 is added to all the data to account for electroosmotic flow in the agarose 5 . Theoretical prediction for soft spheres (red/dashed line) has a core radius a = 10 nm and a PEG grafting density 1.6 nm−2 (aggregation number 2010). Blue/solid lines are the Henry mobility with (bare) sphere radius a = 10 + 12.5 = 22.5 nm and PEG grafting density 0.23 nm−2 (aggregation number 289, lower), as specified by Hanauer et al. 5 , and 1.6 nm−2 (aggregation number 2010, upper), as adopted for the soft-sphere model. (bottom) Streamlines for CM-5kPEG-Au NPs undergoing force-free electrophoresis (I = 20 mM).


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pH charge regulation. The present electrokinetic interpretation suggests that PEG grafting densities are considerably higher than inferred by the Henry-GC methodology.

Acknowledgement Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Centre for Self Assembled Chemical Structures (CSACS). Thanks to Prof. N. Tufenkji (McGill University) for use of the Nano ZetaSizer ZS instrument.

Supporting Information Available Materials, experimental methods, and characterization; supplementary tables and figures.

This

material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press, 1989. (2) Cheng, T.-L.; Chuang, K.-H.; Chen, B.-M.; Roffler, S. R. Analytical measurement of PEGylated molecules. Bioconjugate Chem. 2012, 23, 881–899. (3) Doane, T. L.; Chuang, C. H.; Burda, C. Nanoparticle ζ -potentials. Acc. Chem. Res. Chem. Res. 2012, 41, 2885–2911. (4) Lofton, C.; Sigmund, W. Mechanisms controlling crystal habits of gold and silver colloids. Adv. Funct. Mater. 2005, 15, 1197–1208. (5) Hanauer, M.; Pierrat, S.; Zins, I.; Lotz, A.; Sönnichsen, C. Separation of nanoparticles by gel electrophoresis according to size and shape. Nano Lett. 2007, 7, 2881–2885.


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scaled electrophoretic mobility

Graphical TOC Entry scaled electrostatic potential

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scaled radial distance from f core

ionic strength


Electrophoretic Interpretation of PEGylated NP ... - ACS Publications

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