Frequency Dependence of Dynamic Properties of Polyethylene Glycol

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Frequency Dependence of Dynamic Properties of Polyethylene Glycol Molecules on Oscillating Solid-Liquid Interface Minoru Yoshimoto, Mutsuo Tanaka, and Shigeru Kurosawa J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b03133 • Publication Date (Web): 13 Jul 2017 Downloaded from http://pubs.acs.org on July 14, 2017

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The Journal of Physical Chemistry

Frequency Dependence of Dynamic Properties of Polyethylene Glycol Molecules on Oscillating Solid-Liquid Interface

Minoru Yoshimoto,†,* Mutsuo Tanaka‡,* and Shigeru Kurosawa‡ †

Department of Information Science and Biomedical Engineering, Graduate School of Science and Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima 890-0065, JAPAN ‡

Health Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Central 6, 1-1-1 Higashi, Tsukuba, Ibaraki 305-8566, JAPAN

* To whom correspondence should be addressed

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ABSTRACT Frequency dependence of the resonant length of polyethylene glycol (PEG) on a solid-liquid interface oscillating at megahertz was studied by a quartz crystal microbalance (QCM). A QCM frequency and a number-average molecular weight (Mn) of PEG were changed systematically. This attempt revealed that, in all the frequencies used, the series-resonant frequency shift, ∆F, of the QCM against the square root of the density-viscosity product of the PEG solutions was linear and had intercept. In addition, the systematical analysis made it clear that the ∆F slope was able to be explained by the Debye process and the ∆F intercept became constant above the resonant length. These results led to the novel findings that the resonant length of the PEG molecules formed by physisorption was equal to that in the region of the viscous penetration depth and that the relationship between resonant length and QCM frequency was able to be explained by

τ ∝ M νn , where τ is a relaxation time of molecule and ν is a constant. Especially, we found that the resonant length of physisorption was smaller than that of chemisorption but that the frequency dependency was equal to each other.


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■ INTRODUCTION Adsorption of soft matter onto a solid-liquid interface plays important roles in many physical, chemical, and biological processes.1-36 The adsorption is classified into two types. One is chemisorption, and the other is physisorption. In chemisorption, molecules are strongly bound to an interface by covalent attachment. Therefore, chemisorption is employed widely in fields related to a thin coating of soft matter, including the field of biosensors.18-22 This is based on increasing interest in the interaction of biomolecules, such as protein, DNA, lipid, polymer, and so on. A biosensor is an analytical device constructed from a biological sensing element integrated within a signal transducer. In many cases of biosensors, a self-assembled monolayer (SAM) has been used to prevent adsorption of soft matter onto a solid-liquid interface.23-31 Alkanethiols are known to form SAMs through chemisorption on a gold surface with bond formation. Therefore, the alkanethiols are usually employed to fix antibody, enzyme and other protein receptors. In addition, those are used to prevent nonspecific adsorption of soft mater onto a solid-liquid interface by physisorption. Physisorption relies on weak forces such as Van der Waals forces, hydrophobic and electrostatic interactions, and hydrogen bonding. The phenomenon is a key mechanism influencing physical, chemical, and biological processes at a solid-liquid interface.32-36 On the other hand, physisorption causes universally nonspecific adsorption of soft matter onto a solid-liquid interface. Therefore, nonspecific adsorption, which often leads to unexpected results, is one of hot topics in the field of biosensors.18-22 However, beside chemisorption, the investigation of physisorption is relatively scarce. Recently, much effort has been devoted to obtaining comprehensive understanding of properties of soft matter molecules on the solid-liquid interface.18-36 This has been based on development of QCM (quartz crystal microbalance) technique as rheometer. The QCM measurement is supported by inverse piezoelectric effect of quartz crystal.37 This effect leads to the oscillation of thickness-shearing mode at megahertz. The QCM can operate not only in air but also 3 ACS Paragon Plus Environment


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in liquid. Therefore, the QCM with the frequency and dissipation monitoring allows real-time, quantitative analysis of adsorption and binding onto a solid-liquid interface with sensitivity as high as subnanogram. However, the theory of the QCM rheometer is based on the Voigt model with a spring and a dashpot in parallel, without considering an exact model. In other words, the present analysis does not derive from exact physical properties of soft matter molecules on an oscillating solid-liquid interface. The investigation of dynamic properties of soft matter molecules on a solid-liquid interface oscillating at megahertz is very important from theory and experiment. Generally, the motion of thickness moving with an oscillating plate is important from the point of physical and chemical views. It is called viscous penetration depth in a Newtonian liquid. On the other hand, in soft matters, it is called resonant length, corresponding to their inherent relaxation times. Recently, we have reported the dynamic properties of a chemisorbed soft matter on an oscillating solid-liquid interface, where the self-assembled monolayers of mercapto oligo(ethylene oxide) methyl ethers, HS(CH2CH2O)nCH3, was employed as a model of soft matters.38 The report showed that the resonant length of HS(CH2CH2O)nCH3 was estimated as 8.8 nm by using the Voigt model. On the other hand, on the basis of the Debye Process, the logarithm of the resonant length linearly decreased with the logarithm of 2πF, where F is the frequency of the QCM, and varied from 17.3 (9 MHz) to 12.4 nm (81 MHz). That is, we found that the thickness of soft matter moving with the oscillating plate was approximately 10 to 20 nm on the order of 10 MHz. We have also reported that the resonant length of physisorbed soft mater on a solid-liquid interface oscillating at 9 MHz was ca. 5.4 nm.39 That is, the resonant length of physisorbed soft matter was smaller than that of chemisorbed one. On the other hand, the frequency dependence of dynamic properties of physisorbed soft matter on an oscillating solid-liquid interface remains to be examined in detail. In the present paper, we examined in detail the dynamic properties of the PEG molecules as a model of soft matter. Especially, we focused on the frequency dependence of resonant length 4 ACS Paragon Plus Environment

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for PEG molecule on the solid-liquid interface oscillating at megahertz. First, we report frequency dependence of the resonant length in the region of the viscous penetration depth and the physisorbed thin layer. Next, we discuss the difference of frequency dependence between the resonant lengths of the physisorbed and chemisorbed molecules on an oscillating solid-liquid interface.

■ EXPERIMENTAL SECTION Sample Preparation Water with the specific resistance of 18.2 MΩ·cm was prepared with a Milli-Q apparatus (Millipore, Tokyo, Japan). The PEG solutions were employed using PEG-200 (Mn = 246 g/mol, Mw/Mn = 1.05), PEG-600 (Mn = 576 g/mol, Mw/Mn = 1.04), PEG-2K (Mn = 1912 g/mol, Mw/Mn = 1.01), PEG-6K (Mn = 8798 g/mol, Mw/Mn = 1.00), PEG-20K (Mn = 21335 g/mol, Mw/Mn = 1.00), PEG-500K (Mn = 4×105 g/mol), and PEG-2M (Mn = 1.75×106 g/mol) (Wako, Japan), where Mn and Mw mean the number-average molecular weight and the weight-average molecular weight, respectively. The values of Mn and Mw of the PEG molecules were measured by MALDI-MS (Shimazu AXIMA-CFR plus). In this case, we were not able to exactly estimate the values of Mn and Mw of PEG-500K and PEG-2M because those indicated the broad distribution of the molecular weight. However, the values of Mn measured by MALDI-MS are approximately equal to the average molecular weight depicted by Wako. Therefore, in PEG-500K and PEG-2M, we used the average molecular weight depicted by Wako as the values of Mn. The PEG molecules were dissolved by a magnetic stirring bar at a room temperature, and after two days of the preparation, the PEG solutions were used for the measurements of QCM analysis. The viscosity and density of the PEG solutions were measured by a Cannon-Ubbelohde viscometers and a pycnometer, respectively.39

QCM Measurement 5 ACS Paragon Plus Environment


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9 and 5 MHz AT-cut QCMs (Nihon Dempa Kogyo, Tokyo, Japan) with a pair of gold electrodes were employed in all experiments. The QCMs had a mirrored surface. In all experiments, one side of the QCM was sealed with a blank quartz crystal casing, maintaining it in an air environment (Figure S1b, Supporting Information). We used an impedance analyzer (Agilent Technologies 4294A) for resonant-frequency properties of the QCM. A 16047E test fixture (Agilent Technologies) was employed for a direct impedance measurement (Figure S1a, Supporting Information). The value of the source-power level was adjusted to 0.237 V. The impedance and phase data associated with 801 frequency data points centered at the frequency of the minimum impedance were recorded on a personal computer. The values of the resonant-frequency shift, ∆F, at 5, 9, 15, 27, 45, 63, and 81 MHz were calculated using admittance analysis.19,

21

The frequency span of the impedance analyzer was changed

depending on a QCM frequency. The cell volume was 8 ml. The cell had a water jacket to keep the temperature constant. The QCM was mounted level with the water surface and the immersion depth of the QCM was set at 0.5 cm (Figure S1a, Supporting Information). The cell temperature was adjusted to 25 ± 0.1˚C. For physical equilibrium between PEG molecules and gold electrode, the QCM immersed into the PEG solutions was left for 3 h prior to the onset of the impedance measurement. The PEG solutions were stirred by a stirring bar. The stirring bar was stopped when the impedance properties of the QCM were measured.

■ RESULTS AND DICUSSION We focus on the frequency dependence of resonant length of the PEG molecules. Consequently, we used the 1st (5 MHz) and 3rd (15 MHz) overtones of the 5 MHz QCM, and the 1st (9 MHz), 3rd (27 MHz), 5th (45 MHz), 7th (63 MHz), and 9th (81 MHz) overtones of the 9 MHz QCM. In order to study the frequency dependency of resonant length of the PEG molecules, we 6 ACS Paragon Plus Environment

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employed the value of

ρ Lη L − ρW ηW as the index for concentrations of the PEG solutions,

where ρ and η are the density and viscosity and the subscripts of L and w indicate the PEG solution and pure water, respectively. The index is frequently used in a Newtonian liquid. That is, based on the difference between a Newtonian liquid and a PEG solution, we can discuss the dynamic properties of the PEG molecules with molecular weight on an oscillating solid-liquid interface. The experiments were carried out using seven types of harmonic frequencies of the QCM. However, because the tendencies of ∆F were the same in all the harmonic frequencies, we discuss the case of 5 MHz as an example in the following sections.

Behavior of ∆F in PEG solutions Figure 1 shows the result of ∆F versus

ρ Lη L − ρW ηW in 5 MHz. In all the PEG

solutions, the experimental data of ∆F change linearly with

ρ Lη L − ρW ηW . Furthermore, the

values of ∆F slope of PEG-200 and PEG-600 are approximately equivalent to that of a Newtonian liquid. On the other hand, the values of ∆F slope of PEG-2K, PEG-6K, PEG-20K, PEG-500K, and PEG-2M are smaller than that of a Newtonian liquid. These differences derive from the viscoelasticity of the PEG solutions. These tendencies are the same in 9, 15, 27, 45, 63, and 81 MHz (Figure S2, Supporting Information). We also analyzed the ∆F data using the least-squares method. As an instance, we show the data of the PEG-6k solution in Figure 2. ∆F has the intercept due to the physisorbed thin layer formed on the gold electrode.37, 39 The results of all the PEG solutions are summarized in Figure 3. The value of the ∆F slope decreases rapidly in the region from PEG-600 to PEG-20K, which is based on the behavior of the PEG molecules in the region of the viscous penetration depth.39, 40 On the other hand, the ∆F intercept is detected in all the PEG solutions, which increases with Mn and has the constant value of about 17 Hz above PEG-6K. The results of the other six frequencies are depicted in Figure S3 (Supporting Information). The tendencies are the same as those of 5 MHz. 7 ACS Paragon Plus Environment


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The schematic diagram of the solid-liquid interface of the QCM is shown in Figure S4 (Supporting Information). We discuss the ∆F slope first and, next, the intercept.

Physical meaning of ∆F slope in PEG solutions Figure 3 indicates that the values of ∆F slope in 5MHz decrease rapidly in the region between PEG-600 and PEG-20K. The rapid shift derives from the dynamic behavior of the PEG molecules in the region of the viscous penetration depth on an oscillating solid-liquid interface.19, 21 We examine the rapid shift from the point of view of the Debye process. The Debye process expresses a single-time relaxation process of a molecule.41 In the process, the dynamic compliance, J*(ω), is written by

J * (ω ) =

∆J 1 + iωτ

(1)

where ∆J is the relaxation strength of the molecule, τ is the relaxation time of the molecule and ω is the angular frequency of the oscillating plate. Separation into the real and imaginary parts gives

J * (ω ) = J ′ − iJ ′′ =

∆J 1 + ω 2τ 2

−i

∆Jωτ 1 + ω 2τ 2

(2)

Equation 2 shows that the imaginary part has the maximum value at ω = 1/τ. In other words, the molecule resonates with the oscillating plate at the frequency and has the maximum energy loss. On the other hand, the real part shows

∆J ∆J at the point. That is, is related to the resonant length. 2 2

Hence, we adapt this concept for the present experimental data. In the present experiments, the values of ∆F slope and Mn correspond to J’ and ωτ of eq 2,


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respectively. Consequently, we fitted the real part of eq 2 to the experimental data of the ∆F slope (Figure 3). This attempt led to the result that the value of ∆F slope corresponding to

∆J was 2

3.80×103 Hz/(gcm-2s-1/2). Next, we discuss the relationship between

the Mn value for

∆J and the resonant length. From Figure 3, 2

∆J = 3.80 × 10 3 Hz/(gcm - 2 s -1/2 ) , M ∆J , is about 4.28×103 g/mol. Moreover, the 2 2

relationship between molecular weight, M, and the gyration radius, Rg, is written as follows:42

Rg = 0.215M 0.583±0.031

(3)

If M ∆J is regarded as M, the Rg is 28.2 Å ( = 0.215M ∆0.J583 ).41-43 In other words, the PEG length of 2

2

56.4 Å (= 2Rg) in diameter moves with the oscillating QCM of 5 MHz in the region of the viscous penetration depth, where τ is 3.18×10-8 s on the basis of ω = 1/τ described in eq 2. The fitting results of the other six frequencies are also illustrated in Figure S3 (Supporting Information). The results show that the experimental data of ∆F slope in all the frequencies follows the Debye process. The resonant length of other six frequencies was also estimated in the same way as 5MHz. The results are shown in Figure 4. We discuss in detail the results of frequency dependence of the resonant length in the section of “Frequency dependence of resonant length”.

Physical meaning of ∆F intercept in PEG solutions The ∆F intercept derives from the PEG thin layer formed by physisorption.19, 37 Figure 1 shows that the ∆F values of 5 MHz vary linearly with ρ Lη L − ρ W ηW . Because of the linear, the intercept value of ∆F is constant. In other words, the properties of the PEG thin layer are



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independent of the bulk concentration in the present concentration range. Figure 3 illustrates the ∆F intercept as a function of Mn. The values of the ∆F intercept increase with Mn and become constant above PEG-6K. Here, we exactly calculate the starting molecular weight above which the ∆F intercept becomes constant. We discuss the properties of the thin layer on the basis of the following equation proposed by Voinova et al.:37

∆f V = −

2 η η 2ω q2   η1  1   + h − 2 h ρ ω  2 2 q 2 2 2   2 2πρ q hq  δ 1  δ 1  µ 2 + ω qη 2  

1

2 η µ 2ω q   η1  1   ∆DV = + 2 h  2 2 2   2 ω q ρ q hq δ 1  δ 1  µ 2 + ω qη 2  

2

δ1 =

2η1

(4)

(5)

(6)

ρ1ω q

where ∆fV is the series resonant-frequency shift, ∆DV is the energy dissipation shift, h is the thickness, δ is the penetration depth, and the subscripts of q, 1 and 2 denotes quartz, Newtonian liquid and thin layer, respectively. These equations describe the properties of the one thin layer formed on the QCM in a Newtonian liquid. In the present study, the values of the ∆F intercept are those at the infinite dilution. In other words, the present experimental data of the ∆F intercept mean the values of the thin layer immersed in pure water. Pure water obeys the properties of a Newtonian liquid. Accordingly, the present experimental data of the ∆F intercept can be discussed on the basis of eq 4. Equation 4 indicates that the ∆fV intercept derives from the second and third terms with the thin layer. Figure 3 shows that the values of ∆F intercept become constant at the thickness of more than PEG-6K. This result suggests that the shift of ∆F intercept is mainly caused by that of the 10 ACS Paragon Plus Environment

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thickness of the PEG thin layer. Hence, equation 4 leads to the assumption that, because the ∆fV 2

η 2ω q η  intercept in the PEG thin layer is mainly dependent on h2, the value of ρ 2ω q − 2 1  2 2 2  δ 1  µ 2 + ω qη 2 2

is constant for the variation of Mn. In other words, the density, viscosity and elasticity of the thin layer due to Mn are constant. From the above assumption, in the infinite dilution at 25 ˚C, we estimated that the elasticity of the thin layer was 3.00×104 N/m2 as the constant value. The ∆fV intercept as function of Mn (h2 calculated from eq 3) is depicted in Figure 3. The theoretical value is consistent with the experimental data, where h2 is set at the constant value above 4.28×103 g/mol of Mn. This result supports the assumption to the effect that, in the PEG thin layer, the shift of the ∆fV intercept against Mn is mainly caused by h2. Furthermore, we can confirm that the molecular weight of 4.28×103

g/mol is the starting molecular weight above which the ∆F intercept becomes constant. That is, the resonant length of the PEG thin layer is 56.4 Å. Consequently, the shift of the ∆F intercept is mainly related to that of the thickness of the PEG thin layer and the ∆F intercept becomes constant above the resonant length of the PEG molecule. The thickness of the PEG thin layer above the resonant length does not affect the ∆F intercept. The value of 56.4 Å is also equal to the resonant length in the region of the viscous penetration depth. The results of other six frequencies are also illustrated in Figure S3 (Supporting Information). The results show that, in all the frequencies, the experimental data of ∆F intercept follows the tendencies of 5 MHz. Moreover, in the same way as 5 MHz, the resonant length of the PEG thin layer formed by physisorption are equal to that of the region of the viscous penetration depth in all the frequencies. We discuss in detail the results of frequency dependence of the resonant length in the next section.

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The resonant length of the PEG thin layer formed by physisorption is the same as that in the region of the viscous penetration depth. Therefore, we can discuss simultaneously the frequency dependence of both resonant lengths in the PEG thin layer formed by physisorption and in the region of the viscous penetration depth. Generally, in polymer, the relationship between τ and Mn is given by

τ ∝ M νn

(7)

where ν is a constant.41, 43 According to eq 7, log M ∆J was plotted against log ω, where ω is the 2

angular frequency of QCM and ω is 1/τ. Figure 4a shows that log M ∆J decreases linearly with log 2

ω . This result indicates that the relationship between M ∆J and ω follows eq 7. Next, we 2

calculated the resonant length, RPhys (m), corresponding to M ∆J by using eq 3. Those logarithm 2

values also decrease linearly with log ω (Figure 4b). As a result, we can obtain the following equation using the least-squares method:

log RPhys = −7.054 − 0.158 log ω

(8)

This equation indicates the frequency dependence of resonant length for the PEG molecules in both the viscous penetration depth and physisorption.

Comparison between chemisorption and physisorption In the previous paper, we have proposed the following relationship for frequency dependence of the resonant length, RChem (m), in chemisorption using the self-assembled 12 ACS Paragon Plus Environment

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monolayers of HS(CH2CH2O)nCH3, which is called OEG:38

log RChem = −6.591 − 0.151 log ω

(9)

The value calculated using eq 9 is shown in Figure 4b. Equations 8 and 9 indicate the novel relationship between chemisorption and physisorption. That is, the resonant length of physisorption is smaller than that of chemisorption but the frequency dependency (slope) is equal to each other. In eqs 8 and 9, the slope and the intercept derive from the molecular structure and bond form, respectively. This result may imply that an increase in slip occurs on an oscillating solid-liquid interface in physisorption. In the present paper, we propose the novel model between physisorption and chemisorption of the compounds based on ethylene glycol, i.e., eqs 8 and 9. In order to obtain further insight, we write the format of eqs 8 and 9 as

log R = a − b log ω

(10)

In the equation, R is the resonant length of soft matter, a is a parameter in relation to bonding conditions and b is related to the physical properties of soft matters. The present results suggest that, in homologous soft matters, a changes depending on physisorption and chemisorption but b is constant. The model will be universally applied to other soft matters. The experimental and theoretical studies of the model are now underway in our research group.

■ CONCLUSIONS We have investigated the frequency dependence of resonant length of the PEG molecules on an oscillating solid-liquid interface using the QCM. This study indicates the following novel 13 ACS Paragon Plus Environment


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findings.

1. The resonant length of the PEG molecules formed by physisorption is equal to that in the region of the viscous penetration depth. In both cases, the relationship between resonant length and frequency can be described by eq 8.

2. The resonant length of physisorption is smaller than that of chemisorption but the frequency dependency is equal to each other.

In the present region of frequency, the Debye process was applied to the analysis of dynamic properties of the PEG molecules. This attempt revealed the frequency dependence of resonant length of physisorption and chemisorption. The results presented in this paper may give the new directions of PEG molecules in sensor, biochemistry, medical science, and so on. Moreover, the new findings of the relationship between physisorption and chemisorption may bring innovation to the fields of the polymer and oligomer sciences in relation to a solid-liquid interface oscillating at megahertz.

■ ASSOCIATED CONTENT Supporting Information. Schematic illustration of an experimental apparatus (Figure S1), the concentrations (wt%) of PEG solutions used in the present paper, ∆F versus

ρ Lη L − ρ W ηW of

the PEG solutions in six types of the QCM frequencies (Figure S2), slope and intercept of ∆F versus Mn of the PEG solutions in six types of the QCM frequencies (Figure S3), schematic diagram of the solid-liquid interface on the QCM (Figure S4).


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Figure captions Figure 1 ∆F versus

ρ Lη L − ρW ηW of the PEG solutions in the 5 MHz QCM. ●: PEG-200, ■: PEG-600,

▲: PEG-2K, ▼: PEG-6K, ○: PEG-20K, ∆: PEG-500K, □: PEG-2M. The error bar represents a standard deviation. The measurements were repeated six times. The solid line shows a Newtonian liquid.44 The concentrations (wt%) of PEG solutions used in the present paper is shown in Supporting Information.

Figure 2 ∆F versus

ρ Lη L − ρW ηW

of the PEG-6K solutions in the 5 MHz QCM. The error bar

represents a standard deviation. The measurements were repeated six times. The solid line shows the value calculated by using the least-squares method.

Figure 3 Slope and intercept of ∆F versus Mn of the PEG solutions in the 5 MHz QCM. ●: ∆F slope, ○: ∆F intercept. The mark (■) on the y-axis shows the slope value of a Newtonian liquid. The solid line indicates the value calculated from the real part of eq 2 by using the nonlinear least-squares method. The dashed line shows the value calculated from eqs 3 and 4 with η1 = η2 = 8.90×10-4 Ns/m2, ρ1 =

ρ2 = 997 kg/m3, ρq = 2648 kg/m3, hq = 3.34×10-4 m, ωq = 3.14 ×107 rad/s, and µ2 =3.0×104 N/m2, where h2 is kept constant above 4.28×103 g/mol.

Figure 4 Frequency dependence of the PEG molecules. (a) Frequency dependence of M ∆J . The Mn values 2

corresponding to the resonant lengths of physisorption are equal to the M ∆J values. 2


(b)

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Frequency dependence of the resonant length in the viscous penetration depth and physisorption. The x axis shows the angular frequency of QCM. The solid lines were calculated by using the least-squares method. The dashed line in (b) shows the value of chemisorption evaluated from eq 9.



2000

1500 ∆F [Hz]

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1000

500

0 0.0

0.2

0.4 1/2

0.6 1/2

0.8

1.0

-2 -1/2

(ρLηL) -(ρWηW) [gcm s

] Figure 1

22

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1500

1000 ∆F [Hz]

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500

0 0.0

0.1

0.2

1/2

0.3 1/2

0.4

-2 -1/2

(ρLηL) -(ρWηW) [gcm s

] Figure 2

23



40 8000 30 6000 20 4000 10

2000

0 102

Intercept of ∆F [Hz]

Slope of ∆F [Hz/(gcm-2s-1/2)]

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0 10

3

10

4

10

5

10

6

10

7

Mn [g/mol] Figure 3 24


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105

100

(b) Resonant Length [nm]

(a)

M ∆J/2 [g/mol]

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104

103

102 107

108

10

1

0.1 107

109

ω [rad/s]

108

109

ω [rad/s] Figure 4

25



100 Resonant Length [nm]

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Chemisorption 10

Physisorption 1

0.1 107

108

109

ω [rad/s]

TOC Graphic 26


Frequency Dependence of Dynamic Properties of Polyethylene Glycol

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