Chain Dynamics of Ethylene Oxide Oligomer Melts. An Ultrasonic

Article pubs.acs.org/JPCB

Chain Dynamics of Ethylene Oxide Oligomer Melts. An Ultrasonic Spectroscopy Study Elke Wald and Udo Kaatze* Drittes Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany S Supporting Information *

ABSTRACT: Between 0.2 and 2000 MHz, the ultrasonic attenuation spectra of oligomer melts have been measured at 25 °C. The oligomers comprise ethylene glycols with eight different mean degrees n ̅ of polymerization (1 ≤ n ̅ ≤ 13.2), a poly(propylene glycol) (n ̅ = 16.9) and three poly(ethylene glycol) dimethyl ethers (3 ≤ n ̅ ≤ 10.3). The complexity of the ultrasonic spectra increases with n ̅ . The spectra have been analyzed assuming up to four discrete relaxation terms. The relaxation times of these terms are assigned to the first odd modes in the oligomer conformational dynamics and are discussed in the light of the Rouse spring-and-dashpot and the Tobolsky torsional oscillator models of chain variations. The relaxation amplitudes indicate that not only the shear viscosity but also the volume viscosity is involved in ultrasonic relaxation processes. Comparison of relaxation times and shear viscosities for the ethylene glycols with the corresponding data for dimethyl ethers and also n-alkanes reveals significant effects of association.

been also reported.15,16 Other examples are the PEG-mediated enhancement of the solubility and lifetime of a variety of biomolecules as well as drugs in the blood circulation.14 Much interest is directed toward PEG modified biocompatible surfaces,17−21 and to the control of particle aggregation,22 as well as to cosmetics formulation23 employing this polymer. A quite different usage is to be found in the advancement of water flow through pipes in which the remarkable drag-reducing properties of long-chain poly(ethylene oxides) are utilized.10,24 In order to investigate the chain dynamics of poly(ethylene glycol) oligomers in their melts, with some relevance to their mixtures with water, we have performed broadband ultrasonic attenuation measurements, covering the frequency range from 200 kHz to 2 GHz. Ultrasonic fields couple to the changes in the isentropic molar volume accompanying chain conformational variations and provide thus informative relaxation phenomena in the attenuation spectra. The quantity of possible modes of chain variation increases with the length of polymer chains. So as to enable the study of a manageable number of modes, we have first focused on oligomers, for which likewise acronym PEG is commonly used in the literature. Ultrasonic attenuation data of polymer systems have already been reported several times. Often, however, the measurements were restricted to a single frequency25−28 or a limited number of adjacent frequencies29 so that evaluation of spectral properties is impossible. Attenuation data over moderate frequency bands30−34 were partially indeed assigned to chain conformation changes of the polymer molecules. Yet the spectra were too small-band to allow for reliable conclusions

1. INTRODUCTION The detailed knowledge of the delicate interplay of the structure, hydration, and chain dynamics is a key element for better understanding the properties and functions of hydrosoluble polymers. A wide variety of experimental techniques is thus applied to investigate relevant aspects of polymeric systems,1,2 including the characterization of isolated low-weight polymer chains on a surface.3,4 In this context amphiphilic poly(ethylene oxide) or poly(ethylene glycol) (PEG) has attracted considerable interest. The comparatively simple synthetic polymer adopts a helical conformation in the crystalline state which, to some extent, is retained even in dilute aqueous solution.5,6 It is believed that hydrogen bonding between PEG and water promotes this topology and fits the polymer into the water structure without major distortions of its tetrahedral hydrogen network. The compatibility of the polymer conformation with the hydrogen network structure of water seems to be the main reason for the complete miscibility of PEG at temperatures below 100 °C.7−9 Depending on the degree of polymerization, a miscibility gap may exist at even higher temperatures.10 The exceptional water solubility of PEG is accentuated by the insolubility of similarly structured polyether, especially of poly(methylene glycol) with its superior content of hydrophilic ether groups.8 Because of the perfect water solubility PEG is considered a favorable model for fundamental studies of polymer behavior, providing a basis for insights into more complex systems, such as biopolymers. In addition to this aspect, PEG receives remarkable interest due to its wide range of biotechnological and industrial applications. PEG is used to crystallize11,12 and purify proteins13 and reduced immunogenicity and antigenicity of proteins is reached by modification with this polymer.14 Modification of natural and synthetic membranes by PEG has © 2014 American Chemical Society

Received: August 18, 2014 Revised: November 3, 2014 Published: November 3, 2014 13300

dx.doi.org/10.1021/jp508329s | J. Phys. Chem. B 2014, 118, 13300−13311

The Journal of Physical Chemistry B

Article

Table 1. Oligomer Molar Weights (M), Mean Weights (M̅ ), and Number Averaged Weights (M̅ meas) of the Particular Batch, as Well as Mean Degree n̅ of Polymerization, Specified (wc) and Measured (wcmeas) Water Content, Density ρ, Low-Frequency Sound Velocity cs, and “Static” Shear Viscosity ηs0 at 25 °Ca oligomer acronym ethylene glycol EG triethylene glycol TrEG poly(ethylene glycol) PEG 200 poly(ethylene glycol) PEG 300 poly(ethylene glycol) PEG 400 I poly(ethylene glycol) PEG 400 II poly(ethylene glycol) PEG 400 III poly(ethylene glycol) PEG 600

200 300 400 I 400 II 400 III 600

Poly(propylene glycol) 1000 PPG 1000 triethylene glycol dimethyl ether TrEGDME poly(ethylene glycol) dimethyl ether 250 PEGDME 250 poly(ethylene glycol) dimethyl ether 500 PEGDME 500 a

M,M̅ M̅ meas g/mol

n̅

wc wcmeas, % w/w

HO−CH2−[CH2−O−CH2]n‑1−CH2−OH 62.07 1 3 the α/ν2 spectra do not approach a frequencyindependent value at low frequencies but increase slightly at decreasing ν. Initially it is not clear whether another lowfrequency relaxation exists in those oligomer melts or whether this feature is due to an experimental artifact. For this reason the low-frequency part of the spectra has been re-evaluated as described below. 3.2. Analytical Description. If relaxations reflect transitions between well-defined states, such as different conformers of molecules, the frequency normalized attenuation spectra can be represented by a sum of Debye-type relaxation terms,66−68 and the frequency independent background contribution B′ (eq 1)

Figure 1. Frequency-normalized ultrasonic attenuation spectra for EG (□), TrEG (■), PEG 300 (○), and PEG 600 (●) at 25 °C. The dashed and dot-and-dash lines indicate the Stokes contributions (eq 2) to the PEG 300 and PEG 600 spectra, respectively. The inset reveals α/ν2 data at 1.5 MHz as a function of mean degree n ̅ of polymerization. Experimental values (⧫) are compared to predictions from eq 2 in which volume viscosity is neglected (◊).

⎛ c (ν ) ⎞ I ADi τDi α (ν ) ⎜ s2 ⎟ ∑ 2 = π + B′ 2 2 ν ⎝ cs ∞ ⎠ i = 1 1 + (ωτDi)

relaxations appear in the spectra in the range below the common high-frequency relaxation region. Furthermore, the attenuation coefficient at low frequencies increases with degree of polymerization. This result is also illustrated by the inset to Figure 1 where (α/ν2) data at 1.5 MHz are shown to rise monotonously with n ̅ . Also presented in that figure is the Stokes contribution

⎛α⎞ 8π 2 ⎜ ⎟st = η 2 ⎝ν ⎠ 3ρcs3 s0

(3)

Herein, cs∞ = limv→ cs(v), ω = 2πν, and ADi as well as τDi, i = 1, ..., I, are relaxation amplitudes and relaxation times, respectively. I is the total number of Debye terms. Written to apply to the frequently considered attenuation per unit wavelength, αλ(ν) = α(ν)λ, eq 3 reads ⎛ c (ν ) ⎞ 2 I ADi ωτDi S(ν) = αλ(ν) = ⎜ s ⎟ ∑ + Bν ⎝ cs ∞ ⎠ i = 1 1 + (ωτDi)2

(2)

to the frequency normalized attenuation where ηso denotes the “static” shear viscosity measured at low shear rates (Table 1). This contribution increases likewise with n ̅ and points at a nearly constant ηv/ηs ratio. Toward high frequencies the (α/ν2) data fall below the Stokes value (Figure 1). Evidently α/ν2 values smaller than (α/ν2)St are possible only if the shear viscosity depends on frequency, i.e., if the actual viscosity ηs(ν) at high frequency ν is smaller than ηs0. Hence the PEG spectra clearly demonstrate that the relaxation behavior is not restricted

(4)

where λ = cs(ν)/ν and B = B′cs. Spectral function S(νn, P1, ..., PJ) has been fitted to the experimental spectra using a Marquardt algorithm69 to minimize the variance χ (P1 , ..., PJ) =

1 N−J−1

⎛ αλ(νn) − S(νn , P1 , ..., PJ ) ⎞ ⎟ Δαλ(νn) ⎠ n=1 ⎝ N

∑⎜

(5)

Table 2. Parameter Values of Relaxation Function S(ν) Defined by Eq 4, as Resulting from the Nonlinear Least-Squares Regression Analysis of the Spectra Measured at 25 °Ca oligomer EG TrEG PEG 200 PEG 300 PEG 400 I PEG 400 II PEG 400 II* PEG 400 III PEG 600 PEG 600* PPG 1000 PPG 1000* TrEGDME PEGDME 250 PEGDME 500

τD1, ns ± 10%

AD1, 10−3 ± 10%

6.4

11.4

14.2

9.7

27.2

6.9

τD2, ns ± 10%

AD2, 10−3 ± 10%

3.0 6.5 6.2 1.0 6.3 12.8 2.4 18.2 3.7

19.1 12.4 12.2 41 14.0 11.5 20 12.0 28

4.6

3.6

τD3, ns ± 10%

AD3, 10−3 ± 10%

τD4, ns ±10%

AD4, 10−3 ± 10%

B, ps ±10%

0.59 1.31 0.50§ 1.02 0.91 0.25§ 0.99 1.49 0.53 1.47 0.81 0.48 0.54 1.12

39 29.6 53 43.8 49 56 42.3 35.3 62 83 103 1.9 6.0 14.6

0.053 0.102 0.145 0.130 0.136 0.126 0.117 0.119 0.177 0.142 0.197 0.177 0.044 0.081 0.104

316 457 424 449 505 517 476 578 434 450 737 730 113 87 155

150§ 150§ 165 150§ 147 150 150§ 150§ 171 150§ 162 150§ 50§ 74 127

Parameter sets marked by ∗ result if the existence of an additional relaxation term is presumed (section 5.1.1.) Data marked by § have been fixed at their probable value in the final fitting procedure. Global errors are given to characterize the uncertainties of the parameter values. Individual errors may be smaller but also larger. The latter is particularly true with the dimethyl ethers, for which the uncertainty in the parameters of the highfrequency relaxation term may even reach 100%. a

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In this equation, P1, ..., PJ are the parameters of S, ν1, ..., νN are the frequencies of measurement, and Δαλ = λΔα. The parameter values obtained from the nonlinear least-squares regression analysis are collected in Table 2. In this analysis of spectra any dispersion in the sound velocity has been neglected by putting cs(ν) = cs∞ in eq 4. It is only briefly mentioned that other relaxation functions have been alternatively considered. The distribution in the molecular weights of several oligomers supports a description of the spectra in terms of continuous relaxation time distributions. For that reason we have sought a representation of the αλ(ν) spectra by a sum of Cole−Cole relaxation terms.70 It turned out, however, that the width of the underlying relaxation time distribution function is negligibly small. For that reason the number of adjustable parameters in the fitting procedure has been minimized by taking only Debye relaxation terms into account. Adequateness of this relaxation model is demonstrated by Figure 2, where the subdivision of the αλ(ν) spectrum of PEG 600 into three Debye terms and also the sum of these terms is shown.

Figure 3. Frequency-normalized attenuation spectrum for PPG 1000 at 25 °C with (+) and without (○) correction for sound velocity dispersion. The inset shows the frequency-dependent sound velocity of the same oligomer (●) as calculated according to eq 6.

4. THEORY 4.1. Rouse Spring-and-Dashpot Model. The Rouse model considers the polymer chains to be composed of segments the end-to-end distances of which follow a Gaussian distribution.44,71 Well adapted for polymer melts, in which the repulsion between two segments of the same chain agrees with that between segments of different chains, the model neglects excluded volume interactions. The end points of each segment are represented as beads which are connected by harmonic springs. Each bead is subjected to a random thermal force and a drag force. Neglecting also inertia effects and taking anisotropy into account, the deviation of the j-th bead (j = 1, ..., z) from its equilibrium position is given by the following equations of motion ζ dxj/dt = D[xj + 1(t ) − xj(t ) + xj − 1(t ) − xj(t )], j = 2, ..., z − 1

Figure 2. Ultrasonic excess attenuation per wavelength, (αλ)exc = αλ − Bν, versus frequency ν for PEG 600 at 25 °C. Dashed, dotted as well as dot-and-dash lines indicate the subdivision of the spectrum into three Debye relaxation terms. The full line represents the sum of these terms.

I

∑ i=1

⎤−1/2 ⎥ 1 + (ωτDi)2 ⎥⎦

ζ dx1/dt = D[x 2(t ) − x1(t )]

(8)

ζ dxz /dt = D[xz − 1(t ) − xz(t )]

(9)

where damping coefficient ζ considers the viscous friction of the beads and where the spring constant D, according to

3.3. Consideration of Dispersion and Secondary Relaxation Term. Because of the large ultrasonic attenuation of larger oligomers their sound velocity is subject to a considerable dispersion, as mentioned in section 2.3.2. In order to examine the impact of the dispersion on spectral function S(ν), the frequency-dependent sound velocity has been calculated according to the relation cs(ν) ⎡ 1 = ⎢1 + ⎢⎣ cs ∞ π

(7)

D = 3kBT /σ 2

(10)

is related to the mean end-to-end distance σ of the segments. Here kB is Boltzmann’s constant and T the absolute temperature. The system of differential equations can be solved by a set of z time-dependent relations with relaxation times −1 ⎡ 4D ⎛ pπ ⎞⎤ τp = ⎢ sin 2⎜ ⎟⎥ ⎝ 2z ⎠⎦ ⎣ ζ

ADi

(6)

(11)

in which p, p = 1, ..., z − 1, denotes the mode number (p = 0 corresponds with a simple translational motion of the complete chain and is not considered here). In Figure 4, the first three modes in the motions of a polymer chain of 12 beads are exemplified. Shown is the relaxation of the beads (masses) after a disturbance at t = 0 when equidistant equilibrium positions of the beads are assumed. For sufficiently long chains and low mode numbers (p ≪ z) eq 11 can be linearized

using the ADi and τDi data of Table 2. For PPG 1000, the resulting sound velocity spectrum is displayed in the inset of Figure 3. The main part of that figure demonstrates the small effect in the attenuation spectrum resulting from consideration of the dispersion in cs. Inclusion of this effect in the evaluation of the attenuation spectra leads to marginal changes in the ADi and τDi values. Therefore, the originally derived sets of parameters (Table 2) will be discussed in the following. Some spectra reveal an additional relaxation term at low frequencies. As discussed in the Supporting Information this term has been related to impurities of the PEG samples. Its parameter values are therefore omitted from Table 2.

τp = 13304

ζ z2 = τ1p−2 Dπ 2 p2

(12)



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Figure 4. Illustration of the time behavior of the first three modes in the Rouse model for 12 masses. Full lines demonstrate the relaxation into equilibrium positions (dashed lines) after a relevant displacement at time t = 0.

so that the relaxation time of the pth mode depends on the largest relaxation time τ1 and the mode number p only. 4.2. Damped Torsional Oscillator (DTO) Model. On the basis of the Rouse model the chain segmental motions have been alternatively considered42,43 by replacing segments with Gaussian distribution of end-to-end distances by such with independent angle of rotation around the molecular axis. In this model the angels of torsions follow equations equivalent to those for the positions of beads in the Rouse model. Since the correlation of angles along a chain decays stronger than that of bead positions, the chain segments in the torsional oscillator model may be significantly shorter than those in the Rouse model. Proceeding from the idea of a two state torsional oscillator, with the states separated from one another by the Gibbs free energy barrier ΔG#, the relaxation rate of the first normal mode of oscillations has been calculated as42 ⎛ π ⎞ −ΔG # / RT τ1−1 = 2νt0 sin 2⎜ ⎟e ⎝ 2nt 0 ⎠

Figure 5. Relaxation times τDi at 25 °C of the Debye relaxation terms in the oligomer spectra (eq 4) of ethylene glycols (full symbols) and PPG (open symbols) versus the total number of carbon and oxygen atoms per chain. The inset shows the data following on the assumption of at most three Debye terms per spectrum. The data in the main part of the diagram result if with the longer oligomers allowance is made for an additional (high-frequency) relaxation term. Key: (●, ○) lowest mode (i = 1); (■, □) second lowest mode (i = 2); (⧫, ◊) next higher mode (i = 3); (▲, Δ) highest mode (i = 4) to be extracted from the ultrasonic spectra. Full and dashed lines represent power laws with exponents given in the text (section 5.1.1.). The dotted line shows power law behavior with theoretical slope m = 1, assuming that only the τD4 value for PEG 400 is left within the fourth series of relaxation times (section 5.1.2). For the lowest mode the correspondent Debye relaxation times of the oligomers are indicated.

4) of the Rouse model. Correspondingly, the second largest of τDi values of the spectra may be assigned to a higher mode in the oligomer chain dynamics, etc. However, the slope in the dependency of the smallest Debye relaxation times upon z2 differs significantly from that of both other series. This result seems to indicate that, even though, within their limits of experimental uncertainties, the broadband ultrasonic spectra of the larger oligomers are indeed analytically well represented by a sum of three relaxation terms, another relaxation may exist at high frequencies. For that reason, those spectra have been again analyzed assuming four Debye terms. The relaxation times following thereby are those presented in Table 2 and also in the main part of Figure 5. Now they reveal four series. In order to accentuate their correlations the relaxation times corresponding with a series are denoted by τi, i = 1, ..., 4, as indicated in the figure. The first three series display slopes m = d[log(τp)]/ d[log(z2)] close to 1, namely m = 1.18 ± 0.04, τ1; = 1.2 ± 0.1, τ2; and =0.99 ± 0.03, τ3. Slope m for the group of now smallest relaxation times is as small as 0.31 ± 0.06, thus suggesting that the description of the ultrasonic spectra of the largest oligomers by four relaxation terms is still incomplete. In fact, the trend in the total number of Rouse modes per oligomer points at a larger number of Debye terms with the PEG 600 and PPG 1000 spectra. However, since these Debye terms are expected to be located at and above the highest frequency of measurement, they cannot be identified in the experimental spectra. Nevertheless, the dependencies of the relaxation times in the ultrasonic spectra upon the chain lengths of oligomers report a close relation to the Rouse polymer dynamics of the molecules. Likely torsional oscillations may be also involved. 5.1.2. Mode Order and Length of Segments. We now proceed from the idea that the smallest relaxation times derived

(13)

and the damping coefficient as ζt = 8π 2νt0meΔG

#

/ RT

(14)

In these equations, νt0 denotes the characteristic frequency of rotation about either equilibrium position, nt0 is the number of individual torsional oscillators per chain, R is the gas constant, and m the mass of a bead. Hence the damping is not due to viscous friction as in the Rouse model but instead to intramolecular hindrance of rotation of a segment.

5. DISCUSSION 5.1. Relaxation Times. Both the spring-and-dashpot (section 4.1) and the damped torsional oscillator (section 4.2) model predict identical dependencies of the relaxation times upon the mode number p and the chain length z of the molecules. For that reason the relaxation times of the polymer melts will be first discussed with a view to their relation to p and z without discriminating between the models. For simplicity we shall use term “Rouse model” to subsume both approaches. 5.1.1. Dependence upon Chain Length. In the inset of Figure 5 the relaxation times, assuming at most three Debye terms (I = 3, eq 4), are displayed versus the squared number z of beads within the oligomer chains, where z is simply identified with the mean number of carbon and ether oxygen atoms per oligomer. The data may be taken to define three groups of τDi values. The largest τDi values of each oligomer, including the relaxation time of the single relaxation for EG, form one series. They may be identified with the lowest mode (1st mode, Figure 13305



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least 3 modes whereas PEG 200 (3n ̅ + 1 = 13.3) is capable of at most four modes since the fifth one is definitely missing in the spectra. This consideration leads to a mean length of 3.3 groups per segment, in nice conformity with the predictions of the DTO model.42 We only mention that the Rouse spring-anddashpot model emanates from the assumption of significantly longer segments (ls ≈ 10). This feature is a strong indication of torsional oscillations to contribute noticeably to the oligomer modes. We also mention that the evaluation of ultrasonic spectra of n-alkanes (CH3(CH2)n‑2CH3) in terms of the DTO model showed the best agreement of the Gibbs free energy of activation (eq 14) with activation enthalpies and entropies from temperature dependent measurements if a segment length ls of about three groups (nt0 = n/3; eq 14)) was assumed.73 From the assumption of a mean segment length of 3.3 groups result about 12 and 16 segments per PEG 600 and PPG 1000 molecule, respectively. As mentioned before, however, the higher modes resulting from the motions of these segments cannot be observed in the frequency range of the ultrasonic spectra because the associated relaxation frequencies exceed the frequency range of measurements. 5.1.3. Experimental Results and Nonlinearized Predictions. Surprisingly, even the relaxation times for very short oligomers follow the linearized dependency upon the number of masses or individual torsional oscillators per chain, whereas the Rouse model predicts significant deviations at p ≈ z. For that reason the true Rouse relation for first mode (p = 1) is depicted along with the linearized relation in the inset of Figure 7. In fact

from the spectra of PEG 600 and PPG 1000 suffer from the existence of unresolved relaxation terms at higher frequencies. Therefore, only the τD4 value for PEG 400 is left within the fourth series of relaxation times. As we do not know the other relaxation times within this series, a straight (dotted) line with theoretical slope m = 1 has been assumed to represent it in Figure 5. Assigning the four series of relaxation times to the first modes in the motions of oligomer chains, according to eq 12 ratios τ1/τi, i = 2, 3, 4, should yield the squared numbers of the higher modes. For the present series of oligomers τ1/τ2 = 7.5− 8.0, τ1/τ3 = 24.1−29.4, and τ1/τ4 = 55.3 follows, corresponding with (τ1/τ2)1/2 ≈ 3, (τ1/τ3)1/2 ≈ 5, and (τ1/τ4)1/2 ≈ 7, respectively. Hence, if the ultrasonic spectra are discussed in terms of the Rouse model of conformational variations, the even modes appear to be absent. Taking this feature into account and regarding also the above experimental finding of relaxation times proportional to z2.4, τi/ z2.4 is plotted as a function of p2 in Figure 6. The data condense roughly on one line with slope d ln(τi/z2.4)/d ln(p2) = −1 and thus reveal a passable agreement with the prediction of eq 12 at p = 1, 3, 5, 7.

Figure 6. Dependency of relaxation times τi upon mode number p. Key: (⧫) EG; (diabolo) TrEG; (▲) PEG 200; (▼) PEG 300; (■) PEG 400; (●) PEG 600.

Missing evidence from even modes in the ultrasonic spectra may reflect either nonexistence of volume changes associated with the relevant conformational changes or blocking of those modes. In the framework of the spring-and-dashpot model missing volume changes are plausible, as illustrated by the second mode in Figure 4: a reduction in the length of the upper half of the molecule is compensated for by an increase in the lower half. A pure mode of sequenced elongation and shortening, however, seems unlikely. Rather torsional oscillations associated by less evident volume changes are also involved. Deadlock of even modes is even less plausible. Hydrogen bonds between the oligomers fluctuate rapidly, with dielectric relaxation times on the order of 0.15 ns for PEG 300 to 600,72 and thus impede the molecular dynamics rather than blocking specific modes at all. Also the fact that even modes are missing independently of the lengths of the oligomers contradicts a noticeable effect of deadlock. Hence further studies are seemingly necessary in order to investigate the volume changes associated with torsional vibrations and to definitely comment on the assignment of the ultrasonic relaxation terms to certain modes in the molecular dynamics of oligomers. The lengths ls of segments can be estimated from the total number 3n ̅ + 1 of groups per molecule and the corresponding number p of modes. Specifically TrEG (3n ̅ + 1 = 10) shows at

Figure 7. Debye term amplitudes ADi at 25 °C versus corresponding relaxation times τDi, i = 1, ..., 4. Symbols as with Figure 5. The full line represents power law behavior with exponent −1/2. The dotted line is given to guide the eyes at τDi < 0.2 ns. The inset shows the Rouse relation for relaxation times (eq 11) in its original form g1 = [4 sin2{πls/(2z)}]−1 (dotted line) and as linearized version g1 = [z/(πls)]2 (full line). Here ls (= 3.3) is the segment length.

noticeable deviations emerge at molecular length (3n ̅ + 1)/ls ≈ 1 only, i.e., for molecules constituting just one segment. A reason for the experimental finding of small-molecule relaxation times too short with a view to the Rouse model may be the neglect of the chain-length dependence in the friction coefficient ζ (eqs 11, 12) or Gibbs free energy of activation ΔG# (eq 13) in these considerations. Since with larger polymers the segmental variations are small as compared to the molecular lengths, such variations occur always in similar surroundings. Therefore, ζ and ΔG# will be largely independent of (3n ̅ + 1)/ls. In melts of small oligomers, however, the molecular dynamics may benefit from the rapidly fluctuating environment and may thus reveal shorter relaxation times than predicted by models for long-chain molecules. 13306



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Ai ∼ z −mpk

(16)

Y ↔ Y*

Other reasons for the deviations of the experimental relaxation times from the nonlinearized predictions of the Rouse model could be discretization errors which become more noticeable with oligomers composed of some few groups only than with larger molecules. 5.2. Relaxation Amplitudes. Unfortunately, no theoretical model for the coupling of the Rouse modes to a sonic field is available. We thus first restrict ourselves to the presentation of experimental evidence. 5.2.1. Correlation between Debye-Term Amplitudes and Relaxation Times. In Figure 7 the amplitudes ADi of the Debye relaxation terms are shown as a function of relaxation times τDi. Remarkably, at τDi > 0.2 ns the amplitudes decrease with the relaxation times, almost following a square root dependence ADi ∼ τDi−0.5. At first glance, because the corresponding relaxation frequencies (2πτDi)−1 are larger than 0.8 GHz, deviations at τDi < 0.2 ns may be considered to reflect imperfect consideration of the ultrasonic spectra at their high-frequency limits. Reconsideration of the evaluation procedures, however, clearly demonstrated that neither insufficient provision for experimental uncertainties in the ultrasonic attenuation data nor disregard of high-frequency relaxation terms in the analytical description can account for the strong deviations of the amplitudes at τDi < 0.2 ns from the general trend at larger relaxation times. For lack of definite information about the extraordinarily large relaxation amplitudes at τDi < 0.2 ns let us focus on the empirical relation ADi ∼ τDi−0.5. As discussed before, our measurements yielded also τi ∼ z2.4p−2.2, whereas Rouse theory predicts τi ∼ z2p−2. Hence we expect power law behavior

according to68 A=

⎤2 πcs ∞2ρ Γ ⎡ aV ∞ ⎢ ΔH − ΔV ⎥ RT ⎢⎣ ρCp ∞ ⎥⎦

(17)

are related to the reaction enthalpy ΔH and the isothermal reaction volume ΔV. Further parameters in eq 17 are the thermal expansion coefficient aV∞ and the specific heat at constant pressure C p∞ at frequencies well above the corresponding relaxation frequency, as well as the stoichiometry factor Γ given by Γ−1 = [Y]−1 + [Y*]−1. Here brackets denote concentrations. Often the isothermal volume change predominates in the isentropic reaction volume ΔVS = ΔV − aV∞ΔH/ (ρCp∞). On that condition consideration of the experimental findings in terms of eq 15 suggests ΔV to decrease with chain length of oligomers but to increase with mode number. Alternatively, effects of partial compensation of ΔV by the enthalpy term in ΔVS may become important and also a variation in equilibrium constant K of the monomolecular reaction (eq 16) and thus of stoichiometry factor Γ = Kc/(K + 1)2. Here c = [Y] + [Y*] is the total concentration of the species under consideration. Unfortunately, most of the relevant parameters are unknown so that presently the numerical evaluation of the amplitude data in terms of eq 17 is impossible. In his model71 Rouse did not consider the frequency dependence in the volume viscosity ηv (eq 1). He calculated relaxation amplitudes for the shear viscosity η s only. Analogously applied to the ultrasonic spectra and neglecting the small effect from the dispersion in the sound velocities, amplitudes

(15)

of the corresponding Debye amplitudes, with m = 1, ..., 1.2 and k = 1, ..., 1.1. To check this behavior a master plot of the Ai data at τi > 0.2 ns is given for the first three modes in the inset of Figure 8. It impressively demonstrates consistency of the results and yields m = 1.06 ± 0.3 and k = 1.00 ± 0.09. 5.2.2. Attempts toward Molecular Interpretation. Relaxation amplitudes A associated with uncoupled monomolecular equilibria

Ap =

4π NkBT 3ρcs 2

(18)

are predicted for modes p. Here N = ρNA/M̅ denotes the number density of molecules in the melts, as given by the densities ρ and molar weights M̅ of the oligomers and Avogadro’s constant NA. In Figure 8, amplitudes Ap as obtained from eq 18, along with the relevant experimental ADi data for the first mode of each oligomer, are displayed as a function of 3n ̅ + 1. In that log−log plot representation, both series of amplitude data reveal almost identical slopes. However, the amplitudes based on Rouse theory (eq 18) are generally smaller than those from experiments, even though the mode with smallest amplitudes is considered. Additionally, at serious variance with experiment (Figure 8, inset), eq 18 predicts independency of the Ap values from mode number p. These discrepancies suggest that substantial contributions from volume viscosity exist in the relaxation amplitudes of PEG and PPG ultrasonic spectra and that these contributions have been misleadingly disregarded in the above treatment of polymer chain dynamics. A consideration of α/ν2 data at low frequencies (α/ν2)0 = limv→0α(v)/v2 indicates that not just the volume viscosity contribution is missing in eq 18 for the relaxation amplitudes. Assuming no further relaxation to exist at frequencies below our range of measurements, contribution (α/v2)0,ηs = (α/v2)St (eq 2) to the low-frequency part of the frequency-normalized attenuation spectra can be obtained by analogous application of eq 1 and use of the shear viscosity data measured at small shear

Figure 8. Relaxation amplitude A1 of the first mode in the dynamics of ethylene glycol oligomers and PPG at 25 °C, displayed as a function of the number of carbon and oxygen atoms per chain. Key: (○) experimental data; (◊) calculated according to eq 18. Lines are graphs of power laws with exponents m = d[log(A1)]/d[log(z)] = −1.05 (○) and −0.94 (◊), respectively. The amplitude of EG has been omitted in the regression analysis of experimental data. The inset reveals the relation between amplitudes Ai and mode numbers p, taking proportionality between the amplitudes and z−1.06 into account. Key: (diabolo) TrEG; (▲) PEG 200; (▼) PEG 300; (■) PEG 400; (●) PEG 600. 13307



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rates (Table 1). On the other hand, with the aid of eq 3, that low-frequency part can be estimated as ⎛α⎞ 2π ⎜ ⎟ = ⎝ ν 2 ⎠0, A cs p

P

∑ AP τP + B′ (19)

p=1

from the amplitudes Ap predicted by eq 18. In eq 19, P denotes the number of observed modes in the ultrasonic spectra and τp the relaxation time corresponding with Ap. Together with the low-frequency (α/ν2)0,exp data from the measured spectra, the results from eqs 2 and 19 are listed in Table 3. According to our Table 3. Frequency Normalized Attenuation Coefficient at 25 °Ca oligomer

(α/ν2)exp at 1.5 MHz, 10−15 s2 m−1

(α/ν2)0,ηs0 eq 2, 10−15 s2 m−1

(α/ν2)0,Ap eq 19 10−15 s2 m−1

EG TrEG PEG 200 PEG 300 PEG 400 II PEG 600 PPG 1000

155 370 560 670 820 1220 2450

89 211 290 406 540 785 1580

103 164 218 288 410 563 951

Figure 9. Relaxation times τ1 of the first mode of chain dynamics at 25 °C for ethylene glycol oligomers (●, z = 3n ̅ + 1) and PPG (○), ethylene glycol dimethyl ethers (■, z = 3n ̅ + 1), as well as n-alkanes (⧫, CH3(CH2)n‑2CH3, z = n73). The inset depicts the frequencynormalized attenuation spectrum of n-pentadecane at 25 °C. Symbols show the experimental data,73 the full line is the graph of the relaxation spectral function with two Debye terms (eq 3 with I = 2). The dotted line indicates the subdivision of the spectrum into two relaxation terms. It represents the sum of the frequency independent background contribution and the high-frequency Debye relaxation term. Since this term appears at the upper limit of the measurement frequency range, no preference for either relaxation model (one Debye term73 or two terms) is deducible from the fit of the spectral function to the experimental data.

a

Following from the experimental spectra at 1.5 MHz, according to eq 2 resulting from the shear viscosity ηs0 (Table 1) at low shear rate, as derived from the amplitudes Ap (eq 18) using eq 19.

expectations, the experimental values are always larger than those taking solely shear viscosity into account. Yet, with the exception of EG, the (α/ν2)0,Ap values are additionally smaller than the (α/v2)0,ηs values. This result may be taken an indication of an insufficient description of the shear viscosity relaxation amplitudes by the Rouse model. 5.3. Comparison with Structural Variations of nAlkanes. In order to inspect the effects from hydrogen bonding it is interesting to compare the results for ethylene oxide oligomer melts to such for nonassociating n-alkanes. Ultrasonic (1 MHz ≤ ν ≤ 2 GHz) and shear viscosity (20 MHz ≤ ν ≤ 120 MHz) spectra of n-alkanes [CH3−(CH2)n‑2CH3, n = 14, 15, 16, 20] and some of their mixtures have revealed a Debye-type relaxation process which has been discussed in terms of the first mode in the DTO model of chain dynamics.73 Along with the corresponding data for PEG and PPG as well as for the ethylene glycol dimethyl ethers, the relaxation times τ1 of the alkanes are displayed versus number of rotatable groups per molecule in Figure 9. As with the ethylene glycols, power law behavior emerges also with the τ1 data of the alkanes. The exponent for the latter, however, is slightly larger than the exponents for the ethylene glycols and the ethylene glycol dimethyl ethers. This result appears to do not just reflect insufficient representation of the n-alkane spectra by a single relaxation term. Since higher modes likely exist also in the damped torsional oscillations of the alkanes, we have evaluated their ultrasonic spectra assuming a second relaxation term at the upper limit of the measurement range. An example of such representation of alkane spectra is given in the inset of Figure 9, in order to show that a two-Debye-term model fits well to the measured data. The effect in the τ1 values from the consideration of a second high-frequency relaxation term, however, is small and hardly exceeds the limits of experimental uncertainty.

No obvious reason exists for the subdivision of the τ1 data into three groups (Figure 9). Though the comparatively large relaxation times for PEG and PPG are clearly due to the hydrogen bond structure of the melts, the differences between the relaxation times of nonassociated ethylene glycol dimethyl ethers and n-alkanes at identical molecular length are less evident. Supposably, interactions between lone pair electrons of neighboring ethylene glycol dimethyl ether molecules imped the segmental motions and lead thus to a retardation of the conformational dynamics. This view contradicts, however, the pretended reduction of the potential energy barrier for rotation of a −CH2−O−CH2− group as compared to a −CH2−CH2− CH2− group.75 It is interesting to notice that the difference between the shear viscosities of the ethylene glycol dimethyl ethers and the n-alkanes is smaller (Figure 10, inset). As revealed by Figure 10, when the relaxation times are considered as a function of shear viscosity, two distinct groups of oligomers emerge, clearly distinguishing the nonassociating from the hydrogen bonding liquids.

6. CONCLUSIONS At frequencies below 2 GHz, ultrasonic attenuation spectra of ethylene glycol oligomers exhibit up to four relaxation terms with discrete relaxation times. The dependencies of the relaxation times upon the chain lengths suggest the relaxations to reflect modes in the conformer dynamics of the oligomers, as predicted by the Rouse spring-and-dashpot model44,71 and the Tobolsky damped torsional oscillator model42,43 of conformational variations. The ratios of successional ultrasonic relaxation times reveal the existence of only odd modes of variations. Absence of even modes seems to be plausible within the framework of a pure spring-and-dashpot model in which such modes may be associated with marginal or vanishing volume changes and thus hardly couple to the pressure variations of 13308



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AUTHOR INFORMATION

Corresponding Author

*Telephone: +49 551 39 7715. Fax: +49 551 39 7720. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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(1) Mark, J. E., Ed. Physical Properties of Polymers Handbook; Springer: New York, 2007. (2) Pethrick, R. A.; Amornsakchai, T.; North, A. M. Introduction to Molecular Motion in Polymers; Whittles Publishing: Dunbeath, U.K., 2011. (3) McConney, M. E.; Singamaneni, S.; Tsukruk, V. V. Probing Soft Matter with the Atomic Force Microscopies: Imaging and Force Spectroscopy. Polym. Rev. 2010, 50, 235−286. (4) Tress, M.; Mapesa, E. U.; Kossack, W.; Kipnusu, W. K.; Reiche, M.; Kremer, F. Glassy Dynamics in Condensed Isolated Polymer Chains. Science 2013, 341, 1371−1374. (5) Giordano, R.; Magazù, S.; Maisano, G.; Majolio, D.; Migliardo, P.; Vasi, C.; Wanderlingh, U. Diffusive Motion and H-Bond Effects on Liquid Poly(Ethylene Oxide) and on its Aqueous Solutions. Physica B 1995, 231−214, 515−517. (6) Crupi, V.; Jannelli, M. P.; Magazù, S.; Maisano, G.; Majolino, D.; Migliardo, P.; Ponterino, R. Raman Spectroscopic Study of Water in the Poly(Ethylene Glycol) Hydration Shell. J. Mol. Struct. 1996, 381, 207−212. (7) Saeki, S.; Kuwahara, N.; Nakata, M.; Kaneko, M. Upper and Lower Critical Solution Temperatures in Poly(Ethylene Glycol) Solutions. Polymer 1976, 17, 685−689. (8) Kjellander, R.; Florin, E. Water Structure and Changes in the Thermal Stability of the System Poly(Ethylene Oxide)-Water. J. Chem. Soc. Faraday Trans. 1 1981, 77, 2053−2077. (9) Bae, Y. C.; Shim, J. J.; Soane, D. S.; Prausnitz, J. M. Representation of Vapor-Liquid and Liquid-Liquid Equilibria for Binary Systems Containing Polymers: Applicability of an Extended Flory-Huggins Equation. J. Appl. Polym. Sci. 1993, 47, 1193−1206. (10) Branca, C.; Faraone, A.; Magazù, S.; Maisano, G.; Migliardo, P.; Villari, V. Poly(Ethylene Oxide): a Review of Experimental Findings by Spectroscopic Techniques. J. Mol. Liq. 2000, 87, 21−68. (11) McPherson, A. A Brief History of Protein Crystal Growth. J. Cryst. Growth 1991, 110, 1−10. (12) Cudney, B.; Patel, S.; Weisgraber, K.; Newhouse, Y.; McPherson, A. Screening and Optimization Strategies for Macromolecular Crystal Growth. Acta Crystallogr. D 1994, 50, 414−423. (13) Tjerneld, F. In Poly(Ethylene Glycol) Chemistry: Biotechnical and Biomedical Applications; Harris, J. M., Ed.; Plenum: New York, 1992. (14) Harris, J. M., Zalipsky, S., Eds. Poly(Ethylene Glycol)Chemistry and Biological Applications; ACS Symposium Series 680; American Chemical Society: Washington, DC, 1997. (15) Evans, E.; Rawicz, W. Elasticity of “Fuzzy” Biomembranes. Phys. Rev. Lett. 1997, 79, 2379−2382. (16) Rex, S.; Zuckermann, M. J.; Lafleur, M.; Silvius, J. R. Experimental and Monte Carlo Simulation Studies of the Thermodynamics of Polyethyleneglycol Chains Grafted to Lipid Bilayers. Biophys. J. 1998, 75, 2900−2914. (17) Amiji, M.; Park, K. Prevention of Protein Adsorption and Platelet Adhesion on Surfaces by PEO/PPO/PEO Triblock Copolymers. Biomater. 1992, 10, 682−692. (18) McPherson, T.; Kidane, A.; Szeifer, I.; Park, K. Prevention of Protein Adsorption by Tethered Poly (Ethylene Oxide) Layers: Experiments and Single-Chain Mean-Field Analysis. Langmuir 1998, 14, 176−186. (19) Deible, C. R.; Petrosko, P.; Johnson, P. C.; Beckman, E. J.; Russell, A. J.; Wagner, W. R. Molecular Barriers to Biomaterial

Figure 10. Relaxation time τ1 of the first mode of chain dynamics at 25 °C displayed versus shear viscosity ηs for the ethylene glycols (●) and PPG (○) as well as for the dimethyl ethers (□) and n-alkanes (◊73). The inset presents the relevant shear viscosities as a function of z, where z = 3n ̅ + 1 with the oligomers and z = n (n = number of carbon atoms per molecule) with the n-alkanes.73,74

ultrasonic fields. There are, however, indications of the oligomer conformational variations to involve torsional oscillations with less obvious volume changes. The short mean segment length of 3.3 (hydrocarbon and ether oxygen) groups in the oligomer conformational dynamics has been derived from the sonic relaxation times. Among other findings, this result points at torsional oscillations as the predominant modes of structural changes because longer segments are expected for the spring-and-dashpot modes. The amplitudes AD4 of the relaxation terms with relaxation times τD4 < 0.2 ns are extraordinarily large as compared to the amplitudes of the other terms. This property is only partly due to the fact that, with the larger oligomers, further relaxation terms at higher frequencies remained unresolved due to the finite frequency range of measurement. The amplitudes of the other terms follow power law ADi ∼ τDi−1/2, i = 1, 2, 3, corresponding with ADi roughly proportional to the mode order and inversely proportional to the oligomer length. So far no theory for the coupling of the chain conformational dynamics to ultrasonic fields exists. It is nevertheless obvious that the amplitudes cannot be solely explained by relaxation behavior in the shear viscosity of oligomer melts. Volume viscosity relaxation appears to be also involved. The first-mode relaxation times τ1 and the low-shear-rate viscosities ηs of associating ethylene glycol and propylene glycol oligomers, when compared with such of nonassociating ethylene glycol dimethyl ethers and n-alkanes, clearly reveal noticeable effects of hydrogen bonding. Association affects both parameters significantly. Small hydrogen-bonded ethylene glycol (M = 62.07 g/mol) features nearly the same viscosity as poly(ethylene glycol) dimethyl ether 500 but τ1 of the latter exceeds that of the former by a factor of almost 102. Hence the effect of the molar masses in the relaxation times of the first mode is distinctly stronger than the interference of the conformational variations with the hydrogen network dynamics.

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REFERENCES

ASSOCIATED CONTENT

S Supporting Information *

Figure of ultrasonic spectra for PEG 400 without and with water added, attenuation coefficient measurement with cavity resonators, including an iterative calibration procedure, and secondary relaxation term due to sample impurity. This 13309



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(42) Tobolsky, A. V.; Aklonis, J. J. A Molecular Theory for Viscoelastic Behavior of Amorphous Polymers. J. Phys. Chem. 1964, 68, 1970−1973. (43) Tobolsky, A. V.; DuPré, D. B. Macromolecular Relaxation in the Damped Torsional Oscillator and Statistical Segment Models. Adv. Polym. Sci. 1969, 6, 103−127. (44) Rouse, P. E., Jr. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. II. A First-Order Mechanical Thermodynamic Property. J. Chem. Phys. 1998, 108, 4628−4633. (45) Alig, I.; Lellinger, D. Inline-Monitoring mit Ultraschall. Kunststoffe 2000, 90, 96−100. (46) van Durme, K.; Delellio, L.; Kudryashov, E.; Buckin, V.; Van Mele, B. Exploration of High-Resolution Ultrasonic Spectroscopy as an Analytical Tool to Study Demixing and Remixing in Poly(NIsopropyl Acrylamide)/Water Solutions. J. Polym. Sci. B: Polym. Phys. 2005, 43, 1283−1295. (47) Lionetto, F.; Maffezzoli, A. Polymer Characterization by Ultrasonic Wave Propagation. Adv. Polym. Technol. 2008, 27, 63−73. (48) Hochrein, T.; Alig, I. Prozessmesstechnik in der Kunststoffaufbereitung; Vogel Buisiness Media: Würzburg, Germany, 2011. (49) Vijayalakshmi, S. P.; Madras, G. Effect of Temperature on the Ultrasonic Degradation of Polyacrylamide and Poly(Ethylene Oxide). Polym. Degrad. Stab. 2004, 84, 341−344. (50) Mehrdad, A.; Rostami, M. R. Effect of Temperature and Solution Concentration on the Ultrasonic Degradation of the Aqueous Solutions of Polyethylene Oxide. Iran. Polym. J. 2007, 16, 795−801. (51) Duval, M.; Gross, E. Degradation of Poly(Ethylene Oxide) in Aqueous Solutions by Ultrasonic Waves. Macromolecules 2013, 46, 4972−4977. (52) Scholz, E. Karl Fischer Titration; Springer: Heidelberg, Germany, 1983. (53) Eggers, F.; Kaatze, U.; Richmann, K. H.; Telgmann, T. New Plano-Concave Ultrasonic Resonator Cells For Absorption and Velocity Measurements of Liquids below 1 MHz. Meas. Sci. Technol. 1994, 5, 1131−1138. (54) Polacek, R.; Kaatze, U. A High-Q Easy-to-Handle Biconcave Resonator for Acoustic Spectrometry of Liquids. Meas. Sci. Technol. 2003, 14, 1068−1074. (55) Hagen, R.; Behrends, R.; Kaatze, U. Acoustical Properties of Aqueous Solutions of Urea: Reference Data for the Ultrasonic Spectrometry of Liquids. J. Chem. Eng. Data 2004, 49, 988−991. (56) Brai, M.; Kaatze, U. Ultrasonic and Hypersonic Relaxations of Monohydric Alcohol/Water Mixtures. J. Phys. Chem. 1992, 96, 8946− 8955. (57) Kononenko, V. S. Error of Measurements in an Ultrasonic Resonator due to Energy Losses in the Piezoelectric Plates. Sov. Phys. Acoust. 1985, 31, 499−501. (58) Gailus, T. Dissertation. Georg-August-Universität: Göttingen, Germany, 1996. (59) Kaatze, U.; Kühnel, V.; Menzel, K.; Schwerdtfeger, S. Ultrasonic Spectroscopy of Liquids. Extending the Frequency Range of the Variable Sample Length Pulse Technique. Meas. Sci. Technol. 1993, 4, 1257−1265. (60) Kaatze, U.; Lautscham, K.; Brai, M. Acoustical Absorption Spectroscopy of Liquids between 0.15 and 3000 MHz: II. Ultrasonic Pulse Transmission Method. J. Phys. E: Sci. Instrum. 1988, 21, 98−103. (61) Kaatze, U.; Kühnel, V.; Weiss, G. Variable Pathlength Cells for Precise Hypersonic Spectrometry of Liquids up to 5 GHz. Ultrasonics 1996, 34, 51−58. (62) Lautscham, K.; Wente, F.; Schrader, W.; Kaatze, U. High Resolution and Small Volume Automatic Ultrasonic Velocimeter for Liquids. Meas. Sci. Technol. 2000, 11, 1432−1439. (63) Hanke, E.; Schulz, U.; Kaatze, U. Molecular Interactions in Poly(ethylene glycol) − Water Mixtures at Various Temperatures: Density and Isentropic Compressibility Study. ChemPhysChem 2007, 8, 553−560. (64) Bhatia, A. B. Ultrasonic Absorption; Oxford University Press: Oxford, U.K., 1967.

Thrombosis by Modification of Surface Proteins with Polyethylene Glycol. Biomater. 1998, 19, 1885−1893. (20) Jenney, C. R.; Anderson, J. M. Effects of Surface-Coupled Polyethylene Oxide on Human Macrophage Adhesion and Foreign Body Giant Cell Formation in Vitro. J. Biomed. Mater. Res. 1999, 44, 206−216. (21) Ameringer, T. Biocompatible Ultrathin Coatings from Isocyanate Terminated Star PEG Prepolymers; Dissertation, Rheinisch-Westfälische Technische Hochschule: Aachen, Germany, 2006. (22) Poon, W. C. K.; Pirie, A. D.; Haw, M. D.; Pusey, P. N. NonEquilibrium Behavior of Colloid-Polymer Mixtures. Physica A 1997, 235, 110−119. (23) Bailey, F. E.; Koleske, J. V. Poly(Ethylene Oxide); Academic: New York, 1976. (24) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids, Fluid Mechanics; Wiley: Hoboken, NJ, 1987. (25) Pethrick, R. A.; Poh, B. T. Ultrasonic Attenuation and Adiabatic Compressibility of Polyethyleneoxide-Water Mixtures. Br. Polym. J. 1983, 15, 149−153. (26) Esquivel-Sivent, R.; Tan, B.; Abdelraziq, I.; Yun, S. S.; Stumpf, F. B. Absorption and Velocity of Ultrasound in Binary Solutions of Poly(Ethylene Glycol) and Water. J. Acoust. Soc. Am. 1993, 93, 819− 820. (27) Hassun, S. K.; Hussain, K. H.; Hassan, N. A. Visco-Relaxation Studies of Polystyrene Solutions in Different Solvents by Ultrasound. Acta Polym. 1990, 41, 438−441. (28) Kažys, R.; Rekuvienė, R. Viscosity and Density Measurement Method for Polymer Melts. Ultragarsas 2011, 66, 20−25. (29) Milliken, W. J.; Powell, R. L. Finite-Amplitude Ultrasonic Measurements of Polyethylene Oxide and Polyacrylaminde Solutions. J. Acoust. Soc. Am. 1990, 88, 1547−1554. (30) Bauer, H.-J.; Hässler, H.; Immendö rfer, M. Ultrasonic Relaxation of Rotational-Isomeric Equilibria in Polymer Solutions. Disc. Faraday Soc. 1970, 49, 238−243. (31) Lemarechal, P. On the Ultrasonic Absorption in Polystyrene Solution. Chem. Phys. Lett. 1972, 16, 495−498. (32) Ludlow, W.; Wyn-Jones, E.; Rassing, J. The Ultrasonic Relaxation Spectrum Associated with a Three State Conformational Change of Polystyrene in Solution. Chem. Phys. Lett. 1972, 13, 477− 481. (33) Dunbar, J. I.; North, A. N.; Pethrick, R. A. Acoustic Absorption and Dynamic Compressibility Studies of Dilute and Concentrated Solutions of Polystyrene in Toluene and Cyclohexane. J. Polym. Sci. 1977, 15, 262−277. (34) Bae, J.-R.; Kim, J.-K.; Yi, M.-H. Ultrasonic Velocity and Absorption Measurements for Polyethylene Glycol and Water Solutions. Jpn. J. Appl. Phys. 2000, 39, 2946−2947. (35) Eckert, S.; Hoffmann, S.; Meier, G.; Alig, I. Composition Fluctuations in a Non-Critical Binary Polymer Blend Studied by Ultrasonic and Light Scattering Experiments. Phys. Chem. Chem. Phys. 2002, 4, 2594−2603. (36) Fenner, D. B. Ultrasound Measurements of Polystyrene Solutions in the Binodal, Theta, and Good-Solvent Regions. J. Chem. Phys. 1984, 81, 5179−5188. (37) Mirzaev, S. Z.; Kaatze, U. Broadband Ultrasonic Spectrometry of Polystyrene-Cyclohexane Critical Mixtures. Phys. Rev. E 2013, 88, 042316 (9 pp). (38) Sperkach, V. S.; Sperkach, Y. V.; Stribulevich, A. L. Acoustic Spectroscopy of Polyethylene Glycols and Kinetics of Processes of Their Structural reconstruction. Acoust. Phys. 1997, 43, 462−465. (39) Sperkach, V. S.; Sperkach, Y. V.; Stribulevich, A. L. Mechanisms of Structural Relaxation in Polyethylene Glycol. Acoust. Phys. 1997, 43, 618−620. (40) Stribulevich, A.; Sperkach, V.; Sperkach, Y. Rotational Isomeric Relaxation in Poly(Ethylene Glycols) Studied by Acoustic Spectroscopy. Polym. Eng. Sci. 1999, 39, 394−398. (41) Sperkach, V. S.; Sperkach, S. O. Molecular Structure and Relaxation Properties of Poly(Ethyleneglycol)-Ionene Systems. Uzbek J. Phys. 2003, 5, 135−140. 13310



Article

(65) Zorębski, E.; Zorębski, M. A Comparative Ultrasonic Relaxation Study of Lower Vicinal and Terminal Alkanediols at 298.15 K in relation to Their Molecular Structure and Hydrogen Bonding. J. Phys. Chem. B 2014, 118, 5934−5942. (66) Debye, P. Polar Molecules; Chemical Catalog: New York, 1929. (67) Slutsky, L. J. In Methods of Experimental Physics 19: Ultrasonics; Edmonds, P. D., Ed.; Academic: New York, 1981; pp 179−235. (68) Strehlow, H. Rapid Reactions in Solutions; VCH: Weinheim, Germany, 1992. (69) Marquardt, D. W. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math. 1963, 11, 431−441. (70) Cole, K. S.; Cole, R. H. Dispersion and Absorption in Dielectrics. I. Alternating Current Characteristics. J. Chem. Phys. 1941, 9, 341−351. (71) Rouse, P. E. A Theory of the Linear Viscoeleastic Properties of Dilute Solutions of Coiling Polymers. J. Chem. Phys. 1953, 21, 1272− 1280. (72) Hanke, E.; von Roden, K.; Kaatze, U. Hydrogen Network Fluctuations of Associating Liquids: Dielectric Relaxation of Ethylene Glycol Oligomers and Their Mixtures with Water. J. Chem. Phys. 2006, 125, 084507. (73) Behrends, R.; Kaatze, U. Structural Isomerization and Molecular Motions of Liquid n-Alkanes. Ultrasonic and High-Frequency Shear Viscosity Relaxation. J. Phys. Chem. A 2000, 104, 3269−3275. (74) Viswanath, D. S.; Natarajan, G. Data Book on the Viscosity of Liquids; Hemisphere Publishing: New York, 1989. (75) Eliel, E. L.; Wilen, S. H. Stereochemistry of Organic Compounds; Wiley: New York, 1994.

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Chain Dynamics of Ethylene Oxide Oligomer Melts. An Ultrasonic

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