Evaluation of an Atomic Force Microscopy Pull-Off Method for

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Evaluation of an Atomic Force Microscopy Pull-Off Method for Measuring Molecular Weight and Polydispersity of Polymer Brushes: Effect of Grafting Density Diane Goodman,†,‡ Jayachandran N. Kizhakkedathu,‡,§ and Donald E. Brooks*,†,‡,§ Department of Chemistry, Department of Pathology and Laboratory Medicine, and Centre for Blood Research, University of British Columbia, Vancouver, Britsh Columbia V6T 2B5, Canada Received November 6, 2003. In Final Form: May 5, 2004 The accuracy of the molecular weights Mn and polydispersities of polymer brushes, determined by stretching the grafted chains using atomic force microscopy (AFM) and measuring the contour length distribution, was evaluated as a function of grafting density σ. Poly(N,N-dimethylacrylamide) brushes were prepared by surface initiated atom transfer radical polymerization on latex particles with σ ranging between 0.17 and 0.0059 chains/nm2 and constant Mn. The polymer, which could be cleaved from the grafting surface by hydrolysis and characterized by gel permeation chromatography (GPC), had a Mn of 30 600 and polydispersity (PDI) of 1.35. The Mn determined by the AFM technique for the higher density brushes agreed quite well with the GPC results but was significantly underestimated for the lower σ. At high grafting density in good solvent, the extended structure of the brush increases the probability of forming segment-tip contacts located at the chain end. When the distance between chains approached twice the radius of gyration of the polymer, the transition from brush to mushroom structure presumably enabled the formation of a larger number of segment-tip contacts having separations smaller than the contour length, which explains the discrepancy between the two methods at low σ. The PDI was typically higher than that obtained by GPC, suggesting that sampling of chains with above average contour length occurs at a frequency that is greater than their spatial distribution.

Introduction The ability to characterize polymer brushes prepared by the “grafting from” method1 may be a challenge, particularly if the amount of grafted material is below that which is required for conventional methods of molecular weight determination such as multiangle light scattering or gel permeation chromatography (GPC). This is often the case when the grafting is performed on flat surfaces such as inorganic substrates used for sensors2 or polymeric membranes used in filtration and separation applications,3 where the surface area available for grafting is low. In these cases, the grafted polymer properties are commonly estimated from those of the solution polymers that form in parallel with the grafted species. However, such inferences may not always be appropriate given that surface-initiated polymerization can be very different from bulk polymerization. When the brushes are prepared on small spherical particles, which have higher surface area, there is generally sufficient material for analysis; however the polymer must still be released from the surface. This may require chemically cleaving the polymer at the point of grafting, while ensuring that any potential labile linkage along the chain remains intact. A direct method for the determination of molecular weight and polydispersity (PDI ) Mw/Mn) of end-grafted polymers that does * To whom correspondence may be addressed at 2211 Wesbrook Mall, Department of Pathology and Lab Medicine, University of British Columbia, Vancouver, BC V6T 2B5, Canada. Phone: (604) 822-7081. Fax: (604) 822-7635. E-mail: [email protected]. † Department of Chemistry. ‡ Centre for Blood Research. § Department of Pathology and Laboratory Medicine. (1) Zhao, B.; Brittain, W. J. Prog. Polym. Sci. 2000, 25, 677. (2) Dyer, D. J. Adv. Funct. Mater. 2003, 13, 667. (3) Ulbricht, M.; Riedel, M. Biomaterials 1998, 12, 1229.

not require isolating the chains from the surface is clearly advantageous. Advances in single-molecule atomic force microscopy (AFM) over the past few years have led to the development of a technique in which the Mn and PDI derived from adhesion profiles of end-grafted polymers have been accurately determined.4,5 The measurement involves stretching the polymer, which is adsorbed on the AFM tip at the free chain end, away from the grafting surface, and estimating the contour length of the chain from the separation at which the bond ruptures. A distribution of contour lengths is obtained by collecting a large number of force curves laterally over the grafting surface, which are then converted into Mn and PDI using the length and molar mass of the monomer. The stretching of individual polymer tethers6-10 on surfaces of low grafting density σ has received more attention than that of chains within a dense brush,4,11 presumably because of the complexity of analyzing ruptures from multiple chains. When the force curve can be attributed to the extension of a single chain, as is the case with low-density tethers, multiple ruptures represent the detachment of segments from one chain, with the (4) Yamamoto, S.; Tsujii, Y.; Fukuda, T. Macromolecules 2000, 33, 5995. (5) Al-Maawali, S.; Bemis, J. E.; Akhremitchev, B. B.; Leecharoen, R.; Janesko, B. G.; Walker, G. J. Phys. Chem. B 2001, 105, 3965. (6) Haupt, B. J.; Senden, T. J.; Sevick, E. M. Langmuir 2002, 18, 2174. (7) Haupt, B. J.; Ennis, J.; Sevick, E. M. Langmuir 1999, 15, 3886. (8) Ortiz, C.; Hadziioannou, G. Macromolecules 1999, 32, 780. (9) Chatellier, X.; Senden, T. J.; Joanny, J.-F.; di Meglio, J.-M. Europhys. Lett. 1998, 41, 303. (10) Bemis, J. E.; Akhremitchev, B. B.; Walker, G. Langmuir 1999, 15, 2799. (11) Yamamoto, S.; Ejaz, M.; Tsujii, Y.; Fukuda, T. Macromolecules 2000, 33, 5608.

10.1021/la036092y CCC: $27.50 © 2004 American Chemical Society Published on Web 06/19/2004

Characterization of Polymer Brushes

contour length related only to the separation at final rupture. The Mn and polydispersity of poly(dimethylsiloxane) tethers on a flat silicon substrate were recently estimated from single-chain force profiles in this way.5 In the model describing the dynamics of the polymer-tip interaction, contacts could break and form to reduce the tension caused by pulling on the chain. The kinetics in this study were sufficiently slow so that on the time scale of an AFM experiment, segment-tip contacts were able to overcome the energy barrier for detachment many times. In this way, the tension would be transmitted to the nearest monomer as the polymer slides along the tip. When the tension could no longer be reduced by sliding, the chain would stretch and eventually rupture from the tip. The separations at rupture were used as estimates of the contour length. Yamamoto et al.4 were the first to investigate the elastic properties of a grafted chain in a dense polymer brush. They circumvented the difficulty in interpreting a force profile consisting of multiple contacts with multiple chains by studying a copolymer brush that consisted of a long nonadsorbing poly(methyl methacrylate) (PMMA) block and a short adsorbing poly(4-vinylpyridine) (P4VP) block on the free end. Since only contacts from the short P4VP at the free end contributed to the extension profile, the rupture separations could be successfully used to calculate the Mn and PDI. In this study, we obtained a distribution of contour lengths from the extension profiles of a poly(N,N-dimethylacrylamide) (PDMA) brush in good solvent interacting with an AFM tip. In this system, it is indeed possible for the force profile to contain contributions from both multiple contact formations along a single chain, as well as contacts from multiple chains. We find, however, that the probability of forming contacts primarily with chain ends is high when the brushes are forced into an extended structure as a result of excluded volume effects. For most of the grafting densities examined, this effect allowed accurate determination of Mn from the average contour lengths, with poor results obtained only when σ was sufficiently low and a mushroom structure was adopted. Experimental Section Materials and Methods. Styrene (Aldrich, 99%) was first washed with a 1% NaOH solution, dried, then distilled under reduced pressure. N-Isopropylacrylamide (NIPAM; Aldrich, 97%) was purified by crystallization from hexane and stored at -20 °C until used. N,N-Dimethylacrylamide (DMA; Aldrich, 99%) was distilled in a vacuum and stored under argon at -80 °C. 1,1,4,7,10,10-Hexamethyltriethylenetetramine (HMTETA; Aldrich, 97%), CuCl (Aldrich, 99+%) CuCl2, (Aldrich, 99.99%) were used as received. All other commercial reagents were purchased of highest purity from Aldrich and used without further purification. 2-(Methyl 2′-chloropropionato)ethyl acrylate (HEA-Cl) was synthesized as described in our earlier report.12 Water purified using a Milli-Q Plus water purification system (Millipore Corp., Bedford, MA) was used in all experiments. Molecular weights were determined by GPC on a Waters 2690 separation module fitted with a DAWN EOS multiangle laser light scattering (MALLS) detector from Wyatt Technology Corp. with 18 detectors placed at different angles (laser wavelength λ ) 690 nm) and a refractive index detector from Viscotek Corp operated at λ ) 620 nm. A detailed description of the method is given in our earlier report.12 Particle size measurements (i.e., measurements of the hydrodynamic diameter distribution of particle suspensions) were carried out in a temperature-controlled Beckman Coulter N4 Plus particle size analyzer by dynamic light scattering. Aqueous (12) Jayachandran, K. N.; Takacs-Cox, A.; Brooks, D. E. Macromolecules 2002, 35, 4247.

Langmuir, Vol. 20, No. 15, 2004 6239 dispersions of particles were allowed to thermally equilibrate for 5 min, and the measurements were made at 20 °C unless otherwise mentioned. Size analyses were performed using the software supplied by the manufacturer. Aqueous ATRP of DMA from Functionalized Latex. The synthesis of PDMA brushes prepared by aqueous atom transfer radical polymerization (ATRP) from functionalized polystyrene latex has been described in a previous report in detail.12 Briefly, narrowly dispersed polystyrene (PS) seed latex was synthesized by surfactant-free polymerization of styrene in water initiated by potassium persulfate (KPS) at 70 °C for 24 h under argon atmosphere. A copolymer shell of styrene and ATRP initiator monomer 2-(methyl-2′-chloropropionato)ethyl acrylate (HEA-Cl) was added to the PS seed latex by a shell growth polymerization. Following the reaction, the latex was cleaned extensively by dialysis, centrifugation, and resuspension. Characterization involved determination of the solid content, hydrodynamic size, concentration of ATRP initiator, and sulfate charge on the surface by the reported procedure.12 The hydrodynamic diameter of the shell latex was 650 nm. ATRP reactions were performed in a glovebox filled with argon due to the sensitivity of the Cu(I) complex to air. The details of the polymerization are reported in our earlier communication.12 The reaction was allowed to proceed for 24 h and monomer conversion was determined by analyzing the supernatant solution by reverse phase HPLC. The grafted latex was cleaned by 8 to 10 cycles of sequential centrifugation and resuspension in water, NaHSO3 solution (50 mM), and water to remove adsorbed copper complexes until there was no detectable amount of polymer, monomer, or catalyst in the supernatant. The grafted latex was characterized by determining hydrodynamic size, number, and weight average molecular weight (Mn and Mw), radius of gyration of polymer chains, and the amount of polymer grafted by quantitatively hydrolyzing the PDMA chains grafted to the surface as reported earlier.12 The grafting density, σ, was calculated from the surface area of the latex and amount and molecular weight of the polymer grafted as described earlier.12 Synthesis of PDMA Grafted Latex with Different Grafting Densities. The preparation of grafted brushes with varying graft density and constant Mn by time-dependent quantitative hydrolysis of grafted PDMA was as follows. Grafted latex samples (4 g each) were stirred with NaOH (2 g, 2 N) for different periods of time (up to 96 h), and the hydrolysis was stopped by the addition of dilute (6 N) HCl to the suspension. The supernatant was collected quantitatively by centrifugation of the suspension. The graft densities of these cleaved brushes were determined from the amount of polymer released from different samples at various time intervals. The hydrodynamic sizes of the different samples were also determined by particles size analysis. AFM Measurements. Measurements were performed on a Multimode, Nanoscope IIIa controller (Digital Instruments (DI), Santa Barbara, CA), equipped with a fluid cell. Cantilevers were V-shaped silicon nitride with tip radius of 5-40 nm and spring constant of 0.06 N/m as quoted by the manufacturer (DI). The rate of tip-sample approach was typically 500 nm/s. Samples were prepared by drying the latex suspension onto a glass substrate (precleaned in chromic acid), followed by sonication and thorough rinsing with water to remove any latex which had not been strongly physisorbed. To minimize lateral motion during the force measurements, we aimed at preparing a closely packed monolayer of latex. Inspection under an optical microscope suggested regions of monolayer coverage were present, and measurements were made in these areas. Experiments were performed in 10 mM NaCl solution. Force measurements on the nongrafted latex confirmed that at this ionic strength, repulsion between the negatively charged tip and latex was minimized, allowing forces due to the grafted layer to be studied independent of electrostatics. Samples were allowed to equilibrate for 15 min before performing experiments. Data Analysis. The optical sensitivity (V/nm) was calibrated on a hard glass surface by measuring the slope of the linear region of the raw force curve (photodiode (V) vs piezo position (nm)) after contact had been made with the glass. The cantilever deflection (nm) is obtained by dividing the measured photodiode signal (V) by the optical sensitivity. On a hard surface, the change

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Goodman et al. Table 2. Characteristics of PDMA Brushesa

Table 1. Layer Thickness of PDMA Brushes timea (h)

σb (nm-2)

LHc (nm)

Led (nm)

Le/Lce

0 1 5 7 10 15 24 48 96

0.171 0.0806 0.0376 0.0315 0.0304 0.0220 0.0171 0.0125 0.00590

58 40 30 40 42 32 42 49 20

65 48 34 32 30 25 15

0.84 0.62 0.44 0.42 0.39 0.32 0.19

a Length of time chains were hydrolyzed from grafting surface. Grafting density at surface (chains/nm2). c Hydrodynamic thickness measured by particle size analyzer. d Equilibrium thickness measured by AFM. e Contour length calculated from Mn measured by GPC using molar mass (99 g/mol) and size (0.25 nm) of monomer (eq 1).

b

in cantilever deflection (∆d) is equal to the change in piezo or sample position (∆h). The laser and cantilever were not adjusted after calibration, enabling the same optical sensitivity to be used on the grafted polymer samples. Forces were calculated from the measured cantilever deflection by multiplying ∆d by the cantilever spring constant according to Hooke’s law. Tip-sample separations, L, were obtained by subtracting the change in tip deflection from the measured relative sample position according to the method of Ducker et al.13Separation L ) 0 corresponded to the position at which the sample behaved as a hard surface upon further compression, commonly known as the constant compliance regime, with the slope (∆d/∆h) of this region equal to 1. The equilibrium thickness was taken as the separation at which tip deflection first deviated significantly from the baseline, corresponding to a repulsion of ∼0.02 nN. The maximum force applied to the sample, known as the trigger force, was set high enough that the region of constant compliance was reached on all samples. Because a greater force is required to compress a brush with higher grafting density as a result of excluded volume effects, the trigger force generally increased with increasing σ. The trigger value was chosen so that the length of constant compliance was roughly constant for all samples, and therefore the time of contact with the tip following full compression remained constant as well. For σ ) 0.17 and 0.032 nm-2, the lengths of constant compliance were 15 ( 6 and 20 ( 4 nm, respectively. Using a scan rate of 500 nm/s, the estimated time of contact after full compression was between 0.03 and 0.04 s. Force measurements were repeated at various (∼75) locations on each sample. The compression profiles were generally reproducible, particularly for the higher grafting density samples. The compression data were averaged for each sample and the resulting curves were used to obtain the equilibrium thicknesses shown in Table 1. The inaccuracy of the spring constant quoted by the manufacturer does not allow for an accurate estimate of the mean rupture force of a given sample or for direct comparison of the forces of different samples obtained using different tips. The uncertainties associated with the rupture forces in Table 2 represent the standard deviation of the forces measured at various locations on a sample and are not a measure of the total error in the force measurement. As demonstrated by the difference between measured and estimated spring constants,14 the error in force can be quite high (as large as 100%). Most of the results presented in this work, however, focus on the rupture separations, which are not affected by any variation in spring constant. Due to the relatively small diameter of the latex spheres (650 nm), force measurements were obtained at random locations and could not specifically be performed on the uppermost point of the particles. We have considered the effects on the force curve if the region directly between two spheres was probed, as opposed to the uppermost point of a particle. We have included in the Supporting Information (Figure A) a representation (to scale) of (13) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831. (14) Senden, T. J.; Ducker, W. A. Langmuir 1994, 10, 1003.

timeb (h)

σ (nm-2)

LRd D/Rgc (nm)

0 1 3 5 10 15 24 48 96

0.171 0.0806 0.0472 0.0376 0.0304 0.0220 0.0171 0.0125 0.00590

0.375 0.546 0.714 0.800 0.89 1.046 1.183 1.387 2.018

83.1 80.6 88.3 74.7 90.9 74.6 83.4 49.1 24.6

FRe (nN)

Mn

Mw

PDI

0.14 ( 0.14 0.27 ( 0.21 0.33 ( 0.26 0.35 ( 0.29 0.32 ( 0.29 0.25 ( 0.18 0.22 ( 0.16 0.50 ( 0.35 0.14 ( 0.13

33 000 32 000 35 000 29 600 36 000 29 600 33 000 19 400 9 800

49 200 49 800 56 400 48 300 56 700 49 500 47 800 36 800 14 700

1.49 1.56 1.61 1.63 1.58 1.68 1.45 1.89 1.50

a GPC results: M ) 30 600, M ) 41 300, and PDI ) 1.35. n w bLength of time chains were hydrolyzed from grafting surface.cRatio

of distance between grafting points to radius of gyration. d Average rupture separation (∼75 force curves). e Average rupture force.

an AFM tip (half angle 35ο) located at (a) the top of a latex particle (diameter ) 650 nm) and (b) directly between two adjacent particles. The representation suggests that the contact area between the tip and the grafted layer when the tip is located between two adjacent particles is significantly higher than when it is at the upper most point of the particle. We therefore expect the compression curves to exhibit larger repulsive forces in the former case than in the latter. Indeed, in ∼1-4% of the force profiles collected, the repulsive compression forces were of above average magnitude and the data from the corresponding decompression profiles were not included in the analysis.

Results and Discussion Hydrolysis of the initial PDMA brush (σ ) 0.17 nm-2) over increasing periods of time produced a series of brushes with decreasing σ. The Mn of the hydrolyzed polymer of the different samples, measured by GPC, agreed well with that of the initial brush suggesting that cleavage was independent of molecular weight. We assume here that hydrolysis of the chains was uniform over the grafting surface. The properties of the brushes, having Mn 30 600 and PDI 1.35, are listed in Table 1. The equilibrium thickness, Le, taken from the separation in the force curve at which a repulsive force was first detected upon compression of the hydrated brush, increases with σ. This observation is consistent with the formation of a more extended brush structure due to the excluded volume in good solvent. The same trend cannot be made for the hydrodynamic thickness, LH, measurements due to significant scatter in the data, although the largest hydrodynamic thickness value is found for the sample with the highest σ. Propagation of the errors associated with the diameter estimates of the grafted and shell latexes gives rise to a large uncertainty for relatively small HT values. For example, a particle diameter increase from 650 nm ( 10% to 710 nm ( 10% results in a HT of 30 nm ( 160%. A typical force curve is presented in Figure 1, showing both the compression and extension of the brush. The repulsive force increases monotonically as the brush is compressed due to excluded volume effects. The entire range of data including the high compression region is displayed in the inset. Zero separation corresponds to the fully compressed brush, which behaves as a hard surface upon further approach. We believe that L ) 0 nm corresponds to the point where the AFM tip nearly contacts the latex surface. On the basis of the rigidity of polystyrene (compressive modulus ) 3000 MPa)15 and force profiles obtained on nongrafted latex, we do not expect indentation of the particle to contribute significantly to the force (15) Polymer Handbook; Brandup, J., Immergut, E. H., Eds.; John Wiley & Sons: New York, 1989.

Characterization of Polymer Brushes

Figure 1. Typical force profile showing advancing (compression (O)) and receding (stretching (/)) directions of a PDMA brush (Mn 30 600, σ ) 0.17 nm-2). Solid arrows depict rupture events corresponding to LR. Dotted arrow indicates onset of repulsion corresponding to equilibrium thickness Le of polymer layer.

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102 nm. All values of LR are taken as the distance from the point of grafting (L ) 0 nm) to the point of detachment from the AFM tip, irrespective of whether additional rupture events are detected at smaller separations. The rupture separation corresponds to the length of the fully stretched chain segment located between the point of grafting from the latex and the point of rupture from the AFM tip. It is not known whether the two rupture events observed in Figure 1 represent the detachment of a single chain at different points along its length, or two chains with different contour lengths. We do not distinguish between these different scenarios and include all rupture events from the ∼75 profiles taken per sample in the data set. By comparing the values of LR with the distribution of contour lengths calculated from the GPC data, we can assess the probability of measuring bond rupture at the free chain end. We use the following expressions5 to calculate Mn and Mw from LR

Mi ) Mmon Mn,R )

Mw,R )

Mi

∑i N˜

∑i Mi2

PDIR )

Figure 2. Comparison of Mn and Mw determined by rupture separations from AFM and GPC.

profiles. We have also considered in an earlier report16 the possibility of an offset distance caused by an incompressible layer of polymer between the tip and the latex. On the basis of a comparison of the grafting densities used here and those in other reports,4,17 and the small offset distances that were found at such grafting densities using sharp silicon nitride probes, we believe such an offset distance in this case is negligible. In the main figure (Figure 1), the data from the inset is expanded in the region of -0.2 to 0.4 nN. A repulsive force is first detected in the compression profile at a separation of 65 nm. Assuming that zero separation occurs at the point where the tip contacts the latex surface, the onset of repulsion corresponds to the equilibrium thickness Le of the polymer layer. There is no hysteresis observed in the advancing and receding force profiles from 0 to ∼50 nm separation, suggesting that the brush returns to its equilibrium structure at each data point following initial decompression. Upon further extension, chains adhering to the AFM tip are stretched, until eventually breaking free from the AFM tip at some distance, LR, referred to as the rupture separation. In the example, two rupture events occurred with LR(1) ) 65 nm and LR(2) ) (16) Goodman, D.; Kizhakkedathu, J. N.; Brooks, D. E. Langmuir 2004 20, 2333. (17) Vermette, P.; Meagher, L. Langmuir 2002, 18, 10137.

LR l

∑i

(1)

(2)

(3)

Mi

Mw,R Mn,R

(4)

where Mmon is the molar mass of a dimethylacrylamide monomer (99 g mol-1), N ˜ is the total number of chains, and l is the length of a monomer unit, taken from the projected C-C bond length to be 0.25 nm. The subscript R denotes molecular weights calculated from the rupture separations, which are to be distinguished from Mn and Mw measured by GPC. The values are listed in Table 2. The molecular weights estimated from the contour lengths for each sample of different grafting density are plotted in Figure 2 against D/Rg, the ratio of the distance between grafting points to radius of gyration. The Mn,R values obtained here compare well with the Mn from GPC (solid line) for D/Rg between 0.375 and 1.18. There is less agreement between the two methods as the distance between chains is increased, with the AFM technique underestimating Mn. Like Mn,R, Mw,R is roughly constant for samples having a D/Rg less than 1.18, although it is consistently larger than the value determined by GPC. For larger D/Rg, the Mw,R also decreases. The discrepancy between Mn,R and Mn at higher D/Rg may be explained by comparing the distribution of contour lengths obtained from the two methods in Figure 3, particularly in the region of below average length. For σ ) 0.171 nm-2, the highest probability of length measured by the AFM technique approximated the average contour length measured by GPC, with only a small contribution from ruptures with LR values below that of the shortest chain observed by GPC. In contrast, the number of short segments contributing to the distribution for σ ) 0.0125 nm-2 is significantly larger, suggesting that the probability of a segment detaching from the AFM tip at a point along the chain rather than at the end increases as the grafting

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Goodman et al.

Figure 4. Representation of the most probable interaction of an AFM tip with polymer at (A) high and (B) low grafting density. The tip approaches the grafted layer (I) with the asterisk denoting the most probable initial point of contact. Upon compression (II), the tip may form additional contacts along a single chain as well as with multiple chains. With sufficient extension (III), the chain ruptures from the tip at the contact point denoted by the asterisk, at a separation LR.

Figure 3. Comparison of contour length distributions obtain by AFM and GPC for high (σ ) 0.171 nm-2) and low (σ ) 0.0125 nm-2) grafting densities. Distributions are normalized so that ∑ifi∆L ) 1, where fi is the frequency of AFM rupture event.

density decreases. Indeed, this agrees with advanced brush models18-20which predict the ends to be distributed throughout the brush, in contrast to the earlier steplike volume fraction profile of Alexander21 and de Gennes22 that assumed all chain ends were located at the outer edge of the brush. An explicit expression for the end density distribution away from the grafting surface for a compressed brush, given by Milner et al.,19 predicts the end density at the outer layer of the brush to increase with σ, supporting our interpretation. Our model for the interaction of the tip with the polymer brush at (A) high and (B) low grafting density is depicted in Figure 4. The equilibrium brush structure of the polymer layer before contact with the tip is show in (I). The asterisk denotes the most probable initial point of contact, located at the outer edge of the layer. In the high σ (case A), the asterisk is located at the chain end due to the highly extended equilibrium structure. Upon compression (II), the tip may form additional contacts along a single chain as well as with multiple chains. For clarity, we do not show contact with multiple chains in the low σ representation (B). Note that at low σ, the tip may never contact the chain end, which is buried within the coil. With sufficient extension (III), the chain ruptures from the tip at a separation LR. We emphasize that although other contacts may be formed during compression (II), the data (18) Murat, M.; Grest, G. Phys. Rev. Lett. 1989, 63, 1074. (19) Milner, S.; Witten, T.; Cates, M. Macromolecules 1988, 21, 2610. (20) Lai, P. Y.; Binder, K. J. Chem. Phys. 1991, 95, 9288. (21) Alexander, S. J. Phys. 1977, 38, 983. (22) de Gennes, P. G. Macromolecules 1980, 13, 1069.

in Figure 3 suggest that the asterisk represent the most probable contact at which rupture is measured. The rupture separation measured at high and low grafting densities is related to the contour length Lc by (A) LR ∼ Lc and (B) LR < Lc, respectively. This representation differs from other models where the tip is thought to slide down the length of the polymer upon extension and rupture at the chain end.5 It should be noted that the accurate determination of Mn in this study could, in part, be due to the spherical nature of the brush. Theoretical models predicting the distribution of chain ends in brush layers have shown different behaviors for chains grafted to flat versus curved surfaces,23 with the distribution being dependent on the radius of curvature R. A depletion layer near the grafting surface is predicted for spherical brushes, where the probability of finding chain ends in the region is extremely low. Subsequently, there is a higher probability of locating chain ends away from the grafting surface in spherical than flat brushes. Any effect due to curvature in this system, however, is expected to be small, since even the brush with the largest thickness had a relatively low Le/R ratio (0.095). From the ratios of equilibrium thickness to contour length Le/Lc in Table 1, we see that the thickness of the highest σ brush at equilibrium is ∼84% of its fully extended structure. Perhaps surprisingly, the contribution from short segments only affected the distribution for samples with σ ) 0.0125 nm-2 and lower. Following the trend in Le/Lc, these samples would have thicknesses less than 19% of the fully extended length. The technique, enabling determination of Mn over a large range of grafting densities, is therefore not limited to tightly packed brushes where the equilibrium thickness of the brush approaches the contour length. (23) Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214.

Characterization of Polymer Brushes

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Figure 5. Effect of grafting density on equilibrium thickness of PDMA brushes measured by AFM (Mn ∼ 30 600, PDI ) 1.35). Dotted line corresponds to best fit of linear region (0.0171 nm-2 < σ < 0.170 nm-2) with slope of 0.45.

To establish whether the time the tip remained in contact with the fully compressed sample had any affect on the molecular weights measured by this technique, we repeated the experiment on sample σ ) 0.0125 nm-2 at two different trigger forces. On the basis of the length of the constant compliance regimes and the compression rate (500 nm/s), the estimated contact times for these measurements were 0.03 (trial 1) and 0.06 s (trial 2). The effect of contact time on the average rupture separation Mn,AFM and Mw,AFM is presented in Figure B of the Supporting Information. The results suggest that over this range of time there is no detectable effect. Similarly, to establish the effect of rate on the measured molecular weights, we performed the experiment at 500 nm/s (trial 1) and 2000 nm/s (trial 3). The results are presented in Figure B of the Supporting Information). The constant molecular weight values suggest that there is no detectable dependence in the range of rates examined. We note, however, that the molecular weights are significantly lower than those presented in Figure 2 for sample σ ) 0.0125 nm-2. The data in the Supporting Information were obtained using a tip from a different wafer than the one used to obtain the data in Figure 2. We expect that differences in the tip geometry will affect the results, particularly at low grafting density where the distance between chains at the grafting point approaches the tip radius. The trend of measuring higher molecular weights on more extended brushes was consistent, irrespective of the tip used. To establish the density regime in which the technique could be successfully applied, we compared the dependence of Le on σ, shown in Figure 5, with theoretical predictions. Scaling theory predicts that for a moderately dense brush in good solvent, Le varies with σ, and the degree of polymerization, N, according to

Le ∝ Nσn

Experimental investigations of the power law by measuring the thickness of different brushes using neutron reflectivity and neutron scattering24-28 have produced ambiguous results. While brush heights have been found to obey the predicted scaling law,26 others have found different values of the exponent n.24,27,28 The discrepancy has, in some cases, been attributed to the uncertainty in grafting density of brushes prepared by the adsorption of block copolymers, which must be inferred from experimental data, such as by fitting a model density profile to the reflectivity curve. Bijsterbosch et al.24 measured the thickness of PS-PEO block copolymer at the air/water interface for various PEO block lengths and found the height to scale as H ∼ σ0.41. Numerical SCF calculations by Currie et al.,29 however, have found that the exponent is roughly constant for long chains but may vary for short chains, leading to erroneous predictions when combining H and σ results for all molecular weights in one plot. When the thicknesses for different molecular weights were plotted independently against σ, the fitted power law resulted in an exponent of 0.33 for N ) 700 and 0.35 for N ) 450. Yamamoto et al.11 have measured by AFM the equilibrium thickness of poly(methyl methacrylate) brushes, having Mn ∼ 56 700, in good solvent as a function of grafting density (0.07 < σ (chains/nm-2) < 0.7) and found n ∼ 0.4. This is comparable to the exponent that we measured, despite the slightly lower Mn and σ values used here. The value of n, which was clearly greater than the predicted 1/3, was attributed to higher order virial interactions,30 which must be accounted for when using such large grafting densities. The value of their exponent also appeared to increase with σ; however this result could not be explicitly established due to an insufficient number of high σ data points. Similarly, we could not fit the data above and below σ < 0.0171 nm-2 to a single exponent power-law. This transition point corresponds to the same grafting density at which Mn,R and Mn disagreed, presumably due to a reduction in accessible free chain ends. The effect of curvature should again be addressed, as it has been found that, like the chain end distribution, the height of a spherical brush depends on the radius of curvature R. For spherical geometry, Wijmans and Zhulina23 have found the brush height to scale roughly as

H ≈ (σN3R2)1/5

(6)

This equation, however, does not describe our results, predicting a lower exponent when curvature effects are significant. On the basis of the scaling results obtained, we expect the method of Mn,R determination from rupture separations to be applicable to a moderate to highly dense brush regime. The rupture technique was less successful at accurately predicting the PDI, giving Mw,R values that were consistently overestimated. This is likely due to a bias in the AFM technique toward very long chains. Very long chains have the advantage of reaching and adsorbing to the tip

(5)

where n ) 1/3. The exponent has been predicted using different approaches, first by Alexander21 and de Gennes22 using global energy balance approximations, and then using analytical self-consistent mean-field (SCF) equations.19 The linear logarithmic plot obtained in Figure 5 indeed suggests a power-law relationship; however a nonlinear least-squares fit (R2 ) 0.996) performed on the data for 0.0171 < σ < 0.171 gave an exponent n of 0.45, which was higher than the predicted value.

(24) Bijsterbosch, H. D.; de Haan, V. O.; de Graaf, A. W.; Leermakers, F. A. M.; Cohen Stuart, M. A.; van Well, A. A. Langmuir 1995, 11, 4467. (25) Field, J. B.; Toprakcioglu, C.; Ball, R.; Stanley, H.; Dai, L.; Barford, W.; Penfold, J.; Smith, G.; Hamilton, W. Macromolecules 1992, 25, 434. (26) Auroy, P.; Auvray, L.; Leger, L. Macromolecules 1991, 24, 2523. (27) Kent, M. S.; Lee, L. T.; Farnoux, B.; Rondelez, F. Macromolecules 1992, 25, 6240. (28) Richards, R. W.; Rochford, B. R.; Webster, J. R. P. Polymer 1997, 38, 1169. (29) Currie, E. P. K.; Leermakers, F. A. M.; Cohen Stuart, M. A.; Fleer, G. J. Macromolecules 1999, 32, 487. (30) Lai, P.-Y.; Halperin, A. Macromolecules 1991, 24, 4981.

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Goodman et al.

Table 3. Effect of Solvency on Molecular Weight Estimation of PDMA Brusha water (10 mM NaCl) methanol ethanol (100%) 2-propanol a

Mn,AFM

Mw,AFM

33 000 36 800 40 800 34 500

49 200 50 300 59 800 44 700

Brush properties: σ ) 0.17 nm-2, Mn,GPC ) 30 600, Mw ) 41 300.

even when they are not being directly compressed, resulting in an over-representation in the retraction force profile compared to their average spatial frequency. From the GPC distribution, we see that a small fraction of chains having a much higher than average molecular weight are present. According to the distribution profiles in Figure 3, the relative contribution to the force profiles from very long chains (>300 nm) is greater than that from intermediate chains, when compared with the actual frequencies obtained by GPC. If we recalculate the molecular weight after having removed the two largest values of LR, which contribute only ∼1% of the total number of rupture events sampled, we obtain a Mn,R ∼ 31 000, Mw,R ∼ 41 000 and PDIR ∼ 1.31, which is much closer to the value measured by GPC (PDI ) 1.35). The number of chains being compressed by a tip of radius 15 nm may range from 4 to 120 (for 0.0059 < σ < 0.171 nm-2), assuming a uniform grafting density. While the probability of a chain adhering to the tip is higher for those polymers being directly compressed, nearby chains may adhere anywhere along the surface of the tip. In the study performed by Al-Maawali et al.,5 a better correlation was found between PDIR determined using the AFM technique and GPC, and it was therefore assumed that uncertainty due to tip shape convolution is minor. The good agreement between the PDI measured by the two methods in the reference system was only found for surfaces prepared by a grafting reaction in which the polymer concentration was low. Poor agreement found using high polymer concentrations was attributed to a preferential adsorption and reaction of long chains. Although the σ values were not provided, the surfaces prepared in the referenced system were presumably of much lower grafting density than those used here. Polydispersity also affects the distribution of chain ends, with the ends of longer chains located farther from the grafting surface.31 This effect may also alter the distribution of contour lengths in favor of longer chains, so that the separation at rupture more closely approximates their actual contour length than does that for shorter chains. To examine the effect of solvency, we obtained force profiles of the highest grafting density brush (σ ) 0.17 nm-2) in various alcohols (methanol, ethanol (100%), 2-propanol). We chose the alcohols since they are poorer solvents than water for the grafted PDMA and are also poor solvents for polystyrene. Compression profiles of the nongrafted polystyrene shell latex (data not shown) suggest that the latex behaves as a hard incompressible surface in methanol (similar to in water at high salt concentration). Differences in the data obtained in the various solvents should therefore reflect changes in the structure of the polymer brush layer with negligible contribution from the underlying latex. The Mn,AFM and Mw,AFM values calculated from the rupture separations are listed in Table 3. The values measured in the alcohols were similar to those in water and did not display a trend upon decreasing the solvent (31) Currie, E. P. K.; Wagemaker, M.; Cohen Stuart, M. A.; van Well, A. A. Macromolecules 1999, 32, 9041.

Figure 6. Effect of solvent quality on compression profiles of PDMA brush for σ ) 0.17 nm-2. Curves are average of data obtained at 75 locations on the sample.

Figure 7. Distribution of rupture forces for PDMA brushes of two representative grafting densities. Data are normalized so that ∑ifi∆FR ) 1.

quality for the alcohols tested. According to our model, decreasing the solvency would have the same effect as decreasing the grafting density (i.e., a more collapsed equilibrium structure would result in a decrease in average rupture separation). To further investigate the structure of the polymer layer, we present in Figure 6 the compression profiles obtained in the different solvents. Each profile represents the average profile obtained at ∼75 different locations on the sample in a given solvent. At large separations (small compression), the force profiles in all solvents are similar. Upon further compression, the brush exerts a smaller repulsive force in the alcohols than in water. In this region the profiles in the different alcohols are remarkably similar to one another but are notably different from that in water. According to our data and model, the most probable point of rupture is that where initial contact between the tip and polymer layer is formed (see Figure 4). The superposition of the compression data at large separations coupled with the agreement of average rupture separations in various solvents supports our model for tip-polymer interaction. In all samples, we measured the forces at the point of rupture FR and examined their distribution. The values were averaged for each sample and are listed in Table 2. The mean forces ranged between 0.14 and 0.33 nN, and histograms showing the force distribution of two repre-

Characterization of Polymer Brushes

sentative samples, σ ) 0.171 nm-2 and 0.022 nm-2, are presented in Figure 7. Due to the uncertainty in the spring constant, and the use of different tips throughout this work, we do not compare the mean forces of the different samples directly and note that there is a large uncertainty associated with these values. We simply present the data to illustrate the distribution of forces over the surface. Conclusions Agreement between the Mn values determined by AFM and GPC confirms the validity of the rupture technique over a range of brush densities. The failure of the method to accurately predict Mn at low grafting densities provides a lower limit above which the technique could be successfully applied. In addition to the practical implications of the study, a correlation could potentially be made between the distribution of chain ends in polymer and the difference between the rupture separations measured by AFM and the contour lengths measured by GPC. A powerlaw relationship between the layer thickness estimated

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by AFM and the grafting density, with an exponent of 0.45, was observed only over the density regime for which the Mn could accurately be measured by the rupture method. Acknowledgment. We thank the Canadian Institutes of Health Research, the Canadian Blood Services, Natural Sciences and Engineering Research Council of Canada, and Canada Foundation for Innovation for financial support and Raymond Norris-Jones for help with HT measurements. Supporting Information Available: Representation of conical AFM tip compressing the spherical brush at the uppermost point of the particle and directly between two particles and graph of effect of rate and contact time on molecular weight measured using AFM rupture separations. This material is available free of charge via the Internet at http://pubs.acs.org. LA036092Y