Contact Studies of Weakly Compressed PEG Brushes with a Quartz

Kenworthy et al.,5 using numerical equations developed by Milner et al.,24 showed that a polydispersity index of 1.02 can add about 5 nm to the brush ...
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Langmuir 2006, 22, 9225-9233

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Contact Studies of Weakly Compressed PEG Brushes with a Quartz Crystal Resonator David A. Brass and Kenneth R. Shull* Department of Materials Science & Engineering, Northwestern UniVersity, 2220 Campus DriVe, EVanston, Illinois 60208-3108 ReceiVed June 21, 2006. In Final Form: August 3, 2006 A quartz crystal resonator was used to characterize the contact of an elastomeric polymer membrane with a grafted poly(ethylene glycol) (PEG) brush in an aqueous environment. A two-layer model of the acoustic impedance of the system was used to measure the brush thickness before and after contact with the membrane. This model was further extended to include multiple layers, allowing characterization of other monomeric density profiles along the brush thickness. The polymer brush maintains a hydrated layer between the membrane and the quartz crystal surface, the thickness of which could be determined to within 1 nm. We show that the technique is very well suited for studying the properties of highly hydrated layers with thicknesses between 0 and 100 nm at low contact pressures corresponding to a very weak compression of the PEG brush.

1. Introduction Soluble polymers are often chemically attached to surfaces in order to eliminate adhesive interactions in liquid environments. These swollen polymer “brushes” create a repulsive potential that operates over length scales comparable to the dimensions of the polymer molecule.1,2 These repulsive interactions are important in a variety of applications, including colloid stabilization and fouling protection, and have been probed by a variety of experiments, including direct measurement with the surface forces apparatus2-4 and osmotic swelling of the lipid bilayer phase to which brushes were attached.5 These techniques have greatly enhanced our understanding of surface interactions at solid/liquid interfaces, but they are somewhat difficult to implement. Many of these experiments also probe the brush structure at very high contact pressures, often in the megapascal regime. The regime of very low contact pressures is more difficult to access by direct force methods but is important in many practical applications. Antifouling coatings, for example, resist the attachment of diffusing species that are not brought into contact with the surface at high pressures. Quantitative methods for assessing interfacial structure at relatively low contact pressures are needed in order to make a better connection to the function of these structures. In this investigation, we show that this goal can be accomplished with the aid of a highly sensitive quartz crystal resonator. The resonator is identical to resonators that are often used to measure the thicknesses of thin films and is referred to as a quartz crystal microbalance (QCM) for this reason.6 In our experiments, a membrane expansion apparatus, schematically shown in Figure 1, is used to bring an elastomeric membrane into contact with a poly(ethylene glycol) (PEG) brush that is grafted to the gold electrode surface of an AT-cut quartz crystal. During the contact experiment, the membrane flattens, establishing a region of intimate contact with the underlying (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, England, 1989; p 525. (2) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992; p 450. (3) Heuberger, M.; Drobek, T.; Voros, J. Langmuir 2004, 20, 9445-9448. (4) Drobek, T.; Spencer, N. D.; Heuberger, M. Macromolecules 2005, 38, 5254-5259. (5) Kenworthy, A. K.; Hristova, K.; Needham, D.; McIntosh, T. J. Biophys. J. 1995, 68, 1921-1936. (6) Buttry, D. A.; Ward, M. D. Chem. ReV. 1992, 92, 1355-1379.

Figure 1. Schematic representation of the membrane contact experiment: (a) out-of-contact phase; (b) contact phase.

brush. The radius, a, of the resultant circular contact region is typically between 1 and 3 mm. The pressure required to initiate contact is determined by the elastic modulus and thickness of the membrane and is measured directly by a pressure sensor. Information about the brush structure is obtained from the difference between the QCM response for the out-of-contact phase of the experiment (Figure 1a) and the in-contact phase of the experiment (Figure 1b). The vibrating quartz crystal is sensitive to the overall acoustic impedance of the composite medium with which it is in contact. In the out-of-contact case, the simplest model for the composite medium treats it as a two-layer system consisting of a polymer brush of thickness h, followed by a bulk layer of water. In the contact case a central, circular region of radius a has been replaced by a two-layer system consisting of a polymer brush and the membrane. Each of these models can be refined by subdividing the brush layer, thereby accounting for the detailed concentration profile within the polymer brush. An interpretation of the results requires that a set of control experiments be performed using bare gold electrodes without the brush layer. A consequence of the acoustic coupling of the brush with the contacting polymer membrane is that the sensitivity of the QCM response to the brush thickness is greatly enhanced. This enhancement can be attributed to the nature of acoustic wave propagation in a stratified, viscoelastic medium. General

10.1021/la061793r CCC: $33.50 © 2006 American Chemical Society Published on Web 09/20/2006

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background information needed to interpret these experiments is described in section 2.1, which is followed by a background section on polymer brushes in section 2.2 and also includes a discussion of the two-layer and multilayer models. The experiments involving a grafted poly(ethylene glycol) brush are described in section 3, and the basic results are described in section 4. These results are discussed in section 5, where we compare to theoretical predictions and discuss some of the advantages of this type of measurement.

2. Background 2.1. Quartz Crystal Microbalance. The piezoelectric character of the AT-cut quartz causes it to oscillate in a shear mode during the application of an ac electric field across the crystal thickness.7 The mechanical resonance of the crystal is sensed as a peak in the admittance spectrum of the crystal that is measured by a network analyzer. From a Lorentzian fit to the conductance peak in the vicinity of the resonance, we obtain a complex resonance frequency, (f/n ) fn + iΓn). Here fn is the frequency corresponding to the maximum in the conductance at the nth harmonic, and Γn is the half width at half-maximum of the resonance peak, which is a measure of the energy dissipation. (Note that starred quantities are complex throughout this article). If the quartz crystal is placed in contact with a viscoelastic material, then a shear wave propagates into this material. We use the subscript j to indicate the type of material under consideration (j ) w for water, etc.). The decay length, δj, of this shear wave is given by the following expression8

δj )

( ) |η/j | 2πfnFj

1/2

1 sin(Φj/2)

(1)

where η/j is the complex viscosity, Fj is the density, and Φj is phase angle of the fluid at the resonance frequency fn at which the crystal is oscillating. We use the third harmonic in all of our experiments (n ) 3, fn ) 15 MHz) because the measurements are found to give more reproducible results than those performed at other harmonics. For water at this frequency, a decay length, δw, of 150 nm is obtained. This small value of the decay length is responsible for the surface sensitivity of the technique. As the membrane is pressurized and brought into contact with the electrode surface of the QCM, it does not affect the response of the QCM until its separation from the quartz crystal surface is comparable to δw. If the contact is uniform across the entire surface of the crystal, then the following simple relationship exists between the shift in the complex frequency, ∆f/C, and the related change in the complex impedance, ∆Z*,9

∆f/C i∆Z* ) f1 πZq

(2)

where f1 is the fundamental frequency of 5 MHz and Zq is the acoustic impedance of quartz (8.84 × 106 kg m-2 s-1). The subscript C in our notation for ∆f/C is a reminder that eq 2 assumes complete, uniform loading of the entire crystal, which is generally not obtained in our membrane contact experiment. Instead, the membrane makes contact with the QCM surface only over a circular region of radius a and total area A (A ) πa2) at the center of the crystal. The remainder of the membrane is (7) Nye, J. F. Physical Properties Of Crystals: Their Representation by Tensors and Matrixes; Clarendon Press: Oxford, U.K.; 1985; p 327. (8) Bandey, H. L.; Martin, S. J.; Cernosek, R. W.; Hillman, A. R. Anal. Chem. 1999, 71, 2205-2214. (9) Johannsmann, D. Macromol. Chem. Phys. 1999, 200, 501-516.

far from the surface (relative to δw) and does not contribute to the measured change in the complex frequency shift. This geometric effect, often referred to as energy trapping, can be modeled empirically by introducing a Gaussian sensitivity factor relating the measured coverage-dependent frequency shift, ∆f/A, to the value given by eq 210

{

∆f/A ) ∆f/C 1 - exp

( )} -2βA Ao

(3)

where Ao ) 31.67 mm2 is the area of the smaller of the two electrodes and β is a Gaussian normalization parameter determined independently as described below. Note that ∆f/A ) 0 for A ) 0 and ∆f/A ) ∆f/C for A . A0. 2.2. Concentration Profiles of the Grafted PEG Layer. The properties of end-grafted polymer layers are determined by the relative values of two different length scales. The first of these is Rg, the unperturbed radius of gyration of the polymer in the melt state. The melt value of the root-mean-square end-to-end distance, equal to x6Rg, is an equivalent parameter that can also be used to specify this length scale. The second relevant length scale is the average distance between grafting points, dg, which is one measure of the surface concentration of grafted molecules. We use z*, the integral of the brush volume fraction equal to the dry thickness of the grafted PEG layer, as a measure of the overall brush coverage. The number of grafted chains per unit area, ∑, is given by z*/Vbr, where Vbr is the molecular volume of the grafted molecule. From this we obtain dg ) ∑-1/2 ) (Vbr/ z*)1/2. The grafted PEG layers in our experiments have a molecular weight of 5700 g/mol with Rg ) 2.5 nm11 and Vbr ) 8.6 nm3. By polymer standards, these molecules are not very long, but they are long enough that the statistical ideas used to describe the behavior of grafted polymer layers can be applied. The layers used in our experiments have dry thicknesses of approximately 7.4 nm, giving dg ) 1.1 nm. Because the average distance between grafting points is substantially less than Rg, the chains overlap strongly, forming brushes that are forced to extend away from the surface. We form the grafted layers initially as “melt” brushes by annealing a layer of undiluted, thiol-terminated PEG that had been spuncast onto the gold electrode surface of a quartz crystal. The brush coverage is kinetically limited by the penetration of end groups through the existing brush layer.12 The value of ∼3Rg that is obtained for z* is consistent with experiments performed on other melt brushes in our laboratory. When the melt brushes are immersed in water, a good solvent for PEG, the polymer molecules extend into the aqueous phase to form a wet brush with a thickness, h, that is much larger than z*. The simplest concentration profile is a simple step function profile, assumed in the early scaling theories of brush formation put forth by Alexander13 and de Gennes.14 This concentration profile, illustrated in Figure 2, can be written in the following mathematical form: (10) Nunalee, F. N.; Shull, K. R. Langmuir 2006, 22, 431-439. (11) Brandup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; John Wiley & Sons: New York, 1989. (12) Shull, K. R. Macromolecules 1996, 29, 2659-2666. (13) Alexander, S. J. Phys. 1977, 38, 977-981. (14) de Gennes, P. G. AdV. Colloid Interface Sci. 1987, 27, 189-209.

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Z/1,2 ) Z/1

iZ/1tan(k/1d1) + iZ/2tan(k/2d2) Z/1 - Z/2tan(k/1d1) tan(k/2d2)

(6)

where k/j ) 2πfnFj/Z/j is the complex wavenumber for the acoustic shear wave in layer j and dj is the corresponding layer thickness. In the out-of-contact configuration for our experiments, the first layer is a PEG brush (1 f br, d1 f h) and the second layer is a semi-infinite expanse of water (2 f w, d2 f ∞). In this case, eq 6 simplifies to

Figure 2. Step function (dashed line) and parabolic (solid line) forms of the brush profile.

φpeg(z) )

φ0 for z < h 0 for z > h

φ0 )

z* h

(4)

Here z is the distance from the grafting surface, and φpeg is the local PEG volume fraction in the brush layer. Although many of the properties of polymer brushes can be accurately modeled with this simple step function picture, one expects that the inclusion of a more realistic profile might be important in our experiments. To probe this issue in more detail and to illustrate the extension of our approach to arbitrary concentration profiles, we consider the parabolic profile suggested originally by Milner, Whitten, and Cates:15

3 z* φ0 ) 2 h

z h

2

(5)

Note that in this case φ0, the concentration of polymer at the grafting surface, is larger than the average concentration in the brush by a factor of 1.5. More detailed treatments provide an even more realistic description of the brush profile.12,16 The difference between the parabolic and step profiles is sufficient, however, to illustrate the degree to which the details of the profile affect the measured brush thickness. The QCM response for the step function profile can be understood by applying a series of two-layer models, as described in the following subsection. The parabolic profile requires the development of a generalized multilevel model17 described in section 2.2.2. 2.2.1. Step Function Profile: Two-Layer Modeling. To understand the long-time values of ∆f/C that characterize the equilibrated Au/brush/membrane system, we begin with a general model for the acoustic impedance of two layers of arbitrary thickness and linear viscoelastic response, which are placed in contact with the QCM electrode surface. Each of these layers has / a complex shear modulus, G/j , and an acoustic impedance, Zj

) xfnFjG/j , where j ) 1 for the layer in contact with the Au electrode surface and j ) 2 for the subsequent layer. The overall acoustic impedance for this configuration is given by the following expression9 (15) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610-2619. (16) Orland, H.; Schick, M. Macromolecules 1996, 29, 713-717. (17) Domack, A.; Prucker, O.; Ru¨he, J.; Johannsmann, D. Phys. ReV. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56, 680-689.

(7)

For the in-contact configuration, the second layer corresponds to the membrane (2 f m, d2 f dm): / iZ/brtan(k/brh) + Zm,0 / Zbr,m ) Z/br / / Zbr + iZm,0 tan(k/brh)

(8)

Note that because of the applied pressure of the membrane, h for the contact case will not necessarily be the same as h for the noncontact case. Although our analysis does not require that these two thicknesses be the same, the applied pressures are low enough that the brush is not substantially compressed by contact with the membrane. We return to this issue in section 5.4. / / We have defined the membrane impedance, Zm,0 , as Zm,0 )i / / 18 Zmtan(kmdm). The well-known Sauerbrey equation

∆fj ) -

{ ( )}

φpeg(z) ) φ0 1 -

iZ/brtan(k/brh) + Z/w Z/br,w ) Z/br / Zbr + iZ/wtan(k/brh)

2djf1fnFj Zq

(9)

/ is obtained by taking ∆Z* ) Zm,0 in eq 2, with k/mdm , 1. Note that in our notation quantities with single subscripts represent bulk material properties, whereas properties with 2 subscripts also involve geometrical parameters, such as the film thickness. / Also note that the expression for Zm,0 can be obtained from eq 6 by setting 1 equal to m and d2 equal to zero. Finally, the overall change in acoustic impedance associated with contact with a preexisting brush is given by the following expression:

∆Z/cont ) Z/br,m - Z/br,w

(10)

Values for ∆Z* are related to measured values of the complex frequency change by eq 2, provided that eq 3 is used to account for the nonuniform loading of the quartz crystal. 2.2.2. Parabolic Profile: Multilayer Modeling. Because the two-layer model cannot describe the parabolic profile of grafted polymer brushes, a generalized model that can be used for an arbitrarily large number of discrete layers is used instead. We use the matrix formulation described by Domack et al.,17 modified to account for the presence of the membrane. As with the simpler two-layer model, we number the layers from the layer that is directly in contact with the QCM (layer 1) toward the outer surface (layer N). The total impedance of the loading material is described by

1 - r* Z/cont ) Z/1 1 + r*

(11)

Here Z/1 is the acoustic impedance of layer 1, and r* is the (18) Sauerbrey, G. Z. Phys. 1959, 155, 206.

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reflection coefficient describing the amplitude ratio for the forward and reflected waves at the QCM surface

r* )

u/-

(12)

u/+

where u/+ is the amplitude of the wave traveling away from the QCM surface and u/- is the amplitude of the wave traveling toward the QCM surface. The phase information contained in u/+ and u/- requires that these be complex quantities. In our model, there are a total of N layers. The boundary condition for the last layer (layer N) is that the shear wave is reflected at the top surface of layer N (typically the membrane layer). At the top of layer N, defining the “top” as the side of the layer that is farthest from the crystal surface, the complex amplitudes of the forward and reverse waves are equal to one another. This boundary condition can also be used for the case where layer N is a layer with a thickness that greatly exceeds the decay length of the shear wave in that layer (a layer of bulk water, for example). The values of u/+ and u/- are represented as components of a vector that is related to the displacement vector at the top of the Nth layer by the following matrix transformation:

[]

[]

u/+ 1 / / / / / / / / ) [L1]‚[S1,2]‚[L2]‚[S2,3]‚‚‚[LN-1]‚[SN-1,N]‚[LN]‚ 1 u-

(13)

Here [L/j ] is the propagator of the shear wave through the jth layer

[L/j ]

[

cos(k/j dj) + i sin(k/j dj) 0 ) cos(k/j dj) - i sin(k/j dj) 0

]

(14)

and [Sj/- 1,j] is the propagator of the shear wave through the interface between layer j - 1 and layer j:

[Sj/- 1,j] )

[

/ / / / 1 1 + Zj /Zj - 1 1 - Zj /Zj - 1 2 1 - Z/j /Zj/- 1 1 + Z/j /Zj/- 1

]

(15)

It is often convenient to replace the two terms involving layer N (the combination of SN/ - 1,N‚L/N) with a termination matrix, [T/N] so that eq 13 is transformed to the following:

[]

[]

u/+ 1 / ) [L/1]‚[S/1,2]‚[L/2]‚[S/2,3]‚‚‚[LN-1 ]‚[T/N]‚ 1 u/-

(16)

The following expression for the termination matrix is derived in the Appendix:

[T/N] )

[

/ 1 + ZN,0 /ZN/ - 1 0 / /ZN/ - 1 1 - ZN,0 0

]

(17)

To obtain the full multilayer acoustic impedance of the brush and the parabolic concentration profile of the brush, the total system is broken up into 76 layers (N ) 76). The PEG volume fraction in each of the first 75 layers is obtained from eq 5, with z corresponding to the midpoint of the layer. The acoustic impedance, Z/j , for these 75 layers is assumed to be equal to the acoustic impedance of a bulk PEG solution of the same concentration, measured directly as described in section 3.2. If the brush is immersed in water, then the top layer corresponds / ) Z/w. If the brush is in contact to a thick layer of water, and ZN,0 / / with a membrane, then ZN,0 ) Zm,0. The method for determining

Figure 3. (a) Measured values of ∆fbr (b) and ∆Γbr (0) obtained from measurements with PEG solutions in water. The dashed lines are the polynomial fits given by eq 18. (b) Decay length as a function of PEG volume fraction, obtained from eqs 18 and 22. / is the same as that for the two-layer model described in Zm,0 section 2.2.1.

3. Experimental Section 3.1. Sample Preparation. A poly(ethylene glycol) brush was produced on the gold electrode surface of an AT-cut quartz crystal (5 MHz fundamental resonance frequency, Maxtek, Inc.). The brush was formed by spin coating a warm 1-butanol solution of thiolterminated PEG (5700 g/mol, Nektar Therapeutics) directly onto the electrode surface. After annealing and rinsing in water, a PEG brush with a dry thickness, z*, of 7.4 nm was formed, as determined by ellipsometry. The excess thiol-terminated PEG applied during spin coating was found to dewet from the surface of the underlying brush and was easily removed by rinsing. 3.2. PEG Calibration Experiments. The acoustic impedance for the brush layer (two-layer model) or sublayer within the brush (multilayer model) was assumed to be the same as the acoustic impedance of the calibration solution of the same composition. Our use of a PEG solution as a calibration is justified by the fact that the high-frequency oscillation of the QCM accesses the higherorder Rouse modes characteristic of local motions in the polymer, which are insensitive to polymer molecular weight and to the details of the grafting. These calibrations were performed by flooding the QCM surface with PEG solutions (6800 g/mol, Scientific Polymers Products, Inc.) with polymer volume fractions ranging from 0 to 0.6. Measured values of ∆fC and ∆ΓC for these solutions, which we refer to as ∆fbr and ∆Γbr to indicate that these are values corresponding to the brush, are shown in Figure 3a. The dashed lines in these Figures are the following polynomial fits: ∆fbr (kHz) ) -1.318 - 6.5659φpeg + 11.809φpeg2 - 23.499φpeg3 ∆Γbr (kHz) ) 1.318 + 9.4744φpeg - 6.06φpeg2 + 22.855φpeg3 (18) A variety of information can be obtained from these measured data, including the concentration dependence of the acoustic impedance, Z/br, the complex shear modulus, G/br, and the corre-

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sponding phase angle, Φbr, and the decay length of the acoustic shear wave:9 Z/br )

-iZqπ (∆fbr + i∆Γbr) f1

(19)

(Z/br)2 Fbr

(20)

G/br )

(

Φbr ) 2 tan-1 -

δbr )

∆fbr ∆Γbr

)

-Zq (∆fbr2 + ∆Γbr2) 2Ffnf1 ∆fbr

(21)

(22)

In these expressions, Fbr is the density of the brush, obtained from the densities of pure PEG (Fpeg ) 1.1 g/cm3) and water (Fw ) 1.0 g/cm3) by ignoring any volume change on mixing of PEG and water: Fbr ) φpegFpeg + (1 - φpeg)Fw

(23)

The viscosity for pure water agrees with the measured viscosity to within 10%, a level of agreement that is typical in these types of experiments.19 The decay length of the shear wave increases with increasing PEG volume fraction, as shown in Figure 3b. 3.3. Membrane Expansion Experiments. The brush-coated QCM was immersed in water at room temperature, and a thin elastomeric membrane was brought into contact with it, using pressure (typically between 500 and 1200 Pa) to expand the membrane as illustrated in Figure 1. The pressure is applied through the use of a syringe pump that infuses and extracts air at a rate of 2 mL/h. This membrane was made by spin coating a toluene solution of a triblock copolymer with polystyrene end blocks and an ethylene/butene midblock (178 g/mol, 30 wt % styrene) onto a salt crystal, floating the resultant 1 µm film onto water and transferring this film to the end of a cylindrical expansion chamber. The membrane impedance was measured directly by spin coating the same solution directly onto the QCM and determining its response. The QCM response was monitored by a network analyzer as described previously,20 and a pressure sensor was added to monitor the pressure difference across the membrane. Contact images between the membrane and the QCM surface were obtained using a video camera placed above the transparent expansion chamber.

Figure 4. Measured values of (a) ∆fA and (b) the pressure for a polymer membrane that is brought into contact with a PEG brush with z* ) 7.4 nm. The solid line in part a represents the fit to eq 3, with β ) 2.05.

Figure 5. Time dependence of ∆fC obtained from the expansion data in Figure 4a.

4. Membrane Contact Results Measured values of the frequency shift, ∆fA (the real component of ∆f/A), are plotted as a function of A/A0 for the PEG-thiol-modified gold surfaces in Figure 4a. In these experiments, the membrane contact area initially increases as the membrane is inflated. The inflation process is then stopped, and ∆fA settles down to a value that is characteristic of the equilibrated membrane/gold interface. The pressure is then decreased in order to peel the membrane away from the electrode surface of the QCM. Because this peeling process decreases the area of contact while leaving the remaining contact area unaffected, we can use the withdrawal phase of the experiment to obtain values for β by assuming a constant value of ∆fC, which along with β is used to fit eq 3 to the measured values of ∆fA. We can then use these values of β (1.45 for the unmodified gold surface and 2.05 for the PEG-modified surface) to obtain a timedependent value of ∆fC from the expansion portions of the curve. The results of these calculations, shown in Figure 5, indicate that several minutes are required for an equilibrated Au/membrane (19) Johannsmann, D. Macromol. Chem. Phys. 1999, 200, 501-516. (20) Nunalee, F. N.; Shull, K. R.; Lee, B. P.; Messersmith, P. B. Anal. Chem. 2006, 78, 1158-1166.

interface to be obtained. Equilibration occurs as the water diffuses through the thin membrane layer, where it is in equilibrium with the enclosed water vapor, and is expelled laterally from the contact region. A further explanation of this process is described in section 5.3.

5. Discussion 5.1. Out-of-Contact State. The dry brush was found to produce a 160 Hz shift in the resonance frequency of the quartz, which on the basis of the Sauerbrey equation (eq 9) corresponds to z* ) 8.5 ( 1.6 nm, in good agreement with the ellipsometrically determined result of a 7.4 nm corresponding to an average distance between grafting points, dg, of 1.1 nm. At this coverage, the PEG chains are much closer together than the Flory radius and are well into the brush regime. Values of ∆ΓC and ∆fC for different experiments are summarized in Table 1. These data illustrate the advantages of our method of measuring brush properties by making contact with a high-impedance membrane, a variant of a general idea that has been previously shown by Johannsmann.9 In a more conventional QCM experiment, the brush thickness is obtained

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Table 1. Measured Changes in the Complex Resonance Frequency for Different Experimental Conditions

a

notation for measured ∆f/C ) ∆fC + i∆ΓC

∆fC(Hz)

∆ΓC(Hz)

-138a -1369 -1658

0 1270 1273

∆f/m,0 ∆f/br / / - ∆fbr,w ≡ ∆f/cont ∆fbr,m

-17662 -2508 -13 261

349 3480 -875

∆f/m,0 - ∆f/w ≡ ∆f/cont

-15 938

-320

initial state

final state

bare QCM in air bare QCM in air dry brush in air (z* ) 7.4 nm) bare QCM in air bare QCM in air wet brush in water (z* ) 7.4 nm) bare QCM in water (z* ) 0)

dry brush (z* ) 7.4 nm) bare QCM in water wet brush in water

∆f/br,0 ∆f/w ∆f/br,w

membrane in air PEG solution (23% PEG) wet brush brought into contact with membrane bare QCM brought into contact with membrane

- ∆f/br,0

Calculated from eq 9.

Figure 6. Calculated frequency shift created by the addition of a PEG brush with z* ) 7.4 nm to the surface of a quartz crystal that is submerged in water. The dashed line represents an experimentally found value for the brush.

from the effect of the brush on the response of the QCM when it is immersed in water.3,17,21 Experimentally, this is done by measuring the change in complex frequency obtained when an originally dry brush is immersed in water and comparing this frequency shift to the shift obtained in the absence of the polymer brush (i.e., by the quantity ∆f/br,w - ∆f/w). From the data in Table 1, we see that the brush is responsible for a 427 Hz decrease in the resonance frequency (∆fC) and a negligible change in the bandwidth (∆ΓC). Figure 6 shows a plot of the calculated value of ∆f/br,w - ∆f/w as a function of h for both the step function and parabolic brushes. The measured 427 Hz frequency shift is consistent with h ) 32 nm for the parabolic brush. The step function underestimates the thickness by a couple of nanometers, but is a very good approximation in this concentration region. There is an approximately 30 Hz error in the frequency shift from the removal and replacement of the quartz crystal into its holder, which corresponds to a 3 nm variation in thickness for the brush. Our procedure here is identical to the methodology employed by Domack et al., who used the QCM to study brushes with much larger values of h, where a more substantial QCM response was obtained.17 We obtain a greatly enhanced sensitivity to smaller brush thicknesses by utilizing the membrane, as describe in the following subsection. 5.2. In-Contact State. In the contact experiments, the z dependence of the acoustic impedance of the brush layer is given by the combination of eqs 18 and 19, where φpeg is given by eq 4 for the step function profile and by eq 5 for the parabolic profile. From the approach outlined in section 2.2, we calculate the relationships between ∆fcont and h and between ∆Γcont and h. These relationships are shown in Figures 7 and 8. Separate curves are shown for z* ) 0 and for the step function and parabolic (21) Zhang, G. Macromolecules 2004, 37, 6553-6557.

Figure 7. Calculated frequency shift for PEG layers of different dry thicknesses induced by contact with the membrane as a function of the confined thickness of the brush. Calculations for a bare QCM substrate (z* ) 0) and for parabolic and step function brushes with z* ) 7.4 nm are included. The experimentally measured values of ∆fcont for z* ) 0 and 7.4 nm are indicated as well.

Figure 8. Calculated dissipation shift for PEG layers of different dry thicknesses induced by contact with the membrane as a function of the confined thickness of the brush. Calculations for a bare QCM substrate (z* ) 0) and for parabolic and step function brushes with z* ) 7.4 nm are included.

profiles with z* ) 7.4 nm. There is no distinction between parabolic and step function profiles for z* ) 0 because in this case h represents the thickness of a layer of pure water that exists between the membrane and the QCM electrode surface. For the parabolic profile, the maximum sensitivity, which we can express in terms of the slope of the plot of ∆fcont versus h, is in the range of 0.5 kHz/nm, which is 50 times the sensitivity for brushes in the noncontact geometry (Figure 6). Furthermore, the measured values of ∆fcont correspond to portions of the curve where a high sensitivity to h is obtained. Values for h are accurately obtained from the measured values of ∆fcont, indicated by the dashed lines in Figure 7. The measured values of h are given by the intersections of these dashed lines with the calculated relationships between ∆fcont and h. When the PEG brush is not present (z* ) 0), we obtain h ≈ 0, indicating

Contact Studies of Weakly Compressed PEG Brushes

Langmuir, Vol. 22, No. 22, 2006 9231

Figure 10. Measured values of the osmotic pressure for PEG-8000 solutions in water, taken from ref 22 (b), and the prediction of eq 24, with χ ) 0.4 (line).

Figure 9. Calculated frequency shift for PEG layers with (a) z* ) 7.4 nm and (b) z* ) 25 nm in contact with membranes with thicknesses of 1 µm, 1.3 mm, and 2.5 mm. The dashed vertical lines are drawn at h/z* ) 4, corresponding to the experimental situation for the brushes with z* ) 7.4 mm. Parabolic brush profiles are assumed in all cases.

that the membrane is able to establish intimate contact with the gold surface. For a brush with z* ) 7.4 nm, we obtain h ) 28 ( 0.4 nm, corresponding to a parabolic profile with a maximum PEG volume from a fraction of 1.5z*/h ) 0.4. Typical measured values for ∆Γcont are not shown because of scatter that was obtained at equilibrium. Also, the measured values for ∆Γcont follow the same qualitative trend as the model curves shown in Figure 8 with ∆Γcont going through a maximum as the membrane is brought into contact with the brush layer. The sensitivity of the membrane apparatus to brushes with different thicknesses can be altered by changing the thickness, and hence the acoustic impedance, of the membrane that is used. / for membranes with We modeled this effect by measuring Zm,0 Saurbrey thicknesses of 1, 1.3, and 2.5 µm. The procedure outlined in section 2.2.2 was then used to calculate the relationship between ∆fcont and h for parabolic brushes in contact with these three membranes. The results of these calculations are illustrated in Figure 9 for brushes with z* ) 7.4 and 25 nm. For maximum sensitivity, experiments should be designed so that the measured value of h corresponds to a region of maximal slope in a plot ∆fcont and h. An inspection of the curves in Figure 9 indicates that smaller values of h are most effectively probed by thicker membranes. This effect involves a complicated interplay between many different factors but can be qualitatively understood in terms of the decay length of the shear wave and the impedance mismatch between the brush and membrane layers. Larger values of z* shift the range to larger values of h because the enhanced PEG concentration in the brush increases the decay length of the shear wave (Figure 3b). An increased PEG concentration also increases the acoustic impedance of the brush layer, however, decreasing the impedance mismatch between the membrane and the PEG layer and reducing the tendency for the shear displacement to be confined within the brush layer itself. By increasing the acoustic impedance of the membrane, the decay of the shear wave within the brush layer is enhanced. The brush

layer in these situations behaves as a low-impedance boundary layer, which is able to substantially reduce the overall impedance that would otherwise be associated with the membrane. By appropriate impedance matching, the strength of this effect can be made to vary quite substantially in the appropriate regime of brush thickness. Our results indicate that highly hydrated layers with thicknesses between 0 and 100 nm can be studied by an appropriate choice of the membrane. 5.3. Comparison to Predicted Brush Profiles. As described above, a variety of theoretical approaches have been used to calculate the expected brush thickness. For comparison purposes, we use the most quantitative theory at our disposal, which is a numerical self-consistent-field (SCF) theory.22 Details of our implementation of this theory have been described previously.12 Here we briefly describe the procedure used to determine the three parameters that are needed as input parameters in order to specify the brush thickness completely. The first of these is the Flory-Huggins interaction parameter, χ, used to describe the PEG/water interactions. In this model, the osmotic pressure of a solution of high molecular weight polymer is given by the following expression

Π)

RT {-ln(1 - φp) - φp - χφp2} Vs

(24)

where R is the gas constant, Vs is the molecular volume of the solvent, and φp is the polymer volume fraction. In Figure 10, we compare the prediction of eq 24 for χ ) 0.4 to the osmotic pressure data obtained by Stanley et al.23 for aqueous PEG solutions with a molecular weight of 8000. The good agreement indicates that this simple model of the solution thermodynamics accurately accounts for the solvent interactions with the brush. The other parameters needed for the fitting are the degree of polymerization and the statistical segment length for the brush. For the degree of polymerization, N, we use the volume of the PEG molecule, normalized by the volume of a water molecule (30 Å3). Assuming a density of 1.1 g/cm3 for PEG, we obtain a degree of polymerization, N, of 280 for the PEG molecule with a molecular weight of 5700. Finally, the statistical segment length of each of these 280 units is chosen so that the correct chain dimensions are obtained in the melt state: the radius of gyration for PEG 5700 is 2.5 nm.11 Setting this equal to a(N/6)1/2 gives a ) 3.6 Å. The brush profile obtained from the numerical SCF calculation is shown in Figure 11, where it is compared to a parabolic profile (22) Matsen, M. W. J. Phys.: Condens. Matter 2002, 14, R21-R47. (23) Stanley, C. B.; Strey, H. H. Macromolecules 2003, 36, 6888-6893.

9232 Langmuir, Vol. 22, No. 22, 2006

Brass and Shull

P)

Figure 11. Parabolic profile of eq 5, with z* ) 7.4 nm and h ) 25 nm (symbols) and the calculated SCF profile for a monodisperse PEG 5700 brush (solid line). The position of the arrow corresponds to the measured brush thickness of 28 nm.

( )

with h ) 25 nm. In both cases, we use the experimental value of 7.4 nm for z* as a constraint on the fit. The measured value of 28 nm for h is a bit larger than the value of 25 nm that is obtained from the SCF theory, a result that can quite likely be attributed to brush polydispersity. Kenworthy et al.,5 using numerical equations developed by Milner et al.,24 showed that a polydispersity index of 1.02 can add about 5 nm to the brush thickness for brush parameters and applied pressures that are similar to those used in our experiments. 5.4. Distinctive Features of the Membrane Expansion Technique. The brush thickness determined by a mechanical method such as the one we employ does not directly correspond to a free brush but to a brush that is slightly compressed by the applied pressure. This pressure is constant over the flattened region of the membrane and is controlled by the system geometry and by the elastic properties of the membrane. The overall picture is illustrated schematically in Figure 12, which shows an undeformed membrane of thickness t, Young’s modulus E, and radius Rm that is pressurized to bring it into adhesive contact with a flat surface. The distance between the undeformed membrane and the flat surface is δ. In part a, the pressure is enough to just initiate contact with the membrane. For small strains (small δ/Rm), the pressure required to initiate contact is given by the following expression:25 (24) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1989, 22, 853-861.

3

(25)

The first term corresponds to the work done against the membrane pretension T0, which in the absence of elasticity is equal to twice the surface energy of the membrane. The second term, originating from the elastic deformation, is the dominant term in our experiments. With our experimental values of δ ) 1 mm, Rm ) 3.75 mm, and t ) 1 µm, a measured contact pressure of 640 Pa corresponds to a membrane elastic modulus of 31 MPa. Thinner membranes will give rise to lower pressures and provide a means for controlling the contact pressure of the brush. When the brush comes into contact with the surface as illustrated in Figure 12b, the pressure remains constant throughout the contact region, provided that the contact radius, a, is much larger than the membrane thickness, as is the case in our experiments. Geometric factors, viscoelasticity, and large strain effects modify the contact pressure of the membrane, but eq 25 provides a very useful estimate of the contact pressure. Compression of the brush requires that water either be forced laterally to the region outside the contact area or through the membrane itself. The rate of water transport is determined by the permeability of the membrane, Pm, which for both polystyrene and poly(ethylene-butene) is close to 3.5 × 10-15 mol/cm Pa s.11 This value of the permeability can be converted to the velocity, the rate at which a thin layer of water trapped beneath the membrane will decrease in thickness,

Vw )

Figure 12. Schematic representation of the membrane (a) just prior to contact and (b) while in adhesive contact with the surface.

( )

4T0 δ 4Et δ + Rm Rm Rm Rm

RmVw∆P t

(26)

where ∆P is the pressure difference across the membrane and Vw is the molar volume of water. For t ) 1 µm and ∆P ) 1200 Pa, we obtain Vw ) 7.5 nm/s. This permeation rate is sufficient for excess water that is trapped beneath the membrane to escape through the membrane itself during the experiment. Because the enclosed volume behind the membrane is saturated with water vapor, the brush layer does not dry out during the experiment. The characteristic pressure required to compress the brush substantially is kBT/dg3, which we define as P0. For our brushes, with dg ) 1.1 nm, we have P0 ) 2.4 × 106 Pa. The scaling prediction for the relationship between the applied pressure given by the following relationship between h and the osmotic pressure, Π, of the brush is as follows5

[( ) ( ) ]

Π ) P0

h0 h

9/4

-

h h0

3/4

for h < h0

(27)

where h0 is the thickness where the osmotic forces first become measurable. Equation 27 applies for cases where the applied pressure is comparable to P0, whereas in our experiments the applied pressure is about a factor of 1000 lower than P0. The values of h measured in our experiments, therefore, correspond to a brush that is only very weakly compressed. We are sensitive to the outer portions of the brush, where eq 27 cannot be applied in any quantitative sense. This is the region of the brush where polydispersity effects are most important,24 so it is not surprising that we obtain a value for the thickness that is several nanometers larger than the SCF prediction. 5.5. Brush/Membrane Adhesion. A final comment concerns the nature of the membrane contact with the brush or with the bare gold electrode surface. The membrane contact angle, θ, (25) Wan, K.-T.; Guo, S.; Dillard, D. A. Thin Solid Films 2003, 425, 150162.

Contact Studies of Weakly Compressed PEG Brushes

illustrated in Figure 12b, is a measure of the adhesion between the membrane and the surface with which it is forced into contact. In the absence of adhesion, θ ) 0. Although we have not analyzed the data in sufficient detail to quantify the contact angles, it is clear that the membrane contact angles against the PEG brushes are quite low. For the brush case, the membrane contact radius increases continuously as the pressure increases, a result that is consistent with a contact angle of zero and the absence of adhesion. In the absence of the PEG layer, the membrane contact radius increases discontinuously when contact is made with the surface, a clear signature of adhesion between the membrane and the electrode surface. Differences in adhesion are also responsible for the behavior that is observed when the pressure is decreased and the membrane is peeled from the surface. In the absence of a PEG brush, a negative pressure of approximately 1200 Pa is needed to pull the membrane from the bare gold surface. For the brush-coated surface, a much smaller hysteresis in the pressure is observed (Figure 4b). This hysteresis can be attributed largely to the creep of the membrane itself and is not solely due to adhesive interactions between the membrane and the surface.

6. Conclusions We have shown that quartz crystal resonators, operating in the MHz frequency regime, are very sensitive to the nature of the contact between a polymeric membrane and the polymer-modified electrode surface of the crystal. For the PEG brushes that we have studied, the membrane enhances the sensitivity to the brush thickness by a factor of 50. Coupling of the shear wave to the membrane depends strongly on the thickness of the lowimpedance, hydrated brush layer between the electrode surface and the membrane and is responsible for this enhanced sensitivity. By an appropriate choice of the membrane, brush thicknesses as large as ∼100 nm can be studied. Several factors are responsible for the ability of this technique to provide useful information about polymer-mediated interactions at solid/liquid interfaces: (1) The contact pressure is controllable and can be quite low so that the mechanical stress used to interrogate the interfacial structure does not distort it significantly. (2) The interface is able to equilibrate by the diffusion of water through the membrane. (3) The sensitivity to different thicknesses of the hydrated layer can be adjusted by changing the acoustic impedance of the membrane. (4) Simultaneous measurement of adhesive interactions is possible by monitoring the details of the membrane shape as it is brought into and out of contact with the polymer-modified electrode surface of the quartz crystal. Acknowledgment. This work was supported by grants from the Human Frontier Science Program, NIH (R01 DE14193), and NSF (DMR-0525645). We have also benefited greatly from a series of helpful interactions with Professor D. Johannsmann. We also thank Dr. Anny Flory for helping to establish our pressuresensing capabilities.

Langmuir, Vol. 22, No. 22, 2006 9233

7. Appendix: Derivation of [T/N] In eq 15 for [SN/ - 1,N], the product of all of the terms that do not involve the properties of layer N will give a complex 2 × 2 matrix, with coefficients defined here as A*, B*, C*, and D*. With this definition, we can rewrite eq 13 as follows:

[][ ]

[]

u/+ A* B* / 1 ) [S ][L/ ]‚ C* D* N - 1,N N 1 u/-

(28)

with

[SN/ - 1,N] )

[

/ / / / 1 1 + ZN/ZN - 1 1 - ZN/ZN - 1 / / 2 1 - ZN/ZN - 1 1 + Z/N/ZN/ - 1

]

(29)

and

[L/N] )

[

cos(k/NdN) + i sin(k/NdN) 0 cos(k/NdN) - i sin(k/NdN) 0

]

(30) Combining eqs 28-30 gives the following

[]

u/+ ) 2 cos(k/NdN) × u/-

[

/ (A* + B*) + (A* - B*)(Z/N/ZN-1 )i tan(k/NdN) / / (C* + D*) + (C* - D*)(ZN/ZN-1)i tan(k/NdN)

]

(31)

/ with ZN,0 ) iZ/Ntan(k/NdN). This same result can be obtained by the following matrix multiplication:

[ ] [ ][

][ ]

/ / u/+ A* B* 1 + Zj,0/ZJ - 1 0 1 / ) / / C* D* u1 - Zj,0/ZJ - 1 1 0

(32)

Because we are interested only in the ratio of u/+ and u/-, we can eliminate multiplicative factors that have an equal effect on both of these terms (i.e., the factor of 2 cos(k/NdN)). By comparing eq 28 with eq 32, we see that we can replace the [SN/ - 1,N]‚[L/N] term with the following terminal transfer matrix, [T/N]:

[T/N] ) LA061793R

[

/ 1 + Zj,0 /ZJ/ - 1 0 / /ZJ/ - 1 1 - Zj,0 0

]

(33)