A colorimetric method for the molecular weight determination of polyethylene glycol using gold nanoparticles
© Ling et al.; licensee Springer. 2013
Received: 2 November 2013
Accepted: 5 December 2013
Published: 20 December 2013
A gold nanoparticle (AuNP)-based colorimetric method was developed for the molecular weight (MW) determination of polyethylene glycol (PEG), a commonly used hydrophilic polymer. Addition of a salt solution to PEG-coated AuNP solutions helps in screening the electrostatic repulsion between nanoparticles and generating a color change of the solutions from wine red to blue in 10 min in accordance with the MW of PEG, which illustrates the different stability degrees (SDs) of the AuNPs. The SDs are calculated by the absorbance ratios of the stable to the aggregated AuNPs in the solution. The root mean square end-to-end length (〈h2〉1/2) of PEG molecules shows a linear fit to the SDs of the PEG-coated AuNPs in a range of 1.938 ± 0.156 to 10.151 ± 0.176 nm. According to the Derjaguin-Landau-Verwey-Overbeek theory, the reason for this linear relationship is that the thickness of the PEG adlayer is roughly equivalent to the 〈h2〉1/2 of the PEG molecules in solution, which determines the SDs of the AuNPs. Subsequently, the MW of the PEG can be obtained from its 〈h2〉1/2 using a mathematical relationship between 〈h2〉1/2 and MW of PEG molecule. Applying this approach, we determined the 〈h2〉1/2 and the MW of four PEG samples according to their absorbance values from the ordinary ultraviolet–visible spectrophotometric measurements. Therefore, the MW of PEG can be distinguished straightforwardly by visual inspection and determined by spectrophotometry. This novel approach is simple, rapid, and sensitive.
KeywordsGold nanoparticles Polyethylene glycol Molecular weight determination Colorimetric method Spectrophotometry
Polyethylene glycol (PEG) is a synthetic hydrophilic polymer, which is widely used as an emulsifier and surfactant in cosmetics, foodstuffs, and pharmaceutical products [1, 2]. The molecular weight (MW) of PEG has a significant impact on its properties and applications [1, 3, 4]. In the case of PEG-functionalized drugs, in particular, an increase in the MW of PEG leads to reduced kidney excretion, resulting in a prolonged blood circulation time of the drug . A variety of analytical techniques, such as size exclusion chromatography (SEC) with preferably a universal detector , nuclear magnetic resonance spectroscopy , and matrix-assisted laser desorption ionization time-of-flight mass spectrometry , have been used to determine the MW of PEG polymer. However, these powerful techniques require the use of sophisticated instruments and complicated protocols. Besides, the instruments are not as readily available in many laboratories.
Gold nanoparticle (AuNP)-based colorimetric assays have attracted considerable attentions in detection applications with regard to their simplicity and versatility [7, 8]. This colorimetric assay can be easily observed by visual inspection, which avoids the relative complexity inherent in conventional detection methodologies . Because of the electrostatic repulsion resulting from the negative charges on the surfaces, AuNPs are highly stable in the absence of added salts. The addition of electrolytes to gold sols results in the reduction of charge repulsion and as a consequence nanoparticle aggregation. Nonetheless, AuNPs can be stabilized even at high salt concentrations by adsorbing proteins or other hydrophilic polymers (protecting agents) onto their surfaces . They bind the macromolecules by noncovalent electrostatic, stable adsorption . PEG polymer is one of the most often used stabilizers, as it possesses the advantage of a chemically well-defined composition that ensures the reproducibility of its performance. Moreover, PEG dissolves rapidly and therefore can be prepared just prior to use.
At high salt concentrations, the stability of PEG-coated AuNPs depends upon the MW of PEG . The stabilization of the fully coated AuNPs is due to the steric repulsion effect, which is dependent on the thickness (t) of the PEG adlayer and the conformation of the adsorbed PEG molecules [10, 13, 14]. The adsorbed PEG forms a single protecting layer on the surface of the nanoparticle, because of the resistance of the polymer coil to compress and to release both bound and free water from within the hydrated coil [15–17]. Under the complete coverage of the surface condition, PEG molecules are in direct competition for the adsorption sites on the AuNP surface . Therefore, the adsorbed linear PEG molecules form typical loops and tail conformations [13, 18]. The value of t is roughly equivalent to the size of the PEG molecule as a free molecule in solution under the condition [13, 18]. The root mean square end-to-end length (〈h2〉1/2) is commonly used to specify the size of a linear polymer molecule.
Herein, enlightened by the above facts, we developed a simple and reliable colorimetric method for the MW determination of PEG in aqueous solution using citrate-reduced AuNPs. This method is based on the different stability degrees (SDs) of the AuNPs, which are fully coated by different MW (〈h2〉1/2) of PEG, after screening the electrostatic repulsion between nanoparticles. The SDs of the AuNPs are monitored by ultraviolet–visible (UV–vis) spectrophotometry, which exploits the strong sensitivity of the localized surface plasmon resonance spectrum to the aggregation of AuNPs. In this study, the SDs are calculated by the absorbance ratios of the stable to the aggregated AuNPs in solution. The nanoparticles exhibit greater stability upon an increase in the MW (〈h2〉1/2) of PEG. Of the systems tested, the 〈h2〉1/2 of PEG molecules was found to exhibit a good linear correlation to the SDs of the AuNPs in a specified range. As a result, we can obtain the 〈h2〉1/2 of PEG from the SDs of the AuNPs and then estimate the corresponding MW using a mathematical relationship between the 〈h2〉1/2 and MW of PEG molecule. So far, there is no report on nanomaterial-based methods for the MW determination of polymers. This AuNP-based determination method offers simplicity, convenience, and sensitivity, and can be accomplished in minutes without sophisticated instruments or training overhead.
Hydrogen tetrachloroaurate (III) trihydrate (HAuCl4 · 3H2O) and four PEG samples (SPEG 1,450 to 10,000) were purchased from Sigma-Aldrich (St. Louis, MO, USA). Ten PEG samples (APEG 400 to 20,000) were purchased from Alfa Aesar (Tianjin, China). Trisodium citrate dihydrate (Na3C6H5O7 · 2H2O), sodium azide (NaN3), and sodium chloride (NaCl) were purchased from Sinopharm Group Chemical Reagent Co., Ltd. (Shanghai, China). All chemicals were analytical grade reagents and used without further purification. All water was deionized by reverse osmosis and further purified using a Milli-Q Plus system (Millipore, Billerica, MA, USA) to 18.2 MΩ cm resistivity. All glassware were cleaned using aqua regia solution (HCl/HNO3 = 3:1, v/v) and subsequently rinsed with a copious amount of Milli-Q treated water.
Citrate-reduced AuNPs were prepared according to the modified method [19, 20]. In brief, 99.00 mL of water and 1.00 mL of 1.0% (w/v) HAuCl4 · 3H2O solution were mixed in a flask. The mixture was then heated under magnetic stirring until it began to boil, and a 1.0% (w/v) Na3C6H5O7 · 2H2O solution (1.80 and 2.25 mL) was quickly added to the solution. After boiling for 20 min, the solutions were cooled to room temperature (25°C) with vigorous magnetic stirring. The prepared AuNP solutions were stored at 4°C until ready for use. The nanoparticle concentrations of the prepared two samples were determined by measuring their extinction at 520 and 524 nm, respectively.
The prepared nanoparticles were characterized using a JEM-2010 FEF transmission electron microscope (TEM; JEOL Ltd., Akishima, Tokyo, Japan). Bright-field images of at least 200 particles deposited onto a carbon-coated copper grid (Xinxing Braim Technology Co., Ltd., Beijing, China) were measured using ImageTool graphics software to approximate the average particle diameter. The optical densities of the two AuNP samples at 520 and 524 nm, respectively, were measured using a Lambda 35 UV–vis spectrophotometer (Perkin Elmer, Waltham, MA, USA).
Colorimetric determination of PEG MW
Fully PEG-coated AuNPs were formed by the addition of 3-mL PEG solution (15 mg/mL) to 1 mL of the as-prepared AuNP solution. Immediately after adding the PEG solution, the suspension was ultrasonicated (KQ-100DY, Kun Shan Ultrasonic Instruments Co., Ltd., Jiangsu, China) for 10 min and then incubated over 16 h with gentle agitation using an orbital shaker at low speed (<1 Hz) to allow the polymer to adsorb to the nanoparticles. The PEG-coated nanoparticles were collected by centrifugation (12,000 rpm, 20 min) and resuspended in water three times to wash out the free PEG molecules and produce the fully coated AuNPs used in subsequent examinations. Subsequently, 1-mL aliquots of PEG-coated AuNP solutions were mixed with a certain volume (40, 50, or 60 μL) of 10.0% (w/v) NaCl solution at room temperature (25°C) for 30 s, followed by recording of their absorption spectra using the Lambda 35 UV–vis spectrophotometer after 10 min.
Chromatographic determination of PEG MW
SEC measurements were performed using a Waters 515 liquid chromatography system configured with an Optilab rEX refractive index (RI) detector (Wyatt Technology, Santa Barbara, CA, USA). Separations were performed using three size exclusion columns (SB804HQ, SB803HQ, and SB802.5HQ, Shodex, Japan) in series. PEG samples (100 μL) were run at 5 mg/mL concentrations in aqueous solution. The running buffer contained 0.05% (w/v) NaN3. A flow rate of 0.5 mL/min was used, and samples were characterized using RI detection (internal temperature 30°C). The columns and the buffers were used at the same temperature.
Multi-angle laser light scattering (MALLS) measurements were used to perform analytical scale chromatographic separations for the absolute MW determination of the principal peaks in the above SEC/RI measurements. MALLS determinations were performed using an 18-angle DAWN HELEOS laser light scattering detector (Wyatt Technology, USA) connected in tandem to the Optilab rEX RI detector (Wyatt Technology, USA), operating with a 50-mW solid-state laser at 658 nm. System and instrument validation was performed based on dextran (GPC Standard 80, Pharmacosmos, Denmark).
Dynamic light scattering measurements
Hydrodynamic radii (Rh) of PEG molecules were measured by dynamic light scattering (DLS) (Nanosizer ZS, Malvern Instruments, Worcestershire, UK) at room temperature (25°C). All PEG samples were dissolved in 81.5 mM NaCl solution to 5 mg/mL concentrations. All PEG solutions were then ultrasonicated for 10 min and filtered through 0.22-μm nylon filters. The zeta potentials of the AuNPs were also measured by DLS at room temperature (25°C).
OriginPro 8.0 software (OriginLab, Northampton, MA, USA) was employed to perform data processing. Each sample measurement was repeated in triplicate, and the data were presented as the mean ± standard deviation.
Results and discussion
Colloidal nanoparticles in a dispersion medium always show Brownian motion and hence undergo frequent collisions with each other. The stability of colloids is thus determined by the interaction between the nanoparticles during such collisions. There are two basic interactions: one being attractive and the other repulsive. When attraction dominates, the nanoparticles will aggregate with each other, and finally, the entire dispersion may coalesce. Conversely, when repulsion dominates, the system will be stable and remain in a dispersed state. This idea was originally proposed by Derjaguin, Landau, Verwey, and Overbeek and is therefore referred as the DLVO theory [13, 21]. The DLVO theory assumes that the behavior of colloidal nanoparticles can be simplified by the interaction potential between two neighboring nanoparticles [13, 21].
M w , R h , R g (Equation 2), and 〈 h 2 〉 1/2 (Equation 3) values of PEG samples used in this study
378 ± 30
0.568 ± 0.027
0.668 ± 0.032
1.636 ± 0.078
521 ± 51
0.672 ± 0.054
0.791 ± 0.064
1.938 ± 0.156
997 ± 77
0.944 ± 0.025
1.111 ± 0.029
2.721 ± 0.072
1,887 ± 20
1.602 ± 0.284
1.885 ± 0.334
4.617 ± 0.818
3,981 ± 82
1.784 ± 0.165
2.099 ± 0.194
5.141 ± 0.475
6,185 ± 165
2.343 ± 0.111
2.756 ± 0.131
6.751 ± 0.320
8,232 ± 162
2.749 ± 0.101
3.234 ± 0.119
7.922 ± 0.291
10,535 ± 907
3.306 ± 0.063
3.889 ± 0.074
9.526 ± 0.182
13,646 ± 1359
3.522 ± 0.061
4.144 ± 0.072
10.151 ± 0.176
19,118 ± 631
4.415 ± 0.015
5.194 ± 0.018
12.723 ± 0.043
1,348 ± 64
1.203 ± 0.097
1.415 ± 0.114
3.466 ± 0.280
4,384 ± 436
2.095 ± 0.045
2.465 ± 0.053
6.038 ± 0.130
8,350 ± 301
2.572 ± 0.299
3.026 ± 0.352
7.412 ± 0.862
10,641 ± 219
3.474 ± 0.214
4.087 ± 0.252
10.011 ± 0.617
The data of the above calculations are listed in Table 1.
with an R2 = 0.9994. This relationship is presented in Additional file 1: Figure S1 and plotted according to the Mw and the 〈h2〉1/2 values of the PEG samples (APEG 400 to 20,000) listed in Table 1. The coefficient υ is 0.5250, which is close to 0.5, establishing the fact that the PEG chains behave much like ideal chains in the solution .
In order to verify the colorimetric method, two sizes of AuNPs were prepared by reducing HAuCl4 with different amounts of trisodium citrate (see ‘Methods’). Through TEM examination, the diameters of the as-prepared AuNPs were measured to be about 16 and 26 nm, respectively (Additional file 1: Figure S2). The zeta potential values of the AuNPs were measured to be −34.6 ± 1.9 mV (16-nm AuNPs) and −30.1 ± 1.5 mV (26-nm AuNPs) by DLS. The nanoparticle concentrations were calculated as 1.316 × 10−9 M (16-nm AuNPs) and 2.804 × 10−10 M (26-nm AuNPs) using the corresponding absorbance values of 0.6471 and 0.6911 at 520 and 524 nm in conjunction with the calculated extinction coefficient for and cm−1 M−1, respectively .
To ensure that the amounts of PEG are able to saturate the AuNP surfaces in the final suspensions of 7.925 × 1011 particles/mL (16-nm AuNPs) and 1.689 × 1011 particles/mL (26-nm AuNPs), we estimated the total surface area simply based on the diameters of the uncoated AuNPs. Thus, the total available surface area in the suspensions was estimated as approximately 6.37 × 10−4 m2/mL (16-nm AuNPs) and 3.59 × 10−4 m2/mL (26-nm AuNPs). We then calculated the amount of PEG needed to cover all nanoparticles with a single monolayer of four typical PEG samples (APEG 400, 600, 6,000, and 20,000) occupying areas dictated by their Rh (Additional file 1: Tables S1 and S2). These numbers were then compared to the total concentration of PEG available in the solution for the bulk concentration used (11.25 mg/mL). This concentration is considered to ensure that there are at least 5 orders of magnitude more PEG molecules than necessary as needed to saturate the nanoparticle surfaces, based on the above calculations.
where C is the electrolyte concentration (M) and z is the valence of the electrolyte.
In this study, we added varying amounts of 10.0% NaCl solution (40, 50, or 60 μL, w/v) to each PEG-coated AuNP solution (1 mL) to screen the electrostatic repulsion between nanoparticles. The electrostatic repulsion originates from the surface underlying the adsorbed polymer layer. The resulting NaCl concentrations were 65.8, 81.5, and 96.9 mM, respectively. The corresponding values of κ−1 were determined to be 1.19, 1.07, and 0.98 nm, which were calculated using the above data and Equation 5. The amount of the salt present in the added 40 μL of 10.0% (w/v) NaCl solution does not ensure complete screening of the electrostatic repulsion. This may be attributed to the fact that the Rh of APEG 400 is 0.568 nm (2Rh < κ−1 = 1.19 nm) and the zeta potentials of the fully coated nanoparticles range from −13.4 (APEG 400, 16-nm AuNPs) to −9.5 mV (APEG 20,000, 16-nm AuNPs) and from −12.6 (APEG 400, 26-nm AuNPs) to −8.4 mV (APEG 20,000, 26-nm AuNPs) after adding NaCl solution. The salt added in a 50-μL amount of 10.0% (w/v) NaCl solution can more adequately screen the electrostatic repulsion as a result of the relatively shorter κ−1 with the zeta potentials ranging from −8.3 (APEG 400, 16-nm AuNPs) to −4.8 mV (APEG 20,000, 16-nm AuNPs) and from −7.8 (APEG 400, 26-nm AuNPs) to −4.4 mV (APEG 20,000, 26-nm AuNPs) after NaCl addition. Likewise, the amount of salt for the addition of 60 μL of 10.0% (w/v) NaCl solution can also screen the electrostatic repulsion. However, the hydrophobicity of soluble polymer increases at a higher concentration of salt [28, 29]. Hence, 50 μL of 10.0% (w/v) NaCl solution was added to 1 mL of PEG-coated AuNP solutions in order to screen the electrostatic repulsion between nanoparticles. In addition, the pH values of the PEG-coated AuNP solutions were maintained at 6.3, even after salt addition. According to the above analyses, the Uelec = 0, under the salt addition condition.
where L is the radial distance from the center of particles, σ p is the surface density of adsorbed chains, kB is the Boltzmann constant, T is the kinetic temperature, Np is the number of segments in the polymer chain, and l is the segment length.
where A * is the effective Hamaker constant and H is the separation distance between the surfaces of the core particles. According to the DLVO theory, when the surface layers just touch (i.e., H = 2 t), the Usteric = 0. The total energy (Utotal) of the net interaction has a deep minimum that is dependent on the value of the UvdW (Additional file 1: Figure S3) [13, 18, 31]. In general, the minimum of the Utotal(dashed line in Additional file 1: Figure S3) determines the stability of fully coated AuNPs, which is dependent on the t value of the adlayer . If the adlayer is thick enough, the minimum becomes so slight that it can be ignored, thus resulting in greater nanoparticle stability, and vice versa . In other words, the t can determine the SDs of the PEG-coated AuNPs.
AuNP-based colorimetric method to determine 〈 h 2 〉 1/2 and M w values of PEG samples
3.398 ± 0.298
1,561 ± 259
3.444 ± 0.411
1,611 ± 362
6.017 ± 0.368
4,621 ± 537
6.096 ± 0.349
4,736 ± 515
8.086 ± 0.279
8,096 ± 532
7.974 ± 0.397
7,893 ± 747
9.903 ± 0.432
11,919 ± 989
10.032 ± 0.387
12,212 ± 897
In summary, a unique colorimetric method was developed to determine the MW of PEG, based on the steric stabilization of PEG-coated AuNPs. Using the ordinary UV–vis spectrophotometry technique, the MW of the PEG samples can be calculated by the absorbance values of the PEG-coated AuNP solutions, after adding salt to screen the electrostatic repulsion between nanoparticles. This strategy offers operational advantages (simplicity, convenience, and sensitivity) over many existing methodologies, which has important implications for the development of nanomaterial-based determination methods. In the future, this colorimetric method can be applied to the MW determination of other soluble macromolecules. This strategy would provide a great advantage to current research areas in polymer science, materials science, and biology.
KL and HJ are Ph.D. holders, and QZ is a professor. All authors are from the Key Laboratory of Biomedical Material of Tianjin, Institute of Biomedical Engineering, Chinese Academy of Medical Sciences & Peking Union Medical College, Tianjin 300192, People's Republic of China.
PEG samples were purchased from Alfa Aesar
Dynamic light scattering
Multi-angle laser light scattering
Weight average molecular weights
Radii of gyration
Size exclusion chromatography
PEG samples were purchased from Sigma-Aldrich
Transmission electron microscope
Root mean square end-to-end length.
We are grateful for the financial support of Major Research Plan of NSFC (90923042, 913231004), NSFC (31271023), and Graduate Innovation Fund of PUMC (2011-1001-024).
- Knop K, Hoogenboom R, Fischer D, Schubert US: Poly(ethylene glycol) in drug delivery: pros and cons as well as potential alternatives. Angew Chem Int Ed 2010, 49: 6288–6308. 10.1002/anie.200902672View ArticleGoogle Scholar
- Kou D, Manius G, Zhan S, Chokshi HP: Size exclusion chromatography with Corona charged aerosol detector for the analysis of polyethylene glycol polymer. J Chromatogr A 2009, 1216: 5424–5428. 10.1016/j.chroma.2009.05.043View ArticleGoogle Scholar
- Daou TJ, Li L, Reiss P, Josserand V, Texier I: Effect of poly(ethylene glycol) length on the in vivo behavior of coated quantum dots. Langmuir 2009, 25: 3040–3044. 10.1021/la8035083View ArticleGoogle Scholar
- Kojima C, Regino C, Umeda Y, Kobayashi H, Kono K: Influence of dendrimer generation and polyethylene glycol length on the biodistribution of PEGylated dendrimers. Int J Pharm 2010, 383: 293–296. 10.1016/j.ijpharm.2009.09.015View ArticleGoogle Scholar
- Bovey FA, Mirau PA: NMR of Polymers. San Diego: Academic Press; 1996.Google Scholar
- Montaudo G, Montaudo MS, Puglisi C, Samperi F: Characterization of polymers by matrix-assisted laser desorption ionization-time of flight mass spectrometry. End group determination and molecular weight estimates in poly(ethylene glycols). Macromolecules 1995, 28: 4562–4569. 10.1021/ma00117a028View ArticleGoogle Scholar
- Daniel M-C, Astruc D: Gold nanoparticles: assembly, supramolecular chemistry, quantum-size-related properties, and applications toward biology, catalysis, and nanotechnology. Chem Rev 2004, 104: 293–346. 10.1021/cr030698+View ArticleGoogle Scholar
- Rosi NL, Mirkin CA: Nanostructures in biodiagnostics. Chem Rev 2005, 105: 1547–1562. 10.1021/cr030067fView ArticleGoogle Scholar
- Zhao W, Brook MA, Li Y: Design of gold nanoparticle-based colorimetric biosensing assays. ChemBioChem 2008, 9: 2363–2371. 10.1002/cbic.200800282View ArticleGoogle Scholar
- Hayat A: Colloidal Gold: Principles, Methods, and Applications. San Diego: Academic Press; 1989.Google Scholar
- Horisberger M: Colloidal gold: a cytochemical marker for light and fluorescent microscopy and for transmission and scanning electron microscopy. Scanning Electron Microsc 1981, Pt 2: 9–31.Google Scholar
- Heller W, Pugh TL: “Steric protection” of hydrophobic colloidal particles by adsorption of flexible macromolecules. J Chem Phys 1954, 22: 1778.View ArticleGoogle Scholar
- Berg JC: An Introduction to Interfaces and Colloids: The Bridge to Nanoscience. Hackensack: World Scientific; 2010.Google Scholar
- Napper DH: Polymeric Stabilization of Colloidal Dispersions. San Diego: Academic Press; 1983.Google Scholar
- Ratner BD, Hoffman AS: Non-fouling surfaces. In Biomaterials Science: Introduction to Materials in Medicine. 3rd edition. Edited by: Ratner BD, Hoffman AS, Schoen FJ, Lemons JE. San Diego: Academic Press; 2013:241–247.View ArticleGoogle Scholar
- McPherson TB, Lee SJ, Kinam P: Analysis of the prevention of protein adsorption by steric repulsion theory. In Proteins Interfaces II. Washington, DC: American Chemical Society; 1995:28–395.Google Scholar
- Liu Y, Shipton MK, Ryan J, Kaufman ED, Franzen S, Feldheim DL: Synthesis, stability, and cellular internalization of gold nanoparticles containing mixed peptide-poly(ethylene glycol) monolayers. Anal Chem 2007, 79: 2221–2229. 10.1021/ac061578fView ArticleGoogle Scholar
- Stuart AC: Lecture Notes Colloid Science. Wageningen: Wageningen University; 2007.Google Scholar
- Taton TA: Preparation of gold nanoparticle-DNA conjugates. Curr Protoc Nucl Acids Chem 2002, 9: 12.2.1–12.2.12.Google Scholar
- Wang Y, Zhan L, Huang CZ: One-pot preparation of dextran-capped gold nanoparticles at room temperature and colorimetric detection of dihydralazine sulfate in uric samples. Anal Methods 2010, 2: 1982–1988. 10.1039/c0ay00470gView ArticleGoogle Scholar
- Ishikawa Y, Katoh Y, Ohshima H: Colloidal stability of aqueous polymeric dispersions: effect of pH and salt concentration. Colloids Surf B 2005, 42: 53–58. 10.1016/j.colsurfb.2005.01.006View ArticleGoogle Scholar
- Lim JK, Majetich SA, Tilton RD: Stabilization of superparamagnetic iron oxide core-gold shell nanoparticles in high ionic strength media. Langmuir 2009, 25: 13384–13393. 10.1021/la9019734View ArticleGoogle Scholar
- Lee H, Venable RM, Mackerell AD, Pastor RW: Molecular dynamics studies of polyethylene oxide and polyethylene glycol: hydrodynamic radius and shape anisotropy. Biophys J 2008, 95: 1590–1599. 10.1529/biophysj.108.133025View ArticleGoogle Scholar
- Squire PG: Calculation of hydrodynamic parameters of random coil polymers from size exclusion chromatography and comparison with parameters by conventional methods. J Chromatogr A 1981, 210: 433–442. 10.1016/S0021-9673(00)80335-0View ArticleGoogle Scholar
- Devanand K, Selser JC: Asymptotic behavior and long-range interactions in aqueous solutions of poly(ethylene oxide). Macromolecules 1991, 24: 5943–5947. 10.1021/ma00022a008View ArticleGoogle Scholar
- Doi M, Edwards SF: The Theory of Polymer Dynamics. Oxford: Clarendon Press; 1988.Google Scholar
- Liu X, Atwater M, Wang J, Huo Q: Extinction coefficient of gold nanoparticles with different sizes and different capping ligands. Colloids Surf B 2007, 58: 3–7. 10.1016/j.colsurfb.2006.08.005View ArticleGoogle Scholar
- Ricker RD, Sandoval LA, Ricker RD, Sandoval LA: Fast, reproducible size-exclusion chromatography of biological macromolecules. J Chromatogr A 1996, 743: 43–50. 10.1016/0021-9673(96)00283-XView ArticleGoogle Scholar
- Jiang X, van der Horst A, van Steenbergen MJ, Akeroyd N, van Nostrum CF, Schoenmakers PJ, Hennink WE: Molar-mass characterization of cationic polymers for gene delivery by aqueous size-exclusion chromatography. Pharm Res 2006, 23: 595–603. 10.1007/s11095-006-9574-4View ArticleGoogle Scholar
- Genz U, D'Aguanno B, Mewis J, Klein R: Structure of sterically stabilized colloids. Langmuir 1994, 10: 2206–2212. 10.1021/la00019a029View ArticleGoogle Scholar
- Roucoux A, Schulz J, Patin H: Reduced transition metal colloids: a novel family of reusable catalysts? Chem Rev 2002, 102: 3757–3778. 10.1021/cr010350jView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.