Molecular Weight Dependence of Viscosity and Shear Modulus of

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J. Phys. Chem. C 2009, 113, 13793–13800

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Molecular Weight Dependence of Viscosity and Shear Modulus of Polyethylene Glycol (PEG) Solution Boundary Layers Ping Wang,†,‡,§ Jiajie Fang,†,‡,§ Sheng Qin,† Yihong Kang,‡ and Da-Ming Zhu*,†,‡ Department of Modern Physics, UniVersity of Science and Technology of China Hefei, P.R. China 230027, and Department of Physics, UniVersity of MissourisKansas City, Kansas City, Missouri 64110 ReceiVed: April 2, 2009; ReVised Manuscript ReceiVed: June 10, 2009

The viscosity and shear modulus of solutions of polyethylene glycol (PEG) with different molecular weights near a solid substrate were studied using a quartz crystal resonator technique. The technique probes the solution boundary layer with a thickness of about the viscous penetration depth adjacent to the surface of the quartz crystal. On the basis of the resonant frequency shift and the dissipation broadening of the quartz crystal resonator, the viscosity and shear modulus of the solution boundary layers as a function of the concentration and the molecular weight of PEG molecules are determined. The results show that, near the semidilute concentration of the solution, the viscosity of the boundary layer increases rapidly; the rise of the viscosity follows a power law with an exponent which depends on the molecular weight of PEG molecules. For solution boundary layers consisting of larger PEG molecules, their shear moduli display a rapid increase above the semidilute concentration also. The implications of these results are discussed. 1. Introduction Although the properties of polymer solutions have been studied for quite a long time, the subject still draws considerable interest due to the important roles played by polymer solutions in many different fields.1-3 Much effort has been devoted toward obtaining a comprehensive understanding of the dynamics of polymer chains in different concentration regimes of the solutions, since the fundamental mechanisms which govern the physical processes in these concentration regimes, which have been traditionally classified as dilute, semidilute, and concentrated solutions, are distinctly different.4-31 In dilute solutions, the polymer molecules move nearly independently of each other, and the self-diffusion coefficient is primarily determined by the size of the polymer molecules; the concentration dependence of the viscosity typically shows an approximately linear dependence. In solutions of semidilute concentrations, the side chains of the polymers start to overlap, making their movements difficult and resulting in more rapid rises of the viscosity of the solutions. Above the semidilute concentration, polymer molecules become entangled with each other, resulting in a stronger viscosity dependence on concentration.4,5 These characteristic behaviors were initially predicted in a scaling theory worked out by de Gennes et al., who provided much insight into the fundamental mechanisms that govern the viscoelastic properties of polymer solutions, and were subsequently verified in a large number of experimental investigations using different techniques.4-31 The shear modulus of a fluid solution is considered, in general, to be always near zero except when the conditions are such that the solution is close to its glass transition where the dynamic shear modulus of the polymer solution may rise significantly.1-3 The viscoelastic behaviors of a polymer solution can be affected by the presence of a boundary or an interface, as * Corresponding author. E-mail: [email protected]. † University of Science and Technology of China Hefei. ‡ University of MissourisKansas City. § These authors contributed equally to this work.

suggested by the same scaling theory.4,5,32 Depending on the interaction between polymer molecules and the boundary, and that between polymer and solvent molecules, a boundary or an interface can attract or repel the polymer molecules, resulting in a profile of polymer concentration which varies in the direction normal to the interface.32-39 Also, the boundary or interface imposes a geometric confinement to polymer molecules, which may cause changes in the configuration of the molecules and affect the mobility of polymers in the solution layer next to the boundary. Therefore, the viscoelastic behaviors of a polymer solution layer next to the boundary might differ significantly from those revealed in bulk solutions. However, unlike the case of polymer films at the solid-vapor interface, experimental techniques that can be used to study polymer layers at the solid-solution interface are scarce, since the interface is sandwiched by two condensed phases so many techniques widely used for studying phenomena on solid-vapor interfaces become difficult to be applied here. As a result, our understanding on the properties of polymer films on solid-liquid interfaces is quite limited. In particular, it is not very clear how the viscoelastic properties of a polymer solution boundary layer are affected by the presence of the boundary or the interface. In order to elucidate these issues about the boundary layers in a polymer solution, we have conducted a study on the viscoelastic properties of polyethylene glycol (PEG) solution boundary layers using a quartz oscillator technique. PEG is a commercially important polymer which has a wide range of applications in many different fields ranging from chemical synthesis to biomedical and clinic research.19,20,37 Many of these applications involve PEG polymers in the forms of thin layers applied to interfaces.19,20,37 Thus, the results obtained in this study will not only have the fundamental importance to our understanding of the polymer solution boundary layers, but also practical implication to the application of the polymer solutions. The results obtained in this study show that the viscosity of polyethylene glycol (PEG) solution boundary layer follows a similar concentration dependence as that for bulk solutions, but the scaling exponent depends on the molecular weight of PEG

10.1021/jp903060q CCC: $40.75  2009 American Chemical Society Published on Web 07/06/2009

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polymers. The shear modulus of low molecular weight PEG solution appears to be zero in the concentration regime covered in this study, but for high molecular weight PEG solutions, the shear modulus rises rapidly as PEG concentration increases to above the semidilute concentration.

A )

κξ + κbξb tanh(ξbhb) κξ - κbξb tanh(ξbhb)

µ κ ) η - i , ω



-

ξ )

2. Quartz Crystal Resonator Technique Quartz crystal resonator technique relies on the measurements of response curves of a quartz crystal oscillator under a frequency sweep of acoustic shear mode excitation applied across the crystal.40-49 The propagation of the excitation in the medium that the crystal is in contact with directly affects the frequency response of the crystal.40-44 In general, the response curve would contain a series of peaks that correspond to the resonances of the crystal at the fundamental frequency and its higher harmonics. The shifts of the resonant peaks and the changes in the width of the peaks are related to mass deposition onto the crystal and the energy dissipation in the deposited mass. Thus, by measuring the shifts of the resonant peaks and the broadening of these peaks, one can probe sensitively the adsorbed mass and the energy dissipation associated with it on a surface. The technique has been used for monitoring thin film growth and studying the properties of thin films on solid surfaces for many years.40-44 Recently, the technique has been increasingly used in studies of adsorption of macromolecules at solid-solution interfaces.45-52 To quantitatively describe the resonant frequency shift and the dissipation factor change in terms of the mass deposited onto a quartz crystal or the properties of the medium that the crystal is in contact with, one often employs a single or multilayer slab model which treats the adsorbed mass as a uniform layer on the interface between the electrode surface of the crystal and a uniform medium which can be vacuum, air, or a liquid.44,50 With the assumption that the adsorbed layer and the medium above it can all be described in terms of a Voightbased viscoelastic model, which considers that the system can be represented by a purely viscous damper and purely elastic spring connected in parallel, and under a no-slip boundary condition for all the interfaces involved, the changes of the resonant frequency and the dissipation factor are related to the viscosity η and shear modulus µ of the adsorbed layer by44

(

∆f ) Im

β 2πFqhq

(

∆D ) Re

)

β πfFqhq

)

(1)

1 - A exp(2ξh) 1 + A exp(2ξh)

ξb )

∆f ≈ -

1 2πFqhq

(



F ηω 2

(3)

(5)

iFbω ηb

õ2

+ η2ω2 + µ µ2 + η2ω2



õ2

µ

1 πfFqhq

(  F ηω 2

õ2

+ η2ω2 - µ µ2 + η2ω2

+ η2ω2 - µ + µ2 + η2ω2



µ

(2)



Fω2 µ + iωη

(6)

where h and F are the thickness and density of the adsorbed layer, respectively; Fq and hq are the density and thickness of quartz crystal, respectively; hb, Fb, ηb, are the thickness, the density, and the viscosity of the medium above the adsorbed layer, respectively. Here, an assumption is made that this medium has zero shear modulus. The resonant frequency shift and the broadening of the resonance peak as a function of the thickness of the adsorbed layer and all the material parameters involved can be derived straight forwardly using eqs 1 and 2. The general behaviors of the dependence of ∆f and ∆D on the adsorbed layer thickness, being illustrated in Figure 1a,b, are that -∆f and ∆D rise (-∆f rises approximately linearly with the layer thickness for small h) until a viscous penetration depth δ ) (2η/Fω)1/2 is reached; after that, -∆f and ∆D reach maxima and then decrease to saturated values. This behavior can be easily understood as due to the interference of the acoustic excitations and their exponential decay in a viscoelastic medium with a decay length δ. Therefore, quartz crystal resonator technique essentially probes only a boundary layer with a depth on the order of δ in a solution which is in contact with surface of quartz crystal, as illustrated in Figure 1c. The saturated values for ∆f and ∆D in the thick limit of an adsorbed layer can be derived straightforwardly from eqs 1 and 2 by considering h f ∞:44,45

∆D ≈

where

β ) κξ

κb ) ηb,

(4)

õ2

+ η2ω2 + µ µ2 + η2ω2

) )

(7)

(8)

These expressions have also been used in determining the viscoelastic properties of homogeneous solutions in contact with a quartz resonator.42-45 However, it should be noted that the materials parameters contained in these expressions are those for the boundary layer of the solutions adjacent to the crystal-solution interface and may differ from those for a bulk liquid. Thus, by measuring the frequency shift and the dissipation factor change, the viscosity and shear modulus of the solution boundary layer of viscous penetration depth δ can be determined. The approach of using eqs 7 and 8 and measuring ∆f and ∆D to determine the viscoelastic parameters of a solution

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Figure 2. Calculated -∆f/∆D as a function of viscosity η and shear modulus µ of a semi-infinite solution covering one side of a quartz crystal. Since the acoustic excitation probes the solution in a region adjacent to the quartz surface with a depth comparable to δ, viscosity η and shear modulus µ should be treated as that of the solution boundary layer adjacent to the quartz-solution interface. (a) -∆f/∆D as a function of viscosity at fixed shear moduli; (b) -∆f/∆D as a function of shear modulus at fixed viscosities.

Figure 1. (a) Resonant frequency shift -∆f versus the layer thickness and (b) dissipation factor change ∆D versus the layer thickness, calculated using eqs 1 and 2. The material parameters used in the calculation are listed in the panels. (c) An illustration of a quartz crystal resonator with one surface in contact with a polymer solution. The acoustic excitation generated on the crystal surface decays exponentially in the solution over a distance of viscous penetration depth δ.

boundary layer has been tested on several simple systems such as glycerol solutions.55,56 In these tests, the shear modulus of these solutions was assumed to be zero; the frequency shifts or the dissipation factor changes measured using a quartz crystal resonator were treated as purely due to the viscosity variations in solutions. In this case, which is referred as in the viscous limit (µ = 0), eqs 7 and 8 are reduced to ∆f = -(Fηω)1/2/23/2πFqhq and ∆D = (2Fη/ω)1/2/ Fqhq, the so-called Kanazawa-Gordon equations.43 Then, the viscosity η of a fluid boundary layer can be determined from either ∆f or ∆D assuming the density of the layer is known. The ratio -∆f/∆D = ω/4π in the viscous limit can be used as a criterion for determining whether a solution boundary layer

can be treated as purely viscous (µ = 0) or not. Figure 2 plots -∆f/∆D as a function of viscosity under constant shear modulus and as function of shear modulus under constant viscosity of the solution calculated using eqs 7 and 8. As can be seen from the figures, for nonzero µ (>0), the ratio -∆f/∆D is always below ω/4π and decreases systematically as the shear modulus increases. Thus, for a sufficiently thick overlayer (the thickness is much larger than the viscous penetration depth) covering the surface of a quartz crystal resonator, any changes to the viscoelastic properties of the boundary layer due to various mechanisms such as concentration increase or adsorption or grafting of molecules onto the solid surface would result in a decrease of the ratio from that determined by ω/4π.52 If the shear modulus of the boundary layer increases with the concentration of the solution, the measured -∆f/∆D would show a systematic decrease correspondingly. Thus, by examining the ratio -∆f/∆D, the viscosity and shear modulus of a solution boundary layer can be determined from the experimentally measured ∆f and ∆D using eqs 7 and 8. 3. Experimental Section The quartz crystal resonator technique used in this study is similar to the quartz crystal microbalance with dissipation monitoring (QCM-D) widely used in chemical and biological research.44-49 The details of the experimental setup have been

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described schematically in a previous publication.51 In brief, polyethylene glycol (PEG) samples were mixed with twice distilled water to solutions with known concentrations. The solution was placed in a home-built liquid flow cell in which a freshly cleaned quartz crystal with a fundamental resonant frequency of 5 MHz and a diameter of 25 mm was installed at the bottom of the cell.51 Gold films were evaporated onto both sides of the surfaces of the quartz crystals as the electrodes for applying an ac driving voltage; one side of the electrode surface was in contact with the solution and was used as the adsorption substrates. The surface of the electrode was examined using an atomic force microscope, and the root-mean-square roughness of the gold surfaces was found to be typically less than 3 nm.46-49 The top of the flow cell was connected to a solution reservoir through a plastic tube; a drain tube was also connected to the top of the cell to allow excess solution flowing out the cell. The temperature of the cell was controlled at 24 °C throughout the experiment, and the temperature stability of the liquid flow cell was typically better than 0.01 °C. The electrodes on the quartz crystal were connected to a network analyzer for measuring the acoustic responses of the quartz crystal. By sweeping the frequency of the driving voltage applied across the electrodes, a series of acoustic resonances were recorded using the network analyzer. The resonant frequencies (fresonant) and the width of the resonant peaks were obtained by fitting the measured response curve. The dissipation factor (D) is determined by dividing the full width at half-maximum (fwhm) of the resonant peak with the resonant frequency (D ) fwhm/ fresonant). With the resonant frequency and the width of a resonant peak measured while the flow cell was empty as references, the shift of the resonant frequency ∆f and the changes of the dissipation factor ∆D as a function of polymer concentration in solution were determined. The polyethylene glycol (PEG) samples with molecular weight ranging from 200 to 20 000 g/mol with a polydispersity index Mw/Mn ) 1.06 were purchased from SCRC (Sinopharm Chemical Reagent Co. Ltd.).57 The samples were used as received without further purification. In the experiments, the liquid cell was initially filled with pure water; the frequency shift (∆f) and the change of the dissipation factor (∆D) of the crystal were measured in air and in pure water to check against the standards established in our previous studies.46-48 After that, small amounts of solutions containing PEG polymer molecules were added to the cell. When the equilibrium of the solution was clearly reached, the frequency shift (∆f) and the dissipation factor change (∆D) were recorded. Then, additional PEG solutions were added to the cell, and the measurements on ∆f and ∆D were repeated in solutions with higher concentrations. The concentration of PEG in the liquid cell is determined on the basis of the initial volume of the pure water in the cell and the subsequent additions of PEG solutions with known concentrations. The measured ∆f and ∆D versus the concentration of PEG in solution are essentially a resonant frequency isotherm and a dissipation factor isotherm. 4. Results and Discussion (a) Molecular Weight Dependence of Viscoelastic Properties of PEG Boundary Layers. Figure 3 displays the measured resonant frequency shift (∆f) and the dissipation factor change (∆D) of the quartz crystal in solutions of different PEG concentrations and for PEGs with different molecular weights. Both the ∆f and ∆D change very little as the concentration of the solution is lower than 20 mg/mL (no effort was directed toward observing possible steplike changes in ∆f and ∆D, which

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Figure 3. (a) Shift of resonant frequency ∆f measured with the quartz crystal resonator as a function of PEG solution concentration, for PEGs with different molecular weight. (b) Changes of the dissipation factor ∆D measured simultaneously with the shift of the resonant frequency as a function of PEG solution concentration for PEGs with different molecular weights. The numbers in the inlet specify the symbols used in the plot and are in the unit of g/mol. The lines are guides to the eye.

correspond to adsorption of PEG molecules onto the gold electrode surface occurring at much lower concentration). As the concentration increases to above 100 mg/mL, both -∆f and ∆D start to rise rapidly. The rises in -∆f and ∆D appear to depend on the molecular weight. The rises in -∆f increase with the molecular weight of PEG up to about 4000 g/mol, and then decrease slightly above that, while the rises in ∆D show monotonic increase with molecular weight of PEGs, indicating that the viscoelastic properties of solutions are affected more strongly by larger polymer molecules. In Figure 4, -∆f/∆D values for PEG solutions with different molecular weight are plotted as a function of the concentration. The figure shows that, for light PEG molecules (200, 400, 1000 g/mol), -∆f/∆D is roughly a constant in the concentration regime covered in this study, indicating that the solution is almost in the viscous limit or the dynamic shear modulus of the solution boundary layer is negligible. However, for heavier PEG molecules (2000, 4000, 10 000, and 20 000 g/mol), the ratio shows a clear decreasing tread as the concentration of PEG solution increases. Such a trend is an indication that at the frequency used the dynamic shear modulus of these PEG solutions becomes nonzero and rises as the concentration increases.

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Figure 4. -∆f/∆D versus the concentration of PEG solutions with PEG of different molecular weights. The dotted lines are guides to the eye. The numbers in the inlet specify the symbols used in the plot and are in the unit of g/mol.

Since the ratio -∆f /∆D obviously indicates that the shear moduli of the solution boundary layers in high molecular weight PEG solutions are not zero, the viscosity and the shear modulus of the PEG solution boundary layers were subsequently calculated using eqs 7 and 8 based on the measured ∆f and ∆D, and the results are displayed in Figure 5. The viscosity and the shear modulus of the PEG solution boundary layers remain negligibly small at the low concentrations. As the concentration increases, the viscosities of PEG solution boundary layers rise; the concentration dependence of the viscosity is more pronounced for solutions with high molecular weight PEGs. The shear modulus of the solution boundary layers for lower molecular weight PEGs remain near zero in the entire concentration regime covered in this study. However, for solutions with higher molecular weight PEGs, the shear modulus increases rapidly as the PEG concentration is above about 100 mg/mL. (b) Scaling Properties of the Viscosity and Shear Modulus of PEG Solution Boundary Layers. As mentioned above, the viscoelastic properties of bulk polymer solutions in the different concentration regimes can be described very well by a scaling theory developed by de Gennes et al.4,5 The scaling theory considers that the dynamic properties of a polymer solution are strongly influenced by the interactions and the geometric configurations of the polymer molecules, which could differ significantly in different concentration regimes. The viscoelastic properties of a polymer solution can change dramatically as the concentration reaches a semidilute concentration c* which represents a crossover between a diluted regime in which polymer coils are well separated and the concentrated regime in which polymer coils overlap. The semidilute concentration c* can be expressed as1-3

c* ) 3M/(4π 〈R2〉3/2Nav)

(9)

where M is the molecular weight, R is the gyration radius of the polymer coil, and Nav is Avogadro’s number. According to the scaling theory, the viscosity dependence on the concentration follows a power law η ∼ cν with ν ) 1.25 as the concentration is just above the semidilute concentration, and with ν = 4.0-4.5 as the concentration is well above c*.6,7,18

Figure 5. (a) Viscosity η of PEG solution boundary layer as a function of solution concentration. (b) Shear modulus µ of PEG solution boundary layer as a function of solution concentration. The lines are guides to the eye. The numbers in the inlet specify the symbols used in the plot and are in the unit of g/mol.

The concentration dependences of a large number of different polymer and macromolecule solutions, after being scaled with the semidilute concentrations, are found to collapse onto a single curve which agrees approximately with that predicted the scaling theory.18 To elucidate if and how the scaling behaviors are affected by the presence of a boundary, the viscosity and the shear modulus of PEG solution boundary layers are plotted in Figure 6 against the concentration scaled with the semidilute concentration c* of the corresponding PEG molecules in the bulk solutions. For the determination of the semidilute concentration c*, an empirical relation between the radius of PEG polymer molecules and their molecular weight was used.54 An interesting feature displayed in the figure is that under such a scaling, the rises of the viscosity and the shear modulus of PEG solution boundary layers are less pronounced for high molecular weight PEG compared to that for low molecular weight PEG solutions. (c) Comparison of Viscosity of Boundary Solution Layers with That in Bulk Solutions. A closer look at the molecular weight dependence of the viscosity of PEG solution boundary layers can be found in Figure 7 which is a log-log plot of the viscosity versus the scaled concentration. It can easily be seen from the figure that under the same concentration the viscosity for a solution composed of larger PEG molecules is larger than that for solutions composed of smaller PEG molecules. In Figure 8, the viscosities of several PEG solution boundary layers with different molecular weights are compared with the

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Figure 8. Comparison of the viscosities of the PEG solution boundary layers obtained in this study with the viscosities of PEG bulk solutions measured using a calibrated Cannon-Ubbelohde viscometer.19 The solid lines are the exponential fits of the data to a power law function η ∼ cν. The dashed and dashed-dotted lines are the fits of the same power law function to the viscosity data of bulk PEG solutions (from Ref. 19) with the same molecular weight.19 The numbers in the inlet specify the symbols used in the plot and are in the unit of g/mol.

Figure 6. (a) Viscosity η of PEG solution boundary layer as a function of the concentration scaled with semidilute concentration c*. (b) The shear modulus µ of PEG solution boundary layers as a function of the concentration scaled with c*. The lines are guides to the eye. The numbers in the inlet specify the symbols used in the plot and are in the unit of g/mol.

Figure 9. Molecular weight dependence of the values of exponent ν obtained by using the power law η ∼ cν to fit experimentally determined viscosity of boundary layers of PEG solutions. Also included are the exponent values of the power law for bulk solutions measured using a calibrated Cannon-Ubbelohde viscometer.19 Clearly, while the values of the exponents for the bulk PEG solution are around 2.2, the exponents for solution boundary layers decrease from about 2.4 for solutions consisting of smaller PEG molecules to about 1.5 for solutions consisting of larger PEG molecules.

Figure 7. Log-log plot of the viscosities of boundary layers of PEG solutions with different molecular weights, obtained in this study. Also included are the data from a previous study in which PEG with Mw ) 10 000 g/mol was used.51 The numbers in the inlet specify the symbols used in the plot and are in the unit of g/mol.

viscosities of corresponding bulk PEG solutions measured using a calibrated Cannon-Ubbelohde viscometer.19 Using a power law η ∼ cν to fit the viscosity data at concentration above about

0.1c* yields the values for the exponent; some of the representative fitting curves are also plotted in Figure 8 while the exponents extracted from the fitting curves are plotted in Figure 9. It is quite clear from the figures that, for solutions composed of large molecular weight PEG, the viscosities of the boundary solution layers differ substantially from those of corresponding bulk solutions, as can be seen in Figure 8; the scaling exponent for large molecular weight PEG solution boundary layer also deviates from that of the bulk solutions, as shown in Figure 9. However, it seems that for solutions consisting of smaller molecular weight PEG (their molecular weight being lower than

PEG Solution Boundary Layers several hundreds g/mol) the viscosities of the solution boundary layers agree quite well with those obtained for bulk PEG solutions.19 A comparison of the molecular weight dependence of the viscosities in bulk solutions and that of the solution boundary layers measured using quartz resonator suggests that the differences between the two are mostly due to the boundary effects. Such effects can be explained in terms of a simple geometric consideration. That is, if polymer coils can be treated approximately as rounded balls, a flat rigid surface in contact with a layer of polymer coils would result in void spaces between the polymer coils and the surface. The volume of the voids, which are presumably filled with solvent, depends on the size of the polymer coils; the larger the coil the larger the void space. Thus, the local concentration of PEG molecules in the first layer might be lower than that in the bulk solution as long as the configuration of polymer coils remains roughly unchanged when adsorbed on the surface. Such a consideration does not include the interactions between polymer coils and the solid substrate. It will be particularly interesting to explore if a different substrate will result in different molecular weight dependence of the viscoelastic properties probed by quartz resonator. Qualitatively, the molecular weight dependence of the scaling exponents for the boundary layer can also be explained in terms of the above-described geometric consideration, since for large PEG molecules the concentration in the region occupied by the first few PEG layers (voids filled with solvent are difficult to be filled with PEG molecules by rearrangements) next to a flat solid surface would be affected less by the concentration increase in the bulk solution in comparison to the case of smaller PEG molecules. A quantitative description of the boundary effects to the scaling behaviors requires more detailed measurements of the viscoelastic properties of polymer solution boundary layers. (d) Shear Moduli of the PEG Solution Boundary Layers. The results plotted in Figure 5b show that the shear moduli of the boundary layers are sensitive to the molecular weight of PEG. For low molecular weight PEG solutions, the shear moduli of the boundary layers remain near zero in the entire concentration regime covered in this work. However, for PEG with molecular weight higher than 1000 g/mol, the shear modulus rises as the concentration increases; higher molecular weight PEG molecules result in a more rapid increase in the shear modulus of the solution boundary layers. This behavior can be qualitatively explained as follows: The atoms and molecules within a PEG polymer coil are connected by chemical bonds, and thus, the polymer coils in the solution can be treated as relatively rigid. For solutions consisting of larger PEG molecules, the polymer coils which are the fundamental entities that resist shear stress, are larger. Once these entities are connected, they would contribute appreciably to the shear modulus of the system. Also, the number of the connections needed for the system to show an appreciable amount of shear resistance is less for systems consisting of large solution molecules. This explanation is consistent with the scaling behavior in µ plotted in Figure 5b which shows that as the concentration approaches the semidilute concentration c* the polymer coils start contacting each other, so that shear modulus rises appreciably with the concentration. At the highest concentration covered in this study, the low molecular weight PEG solutions still have not yet reached the semidilute concentration c*; thus, their shear moduli remain near zero. It is interesting to notice, however, that in a scaled concentration plot (such as Figure 6b), the shear moduli of lower molecular weight PEG

J. Phys. Chem. C, Vol. 113, No. 31, 2009 13799 solutions rise more rapidly than those of the higher molecular weight PEG solutions. The appearance of nonzero shear modulus of PEG solution boundary layers indicates that a boundary layer in a solution might have distinctly different viscoelastic properties compared to the interior part in a bulk solution, due to the interaction of the layer with the boundary and the geometric confinement to the layer. It is interesting to note that boundary layer thickness (viscous penetration depth) probed using the quartz crystal resonator technique in this work is about a few hundred times larger than the PEG coil size. The effects observed in this work may be more pronounced if higher measuring frequency and/or larger PEG polymer molecules were used. More detailed and more quantitative studies of viscoelastic properties of polymer solution boundary layers are needed in order to obtain quantitative understanding on the fundamental properties of boundary layers in a solution. It should be recognized, however, that the subject is closely related to a number of important fundamental and technological issues related to solid-solution interfaces. 5. Conclusions We have studied viscosity and dynamic shear modulus of solution boundary layers of polyethylene glycol (PEG) with different molecular weights near a solid substrate using a quartz crystal resonator technique. On the basis of the resonant frequency shift and the dissipation broadening of the quartz crystal resonator, the viscosity and shear modulus of the solution boundary layers as a function of the concentration and the molecular weight of PEG molecules are determined. The results show that, near the semidilute concentration of the solution, the viscosity of the boundary layer increases rapidly; the rise of the viscosity follows a power law with an exponent which depends on the molecular weight of PEG molecules. The exponents for solution boundary layers with smaller PEG molecules are coincident with that of bulk solutions, while the exponent for larger PEG molecular boundary layers appears to be consistently smaller than that of bulk solutions. The shear modulus of boundary solution layers consisting of smaller PEG molecules remains nearly zero in the concentration range covered in this study, while the shear modulus for solution boundary layers consisting of larger PEG molecules displays a rapid increase above the semidilute concentration. These results indicate that over the acoustic penetration depth (a few hundred nanometers) the viscoelastic properties of solution boundary layers for small PEG molecules are indistinguishable from that of bulk solution, but for large PEG molecules, their viscoelastic properties differ distinctly from that of bulk solutions. Acknowledgment. D.-M.Z. is supported in part by a grant from the Research Corporation, USA and by the University of Missouri Research Board. P.W. and J.F. are supported by fellowships from the Minister of Education of China. References and Notes (1) Teraoka, I. Polymer Solutions; John Wiley & Sons: New York, 2002. Ferry, J. D. Viscoelastic Properties of Polymers; Wiley: New York, 1980. (2) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961. (3) Morawetz, H. High Polymers, 2nd ed.; Macromolecules in Solution; Wiley: New York, 1975. (4) De Gennes, P. G. Macromolecules 1976, 9, 594. De Gennes, P. G. Macromolecules 1981, 14, 1637. De Gennes, P. G. Macromolecules 1980, 13, 1069. (5) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (6) Colby, R. H.; Rubinstein, M. Macromolecules 1990, 23, 2753.

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