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Scaling for Sedimentation and Diffusion of Poly(ethylene glycol) in Water Zhenli Luo and Guangzhao Zhang* Hefei National Laboratory for Physical Sciences at Microscale, Department of Chemical Physics, UniVersity of Science and Technology of China, Hefei, China 230026 ReceiVed: July 9, 2009; ReVised Manuscript ReceiVed: August 7, 2009
The sedimentation and diffusion of poly(ethylene glycol) (PEG) with weight average molecular weight (Mw) from 5 × 102 to 2 × 105 g/mol have been investigated by use of an analytical ultracentrifuge (AUC). The sedimentation velocity analysis based on the Lamm equation gives the sedimentation coefficient (s) and diffusion coefficient (D). Our study reveals that the sedimentation coefficient (s0) and diffusion coefficient (D0) at infinite dilution scale to molecular weight (Mw), namely, s0 ) KsMwR, with Ks ) 6.14 × 10-3 and R ) 0.469 ( 0.008, and D0 ) KDMw-β, with KD ) 1.43 × 10-8 and β ) 0.576 ( 0.007. Introduction
TABLE 1: Characterization Data of PEG Samples
Poly(ethylene glycol) (PEG), also known as poly(ethylene oxide) (PEO) or polyoxyethylene (POE), which is an oligomer or polymer of ethylene oxide with the monomeric unit of -(O-CH2CH2)-, exhibits unusual properties in solution.1 Such a polymer has been used as a model to study the mechanism about water-macromolecule and protein-macromolecule interactions.2,3 Moreover, PEG has found applications in drug delivery,4-6 papermaking,7,8 protein stabilization,9,10 and protein resistance.11,12 However, the origin of its peculiar properties remains largely unclear. Molecular parameters of a polymer including the sedimentation coefficient (s), diffusion coefficient (D), and weight average molecular weight (Mw) are fundamentally important for understanding its dynamics and properties. Furthermore, the D-Mw scaling can be used to evaluate the molecular weight of a polymer based on the measured diffusion coefficient. Theoretically, s, D, and Rh of a polymer scale with Mw as follows13,14
sample
Mw (g/mol)a
Mw (g/mol)b
Mw/Mna
Mw/Mnb
ν (mL/g)
PEG1 PEG2 PEG3 PEG4 PEG5 PEG6 PEG7 PEG8 PEG9 PEG10
630 1520 4400 7000 12400 19800 38700 77000 166000 333000
510 1520 3940 6220 10300 16400 32000 70600 129000 229000
1.06 1.08 1.06 1.11 1.06 1.05 1.07 1.09 1.13 1.36
1.02 1.09 1.03 1.02 1.08 1.11 1.13 1.37
0.841 0.835 0.835 0.833 0.834 0.828 0.830 0.827 0.825 0.822
s0 ) KsMwR
(1)
D0 ) KDMw-β
(2)
Rh,0 ) KRMwγ
(3)
where s0, D0, and Rh,0 are the sedimentation coefficient, diffusion coefficient, and hydrodynamic radius of the polymer at infinite dilution. Ks, KD, and KR are the scaling prefactors, and R, β, and γ are the corresponding scaling indexes. On the other hand, D and Rh can be related by the Stokes-Einstein equation
D)
kBT k BT ) f 6πηRh
(4)
where kB, T, f, and η are the Boltzmann constant, absolute temperature, frictional coefficient, and solvent viscosity, respectively. Obviously, β ) γ in terms of eqs 2-4. The indexes * To whom correspondence should be addressed. E-mail: gzzhang@ ustc.edu.cn.
a Determined by SEC. b Determined by AUC, where - means that AUC cannot give the data.
R and β can describe the chain conformation. For a random coil, R ) 0.5-0.6 and β ) 0.4-0.5. For a compact sphere, R ) 2/3 and β ) 1/3.14 The solution properties of PEG have been investigated by a number of techniques.15-20 Laser light scattering was used to study the diffusion and chain interactions of PEG with Mw from 4 × 104 to 106 g/mol in water.15,19 Unfortunately, PEG with lower Mw was not characterized because of the low scattering intensity. Nuclear magnetic resonance (NMR) was successfully used to study the diffusion of PEG with Mw from 100 to 6 × 105 g/mol.16,18 However, the solvent D2O used in NMR is somewhat different than H2O in nature, particularly in polymer solubility, which may affect the chain conformation.21,22 Thus, the diffusion of PEG in a wide range of molecular weights from an oligomer up to a polymer with high Mw in H2O needs a systematic examination. An analytical ultracentrifuge (AUC) can characterize the sedimentation coefficient, diffusion coefficient, and absolute molecular weight of a sample with molar masses from 102 to 1013 g/mol based on sedimentation velocity (SV) and sedimentation equilibrium (SE) analysis.23-28 In this study, we have investigated the diffusion of PEG with molecular weights from 5 × 102 to 2 × 105 g/mol by sedimentation velocity analysis. Our aim is to understand the scaling for the diffusion of PEG in water. Experimental Section Sample Preparation. All PEG samples (Table 1) were purchased from Polymer Standards Service and used as received.
10.1021/jp906468n CCC: $40.75 2009 American Chemical Society Published on Web 08/26/2009
Sedimentation and Diffusion of PEG in Water
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Each PEG solution was prepared by dissolving the PEG sample in Milli-Q water (Millipore, resistivity ) 18.2MΩ · cm, 25 °C) for 24 h under stirring rate of about 10 rpm. The concentration of the stock solution for SV measurements was 1.0 mg/mL, and that for either size exclusion chromatography (SEC) or partial specific volume measurements was 2.0 mg/mL. SEC Measurements. The Mw and polydispersity (Mw/Mn) of PEG (Table 1) were first measured by SEC (Waters) equipped with a 1515 pump and Ultrahydrogel (Waters) 2000, 1000, 500, and 250 columns at 40 °C. Milli-Q water filtered by a 0.45 µm PVDF film with a flow rate of 1.0 mL/min was used as the eluent. An Optilab Rex (Wyatt) differential refractometer was used as the detector. A volume of 50 µL of PEG solution with a concentration of 2.0 mg/mL filtered by a 0.22 µm PVDF film was injected in the measurements. The data were collected and analyzed by Waters Breeze (version 3.30) software. Measurements of the Partial Specific Volume. The partial specific volume (ν) of PEG in aqueous solution was determined on a DMA4500 densitometer (Anton Paar) at 20 °C (Table 1) based on eq 5
ν)
1 ∆F 1F0 ∆c
(
)
(5)
where F0 and F are the solvent density and solution density, respectively, and c is the solute concentration.29 The concentrations used were 0.3, 0.6, 0.9, 1.2, and 1.5 mg/mL. SV Measurements. SV experiments were performed on a Beckman Optima XL-I analytical ultracentrifuge (Beckman Coulter Instruments) with an An-60 Ti rotor, three cells assembled by sapphire windows, and a double-sector 12 mm length aluminum centerpiece and interference optics detector. All experiments were conducted at 60000 rpm and 20 °C. A volume of 400 µL of PEG solution was loaded for measurement with 420 µL of Milli-Q water as the reference; 900 scans of data for each sample were acquired at a time interval of 2 min for PEG1-6 and 1 min for PEG7-10. The concentrations used were 0.1, 0.4 (0.3 for PEG9 and PEG10), 0.7, and 1.0 mg/mL. In principle, the change of solute concentration with time and radius during sedimentation can be described by the Lamm equation24
[
]
∂c ∂2c 1 ∂c ∂c - sω2 r + 2c )D 2 + ∂t r ∂r ∂r ∂r
[
]
(6)
where c, t, r, and ω are the solute concentration, sedimentation time, radial distance from the rotation axis, and rotor angular velocity, respectively. The s is the sedimentation coefficient in units of S (Svedberg or 10-13 second). On the other hand, s relates to D by the Svedberg equation24
M)
s RT (1 - νF0) D
(7)
where R is the gas constant. Both the diffusion and sedimentation can influence the change of solute concentration as a function of time. Their roles can be extracted simultaneously by solving eq 6. Note that a naive Lamm equation only suits for single species. For polydisperse species, a finite element analysis by SEDFIT software with the continuous c(s) distribution model and continuous c(s,ff0) model can be used to solve the superposition of eq 6 so that s and D are simultaneously
Figure 1. Sedimentation velocity analysis of PEG7 at 60000 rpm and 20 °C, where the concentration is 0.1 mg/mL (every 60th scan loaded). (a) The collected data (discrete points) and fit data (solid line); systematic noise is subtracted by SEDFIT; (b) fit residual.
extracted with a relative error of 5%, where the size heterogeneity can be discriminated from diffusion spreading.30,31 In the continuous c(s) distribution model, s is fit with frictional ratio (f/f0) corresponding to Rh as the variant in a certain range to obtain its distribution, where f0 ) 6πηRh,c and f0 and Rh,c are the frictional coefficient and hydrodynamic radius of a compact particle. D can be obtained from Rh based on eq 4.30 The distribution of D can be obtained by fitting D in a certain range by use of the continuous c(s,ff0) model. On the other hand, the number average molecular weight (Mn) and weight average molecular weight (Mw) can then be measured in terms of the following equations32
RT Mn ) (1 - νF0)
∑ skck,l k,l
∑ Dk,lck,l
(8)
k,l
s
Mw )
RT (1 - νF0)
∑ Dk,lk ck,l k,l
∑ ck,l
(9)
k,l
where ck,l is a fraction of s and D. Thus, we can obtain the polydispersity by SV measurements. Results and Discussion Sufficient data scans are loaded to SEDFIT (version 11.71), and the continuous c(s) distribution model with maximum entropy regularization is chosen to fit s and D. Figure 1 shows a typical sedimentation profile of the PEG sample (PEG7), where the concentration is 0.1 mg/mL. The discrete points are the data with systematic noise subtracted by SEDFIT, and the solid lines represent the best fit (Figure 1a). Figure 1b shows the fit residual. The frictional ratio (f/f0) is 2.78, with a rootmean-square deviation of 0.0043. The weight average s and D are 0.79 S and 3.49 × 10-11 m2/s, respectively. Meanwhile, Mw (3.2 × 104 g/mol) can be obtained based on eq 7. The polydispersity (Mw/Mn) estimated by using the continuous c(s,ff0) model based on eqs 8 and 9 is 1.08.32,33 Likewise, we can characterize other samples. The characterization data of all samples are listed in Table 1. Note that Mw obtained by SV is some smaller than that obtained by SEC. This is understandable because the latter gives a relative molecular weight. The
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Figure 2. Sedimentation coefficient (s) distributions of PEG1, PEG4, PEG7, and PEG10 at 20 °C, where the concentration is 0.1 mg/mL.
Figure 3. Concentration (c) dependence of sedimentation coefficients (s) of PEG1, PEG4, PEG7, and PEG10 at 20 °C.
polydispersity measured by AUC is consistent with that by SEC. This is because Mw relates to s in terms of eq 7. In other words, AUC can characterize the polydispersity of polymers well. Figure 2 shows the typical s distributions of PEG, where PEG1, PEG4, PEG7, and PEG10 with a concentration of 0.1 mg/mL were used. Obviously, PEG1, PEG4, and PEG7 exhibit a narrow distribution, while PEG10 has a broad distribution. As shown in Table 1, the sedimentation coefficient distributions are consistent with the polydispersity indexes characterized by SEC, namely, the sedimentation coefficient distribution reflects the size distribution of individual chains themselves. Note that previous work demonstrates that PEG molecules aggregate even in a dilute solution, characterized by two separate peaks, because the aggregates have a much larger molar mass than the individual chains.34-37 We did not observe such aggregates in our sedimentation velocity experiments even at a concentration as high as 5.0 mg/mL (not shown). This is probably because the binding between PEG chains in the aggregate is weak,35 and the aggregates are destroyed by the centrifugal force. It is known that the sedimentation coefficient and diffusion coefficient have concentration dependence due to the hydrodynamic nonideality originating from the solute-solute and solute-solvent interactions.19,24,38 Figure 3 shows c dependence of s. When Mw is below 6 × 103 g/mol (PEG1-4), s almost does not change with concentration. This is because the lowmolecular-weight PEG molecules separate far from each other and the interactions between them are still weak in the concentration range that we investigated. When Mw is higher than 6 × 103 g/mol (PEG5-10), PEG molecules with larger size are closer to each other, and the interactions between them increase. As a result, PEG molecules restrict each other in their movement. That is why the sedimentation coefficient decreases with the concentration when Mw > 6 × 103 g/mol. Extrapolation of s to zero concentration leads to s0, where the concentration dependence is eliminated. It is obvious that s has linear dependence of c, namely, s ) s0(1 - ksc), where ks is the sedimentation concentration coefficient.
Luo and Zhang
Figure 4. Concentration (c) dependence of diffusion coefficients (D) of PEG1, PEG4, PEG7, and PEG10 at 20 °C.
Figure 5. Weight average molecular weight (Mw) dependence of sedimentation coefficient (s0) at infinite dilution at 20 °C.
Figure 6. Weight average molecular weight (Mw) dependence of the diffusion coefficient (D0) and hydrodynamic radius (Rh,0) at infinite dilution at 20 °C.
Figure 4 shows the c dependence of D. Like s, D slightly varies with c when Mw is lower than 6 × 103 g/mol (PEG1-4) but decreases with c when Mw is higher than 6 × 103 g/mol (PEG5-10). The higher interactions between PEG molecules at higher molecular weight are responsible for the behavior. Extrapolation of D to zero concentration leads to D0. Figure 4 shows that D has linear dependence on c, that is, D ) D0 (1 kDc), where kD is the diffusion concentration coefficient. Figure 5 shows the Mw dependence of s0 in a double logarithmic plot. Obviously, s0 ) KsMR. This agrees with the theoretical prediction (eq 1). From the plot, we have Ks ) 6.14 × 10-3 S and R ) 0.469 ( 0.008. The scaling index R indicates that the PEG chain exists as a random coil in water.14,27,28,39 This scaling can help understand the interactions of PEG and proteins.3,40 Figure 6 shows the Mw dependence of D0. From the double logarithmic plot, we have D0 ) KDMw-β, which is consistent with eq 2. Also, we can obtain KD ) 1.43 × 10-8 m2/s and β ) 0.576 ( 0.007. The scaling index β indicates that water is a good solvent for PEG at 20 °C and that PEG molecules are random coils.14 Figure 6 also shows the Mw dependence of Rh,0.
Sedimentation and Diffusion of PEG in Water The combination of eqs 2 and 4 leads to Rh,0 ) KRMwγ, where KR and γ are 0.015 nm and γ ) β ) 0.576 ( 0.007. The results are consistent with those from light scattering.14,18 However, the β value here is smaller than that from NMR measurements in D2O.15 This should be due to the difference between D2O and H2O. It should be stressed that the scaling relations work for PEG with a wide range of molecular weights from an oligomer up to a polymer. The scaling is different from that for the diffusion of polymers in an organic solvent. A deviation can be observed when the molecular weight is below a critical value in the latter.41-43 Since PEG oligomers are more hydrated and extended in aqueous solutions, the index does not decrease, so that such a deviation cannot be observed.16,19 Conclusion The analytical ultracentrifuge (AUC) study on PEG with a wide range of molecular weights from an oligomer up to a polymer in H2O leads to the following conclusions. AUC is sensitive to measurement of the molecular weight and molecular weight distribution of oligomers and polymers. For PEG with Mw from 5 × 102 to 2 × 105 g/mol, s0 and D0 scale to Mw, that is, s0 ) KsMwR, with Ks ) 6.14 × 10-3 and R ) 0.469 ( 0.008, and D0 ) KDMw-β, with KD ) 1.43 × 10-8 and β ) 0.576 ( 0.007. Acknowledgment. The financial support of the National Distinguished Young Investigator Fund (20474060) and Ministry of Science and Technology of China (2007CB936401) is acknowledged. References and Notes (1) Bailey, F. E.; Koleske, J. V. Poly(ethylene oxide); Academic Press: New York, 1976. (2) Barnes, A. C.; Enderby, J. E.; Breen, J.; Leyte, J. C. Chem. Phys. Lett. 1987, 142, 405. (3) Lu, Y. L.; Harding, S. E.; Turner, A.; Smith, B.; Athwal, D. S.; Grossmann, J. G.; Davis, K. G.; Rowe, A. J. J. Pharm. Sci. 2008, 97, 2062. (4) Pasut, G.; Veronese, F. M. Prog. Polym. Sci. 2007, 32, 933. (5) Fipula, D.; Zhao, H. AdV. Drug DeliVery ReV. 2008, 60, 29. (6) Veronese, F. M.; Mero, A. Biodrugs 2008, 22, 315. (7) Yamauchi, T.; Tamai, N. J. Appl. Polym. Sci. 2003, 89, 2798. (8) Khoultchaev, K. K.; Kerekes, R. J.; Englezos, P. AICHE J. 1997, 43, 2353.
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