Journal of Colloid and Interface Science 418 (2014) 103–112

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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis

Size histograms of gold nanoparticles measured by gravitational sedimentation Colleen M. Alexander, Jerry Goodisman ⇑ Department of Chemistry, Syracuse University, 111 College Place, CST, Rm 1-014, Syracuse, NY 13244-4100, United States

a r t i c l e

i n f o

Article history: Received 15 August 2013 Accepted 27 November 2013 Available online 5 December 2013 Keywords: Sedimentation Gold nanoparticles Histogram

a b s t r a c t Sedimentation curves of gold nanoparticles in water were obtained by measuring the optical density of a suspension over time. The results are not subject to sampling errors, and refer to the particles in situ. Curves obtained simultaneously at several wave lengths were analyzed together to derive the size histogram of the sedimenting particles. The bins in the histogram were 5 nm wide and centered at diameters 60, 65, . . ., 110 nm. To get the histogram, we weighted previously calculated solutions to the Mason–Weaver sedimentation–diffusion equation for various particle diameters with absorption/scattering coefficients and size (diameter) abundances {cj}, and found the {cj} which gave the best fit to all the theoretical sedimentation curves. The effects of changing the number of bins and the wave lengths used were studied. Going to smaller bins would mean determining more parameters and require more wave lengths. The histograms derived from sedimentation agreed quite well in general with the histogram derived from TEM. Differences found for the smallest particle diameters are partly due to statistical fluctuations (TEM found only 1–2 particles out of 103 with these diameters). More important is that the TEM histogram indicates 12% of the particles have diameters of 75 ± 2.5 nm, and the sedimentation histogram shows none. We show that this reflects the difference between the particles in situ, which possess a lowdensity shell about 1 nm thick, and the bare particles on the TEM stage. Correcting for this makes agreement between the two histograms excellent. Comparing sedimentation-derived with TEM-derived histograms thus shows differences between the particles in situ and on the TEM stage. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction There are a number of applications for gold nanoparticles (AuNPs) in which size and size distribution are significant. A broad list includes catalysis [1], assembly [2,3], ‘‘smart’’ particle systems [4], sensors [5], and biomedical applications [6]. It is therefore useful to have one or even multiple means of accurately measuring the size distributions of AuNP samples. In this work, we provide an approach for determining AuNP size distribution, complementary to more conventional techniques such as transmission electron microscopy (TEM) and dynamic light scattering (DLS). By measuring the gravitational sedimentation of polydisperse AuNP in suspension, we are able to generate size distribution histograms reflecting the entire particle population. One application in which AuNP size has been shown to be critical is catalysis. In general, smaller particles exhibit superior catalytic activity due to their greater surface area and enhanced surface effects [7–10]. However, the relationship between AuNP size and catalytic activity is often complex. The catalytic rate of eosin reduction, for example, was found to be strongly dependent ⇑ Corresponding author. Fax: +1 315 443 4070. E-mail address: [email protected] (J. Goodisman). 0021-9797/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcis.2013.11.074

on the diameter of AuNP catalysts ranging from 10 – 46 nm. The kinetics were a complex function of size effects involving surface area, mass per area, and particle concentration [11]. Clearly, the design of AuNP-based catalysts benefits from a complete understanding of AuNP size and size distribution. In many applications, the monodispersity of particles is an important goal for quality control purposes. This is particularly so when linker-functionalized AuNP are assembled into structured arrays. In 2D or 3D arrays, the AuNP sizes can affect the interparticle distance, array uniformity, optical, and electronic properties [3,12]. For example, Kim showed that the interparticle spacing parameter in a 2D array was a direct function of AuNP size, and that spacing and size influenced the optical properties (e.g., kspr) of the array [3]. Gold nanoparticle size is critical for biomedicine in a number of ways. Size-selective accumulation of nanoscale particles in the leaky vasculature of tumor tissue (i.e., passive targeting) occurs via the enhanced permeability and retention (EPR) effect [6]. The AuNP size also strongly determines cellular uptake [13,14], nanoparticle toxicity [15,16], and ligand loading capacity [17]. Optimizing AuNP design for diagnostic or therapeutic purposes requires consideration of these size-dependent properties. For example, we recently developed DNA-capped AuNP for the delivery of the anticancer

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drugs doxorubicin and actinomycin D [18,19]. With an AuNP core size of 11.4 ± 0.6 nm and hydrodynamic diameter, Dh, of 32.9 ± 1.5 nm, the vehicles are within the size range requirements for passive targeting (Dh = 6–200 nm6) and cellular uptake (optimized at Dh = 50 nm13), and their monodispersity permits consistent loading of the drug cargo. The current gold standard for determining the size and shape of AuNP is transmission electron microscopy (TEM). Here, particle size is measured directly from an image of a sample of particles. While the directness of the measure is an advantage, there may be several problems. First, because a relatively small number, typically one or two hundred, particles are measured, there may be only one or two particles in some size ranges. The statistical errors may be large for these. Second, experimental artifacts may affect the sampling process which brings the particles to the TEM stage. Third, the particles are not being measured in situ, and there may be differences between the particles in suspension and the particles on the TEM stage. In previous work, we showed that useful information about the sizes of gold nanoparticles in suspension could be obtained using ordinary gravitational sedimentation [20]. The measured quantity was the ‘‘sedimentation curve,’’ which is the optical density of the sedimenting system, at a fixed height, as a function of time. The contribution of particles of a given size to the optical density is proportional to the absorption/scattering coefficient and the number density of those particles. These measurements see all gold nanoparticles in a sample, and do not require removal of the particles from the system. They are also free from possible effects of shear forces which may result from the high gravitational fields used in ultracentrifugation. These forces could affect the structure and properties of the particles, and small changes would not necessarily be seen in TEM. More important, the density profile from ultracentrifugation is less detailed than from sedimentation under ordinary gravitational fields. Although the measurement time is long, the experiment is simple and unambiguous. In previous work, we calculated [20] sedimentation curves for particles of various diameters by solving the Mason–Weaver equation [21]. Then we showed that, if these curves were weighted with the known size-dependent absorption/scattering coefficients and the relative numbers of particles of various diameters (from TEM measurements), we could reproduce the experimental sedimentation curve for the collection of particles. The agreement was good for three different samples, with particle diameters 65.0 ± 5.2, 82.5 ± 5.2, and 91.8 ± 6.2 nm. In the last case, discrepancies between calculated and measured sedimentation curves pointed to inhomogeneities in the system. The overall good agreement for all three samples showed that the particles sediment individually, without agglomeration We also pointed out that the process could be inverted, that is, the measured sedimentation curve could be analyzed, given calculated sedimentation curves for various sized particles, to give the histogram. As an example, three particle abundances were calculated in reasonable agreement with the TEM results. Since the measured sedimentation curve is essentially featureless, the number of such parameters one can extract is limited. However, it was suggested that much more information could be obtained if the sedimentation curve was measured at several wave lengths. The absorption/scattering coefficient for a particle of a given size depends on the wave length in a known way, so particles of different sizes are weighted differently in the sedimentation curves for different wave lengths. In the present work, we develop this approach, to show that sedimentation curves of AuNP monitored at multiple wavelengths can provide more information about a polydisperse AuNP suspension and generate extremely accurate size histograms. We start from a set of sedimentation curves measured at different wave

lengths, and analyze them in terms of previously calculated sedimentation curves for particles of various sizes to determine up to ten parameters in the size histogram (numbers of particles of various sizes). After summarizing the theory relating sedimentation rate to particle diameter, we show how one generates the theoretical sedimentation curves for particles of given sizes. These curves are derived from solutions to the Mason–Weaver sedimentation–diffusion equations, starting from a uniform density at time zero. We also show how absorption/scattering coefficients are calculated. Then, if Tj(t) is the calculated sedimentation curve for particles of size j and akj is the absorption/scattering coefficient for a particle of size j at wave length kk, T kj ðtÞ  akj T j ðtÞ is the contribution of one such particle to the optical density. The total optical density as a function of time, which is the measured sedimentation curve, is Rj cj T kj , where cj is the concentration of particles of size j. Given a number of sedimentation curves for different kk, the cj can be calculated by determining the cj to give the best fit between the theoretical and the measured sedimentation curves. We show how many histogram numbers can be obtained, and how many different wave lengths should be considered. Finally, calculated histograms are compared with the TEM-derived histogram. Differences between the two which do not disappear when more wave lengths are used point to actual physical effects which make the particles in suspension differ from the bare particles on a TEM grid. These are considered in the Discussion. 2. Theory 2.1. Mason–Weaver equation We consider a cylinder of height h containing a uniform suspension of nanoparticles of a given size at t = 0. If the concentration of the nanoparticles at height x is represented by c(x,t), at t = 0 we have c(x,0) = co, 0 6 x 6 h and c(x,0) = 0 elsewhere. When sedimentation occurs, the concentration of nanoparticles in the medium becomes non-uniform, larger for smaller x. At any time, the concentration obeys the equation of continuity

@c @J ¼ @t @x

ð1Þ

where J is the total current density in the +x-direction (upward, against the gravitational force). The current density must vanish at x = 0 and x = h at all times t > 0. It has been previously shown [21,22] that there are two contributions to J: the sedimentation current density, Jsed, and the diffusion current density, Jdif, such that J = Jsed + Jdif. Then

  q gc @c J ¼ Dm 1  1 D @x q2 kT Here m is the mass of a nanoparticle, g is the acceleration of gravity,

q1 is the density of the suspending fluid (water), and q2 is the density of a nanoparticle and D is the diffusion coefficient. Then the equation of continuity, (1), becomes

      1 @c @ q gc @c @ @c ¼ m 1 1 ac þ ¼ þ D @t @x @x @x q2 kT @x

ð2Þ

where

  q g a¼m 1 1 q2 kT

ð3Þ

We write the diffusion coefficient D as kT/f with the friction factor f = 3pgd where g = 9.0  104 Pa s (viscosity of water at 25 °C) and d is the particle diameter. At 25 °C, D = (4.53  1019 m3/s)/d, and g/kT = 2.382  1021 kg1 m1.

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For example, consider spherical gold nanoparticles of diameter d. The particle mass m is (10.11 kg/m3)d3 and, at 25 °C, the quantity a is (2.284  1024 m4)d3. Thus if d = 40 nm, a = 1.461 m1, f = 3.39  1010 J s m2, and D = 1.213  1011 m2 s1. If the sedimenting particle is the gold core surrounded by a 1 nm-thick layer of density 1.0 containing capping ligands, solvent, ions, etc., D is multiplied by d/(d + 2 nm), m is multiplied by

"  3 # 18:31 1 dþ2 þ 19:31 19:31 d

105

calculations need be done only once. For convenience, each average curve is fitted to a two-parameter logistic curve

T j ðtÞ ¼

1 1 þ ðt=a1 Þa2

ð4Þ

where t is in hours and j designates the particle size. After determining the best values of the parameters a1 and a2 for each diameter dj, we found we could write

a1 ðdj Þ ¼ 2:83  105 ðdj in nmÞ

1:78

ð5Þ

and the particle density q2 is multiplied by

a2 ðdj Þ ¼ 4:79 þ 0:378ðdj in nmÞ

ð6Þ

" #  3 18:31 d 1 þ 19:31 d þ 2 19:31

Fig. 1 shows the Tj(t) for diameters 40 50 60 70 80 and 90 nm (right to left). Each of the Tj curves must be multiplied by the extinction coefficient, which depends on dj as well as on the wave length of light used in the experiment. In the present work, we measure optical density as a function of time for several wave lengths. Thus we define

We do not consider particle diameters below 40 nm. For d = 40 nm, q2 changes from 19.31 to 16.82 and a changes from 1.461 m1 to 1.473 m1, a negligible change, while D changes from 1.213  1011 m2/s to 1.155  1011 m2/s. For larger particle sizes, the relative changes are smaller. Thus these measurements may be sensitive to capping layers through the changed value of D. Eq. (2), sometimes referred to as the Mason–Weaver equation [21], may be solved numerically [20] for c(x,t) with the boundary conditions, J(0,t) = J(h,t) = 0 and J(x,1) = 0 (equilibrium). When equilibrium is reached, J = 0 and oc/ot = 0 for all x. This requires that ac + oc/ox = 0, so that c(x,1) = Aeax where A is a constant. The process generates ‘‘relative absorbance vs. time curves’’, referred to as ‘‘sedimentation curves,’’ which give A(t) for a fixed height x, and also density profiles A(x) for any time t, for particles characterized by particular values of a and D. According to the Beer–Lambert Law, the optical density measured in the experiment at the location of the optical beam is the concentration of particles in solution at that location multiplied by the absorption/scattering coefficient of the particle, which is dependent on the observation wavelength. 2.2. Theoretical curves Since we measure absorbance as a function of time for the height range where the optical beam of the spectrophotometer passes through the cuvette, 2 mm 6 x 6 6 mm (see below), we average the absorbance vs. time curves over this range of x, to generate a library of average curves for all particle sizes. All these

T kj ðtÞ  akj T j ðtÞ

ð7Þ

where akj is the extinction coefficient for particles of diameter dj when the wave length is k(k). The akj have been calculated and reported in graphical form by Jain et al. [23]. For our purposes, it is sufficient to approximate akj for wave lengths near the wave length of maximum extinction by a Lorentzian:

akj ¼ 1þ

b0 

kk b1 b2

ð8Þ

2

The parameters b0 b1 and b2 were found for several particle sizes by fitting the results presented graphically by Jain et al. [23] to Eq. (8). They were found to be linear functions of d, i.e. b0 = 0.766 + 0.0827d, b1 = 515 + 0.412d, and b2 = 46.2 + 0.0774d, where d is in nm. Note that the extinction coefficients need be determined only on a relative scale. The extinction-weighted curves T kj (t) must be combined to give the experimental sedimentation curve measured at wave length k(k). Thus, if the concentration of particles of size dj is cj, the experimental curve S(k), extinction of light of wave length k as a function of time, is to be approximated as

SðkÞ 

X k cj T j

ð9Þ

j

The normalization of the coefficients cj is assured by the fitting process. 2.3. Determination of size histogram The size histogram, i.e. the relative abundances of particles of size j, is defined by the binning used. Here we use bins of width 5 nm, centered at 50, 55, . . . 110 nm. All particles within a bin are assigned the same diameter, the value at the center of the bin. To determine the size histogram, we minimize the sum of the squared deviations between the experimental and theoretical sedimentation curves,



XZ k

Fig. 1. Particle density in window as a function of time for different particle sizes. From right to left, the curves correspond to diameters of 40 50 60 70 80 and 90 nm. Each curve is normalized to 1 at t = 0.

$

X k dt S ðtÞ  cj T j ðtÞ k

%2 ð10Þ

j

with respect to the coefficients cj, which are the relative abundances. The optical densities are measured for each k at intervals of 11.5 min, so that the integrals over time are actually sums of time points. The time points are different for each k by 10 s, and we calculate the T kj for each kk at the relevant time points.

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By setting (@Q/@cm) = 0 we obtain the following linear equations for the {cj}:

XX k X k M mj cj ¼ Dm j

k

ð11Þ

k

Here,

M kmj



X k T m ðt i ÞT kj ðti Þ

ð12Þ

i

and

Dkm 

X k S ðt i ÞT km ðt i Þ

ð13Þ

i

and the sums over i are over time points. Note that the matrices M kmj and Dkm need be calculated only once for use in all the calculations. P k Eq. (11) may be solved by inverting the matrix k M mj . In some cases, this direct solution produces negative values for some of the cj. When this happens, we use an optimization routine like SOLVER to find the set of cj that minimizes Q (Eq. (10)) while maintaining all the cj positive.

Fig. 2. Experimental sedimentation curves for (left to right) wave lengths k = 508, 528, 548, 568, and 588 nm. Each shows optical density for light passing through window for 2 < x < 6 mm. For increasing wave length, the curves get closer together.

3. Experimental section 3.1. Reagents Ascorbate-capped gold nanoparticles possessing a TEM-measured diameter of 82.5 ± 5.2 nm were a gift from Professor M. Maye (Syracuse University). 3.2. UV–Visible Spectroscopy (UV–vis) AuNP absorbance spectra were measured in the 300–900 nm region using a Varian Cary 50 UV–vis spectrophotometer in scan mode, at a scan rate of 300 nm/min, using zero/baseline correction. AuNP were diluted in ultrapure water (18.2 MX cm). 3.3. Transmission Electron Microscopy (TEM) TEM images of AuNP were acquired using a JEOL 2000EX operated at 120 kV with a tungsten filament (SUNY-ESF, N.C. Brown Center for Ultrastructure Studies). Particle diameter was manually analyzed by modeling each individual AuNP as a sphere, using >100 particles per count. ImageJ software was used to give the distribution of AuNP spherical areas, which were then converted to AuNP diameters. 3.4. Sedimentation experiments Sedimentation experiments were performed using a slightly modified 2.5 mL disposable cuvette, as reported previously [20]. The cuvette was modified so that, when placed in the spectrophotometer, the optical beam passes through the AuNP sample only in the range of height from 2 to 6 mm from the base of the internal volume of the cuvette. The AuNP sample was diluted in ultrapure water and transferred to the cuvette. Immediately before data collection, the suspension was pumped 20 times using a pipette to ensure uniformity, and the cuvette was capped. Sedimentation curves were collected by monitoring absorbance as a function of time using a Varian Cary 50 UV–vis spectrophotometer in scanning kinetics mode, using zero correction, at room temperature. Nine wavelengths centered around the measured kmax (548 nm) at 10 nm increments were selected for absorbance monitoring: 508, 518, 528, 538, 548, 558, 568, 578, and 588 nm. The average measurement time for each wavelength was 10 s.

Fig. 3. Absorbance spectra at different times (whole spectra were normalized to begin at Abs = 1 for kmax = 548 nm), revealing a blue shift as a function of time. Inset shows zoomed-in spectra at 120–180 h.

The cycle was repeated every 11.5 min until sedimentation was complete at 140 h. The selection of the wave lengths to be used in the measurements was done as follows. For the particles studied here, the diameter was stated to be about 80 nm. (The electron microscopy measurements found d = 82.5 ± 5.2 nm.) Referring to Eq. (11), we see the maximum optical density occurs for a wave length kk equal to b1 = 515 + 0.412(82.5) = 549 nm. Thus the wave lengths used were 508(10)588 nm, centered on 548 nm (the maximum of the measured kspr). The sedimentation curves were generated by dividing all absorbances (A) by the absorbance for t = 0 and adjusting the result so that A = 1.0 at t = 0 and A approaches 0 as t approaches infinity. The sedimentation curve is thus a plot of relative absorbance vs. time for the part of the system between h = 2 mm and h = 6 mm. Fig. 2 shows five of the nine experimental sedimentation curves for 0 < t < 140 h. Each is a plot of optical density vs. time for a different wave length, measured from the window for 2 < x < 6 mm. Only five plots are shown for clarity, for k = 508, 528, 548, 568, and 588 nm. In fact, the other four were not used in our analysis because all the plots are so similar.

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Fig. 3 shows the measured UV–vis absorbance or optical density spectra at t = 0, 20 h, 40 h, . . . 180 h. The decrease in optical density to zero as the sedimentation front moves through, after 60 h is evident. Moreover, there is a blue shift in the spectra with increasing time, where the overall kmax gradually changes from 548.3 nm at time = 0 h to 543.0 nm at time = 140 h. Beyond 140 h, the spectra were too flat for kmax to be distinguished from noise. The observed blue shift reflects the decreasing average size of particles remaining in suspension as a function of time; the larger particles sediment faster than the smaller. Of course, their absence decreases the total intensity.

4. Results of calculations We obtain the size histogram by minimizing Q (Eq. (10)), the total squared deviation between calculated and experimental sedimentation curves for one or more wave lengths. The calculated curves were obtained as shown above (Eq. (5)). The parameters varied in the minimization were the relative numbers of particles of various sizes. We considered particle diameters of 50, 55, 60, . . . 110 nm, i.e. the bins for the histogram were 5 nm wide and centered at 50, 55, 60, . . . 110 nm. For example, all particles with diameters between 47.5 nm and 52.5 nm were considered together as particles with diameter 50 nm. First we show results of the minimization of Q when only a single wave length is considered. The number of bins considered is 2 3 or 4. The resulting histograms are shown in Fig. 4. The three on the left are derived from the data for 548 nm, and the three on the right from the data for 568 nm. The histogram derived from TEM is shown at the bottom; since we consider only 4 bins (75 80 85 and 90 nm), we have put all the particles with d > 87.5 nm into the fourth bin. Note that the choice of wave length is not important here: the results for 548 nm are quite close to the results for 568 nm. It is also clear that 2 or 3 bins are not enough to get a reasonable idea of the actual particle size distribution (except for the fact that the highest population is for 85 nm). If, on the other hand, 4 bins are used, one of the populations comes out unphysically negative, and the other three do not much resemble the TEM populations. Using more than 4 bins leads to more negative populations. We now consider what can be accomplished if more than one wave length is used. Fig. 5 shows the histograms obtained using measurements for two wave lengths, 548 nm and 568 nm. The calculated histograms (dark bars) are compared with the TEM histogram (light bars). In graphs a b and c, we show results for 4 5 and 6 bins, covering the ranges 75–90 nm, 75–95 nm, and 70–95 nm. It is clear that increasing the number of bins makes the predicted histograms resemble the TEM histogram more. Because the actual histogram involves more than four bins, one cannot hope to get a reasonable representation using fewer than five bins, which requires measurements at several wave lengths. Panels d and e also show 6-bin histograms, but using bins 75–100 (d) or 65–90 (e). It may be that using bins for larger diameters improves the agreement with TEM for larger diameters (compare 5d with 5c) and vice versa (compare 5e with 5c). However, either choice disturbs the agreement in the mid-range. Going to seven bins restores agreement, except for the troubling negative populations. In these results choice of bins is important. To avoid errors associated with bad choice of bins, one should use as many bins as practical In the calculations shown in graph f, 7 bins, from 65 to 95 nm, produced quite good agreement. Remarkably, the ‘‘outliers’’ at 65 nm in the TEM results are detected by the sedimentation measurements. This shows that these very small particles are really present, and are not a statistical fluctuation associated with the small number of particles in these bins (only 103 particles were

Fig. 4. Two-, three-, and four-bin histograms generated from sedimentation measurements at a single wave length (548 nm for the three on the left, 568 nm for the three on the right). The histogram at the bottom comes from TEM.

measured on the TEM grid). One persistent problem is the large calculated negative value at 75 nm in graphs c e and f. It was thought that this was a numerical problem associated with the large change in cj between adjacent bins. To avoid it, we used the SOLVER program, which allows us to carry out a constrained variation, i.e. to minimize the sum of squared deviations between calculated and measured sedimentation curves while maintaining all coefficients positive. Some of the results of these calculations, using data for the same two wave lengths, are shown in Fig. 6. Here, 5, 6, and 7 bins were used. The 5 bins for the first calculation were 75 80 85 90 and 95 nm. The second calculation differs from the first in the addition of a 6th bin, at 70 nm. Since the constrained minimization made the population of this bin equal to zero, the two graphs are identical. The difference between calculated and TEM histograms for the lower particle diameters is explained by the existence of particles with diameters less than 70 nm, as mentioned above. This is similar to what happens for

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Fig. 5. Histograms derived from sedimentation measurements at two wave lengths, 548 and 568 nm. Panels a b and c are for 4 5 and 6 bins, panels d and e are for 6 bins with different choices of bins, and panel f is for 7 bins. Dark bars = sedimentation results, light bars = TEM results.

Fig. 6. Analysis of sedimentation curves for wave lengths 548 and 568 nm using constrained variation. Results are shown for 5 6 and 7 bins (the 5- and 6-bin results are identical because sedimentation and TEM both indicate no particles with d = 70 nm). Dark bars = sedimentation results, light bars = TEM results.

the last bins in going from the 6-bin to the 7-bin calculation. In the former, the minimization puts much too many particles into the 95-nm bin in an attempt to account for particles of higher diameter. The calculations still miss the substantial number of particles reported by TEM for d = 75 nm. All the calculations shown in Figs. 5 and 6 were repeated using the scattering curves for 548 nm and 528 nm. The results resembled the results for 548 and 568 nm, showing that choice of wave lengths is not of great importance, as was already noted for the single-wave length calculations. However, the results so far suggest that one should consider more bins, which in turn requires more wave lengths. We therefore consider using three wave lengths, 528 548 and 568 nm. First we do the unconstrained minimization, producing the results in Fig. 7 (results from only four of the calculations performed are shown, namely those using 5, 6, 8, and 10 bins). Negative coefficients appear in almost all the calculations. The histogram for the 5-bin (diameters 80–100 nm) calculations compares very well with the TEM histogram over the region considered. When the bin at 75 nm is included, the problem appears: the calculations give a small negative number for this bin instead of 12, as measured from TEM. The problem gets worse when more bins are added: in the 8-bin and 10-bin results (3rd and 4th panels), it is 12 and 10. Note, however, that the small particles (d 6 70 nm) have been found in the last two calculations.

We eliminate unphysical negative values by using constrained variation instead of the exact solution to Eqs. (11)–(13). The results are displayed in Fig. 8. The three columns, left to right, show results for six, eight, and ten bins, with the particle diameters shown. The three rows correspond to three choices of wave lengths measured. The top row is for k = 528, 548, and 568 nm, the second row is for k = 548, 568, and 588 nm, and the third row is for k = 508, 528, and 548 nm. In all cases, the largest particle numbers (bins for 80, 85, and 90 nm) are well reproduced. Going from the first row to the second, which means using larger wave lengths, improves things for the larger-sized particles somewhat, but the smallest particles (diameters below 80 nm) are completely missing. This could have been expected: since the smaller particles scatter most at smaller wave lengths, the optical density measured at larger wave lengths is not sensitive to the smaller particles. To pick up the smaller particles, one should go to shorter wave lengths, as is seen in the third row. None of the calculations of Fig. 8 ‘‘finds’’ the particles in the 75nm bin, which have diameters between 72.5 and 77.5 nm. Also, all of the calculations make the number of particles in the 80-nm bin, with diameters between 77.5 and 82.5 nm, too small, and make the number in the 85-nm bin too large. Possible explanations for this include (1) underestimation by TEM of the sizes of the larger particles, and (2) a ‘‘solvation shell’’ on the sedimenting particles. As

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Fig. 7. Results of analysis of data collected at three wave lengths, 528 548 and 568 nm. Shown are results for 5, 6, 8 and 10 bins. No constraints were used, allowing the coefficient for 75 nm to become substantially negative. Dark bars = sedimentation-derived histogram, light bars = TEM histogram.

noted in the Introduction, the main effect of such a shell would be to decrease the diffusion coefficient. The last calculations we present are for four wave lengths, 208 228 248 and 268 nm. Results for 10 bins, from 60 to 105 nm, and for 12 bins, from 55 to 110 nm, are shown and compared with the TEM histogram in Fig. 9. These results are not very different from the results of the analysis using three frequencies (Fig. 8). This suggests that no additional information is to be gained by using more than three frequencies. The problem of finding no particles at 75 nm is still present. The undercounting (compared to TEM) of 80-nm particles and the overcounting of 85-nm particles persist as well. This suggests that the origin of the problem is not numerical, but physical. 5. Discussion In this paper, we explore the possibility of measuring the size histogram for gold nanoparticles using gravitational sedimentation. Most commonly, the size histogram is derived from measurements on a TEM image containing one or two hundred particles. There is a possibility of statistical or sampling errors in this kind of measurement. The sedimentation experiment ‘‘sees’’ all the particles, and measures their properties in situ. It is true that this measurement takes a long time, but the experiment is simple and needs no attention, as optical density is measured and recorded automatically. The time for the experiment can be decreased by measuring optical density closer to the top of the liquid, so the sedimentation front arrives earlier, but one loses information because the sedimentation curves are less spread out. In order to get more information from sedimentation measurements, they should be made simultaneously at several wave lengths, which is easy to do with conventional spectrometers. The number of wave lengths to use depends on how many histogram bins are considered, which in turn depends on the width of the bins and the width of the histogram. We chose 5-nm bins

because they were convenient for the TEM histogram. It is clear from the calculated histograms that making the bins much larger would lose valuable information. On the other hand, making the bins too narrow could lead to numerical oscillations in the results. Also, with only a hundred particles measured, the TEM histogram would not be meaningful if bins narrower than about 5 nm were used. With 5-nm bins, the TEM histogram shows particles of diameters between 65 and 100 nm, corresponding to eight bins. In order to identify statistical errors, more than eight bins should be used. We have gone up to ten and twelve bins. From our results for different numbers of wave lengths, we concluded that, from measurements at a single wave length, no more than four populations could be extracted, that measurements at two wave lengths could give seven populations, that measurements at three wave lengths could give ten populations, and that measurements at four wave lengths gave no new information. All these conclusions could be reached without reference to the TEM results. As far as the choice of wave lengths is concerned, we showed above that it is not of great importance, as long as the wave lengths are not too far from the wave lengths of maximum scattering. The analysis required the absorption/scattering coefficients as functions of particle diameter and wave length. This information was readily available [23] and was put in a convenient parametric form by us (Eq. (8)). Also required were the calculated sedimentation curves for particles of all sizes considered. Having solved the Mason–Weaver equation, we were able to put these sedimentation curves into a convenient form, Eqs. (5) and (6). We emphasize that these calculations are done once and for all. The only change would be if the height of the observation window is changed, and the change is easily made since the full theoretical sedimentation curves, intensity as a function of height and time, are available. From the calculated and experimental sedimentation curves, it is easy to generate the matrices Mk and Dk (Eqs. (10)–(13)), which are used in all the calculations.

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Fig. 8. Results of calculations using three wave lengths and 6, 8 or 10 bins, with all coefficients constrained to be positive. The three columns correspond to 6 8 and 10 bins; the three rows to three choices for the three wave lengths used. Dark bars = histogram derived from sedimentation, light bars = histogram derived from TEM.

The coefficients in the histogram are determined to minimize Q (Eq. (10)), the sum of the squared deviations between the measured scattering curves and the theoretical curves, weighted by the coefficients. The exact solution to this problem is given by Eq. (11). However, the exact solution sometimes gives unphysical negative values for some coefficients. It was found that carrying out constrained variation, i.e. minimizing Q while requiring all coefficients to be positive, produced coefficients which resembled those of the exact solution, but with negative coefficients removed. We now compare the best sedimentation-derived histograms with the results of TEM. In particular, we look at the 10- and 12bin results obtained using three or four wave lengths (Figs. 8 and 9), considered to be our most reliable. It should be remembered that neither our results nor the TEM results should be considered to be the ‘‘true’’ histogram, since the two measurements are not on exactly the same system, and both may be subject to errors. That said, we can note that the two are in good agreement on the general shape of the histogram, the position of the maximum, and so on. It remains to calculate other characteristics of the histograms.

Fig. 9. Results of analysis using four wave lengths, 208 228 248 and 268 nm. Constrained variation, to make all coefficients positive, was used. Left graph used ten bins, from 60 nm to 105 nm diameters in steps of 5 nm; right graph used these plus bins at 55 and 110 nm. Dark bars = sedimentation histogram, light bars = TEM histogram.

For comparison purposes, all the histograms have been normalized to a total of 103 particles. Then various characteristic

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Table 1 Characteristics of four size distributions (histograms) derived from sedimentation, and the TEM-derived histogram, for comparison. The three-k distributions are in the first two rows of the last column of Fig. 8; the four-k distributions are in Fig. 9.

Mean Variance Skewness Kurtosis

3 k’s, 10 bins 1

3 k’s, 10 bins 2

4 k’s, 10 bins

4 k’s, 12 bins

TEM, 12 bins

85.51 376.1 0.0523 2.287

86.63 408.2 0.4770 2.503

84.58 328.2 0.3922 2.124

84.68 416.11 0.4891 1.937

82.48 294.4 0.0040 2.466

Fig. 10. The 4-k, 12-bin histogram derived from sedimentation is corrected for the solvent core which migrates with the particle (dark bars). The corrected histogram is compared with the TEM-derived histogram (light bars), showing excellent agreement.

quantities, listed in the first column of Table 1 have been calculated for the four sedimentation-derived histograms and the TEM histogram. The jth moment of a histogram is defined as

mj ¼ n1

X i

ðX i  MÞj

where M is the mean of the entries Xi Then [24] the variance is the 2nd moment, the skewness is equal to m3/m23/2, and the kurtosis is  equal to m4 =m22  3. The means are well within one standard deviation of each other, the variance and the kurtosis (measure of peakedness) is about the same for all five, and the skewness is small in all cases. However, none of the distributions derived from sedimentation is a really good match for the distribution derived from TEM. The means and the variances for the former are always significantly larger than the mean and variance for TEM. There are important differences between the TEM and sedimentation histograms for small particle diameters, and, most important, for particles of diameter 75 nm (72.5–77.5 nm). For small diameters (less than 70 nm), the number of particles measured by TEM is only one or two, so statistical fluctuations may play a role. That the sedimentation measurements are sensitive to the small fraction of particles which have these small diameters is impressive; it shows that these small particles exist in the original suspension and are not created during the process of transfer to the TEM grid. On the other hand, these particles consistently are assigned smaller diameters in the sedimentation histograms than in the TEM. We suggest that the errors in measuring particle diameters from TEM are larger when the particles are small. The problem at 75 nm is more striking: our histograms consistently show no particles in the 75-nm bin, whereas the TEM shows 12 particles out of 103 fall into this bin. There are at least two explanations for this. It is possible that the TEM measurements underestimate the sizes of the larger particles, so that these particles are actually a few nanometers bigger than reported. A second explanation is that the particles in the sedimenting mixture carry a

‘‘solvation shell’’ of thickness 1–2 nm (a few times the length of an ascorbate molecule). As noted in the Introduction, the biggest effect of such a shell would be to decrease the diffusion coefficient, making these particles sediment more slowly. The density and mass of the sedimenting particles are not much affected. Thus sedimentation measurements point to an actual physical effect: the particles in suspension have a shell containing solvent, citrate, and other ions, and this shell migrates with the gold core. If the solvent shell has a thickness of 1 nm, the effective diameter of a sedimenting particle is increased by 2 nm. The TEM histogram shows only the size of the gold cores. We test this idea as follows: Since our bins are 5 nm wide, and each observed particle diameter is 2 nm bigger than the diameter of the gold core, roughly 2/5 of the particles in each bin should be assigned to the bin preceding it to compare with the TEM histogram. (We leave the first few bins intact.) When this is done for any of the four histograms, the particles with diameter 75 nm of course appear, and, more important, the new histogram compares extremely well with the TEM histogram. This is shown in Fig. 10 for the 4-k, 12-bin histogram. The mean is 82.73, the variance 450.84, the skewness 0.3219, and the kurtosis 2.248. The large variance is because the smallest particles are found to be smaller in the sedimentation than in the TEM. The excellent agreement makes it likely that our idea is correct: sedimentation measurements reflect the solvent shell which moves with the gold core. Thus comparison of the histograms from TEM and sedimentation gives useful information about the nature of the gold particles in situ. Our analysis of sedimentation data to obtain the size histogram of the sedimenting particles requires calculated sedimentation curves (optical density as a function of time and height) for particles of various sizes. From these curves, which need be calculated only once, and the measured sedimentation curve, it is straightforward to extract the size histogram. It must be noted that the calculated curves were obtained assuming spherical particles of known density, and the measured curve is relatively featureless. If the actual particles were not spherical, the experimental sedimentation curve would probably not differ qualitatively from that for spherical particles. Thus our analysis would not detect deviations from spherical shape. Similarly, if agglomeration occurred during sedimentation, it would not be detected unless it changed the shape of the experimental sedimentation curve so much that it could not be represented as a sum of sedimentation curves for spherical particles . On the other hand, there is no limit on the sizes of the particles that can be studied. All that is required is the absorption/scattering coefficients of the particles as a function of particle size and wave length, and some knowledge of the particles’ composition (so particle mass can be calculated as a function of size).

6. Conclusions Sedimentation curves for gold nanoparticles in water were obtained by measuring the optical density of a suspension, initially

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with particle concentration and optical density uniform, over time. As the sedimentation front moved through the optical beam of the instrument, the measured optical density decreased to zero. The spectrophotometer was programmed to measure optical density at nine different wave lengths sequentially every eleven minutes. Up to five of the resulting nine curves were analyzed simultaneously to derive the size histogram of the sedimenting particles in situ. We used previously calculated solutions to the Mason–Weaver sedimentation–diffusion equation for a series of particle diameters, assuming independent spherical particles in solution, and available absorption/scattering coefficients for particles of those diameters. Weighting the calculated sedimentation curves with these coefficients and the size (diameter) abundances {cj} produces a theoretical sedimentation curve for each observation wave length. The {cj} were then chosen to minimize the deviations between the experimental and theoretical sedimentation curves for all wave lengths used. The effects of changing the number of histogram bins and the wave lengths used were studied. It was concluded that at least ten bins were required (i.e. 10 coefficients cj had to be determined), which required at least three wave lengths. Which wave lengths were used was less important. Using smaller bins would mean determining more parameters and require more wave lengths, but it was felt that the additional information gained would not be of much interest. Agreement between the sedimentation-derived histogram and the TEM-derived histogram was very good as far as overall shape, position of the maximum, etc. are concerned (see Figs. 8 and 9 and Table 1). However, there were differences for the smallest particles and for the 75-nm bin. The former are partly due to statistical fluctuations (TEM found only 1–2 particles out of 103 with these diameters) and partly to the difficulty of measuring small sizes. That the sedimentation measurements find these small particles shows that they exist in the original suspension and are not created during transfer to the TEM grid. For the 75-nm bin, the TEM histogram indicates that 12% of the particles have diameters of 72.5–77.5 nm whereas the sedimentation histogram finds none. We argue that this is because sedimentation measures the particles in situ, where the gold cores are covered with a low-density shell about 1 nm thick, whereas TEM shows only the cores. A rough correction for this makes agreement between the two histograms excellent (Fig. 10). Thus we have shown that the size histogram of gold particles in suspension can be accurately measured by monitoring sedimentation in a gravitational field using measurements of optical density

as a function of time. Simultaneous measurement at several wave lengths yields more information (more histogram numbers). Furthermore, comparing with a TEM-derived histogram gives important information about how the particles in situ differ from the simple gold cores seen in TEM. Acknowledgments We would like to thank Professor M. Maye of Syracuse University for helpful discussions, and for donating samples of AuNP to this study. Prof. J. Dabrowiak is to be thanked for his seminal ideas and important comments. Finally, we would like to acknowledge the Department of Chemistry of Syracuse University for support. References [1] A. Sanchez, S. Abbet, U. Heiz, W.-D. Schneider, H. Ha1kkinen, R.N. Barnett, U. Landman, J. Phys. Chem. A 103 (1999) 9573–9578. [2] J.J. Storhoff, A.A. Lazarides, R.C. Mucic, C.A. Mirkin, R.L. Letsinger, G.C. Schatz, J. Am. Chem. Soc. 122 (2000) 4640–4650. [3] B. Kim, S.L. Tripp, A. Wei, J. Am. Chem. Soc. 123 (2001) 7955–7956. [4] N.S. Ieong, K. Brebis, L.E. Daniel, R.K. O’Reilly, M.I. Gibson, Chem. Commun. 47 (2011) 11627–11629. [5] Y. Kim, R.C. Johnson, J.T. Hupp, Nano Lett. 1 (2001) 165–167. [6] E.C. Dreaden, M.A. Mackey, X. Huang, B. Kangy, M.A. El-Sayed, Chem. Soc. Rev. 40 (2011) 3391–3404. [7] M. Stratakis, H. Garcia, Chem Rev. 112 (2012) 4469–4506. [8] X. Zhou, W. Xu, G. Liu, D. Panda, P. Chen, J. Am. Chem. Soc. 132 (2009) 138–146. [9] Y. Mikami, A. Dhakshinamoorthy, M. Alvaro, H. Garcıa, Catal. Sci. Technol. 3 (2013) 58–69. [10] C. Burda, X. Chen, R. Narayanan, M.A. El-Sayed, Chem. Rev. 105 (2005) 1025– 1102. [11] T.K. Sau, A. Pal, T. Pal, J. Phys. Chem. B 105 (2001) 9266–9272. [12] B.J.Y. Tan, C.H. Sow, T.S. Koh, K.C. Chin, A.T.S. Wee, C.K. Ong, J. Phys. Chem. B 109 (2005) 11100–11109. [13] B.D. Chithrani, A.A. Ghazani, W.C.W. Chan, Nano Lett. 6 (2006) 662–668. [14] C.D. Walkey, J.B. Olsen, H. Guo, A. Emili, W.C.W. Chan, J. Am. Chem. Soc. 134 (2012) 2139–2147. [15] E. Boisselier, D. Astruc, Chem. Soc. Rev. 38 (2009) 1759–1782. [16] B. Fadeel, A.E. Garcia-Bennett, Adv. Drug Deliv. Rev 62 (2010) 362–374. [17] S.J. Hurst, A.K.R. Lytton-Jean, C.A. Mirkin, Anal. Chem. 78 (2006) 8313–8318. [18] C.M. Alexander, M.M. Maye, J.C. Dabrowiak, Chem. Commun. 47 (2011) 3418– 3420. [19] C.M. Alexander, J.C. Dabrowiak, M.M. Maye, Bioconjugate Chem. 23 (2012) 2061–2070. [20] C.A. Alexander, J.C. Dabrowiak, J. Goodisman, J. Coll. Interface Sci. 396 (2013) 53–62. [21] M. Mason, W. Weaver, Phys. Rev. 23 (1924) 412–426. [22] P.C. Hiemenz, Principles of Colloid and Surface Chemistry, second ed., Marcel Dekker, New York, 1986 (Section 2.13). [23] P.K. Jain, K.S. Lee, I.H. El-Sayed, M.A. El-Sayed, J. Phys. Chem. B 110 (2006) 7238–7248. [24] L.L. Havlicek, R.D. Crain, Practical Statistics for the Physical Sciences, American Chemical Society, Washington DC, 1988.

Size histograms of gold nanoparticles measured by gravitational sedimentation.

Sedimentation curves of gold nanoparticles in water were obtained by measuring the optical density of a suspension over time. The results are not subj...
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