J. Phys. Chem. 1996, 100, 13687-13695
13687
Dilute Aqueous Poly(ethylene oxide) Solutions: Clusters and Single Molecules in Thermodynamic Equilibrium Marco Polverari and Theo G. M. van de Ven* Paprican and Department of Chemistry, Pulp and Paper Research Centre, McGill UniVersity, Montreal, Canada, H3A 2A7 ReceiVed: January 23, 1996; In Final Form: April 25, 1996X
Dilute aqueous solutions of broad and narrow distribution poly(ethylene oxide) fractions were studied using dynamic (DLS) and static (SLS) light scattering, as well as gel permeation chromatography (GPC). It was found that above a critical self-association concentration, which depends on the molecular weight of the PEO, polymer clusters and free polymer coils coexist in a thermodynamic equilibrium. The polymer clusters can be removed by filtration and begin to reform spontaneously in solution within 30 min after filtration. A steady-state amount of PEO in clusters is attained within about 24 h. At steady-state the narrow-distribution and broad-distribution PEO samples have 7-10% wt/wt and 12-18% wt/wt of the polymer present as clusters in solution, respectively, at PEO concentrations of about 250 mg/L. It was found that the PEO clusters were composed of a few hundred polymer chains per cluster. From DLS the polymer cluster diameters were found to be independent of molecular weight and to decrease, with time, from 0.90 to 0.45 µm. In dynamic light scattering studies of the PEO solutions containing clusters, the light scattered from the clusters is mixed with light scattered from freely dissolved molecules. This mixing of light with different Doppler-shifted frequencies leads to quasi-heterodyning.
1. Introduction A large amount of work has recently been done to understand the unusual properties of poly(ethylene oxide) in solution. Poly(ethylene oxide) (PEO) has attracted considerable interest for its unique properties: unlike structurally similar polyethers, PEO is soluble in water at moderate temperatures for a wide range of degrees of polymerization, from oligomers up to a molecular weight of 7 million.1-3 PEO continues to be of interest on account of its ability to readily adsorb at particle interfaces,4 confer steric stability,5,6 as well as act as a good retention aid, surfactant, and drag reducing agent in industrial processes.7-10 Solution properties of PEO have been extensively studied, particularly in aqueous solution,4,11-16 On the basis of spectroscopic evidence, suggestions have been made that the water solubility of PEO results from a favorable fit with the water lattice and that the polymer has a more ordered structure in aqueous media than in organic media.17-22 Recent papers suggest that the gel-like aggregates or clusters exist in solution. Some authors have found that the efficiency in drag reduction for PEO is enhanced by its propensity for forming such aggregates in solution, even at very low concentrations.10,23-27 Aggregation of PEO in solution has been confirmed by light scattering,28-43 electron microscopy,44 viscometry,28,45 spectroscopic techniques,17,30,46 and sedimentation velocities.37,46 Moreover, the formation of clusters in solution has been found to be dependent on a number of solution properties: solvent type,34,38,41,44 temperature,28,30,35,39,42,43,45 polymer solution concentration,17,33,37,39,44 and the type of salt solution.48-51 The present paper describes dynamic light scattering measurements on dilute solutions of aqueous PEO, and a comparison is made of these data with those from Zimm analysis. GPC studies of aqueous and chloroform PEO solutions are also used to characterize the polymer molecular weights and, where possible, the degree of polymer aggregation. * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, July 15, 1996.
S0022-3654(96)00215-8 CCC: $12.00
The principal aims of this work are to ascertain the presence of clusters in solution as well as to determine the globule size and quantity, and their dependence on solution properties such as polymer solution concentration, polymer molecular weight, polydispersity of the polymer sample, and solvent type. A comparison of narrow-distribution and polydisperse polymer samples is also required for future polymer adsorption studies. 2. Theoretical Background A. Photon Correlation Spectroscopy. Monodisperse Dispersions or Polymer Solutions. The normalized homodyne autocorrelation function obtained from dynamic light scattering measurements, also referred to as photon correlation spectroscopy (PCS), on an ensemble of monodisperse colloidal particles is given by53
g(2)(τ) ) 1 + e-2Γτ
(1)
where τ is the autocorrelation time and Γ is defined as
Γ ) Dq2
(2)
Here D is the translational diffusion constant related to the particle radius by the Stokes-Einstein relationship54
D ) kT/6πηa
(3)
where k is the Boltzmann constant, T is the absolute temperature, η is the viscosity of the suspending medium, and a is the particle radius. The magnitude of the scattering vector, q, is given by
q)
θ 4πn sin λ0 2
()
(4)
where n is the refractive index of the medium, λ0 is the wavelength of the light used, and θ is the scattering angle. © 1996 American Chemical Society
13688 J. Phys. Chem., Vol. 100, No. 32, 1996
Polverari and van de Ven
Equation 1 applies to a monodisperse polymer solution, provided qa < 5. For qa > 5 the internal motions of the polymer coils also contribute to the autocorrelation function.55 Bimodal Dispersions or Polymer Solutions. For a bimodal dispersion or polymer solution consisting of two monodisperse fractions A and B of different sizes, where A is the smallersized fraction, we can define the normalized homodyne autocorrelation function as 2
(2) gBI (τ) ) 1 +
2
IA
(IA + IB)
e-2ΓAτ +
2
IB
e-2ΓBτ + (IA + IB)2 2IAIB -ΓAτ ΓBτ e e(5) (IA + IB)2
where IA and IB is the intensity of scattered light reaching the detector from fraction A and fraction B, respectively. ΓA and ΓB are given by eq 2 with D ) DA and D ) DB, respectively. A PEO solution containing both single molecules and clusters can be analyzed with the aid of eq 5. Polydisperse Samples and the Cumulant Method. For polydisperse systems the autocorrelation function can be analyzed in terms of a cumulant expansion.56 The average value of Γ is given by
Γ h ) ∫ΓG(Γ) dΓ
(7)
where µi is the ith moment of the distribution. B. Static Light Scattering and Zimm Plot Analysis. Static Light Scattering. It can readily be shown that a cluster consisting of N macromolecules, each of molecular weight M, scatters light in the same way as a single macromolecular coil of molecular weight MN ) NM. From fluctuation theory70 it follows that, for a solution of macromolecular clusters, the light scattered at a given angle is given by
N dn 2cMV* Isca ∝ ∆R ) 2 2 4π n dc NA
( )
2
(8)
where ∆R is a fluctuation in polarizability, n the refractive index of the solvent, c the polymer concentration, NA Avogadro’s number, and V* the scattering volume. Equation 8 applies to small clusters and to larger clusters at low scattering angles. When no clusters are formed in solution, N ) 1 and eq 8 equals the classical result for scattering from polymer solutions. Zimm Analysis. The weight average molecular weight, Mw, the z-average rms radius of gyration, Rg ) 〈rg2〉z1/2, and the second virial coefficient, A2, of a dilute polymer solution are related to the angular distribution of scattered light as follows: 57,58
K*c 1 + 2A2c ) R(θ) P(θ)Mw
(9)
where R(θ) is the excess scattering intensity ratio (Rayleigh ratio) of the polymer solution at concentration c at angle θ, K* is an optical constant equal to
(dndc) λ
K* ) 4π2n2
2
NA-1
-4
PEO sample
Mw
Mn
Mw/Mn
S-1 S-2 S-3 S-4 S-5
54.3 145 270 606 783
51.8 142 246 594 691
1.05 1.02 1.09 1.02 1.13
in which λ is the wavelength of horizontally polarized light and P(θ) is the form factor of scattering. This factor describes the angular variations of the intensity due to molecular extension and is equal to
1 16π2 2 2 θ Rg sin )1+ 2 P(θ) 3λ2
()
(10)
(11)
Plotting the left-hand side of eq 9 against a function of angle and concentration, to produce a Zimm plot, provides a graphical method of solving eq 9 for Mw, Rg, and A2. The experimental points are plotted with (K*c/R(θ)) as the vertical axis and sin2(θ/2) + Bc as the horizontal axis, where B is a scaling factor. The data are then extrapolated to c ) 0 and θ ) 0 (i.e., where P(θ) ) 1). According to polymer solution theory59-65 the second virial coefficient is related to the polymer-solvent interaction parameter, χ:
(6)
where G(Γ) is the normalized distribution in Γ-values. The homodyne autocorrelation function then becomes
g(2)(τ) ) 1 + e-2Γh τ(1 + 1/2µ2τ2 - 1/6µ3τ3 + ...)2
TABLE 1: Narrow PEO Molecular Weight Fractions
A2 )
( )( ) 1 -χ 2
F h(z) Fp Ms 2
(12)
where F is the density of the solvent, Fp is the density of the polymer, and Ms is the molecular weight of the solvent (i.e. water). The function h(z) is a correction to the Flory-Huggins (FH) theory,66 for which A2 is given by eq 12 with h(z) ) 1. The parameter z is an “excluded volume” variable.59 Similarly to Gaeckle and Patterson,61 the function h(z) was approximated by the Krigbaum-Carpenter-Kaneko-Roig (KCKR) theory,62 as well as with the Yamakawa-Kurata67 (YK) theory which takes the effects of polydispersity into account. 3. Experimental Section A. Polymers. Narrow-distribution poly(ethylene oxide) fractions were obtained from Scientific Polymer Products (Ontario, N.Y), (set-A) and are presented in Table 1. The molecular weights and polydispersities were provided by the manufacturer. A set of broad-distribution poly(ethylene oxide) fractions was obtained from Aldrich Chemical Co. (Milwaukee, WI), (setB). The Mv values (molecular weights from viscosity) were given by the manufacturer as 100, 200, 300, 600, and 900. These samples are indicated as PEO-1, PEO-2, PEO-3, PEO-4, and PEO-5, respectively. B. Solution Preparation. Freshly double distilled, deionized water was used to prepare aqueous PEO solutions. Analytical grade chloroform was used to prepare the PEO solutions in chloroform. All solvents were filtered three times through a 0.22 µm chromatographic filter prior to use to remove all dust. All glassware was cleaned in a potassium dichromate acid solution, followed by rinsing in distilled, deionized water. The glassware was then dried. Prior to use of the glassware, the glassware was rinsed with the appropriate dust-free solvent twice. C. Instrumentation and Data Analysis. Gel Permeation Chromatography. Approximately 0.3 g/L solutions of PEO were prepared in 50 mL volumetric flasks by weighing. The
Dilute Aqueous Poly(ethylene oxide) Solutions
J. Phys. Chem., Vol. 100, No. 32, 1996 13689
TABLE 2: Dilute PEO Solution Concentrations for GPC, PCS, and SLS PEO sample
solvent type
GPC/PCS solution concn (mg/L)
S-1 S-2 S-3 S-4 S-5 PEO-1
H2O/CHCl3 H2O/CHCl3 H2O/CHCl3 H2O/CHCl3 H2O/CHCl3 H 2O
221 248 242 251 363 262
PEO-2
H 2O
313
PEO-3
H 2O
285
PEO-4
H 2O
244
PEO-5
H 2O
251
ZIMM solution concn (mg/L)
TABLE 3: Comparison of Average Molecular Weights of the Broad Distribution PEO samples by GPC and Zimm Analysis Mw aqueous solution
CHCl3 Mn CHCl3 solution solution
Zimm GPC GPC PEO Mv sample (supplier) analysis analysis analysis 89 380 550 150 293 596 79 204 402 131 218 261 250 340 540
solutions were stirred vigorously on a stirring plate for 48 h at room temperature. The solutions were degassed under vacuum. This provided a stable baseline, enhanced sensitivity, reproducible retention times, and reproducible injection volumes. If necessary, the solutions were filtered through a 0.22 or 0.45 µm filter. A clean filter was used for each solution. A 10 µL sample was immediately injected into the gel permeation chromatograph (GPC). The solutions were prepared from the PEO samples, the concentrations of which are given in Table 2. The solutions prepared from the narrow-distribution PEO fractions in Table 1 were used to calibrate the GPC instruments. The aqueous PEO solutions were analyzed using a Varian DS604 GPC with a 600 Series data system. The system was connected to a WATERS 410 differential refractometer, and the flow was supplied by an LKB Bromma 2248 HPLC pump. The GPC column used was a SHODEX OHpak SB-8M manufactured by SHOWA Denko K. K. The flow was 1 mL/ min at a column pressure of 1.7 MPa at 25 °C. The chloroform PEO solutions were analyzed using a WATERS 150-CV GPC with a MAXIMA 820 GPC analysis data system. The GPC column used was a 7.8 × 300 mm µStyragel HT linear 10 µm column manufactured by WATERS. The GPC was equipped with an on-line viscometer. The rate of data acquisition was 2 points/min for a 20 min duration. The operating temperature was 40 °C. The flow rate was 1 mL/ min. Both GPC’s were calibrated using the narrow PEO standards in Table 1. Dynamic Light Scattering. The same aqueous PEO solutions used for GPC were also used for the dynamic light scattering experiments. The intensity and the diffusion coefficients of the polymers in solution were measured as a function of time. Dynamic light scattering experiments were made using a vertically polarized 50 mW He-Ne laser manufactured by Spectra Physics. The scattering plane was perpendicular to the incident light polarization. The incident wavelength was 632.8 nm. A commercial goniometer (Brookhaven Instruments BI2030) was used with its original integrated optics to measure the scattered light at 90°. A refractive index matching bath of filtered decalin (0.22 µm) surrounded the scattering cell, and its temperature was controlled to 25 ( 0.1 °C. Most autocorrelation functions were analyzed using the cumulant method (eq 7) except when indicated otherwise. For the narrowdistribution PEO samples the differences between the results
PEO-1 PEO-2 PEO-3 PEO-4 PEO-5
100 200 300 600 900
167 216 331 497 572
118 242 356 471 586
117 240 355 469 545
GPC analysis
Mw/Mn
22.7 76.9 112 162 229
5.18 3.12 3.17 2.86 2.37
of eqs 1 and 7 are very small, confirming that the samples are monodisperse. Static Light Scattering. Aqueous solutions of PEO were prepared for each of the broad distribution polymers by dissolving approximately 0.1 g of polymer in a 50 mL volumetric flask. The solutions were stirred vigorously for 48 h on a stirring plate. A dilution series of three or four concentrations ranging from 100 to 500 mg/L was then prepared for each polymer sample. A 10 mL sample of the diluted solutions was immediately filtered through a 0.22 µm filter into a scattering cell. The intensity as a function of angle was immediately taken. If the measured light scattering curve was found to be nonlinear, the sample was filtered again until a linear Kc/R(θ) vs sin2(θ/2) curve was obtained. For experimental purposes K was defined as K* × 10-5. The final polymer concentration in the diluted, filtered sample was determined gravimetrically or by GPC. The final solution concentrations are included in Table 2. Light scattering intensity measurements were made using a commercial light scattering instrument (DAWN Model-F by Wyatt Technology Corp.) at 25 °C. The incident light laser source was a 3 mW horizontally polarized He-Ne of wavelength 632.8 nm. The scattering angles were between 25° and 130°. The refractive index increment was obtained from the literature68 as 0.135 mL/g at 25 °C and at a wavelength of 632.8 nm. The light scattering data were analyzed on the basis of eq 9 using the SKOR-F87 software package supplied by the manufacturer. The Mw, radii of gyration, and second virial coefficients were obtained by plotting the intensity data using the AURORA software package also supplied by the manufacturer. The polymer-solvent interaction parameter, χ, was calculated from eq 12 using for the polymer density the value of Fp ) 1.13 g/mL. 4. Results and Discussion A. Gel Permeation Chromatography. GPC results in chloroform and in aqueous solutions are summarized in Table 3. Table 3 also shows the molecular weights obtained in chloroform, which is a good solvent for PEO. The experimental polydispersities vary between 2.37 for high molecular weight polymers to 5.18 for lower molecular weight polymers. Table 3 also summarizes the number-average and weight-average molecular weights obtained in aqueous and chloroform solutions using GPC and static light scattering. Very good agreement is found for Mw by these different methods in the two solvents. However, large differences exist between these Mw values and the viscosity molecular weights, Mv, cited by the manufacturer for the broad-distribution PEO fractions. Our results show GPC elution peaks due to clusters in aqueous solution. No elution peaks due to clusters were found in a chloroform solution. The columns used showed no adsorption of polymer. This was evidenced by eluting better solvents for
13690 J. Phys. Chem., Vol. 100, No. 32, 1996
Polverari and van de Ven TABLE 4: Elution Times and Peak Areas for Dilute Solutions of Narrow-Distribution PEO Samples elution peak area time (min) (arbitrary units) PEO filter concna latency peak peak 2 sample size (µm) (mg/L) period (h) 1 S-1 S-2 S-3
Figure 1. Typical GPC elution peaks for PEO in water. Experimental data for sample S-2. Peak 1 is due to clusters. Peak 2 is due to free polymer in solution. (a) Solution filtered through a 0.45 µm filter. (b) Solution filtered through a 0.22 µm filter. (c) Solution filtered through a 0.22 µm filter and allowed to reequilibrate for 24 h.
PEO, such as benzene, through the column after the experiments were conducted. (i) Narrow-Distribution PEO Samples. In an attempt to determine the existence and the amount of clusters PEO in solution, the narrow-distribution PEO aqueous solutions (setA) previously used to calibrate the GPC instruments were left undisturbed for a periods of 0.5 and 24 h. At the end of each period, a 10 µL sample was injected into the GPC. The GPC elution diagrams clearly showed a set of two elution peaks regardless of the PEO molecular weight fraction. A typical elution diagram is shown in Figure 1a. Peak 1 is attributed to the clusters while peak 2 is due to the free polymer in solution. When the aqueous polymer solution is filtered through a 0.22 µm filter and the sample is immediately reinjected into the GPC, peak 1 is no longer detected. This behavior is typical of all the narrow distribution PEO samples. Figure 1b shows a typical elution diagram for the filtered PEO system. As can be seen from Figure 1b, the elution time of peak 2 does not change. If this filtered polymer solution is left undisturbed for an additional period of 24 h, peak 1 is detected again. This is shown in Figure 1c. The peak area for the free polymer (peak 2 in Figure 1c) is now found to be smaller than in Figure 1b. This is due to the fact that a portion of the free polymer in Figure 1b has formed clusters to reform peak 1 in Figure 1c. If peaks 1 and 2 for Figure 1c are summed, however, the total area is found to be equal to the peak area in Figure 1b. This behavior is typical of all the narrow PEO fractions in setA. This observation implies that the integrated peak area is directly proportional to the amount of aggregated polymer in solution and therefore it is possible to directly monitor the amount of aggregated polymer in solution using GPC. The results are summarized in Table 4 for the narrow-distribution PEO samples. Regardless of the PEO molecular weight, the elution peak attributed to the PEO clusters has the same elution time for all the narrow-distribution PEO fractions. Since the size of the clusters can range from 0.4 to 0.9 µm (see below), this indicates that the clusters are eluted at the exclusion limit of the GPC column. This exclusion limit corresponds to a molecular weight of about 1.1 million and indicates that the clusters must contain more than 10 polymer chains. The data show that the formation of clusters is thermodynamically driven. If 24 h is considered to be a condition sufficient for equilibrium to be established, the amount of aggregated polymer in dilute polymer solutions (≈250 mg/L) varies between 7 and 10% wt/wt. (ii) Broad-Distribution PEO Samples. The broad-distribution PEO fractions show similar trends as the narrow distribution
a
0.45 0.22 0.22 0.45 0.22 0.22 0.45 0.22 0.22
321 301 301 225 207 207 196 190 190
0.5 0.5 24 0.5 0.5 24 0.5 0.5 24
6.05 7.55 7.35 5.90 7.40 6.05 8.30 8.00 6.05 8.20 6.00 8.70 8.60 6.05 8.55
peak 1
peak 2
% peak 1
0.008
0.128 0.131 0.133 0.102 0.103 0.105 0.098 0.071 0.071
6.25 0 9.02 7.84 0 9.53 3.06 0 7.04
0.012 0.008 0.010 0.003 0.005
For filtered samples measured after filtration.
TABLE 5: Elution Times and Peak Areas for Dilute Solutions Broad-Distribution PEO Fractions elution time (min)
peak area (arbitrary units)
PEO filter size concna latency peak peak peak (mg/L) period (h) 1 2 1 sample (µm) PEO-1
PEO-2
PEO-3
PEO-4
PEO-5
a
N/Fb 0.45 0.22 0.22 N/Fb 0.45 0.22 0.22 N/Fb 0.45 0.22 0.22 N/Fb 0.45 0.22 0.22 N/Fb 0.45 0.22 0.22
262 236 234 234 313 307 304 304 285 275 265 265 244 238 236 236 251 236 233 233
0.5 0.5 0.5 24 0.5 0.5 0.5 24 0.5 0.5 0.5 24 0.5 0.5 0.5 24 0.5 0.5 0.5 24
6.25 6.00 6.00 6.30 5.95 5.95 6.00 6.15 6.10 6.10 6.10 6.05 6.15 6.10 6.10 6.10 6.10 6.10 6.10 6.10
8.55 8.60 8.45 8.50 8.15 8.13 8.00 8.10 7.80 8.00 7.95 7.80 7.40 7.50 7.40 7.50 7.15 7.15 7.20 7.20
0.020 0.007 0.005 0.015 0.020 0.014 0.014 0.019 0.025 0.013 0.011 0.023 0.023 0.018 0.016 0.020 0.025 0.014 0.012 0.021
peak % 2 peak 1 0.104 0.104 0.087 0.079 0.145 0.125 0.138 0.122 0.126 0.125 0.122 0.108 0.128 0.124 0.120 0.116 0.121 0.110 0.106 0.095
16.1 6.3 5.4 15.9 12.1 10.0 9.2 13.5 16.6 9.4 8.3 17.6 15.2 12.7 11.8 14.7 17.1 11.3 10.2 18.1
For filtered samples measured after filtration. b Not filtered.
PEO fractions, with certain exceptions. The broad-distribution PEO aqueous solutions (set-B) previously used for GPC were left undisturbed for periods of 0.5 and 24 h, respectively. At the end of each period, a 10 µL sample was injected into the GPC. The results are shown in Table 5. The peaks attributed to the presence of clusters (peak 1) show an elution time similar to that of the narrow-distribution PEO fractions, but with a larger width. Probably the clusters are polydisperse because the polymer fractions themselves have a broad distribution. In an attempt to determine the percentage of aggregated polymer more accurately, unfiltered PEO samples were injected into the GPC. The unfiltered polymer samples were found to contain between 12 and 17% wt/wt of the free polymer in the form of aggregates. Higher molecular weights do not necessarily show a greater amount of aggregated material. Filtration of the polymer through a 0.45 µm filter significantly reduces the amount of aggregated material in solution, as seen in Table 5. The elution times of peaks 1 and 2 are not affected by filtration and remain the same. If the polymer solution is again filtered through a 0.22 µm filter, the amount of aggregated polymer in solution is found to decrease only slightly. These results seem to show that, for the broad-distribution PEO fractions, it is not possible to remove all the aggregated polymer. As in the case of the narrow-distribution PEO fractions, when the filtered polymer solutions are left undisturbed for a period
Dilute Aqueous Poly(ethylene oxide) Solutions of 24 h, the amount of aggregated polymer in solution is found to increase again. No further increase in the percentage of aggregated polymer is obtained after a 24 h period. This observation indicates that for equilibrium to be re-established a 24 h period is sufficient. Furthermore, it can be seen from Table 5 that the amount of aggregated polymer after 24 h is comparable to the amount of aggregated material in the unfiltered polymer solution. This implies that the formation of clusters is in fact a true equilibrium process. Vigorous stirring of the polymer solutions, when preparing the solutions, does not break down the clusters. Experimental results also showed that, regardless of the time the polymer solution is stirred, the areas and elution times of the GPC aggregate and free polymer peaks (peaks 1 and 2, respectively) do not change. Another conclusion which can be drawn from the results is that for the broad-distribution molecular weight fractions, the amount of aggregated polymer is larger than for the case of narrow-distribution polymer fractions. B. Static Light Scattering. Zimm Analysis. Very few static light scattering studies and Zimm analyses of aqueous PEO solutions have yielded conclusive results.28,32,33,40,41,43 In most cases the molecular weights and the radii of gyration were obtained from the extrapolation of the scattered intensity data at higher angles only (>60°) or by using non linear fitting techniques. This was necessary since the scattered light showed a non linear dependence on the scattering angle and polymer concentration. In most cases the extrapolation from higher angles did not yield the real molecular weight of the polymer. For example, Polik and Burchard43 recently reported a molecular weight (Mw) of 260 000 for a PEO fraction of Mw) 20 000 by using these techniques. These anomalies were attributed to the presence of polymer aggregates in solution. Our own results support these conclusions. The dilute PEO solutions listed in Table 2 were prepared. Figure 2a shows a typical Zimm diagram obtained from these solutions for PEO2. Zimm plots for the other PEO samples show a similar non linear behavior.69 No useful data could be extracted from these diagrams. The solutions were then centrifuged at 60 000 rpm for 1 h. No change was observed in the Zimm diagrams. In an attempt to remove the aggregates from the polymer solution, the stirred solutions were filtered through a 0.22 µm filter. The filtrate was immediately injected into the scattering cell. The scattered intensity as a function of angle was immediately measured. If a nonlinear dependence of the scattering intensity was found, the polymer solution in the scattering cell was removed and filtered again. The intensity as a function of angle was measured again. The procedure was repeated until a nearly linear dependence of scattered intensity as a function of both angle and concentration was obtained. Usually only two filtrations were required. Figure 2b demonstrates the data obtained by employing this experimental technique for PEO-2. The plots of Kc/R(θ) vs sin2(θ/2) are very nearly linear. The slight variation from linearity for the filtered polymer solutions is not due to the presence of clusters but rather to the fact that linearity of Kc/R(θ) vs sin2(θ/2) is only to be expected for random coils having a most probable Schulz-Flory distribution Mw/Mn ) 2.71 For scattering angles between 20° and 130°, the Zimm diagrams of all the PEO samples show a similar “linear” dependence of Kc/R(θ) vs sin2(θ/2) and concentration.69 The results are summarized in Table 6 and Figure 3. Excellent agreement between the molecular weights obtained from GPC and those obtained using this technique was found. The polymer molecular weight fraction and the polymer solution concentration were found to have a strong influence on the solution properties.
J. Phys. Chem., Vol. 100, No. 32, 1996 13691
Figure 2. Typical Zimm plots for (a) PEO solutions with polymer clusters and (b) PEO solutions with no polymer clusters. Results for PEO-2.
TABLE 6: Dilute Solution Properties of PEO from Zimm Analysis PEO Rg sample Mw (nm) A2 (cm3 mol g-2) χ (FH) χ (KCKR) χ (YK) PEO-1 PEO-2 PEO-3 PEO-4 PEO-5
167 216 331 497 572
23 26 28 37 41
1.14 × 10-3 9.88 × 10-4 6.30 × 10-4 6.12 × 10-4 6.01 × 10-4
0.474 0.477 0.486 0.485 0.486
0.470 0.475 0.483 0.483 0.483
0.463 0.465 0.470 0.470 0.470
The radii of gyration for the broad-distribution PEO fractions range between 23 and 41 nm and tend to increase with increasing molecular weight. The radii of gyration were compared to the scaling relation:72
Rg ) k1MwR
(13)
where R ) 0.5 or 0.6, depending on whether or not the polymer chains in solution behave as ideal random coils under theta conditions (χ ) 1/2) or have their dimensions expanded by the excluded volume in a good solvent. The experimental results are in good agreement with those obtained by Cohen-Stuart et al.73 who found that for narrow-distribution PEO fractions in aqueous solution R ) 0.6, but our data are not accurate enough to distinguish between a coefficient of 0.5 or 0.6. The results are plotted in Figure 3. The value of k1 in eq 13 used for our data was 0.015. A good fit to the experimental results was found above a molecular weight of 300 000. The discrepancy between our experimental data and eq 13 is most likely due to the large polydispersity of the PEO samples used, which is largest for the lower molecular weight fractions. Kambe and Honda32,33 obtained values 2 or 3 times as large from their work on dilute PEO solutions.
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Polverari and van de Ven
Figure 3. Comparison of theoretical and experimental Rg values as a function of PEO molecular weight obtained from Zimm analysis: (b) experimental data calculated from eq 9; (s) Rg values calculated from eq 13 with R ) 0.6 and k1 ) 0.015.
The second virial coefficient, A2, and thus the polymersolvent interaction parameter, χ, did not show a pronounced molecular weight dependence. The second virial coefficients varied between 1.14 × 10-3 for PEO-1 to 6.01 × 10-4 cm3 mol g-2 for PEO-5. This trend is in agreement with eq 12 which predicts that A2 decreases with increasing molecular weight. These values are also in good agreement with those obtained by Kambe and Honda32,33 using low-angle light scattering on similar molecular weight polymers. The polymer-solvent interaction parameter calculated with the FH theory, varied between 0.474 and 0.486 as shown in Table 6. Calculating the function h(z) by the KCKR dilute solution theory with the F parameter of that theory being zero,61 the polymer-solvent interaction parameters for the PEO samples were recalculated and found to vary between 0.470 and 0.483. To assess the effect of polydispersity on the value of χ (and h(z)), the Yamakawa-Kurata61 (YK) treatment was also used, resulting in values between 0.463 and 0.47. The results of the various theories are compared in Table 6. It can be seen that the difference between the various approaches is always less than 3%. The values obtained using the YK theory are in very good agreement with those reported by Ataman and Boucher49 for 20 000 molecular weight PEO fractions and with the value of χ ) 0.45 obtained by Cohen-Stuart et al.73 Their results show, in agreement with our observations, that most or all of the molecular weight dependence of the second virial coefficient comes from the function h(z) while the χ-parameter depends little or not on molecular weight and polydispersity. Polymer Solution Concentration and Polymer Molecular Weight Dependence. The dependence of the polymer concentration and polymer molecular weight on the kinetics of cluster formation was studied. The scattering intensities of the filtered polymer solutions used for Zimm analysis (Table 2) were monitored as a function of time. The scattering intensities were found to increase with time. Large changes in scattering intensity were found to begin within 30 min of filtration. The increases in scattering intensities were most pronounced at the low scattering angles ( 13 or 45 for the high and low molecular weight PEO solutions, respectively. The results in Figures 5, a and b, indicate that large clusters are present immediately after the PEO polymer solutions are prepared and that they break down, with time, into smaller, more numerous clusters. It is interesting to note that in this process the total amount of aggregated polymer in solution remains constant. That is, the clusters do not breakup to form free polymer. This can also be seen in the GPC results which show that the area of the aggregated peak (peak 1) remains constant at all times. Size of PEO Molecules. In Figure 6 the apparent PCS polymer coil diameters are plotted as a function of time. The
Figure 5. (a) The calculated PCS cluster diameter (eq 7) and (b) the measured scattered light intensity for different PEO molecular weight fractions as a function of aging time.
PEO solutions were filtered through a 0.22 µm filter and the polymer coil diameter measured at fixed intervals. The PCS sample time was kept constant at 16 µs, the appropriate sample time for the size of freely dissolved PEO molecules. For all the polymer solutions, the apparent PCS polymer coil diameters were found to decrease and eventually reach a steady-state diameter after about 3 h. Since no large clusters are present in solution immediately after filtration, this is likely due to the re-formation of clusters in solution. Furthermore, since the PCS sampling time was kept constant at 16 µs, the autocorrelation function reflects data for only very small diameter particles or polymer coils. In effect, a “sampling window” was created in which only polymer coils or particles whose diameters are smaller than 0.2 µm are detected by PCS. It follows from eq 5 that when ΓB , ΓA for sampling times τ = rA-1 (≈16 µs) the term exp(-2ΓBτ) = 1. This condition is fulfilled at t = 6 h. Since the polymer solution is not at equilibrium, clusters will begin to reform. As the clusters become large (>0.2 µm) they move out of the “sampling window” and are no longer detectable by PCS for our experimental conditions. Eventually, equilibrium is reestablished and PCS will only detect very small clusters and free polymer coils. Thus, rather paradoxically, the decrease in the apparent coil diameter with time suggests that large clusters are re-forming. However, the presence of clusters makes the apparent value of the diameter an unreliable approximation of the true size. Equation 5 for bimodal dispersions or polymer solutions can be used to obtain an accurate ΓA value for the polymer coil. By fitting the experimental autocorrelation functions at equilibrium (i.e., t > 3 h) to eq 5, ΓA and ΓB can be obtained. Equation 5 is greatly simplified by the fact that ΓB ≈ 0 at steady state, since the clusters are large. In Figure 7 the polymer coil diameters obtained from the same autocorrelation function using
13694 J. Phys. Chem., Vol. 100, No. 32, 1996
Polverari and van de Ven leads to a quasi-heterodyne signal74 in which the time scale of the decay in the autocorrelation function is 1/ΓA instead of 1/2ΓA. Quasi-heterodyning occurs when ΓB , ΓA and IB/IA > 30. From the best fit of the autocorrelation function with eq 5 we found that IB/IA varies between 40 and 50, for low and high molecular weight polymers. From eq 14 it follows that
IB x ) N IA 1 - x
(16)
Assuming 20% of the free polymer is present as clusters, eq 16 would imply that the PEO clusters contain between 160 and 200 polymer chains. The Rg values obtained from the Zimm analysis are seen to be smaller than the PCS (bimodal) diameters obtained at t ) 6 h, with the difference being largest at lower molecular weights. This is contrary to the prediction for monodisperse polymers. For a nondraining chain model59
R ) 0.676Rg
Figure 6. Apparent PCS hydrodynamic polymer coil diameter plotted as a function of time: (a) PEO-1; (b) PEO-2; (c) PEO-3; (d) PEO-4. The PCS polymer coil diameters were calculated using eq 7.
(17)
where R is the Stokes or hydrodynamic radius which can be equated with the radius obtained from PCS. Equation 17 is accurate for monodisperse polymer samples but is found to be inaccurate for polydisperse samples. Apparent discrepancy, in the calculated values of R, for polydisperse samples can be ascribed to the effect of polydispersity on the coil size.75 Assuming a Gaussian distribution in molecular weight of the polymer sample and the resulting size of the polymer coil in solution, the hydrodynamic radius for polydisperse samples, Rp, closely corresponds to 2 a2 ∫a p(a) da ) aj(1 + σa) Rp ) ) a a p(a) da
∫
(18)
where σa is the relative standard deviation in coil radius, p(a) is the Gaussian distribution function, and aj is the mean coil radius. The mean coil radius is equivalent to R for monodisperse samples (eq 17). The relative standard deviation in coil radius is defined as σ/aj, where σ is the standard deviation in coil size. The relative standard deviation in molecular weight is related to the polydispersity of the polymer sample and is defined as66
σm )
(
)
Mw σ* ) -1 Mn Mn
0.5
(19)
where here σ* is the standard deviation in molecular weight. For an ideal, monodisperse polymer coil in a θ-solvent (χ ) 1/ ), the radius of gyration of the average coil dimensions 2 increases as (cf. eq 13): Figure 7. Comparison of coil diameters or Rg from dynamic or static light scattering as a function of PEO molecular weight. Results for the cumulant method (1) (eq 7); the bimodal method (3) (eq 5); static light scattering (9) using Zimm analyses (eq 9); and theoretically calculated (0) using eq 25.
the cumulant method and eq 5 are compared to Zimm analysis data. Figure 7 shows that excellent agreement between PCS polymer coil diameters from eq 5 and Rg values from Zimm analysis exists. It can be seen that coil diameter of a PEO molecule obtained from the cumulant method is almost twice that obtained from the best fit to eq 5. This is due to the fact that the clusters act as local oscillators which hardly change the frequency of the scattered light. The mixing of the light scattered from the clusters and from the free coils (which is more frequency shifted)
Rg ) kM0.5
(20)
Substituting eq 17 into eq 20 yields
R ) 0.676kM0.5
(21)
The analagous equation for a polydisperse sample is
R(1 + σa) = 0.676kM0.5 (1 + σm)0.5
(22)
Inserting eqs 19 and 21 into eq 22 results in
[ (
Rp ) 0.676Rg 1 +
)]
Mw -1 Mn
0.5 0.5
(23)
The hydrodynamic radii, Rp, for polydisperse polymer samples
Dilute Aqueous Poly(ethylene oxide) Solutions calculated from eq 25 are in very good agreement to those obtained experimentally using the bimodal method (eq 5), as is shown in Figure 7. The small difference is likely due to a nonGaussian molecular weight distribution. 5. Concluding Remarks From this study, it is evident that aqueous solution properties of PEO are complex and not as simple as proposed in previous studies. Our method shows that a careful analysis and interpretation of the light scattering data must be made to obtain meaningful results. This is especially important in dynamic light scattering. Above a critical polymer concentration, aqueous PEO solutions are not homogeneous: clusters and free polymer coexist in thermodynamic equilibrium. In solutions in which clusters are filtered out it takes about 1 day to reach equilibrium, while in solutions containing clusters, it takes about 1 week to reach equilibrium. The formation of PEO clusters could be due to a “hydrophobic” effect similar to that for micelle formation in aqueous solution or to some crystallization phenomena with a small driving force. Each cluster contains approximately a few hundred polymer chains. The behavior of PEO solutions with clusters is rather different from that of solutions in which clusters are absent. This is, for example, very evident from studies of the absorption of PEO on latex colloids76 or of the ability of PEO to reduce drag.10,23 References and Notes (1) Bailey, F. E., Jr.; Koleske, J. V. Poly(ethylene oxide); Academic Press: New York, 1976. (2) Gaylard, N. G. Polyethers, part 1; Interscience: New York, 1963. (3) Furukawa, J.; Saegusa, T. Polymerization of Aldehydes and Oxides; Interscience: New York, 1963. (4) De Witt, J. A.; van de Ven, T. G. M. AdV. Colloid Interface Sci. 1992, 42, 41. (5) Napper, D. H. J. Colloid Interface Sci. 1977, 58, 390. (6) Tadros, Th. F.; Vincent, B. J. Colloid Interface Sci. 1979, 72, 505. (7) Layec, Y.; Layec Raphalen, M-N. J. Phys Lett. 1983, 44, L-121. (8) Wolff, C. Polymeres et lubrification; CNRS: Paris, 1975. (9) Berman, N. S. Annu. ReV. Fluid Mech. 1978, 10, 47. (10) Schick, M. J., Ed. Nonionic Surfactants; Marcel Dekker: New York, 1967. (11) Saeki, S.; Kuwahara, N.; Nakata, M.; Kanek, M. Polymer 1976, 17, 685. (12) Cox, H. L.; Cretcher, L. H. J. Am. Chem. Soc. 1926, 48, 451. (13) Malcolm, G. N.; Rowlinson, J. S. Trans. Faraday Soc. 1957, 53, 921. (14) Rowlinson, J. S. In Liquids and liquid Mixtures, 2nd ed.; Butterworth: London, 1969; pp 167-170. (15) Karlstro¨m, G. J. Phys. Chem. 1985, 89, 4962. (16) Kjellander, R.; Florin, E. J. Chem. Soc., Faraday Trans. 1 1981, 77, 2053. (17) Brown, W.; Stilbs, P. Polymer 1982, 23, 1780. (18) Assarsson, P. G.; Leung, P. S.; Safford, G. J. Polym. Prepr., Am. Chem. Soc., DiV. Polym. Chem. 1969, 10(2), 1241. (19) Liu, K. J.; Ullman, R. J. J. Chem. Phys. 1968, 48, 1158. (20) Liu, K. J.; Parsons, J. L. Macromolecules 1969, 2, 529. (21) Liu, K. J.; Anderson, J. E. Macromolecules 1970, 3, 163. (22) Tadokoro, H.; Chantani, Y.; Yoshihara, T.; Tahara, S.; Murahashi, S. Makromol. Chem. 1964, 73, 109. (23) Cox, J. E.; Dunlop, E. H.; North, A. M. Nature 1974, 249, 243. (24) Gadd, G. E. Nature 1968, 217, 1040. (25) Laufer, Z.; Jalink, H. L.; Staverman, A. J. J. Polym. Sci., Poly. Chem. Ed. 1973, 11, 3005. (26) Dunlop, E. H.; Cox, L. R. Phys. Fluids 1977, 20, 10(II) S203.
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