Molecular Weight and Polydispersity Estimation of Adsorbing Polymer

The values obtained by AFM−(1) Mn,AFM = (3.8 ± 0.5) × 104, PDI,AFM = 1.3 ± 0.1 and (2) ... Biomacromolecules 0 (proofing), ... Chemical Reviews 0...
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Langmuir 2004, 20, 3297-3303

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Molecular Weight and Polydispersity Estimation of Adsorbing Polymer Brushes by Atomic Force Microscopy Diane Goodman,† Jayachandran N. Kizhakkedathu,‡ and Donald E. Brooks*,†,‡ Department of Chemistry and Department of Pathology and Laboratory Medicine, University of British Columbia, Vancouver, BC, V6T 2B5, Canada Received November 17, 2003. In Final Form: January 27, 2004 We have estimated the molecular weight, Mn, and polydispersity, PDI, of densely grafted poly(Nisopropylacrylamide) (PNIPAM) brushes using a novel atomic force microscopy (AFM) approach. When compression of a polymer brush induced adsorption of multiple chains to an AFM tip, the resulting decompression force profile exhibited a maximum attractive force at a separation, Lm, that decayed to zero with increasing tip-sample separation. We have found that the separation Lm approximates the average contour length, Lc, determined by gel permeation chromatography (GPC). The detection of a decaying attractive force at separations larger than Lc suggests that chains of above average length sequentially break free from the tip as they are stretched away from the grafting surface. The shape of the decompression profile in this region approximately paralleled the cumulative weight fraction of the grafted chains determined by GPC. The fraction of chains of a given molecular weight determined from a single force curve fit a log-normal distribution, having a standard deviation that provided an estimate of the PDI. We have characterized two PNIPAM brushes by this AFM technique as well as by GPC coupled to a multiangle laser light-scattering detector (MALLS). The values obtained by AFM-(1) Mn,AFM ) (3.8 ( 0.5) × 104, PDI,AFM ) 1.3 ( 0.1 and (2) Mn,AFM ) (9.4 ( 1.4) × 104, PDI,AFM ) 1.3 ( 0.1-agreed quite well with the corresponding GPC/MALLS values of (1) Mn,GPC ) 4.77 × 104, PDI,GPC ) 1.33 and (2) Mn,GPC ) 9.49 × 104, PDI ) 1.35. This technique requires only a single force curve to obtain a statistical distribution of contour lengths and provides a novel method for estimating the Mn and PDI of appropriate uniformly grafted dense polymer layers.

Introduction Polymer characterization by atomic force microscopy (AFM) has progressed considerably since the technique was initially developed for imaging.1 The determination of viscoelastic properties,2-4 adhesion interactions,5-7 and steric forces8-10 in polymeric materials by AFM has become fairly common. Recently, two notable independent AFM techniques for determining the molecular weight (Mn) and polydispersity (PDI) of grafted polymer layers have emerged.11-14 One method involves imaging the polymer chains with AFM and counting the number of chains per * Corresponding author. Address: 2211 Wesbrook Mall, Department of Pathology and Lab Medicine, University of British Columbia, Vancouver, BC, V6T 2B5, Canada. Phone: (604) 8227081. Fax: (604) 822-7635. E-mail: [email protected]. † Department of Chemistry. ‡ Department of Pathology and Laboratory Medicine. (1) Binnig, G.; Quate, C. F.; Gerber, Ch. Phys. Rev. Lett. 1986, 56, 930. (2) Weisenhorn, A. L.; Khorsandi, M.; Kasas, S.; Gotzos, V.; Butt, H.-J. Nanotechnology 1993, 4, 106. (3) Domke, J.; Radmacher, M. Langmuir 1998, 14, 3320. (4) Satomi, N.; Tanaka, K.; Takahara, A.; Kajiyama, T. Macromolecules 2001, 34, 6420. (5) Chen, X.; Davies, M. C.; Roberts, C. J.; Tendler, S. J. B.; Williams, P. M.; Davies, J.; Dawkes, A. C.; Edwards, J. C. Langmuir 1997, 13, 4106. (6) Feldman, K.; Tervoort, T.; Smith, P.; Spencer, N. D. Langmuir 1998, 14, 372. (7) Efremova, N. V.; Sheth, S. R.; Leckband, D. E. Langmuir 2001, 17, 7628. (8) Abe T.; Higashi, N.; Niwa, M.; Kurihara, K. Langmuir 1999, 15, 7725. (9) Butt, H.-J.; Kappl, M.; Mueller, H.; Raiteri, R. Langmuir 1999, 15, 2559. (10) Vermette, P.; Meagher, L. Langmuir 2002, 18, 10137. (11) Sheiko, S. S.; da Silva, M.; Shirvaniants, D.; LaRue, I.; Prokhorova, S.; Moeller, M.; Beers, K.; Matyjaszewski, K. J. Am. Chem. Soc. 2003, 125, 6725. (12) Yamamoto, S.; Tsujii, Y.; Fukuda, T. Macromolecules 2000, 33, 5995.

unit area.11 To ensure accurate determination of the number density, a large number of chains from several different images must be counted. The average molecular weight can then be calculated by dividing the mass per unit area, which was determined in the referenced study using the Langmuir-Blodgett technique, by the number density measured by AFM. The other new approach for determining the Mn and PDI of grafted polymers is based on the stretching of individual chains away from the grafting surface with an AFM tip.12-14 Typically, a tethered polymer that is adsorbed onto the tip by the free end will, under moderate tension, exhibit a restoring force that varies nonlinearly with distance. The attractive force increases with chain extension until the gradient eventually exceeds the spring constant of the cantilever, causing a mechanical instability. When the elastic restoring force is greater than the adhesion force of the chain to the tip, the polymer snaps free from the tip, and the deflection signal returns to the baseline. The separation at which the polymer ruptures from the tip is used as a measure of the chain contour length, Lc,i. The molecular weight Mi of an individual chain is calculated using Lc,i, and the size and molar mass of the monomer. Collection of a large number of force curves is required to obtain an accurate statistical distribution of the contour lengths so that the average values of Mn and PDI can be determined. When the grafting density of the polymer is sufficiently low, each force profile represents the stretching of a single chain. Similarly, at higher grafting densities, individual chains can be stretched when the tip-polymer interaction is weak and the probability of a chain physisorbing to the (13) Al-Maawali, S.; Bemis, J. E.; Akhremitchev, B. B.; Leecharoen, R.; Janesko, B. G.; Walker, G. J. Phys. Chem. B 2001, 105, 3965. (14) Goodman, D.; Kizhakkedathu, J. N.; Brooks, D. E.; Langmuir 2003 (submitted).

10.1021/la036164l CCC: $27.50 © 2004 American Chemical Society Published on Web 03/06/2004

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tip during compression is low.12-21 In contrast, when a polymer brush strongly interacts with the AFM tip during compression, as is the case with poly(N-isopropylacrylamide) (PNIPAM), the corresponding decompression profile is quite different from the profile obtained by stretching an individual chain. Discrete bond rupture events are not discerned due to overlapping data from the simultaneous stretching of multiple chains of a nearly continuous range of lengths. The force profile is characterized by a maximum attractive force, which decays to zero with increasing chain extension. The magnitude of the force at any separation is related to the number of chains of a particular contour length adsorbed to the tip. The total number of chains being stretched decreases with increasing extension, as progressively longer chains approach their contour lengths and break free from the tip. The fraction of grafted chains of a given length is determined by analyzing the decaying attractive force, allowing for estimation of Mn and PDI from a single force profile. Experimental Section Poly(N-isopropylacrylamide) and poly(N,N-dimethylacrylamide) (PDMA) brushes were prepared on the surface of polystyrene latex particles with diameters of 584 ( 8 nm and 650 ( 10 nm diameter, respectively, by aqueous atom transfer radical polymerization (ATRP). The synthesis of ATRP initiator functionalized latex and the surface initiated polymerization of PNIPAM and PDMA are described in detail in our earlier reports.22,23 The grafted polymer was cleaved from the surface for analysis by hydrolyzing an ester linkage at the point of grafting. The grafted latex was characterized by determining its hydrodynamic diameter and the molecular weight, molecular weight distribution, radius of gyration of polymer chains and the amount of polymer grafted by quantitative hydrolysis of the grafted chains as reported earlier.22 The grafting density σ was calculated from the amount of the hydrolyzed grafted material measured with a calibrated RI detector, the molecular weight Mn, and the surface area of the latex. The surface areas per particle of the latex particles on which the PDMA and PNIPAM brushes were prepared were (1.33 ( 0.03) × 10-12 and (1.07 ( 0.02) × 10-12 m2, respectively. A 1 µm2 tapping mode AFM image (not shown) obtained in 10 mM NaCl solution shows the latex surface to consist of approximately spherical undulations having an average diameter of 102 ( 10 nm. The measured vertical depth of the features, which depends on the geometry of the square pyramidal shaped tip (35° half angle), was 15 ( 4 nm. Molecular weights were determined by gel permeation chromatography (GPC) on a Waters 2690 separation module fit 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.22 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 a tip radius of 5-40 nm and a (15) Kellermayer, M. S. Z.; Smith, S. B.; Granzier, H. L.; Bustamante, C.; Science 1997, 276, 1112. (16) Oesterhelt, F.; Rief, M.; Gaub, H. E. New J. Phys. 1999, 1, 99121. (17) Ortiz, C.; Hadziioannou, G. Macromolecules 1999, 32, 780. (18) Haupt, B. J.; Ennis, J.; Sevick, E. M. Langmuir 1999, 15, 3886. (19) Bemis, J. E. Akhremitchev, B. B.; Walker, G. Langmuir 1999, 15, 2799. (20) Kienberger, F.; Pastushenko, V. P.; Kada, G.; Gruber, H. J.; Riener, C.; Schindler, H.; Hinterdorfer, P. Single Mol. 2000, 1, 123. (21) Wang, C.; Shi, W.; Zhang, W.; Zhang, X.; Katsumoto, Y.; Ozaki, Y. Nanoletters 2002, 2, 1169. (b) Zhang, W.; Zou, S.; Wang, C.; Zhang, X. J. Phys. Chem. B 2000, 104, 10258. (c) Li, H.; Zhang, W.; Xu, W.; Zhang, X. Macromolecules 2000, 33, 465. (22) Jayachandran, K. N.; Takacs-Cox, A.; Brooks, D. E. Macromolecules 2002, 35, 4247. (23) Kizhakkedathu, J. N.; Norris-Jones, R.; Brooks, D. E. Macromolecules 2004, 37, 734.

Goodman et al. spring constant of 0.06 N/m as quoted by the manufacturer (DI). The rate of tip-sample approach was typically 1.5 µm/s but ranged between 0.30 and 6.0 µm/s. The values are listed in the Supporting Information. The distance of the piezo travel was chosen to be large enough so that the tip was free of adsorbed polymer at the end of the retract cycle and the tip deflection had returned to the baseline. Samples were prepared by drying the latex suspension onto a glass substrate (pre-cleaned 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, our goal was to prepare a closely packed monolayer of latex. Inspection under an optical microscope suggested regions of monolayer coverage were present. Force measurements were performed in these regions. Experiments were performed in a 10 mM NaCl solution, pH adjusted to 7.0 with dilute sodium bicarbonate solution. Force measurements on nongrafted latex confirmed that at this ionic strength, repulsion between the negatively charged tip and latex was minimal, allowing forces due to the grafted layer to be studied independent of electrostatics. Since the structure of PNIPAM is known to be sensitive to salt concentration,24 force measurements as a function of ionic strength were performed and confirmed that in 10 mM NaCl the polymer was in an extended conformation. Samples were allowed to equilibrate for 15 min before performing experiments. The force between the tip and a clean glass substrate was measured before and after probing each sample to check tip quality. A consistent profile was observed with only a snap-tocontact attraction at very small separations and a primary adhesive force upon retract, suggesting that the tip remained free of contaminants. 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 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.25 Separation 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 one. At L ) 0 nm, the tip is believed to be located at approximately the surface of the latex. It is possible for a layer of incompressible polymer to be located between the surface of the latex and the tip, giving rise to an offset distance. However, we believe that in this work, given the use of a sharp tip and grafting densities that have been associated with a small offset distance,12 the contribution from such a layer is negligible.14 From compression profiles obtained on nongrafted shell latex at a high salt concentration (500 mM NaCl), we have found that indentation of the latex by the tip is very small (∼1 nm). Because the forces on the brushes are much greater and occur at larger separations, deformation of the latex is considered negligible. This seems reasonable given that polystyrene is a relatively difficult polymer to compress (compressive modulus ) 3000 MPa).26

Results and Discussion We compare in Figure 1 the force profiles obtained by stretching a PDMA brush (Mn,GPC ) 30 600, σ ) 0.037 nm-2) and a PNIPAM brush (Mn,GPC ) 47 700 and σ ) (24) Park, T. G.; Hoffman, A. S. Macromolecules 1993, 26, 5045. (25) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831. (26) Brandup, J. Immergut, E. H., Eds. In Polymer Handbook; John Wiley & Sons: New York, 1989.

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0.037 nm-2) away from the grafting surface using an AFM tip. The decompression profile of the PDMA brush (Figure 1a) is characteristic of an individual polymer chain physisorbed to the tip.12-21 The attractive force produced by stretching the tethered chain increases monotonically with moderate tension. At sufficiently high extension, the elastic restoring force exceeds the adhesive force between the polymer and the tip and the chain breaks free from the tip at rupture separation LR. The contour length Lc,i of the chain segment between the point of grafting and the point of attachment to the tip is related to LR according to

LR,i ) RLc,i

(1)

where R, the extension ratio, is a measure of the extent to which the chain is stretched before dissociating from the tip. The extension ratio is typically determined by fitting the force curve to a statistical mechanical ideal chain model, such as the freely jointed chain (FJC)16,21a,b or the wormlike chain (WLC)12,13,15 models. The force profile in Figure 1a fits the WLC model in which the force due to stretching a single chain varies with extension L according to

F)

(

)

L 1 1 kT - + q 4(1 - L/L )2 4 Lc,i c,i

(2)

where k is the Boltzmann constant, T is the temperature, q is the persistence length, Lc,i is the contour length of the portion of the chain being extended, and LR/Lc,i is equal to the extension ratio R. From the fitting parameters we obtained values of q ) 0.41 ( 0.04 nm and R ) 0.88. The above model predicts that when L , Lc,i, the force is directly proportional to the extension and the chain behaves as a Hookean spring. The force varies nonlinearly with increased extension and eventually diverges as Lc,i is approached, due to the reduction in entropy associated with fewer possible chain configurations. Many Lc,i values are averaged to obtain a mean contour length Lc. The decompression profile of the PNIPAM brush (Figure 1b) is quite different from that of the PDMA brush (Figure 1a). The attractive force increases with extension and reaches a maximum value Fm at a distance Lm of 90 nm in the example. A decaying attractive force is observed upon further extension until a separation of about 300 nm. This is in contrast to the force profile of a single chain, which exhibits an abrupt jump indicating the rupture of the chain from the tip. The profile in Figure 1b is likely due to the simultaneous stretching of a large number of PNIPAM chains adsorbed to the AFM tip. When the interaction is favorable, compression of the brush can induce adsorption of the polymer to the tip and produce an attractive bridging force.28 The corresponding compression profile of the PNIPAM brush exhibited a long-range attractive force, which was attributed to bridging of the polymer and the AFM tip as previously reported.28 When the number of bridged chains is high, the attractive restoring force can exceed the repulsive osmotic force that is typically observed when a dense polymer brush is compressed. It should be noted that the PDMA and PNIPAM brushes in Figure 1 have the same grafting density (σ ) 0.037 nm-2), indicating (27) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London 1992. (28) Goodman, D.; Kizhakkedathu, J. N.; Brooks, D. E. Langmuir, in press.

Figure 1. (a) Decompression profile of a PDMA brush (Mn,GPC ∼ 30 600, PDIGPC ) 1.35, σ ) 0.037 nm-2) in 10 mM NaCl. Solid line represents fit to wormlike chain (WLC) model with persistence length q ) 0.41 ( 0.04 nm and extension ratio R ) 0.88. (b) Decompression profile of a PNIPAM brush (Mn,GPC ) 47 700, PDIGPC ) 1.33, σ ) 0.037 nm-2) in 10 mM NaCl, pH 7. Dotted line shows nonlinear least-squares fit to eq 6 (see text).

that the profile obtained for the PNIPAM sample is not solely a result of the high density of grafted chains. Many of the single-chain force profiles presented in the literature have been measured on surfaces bearing lower density tethers13,15-16 or physisorbed polymers.18,20-21 Extension of single chains in higher density brushes has been reported as well12,14 and can be fit to the single chain WLC model even when multiple rupture separations in a single force profile are observed. In our case, with the two brush samples having the same σ, the difference in force profiles reflects the higher probability for PNIPAM to adsorb to the silicon nitride tip. A similar force profile with a maximum attractive force that decayed with increased extension when stretching PNIPAM chains physisorbed onto glass with a silicon nitride AFM tip in an aqueous medium was observed by Zhang et al.21b The PNIPAM had a weight average molecular weight Mw of 2.4 × 106 g/mol and a PDI of 1.3. When the polymer was adsorbed onto glass from a dilute solution (6.4 × 10-7 g/mL), individual chains were stretched and the force profile fit an extended FJC model. When the measurement was initially attempted on a surface to which the polymer had adsorbed from a more concentrated solution (2.5 × 10-4 g/mL), the authors could not obtain a clean force-extension trace. The example presented of such a profile had similar features to those

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observed here. Interestingly, single-chain elongation has been performed on other polymers absorbed onto glass from higher concentrations,29 suggesting that the probability of chains adsorbing to a silicon nitride tip during compression is higher for PNIPAM than for these other polymers studied. The maximum attractive force Fm measured was -0.7 ( 0.2 nN for the PNIPAM sample having Mn,GPC ) 47 700 and -1.2 ( 0.4 nN for Mn,GPC ) 94 900. The values are listed in the Supporting Information. The uncertainties represent the standard deviation for the average value and reflect the variation in maximum attractive force. Due to the error associated with the nominal spring constant which can be as large as 100%,30 we do not emphasize the absolute magnitude of the forces but present the data to demonstrate the relative force values measured over the surface. Multiple force measurements obtained at different locations on the sample showed some variation in Fm, as did repeated measurements obtained at the same location; their contributions are included in the above uncertainties. The decompression profile is likely sensitive to the number of chains adsorbed to the tip during compression, which is highly dependent on the local structure of the brush. Self-consistent mean-field calculations have predicted the fraction of bridged chains to be strongly dependent on the distribution of free chain ends.31-33 If the local brush configuration does not return to equilibrium between successive compression-extension cycles, the fraction of bridged chains will vary resulting in nonreproducible force measurements. The bridging fraction is also sensitive to variations in chain configurations over the grafting surface. The overall shape of the force profile, however, was constant and did not depend on the location of the measurement, the number of cycles, or the rate of compression. We have demonstrated the rate dependence of bridging interactions over the range 0.60-12 µm/s upon compression of the PNIPAM brushes in another report.28 Despite the increase in the attractive bridging force observed with decreasing rate of approach, we could not establish a rate dependence for the decompression profiles of the bridged chains over a range of 0.30-6.0 µm/s with respect to the location of the attractive minimum, the maximum separation at which interaction was detected, or the frequency of discrete rupture events. The curves were normalized by dividing by the maximum attractive force Fm in each profile and plotted in Figure 2. The good superposition suggests that the procedure established for determining the distribution of contour lengths from the decompression profile does not depend on the total number of chains adsorbed to the tip. The separation, Lm, corresponding to the point where the attractive force is maximum, is 85 ( 12 nm for the 9 curves presented in Figure 2. This is slightly lower than the average contour length of the brush of 106 nm, obtained by multiplying the number average degree of polymerization (from GPC/MALLS) with the size of a monomer unit (0.25 nm, taken from the projected C-C bond length and assigning two bonds per monomer). The attractive (29) Zhang, W. K.; Xu, Q. B.; Zou, S.; Li, H. B.; Xu, W. Q.; Zhang, X.; Shao, Z. Z.; Kudera, M. Gaub, H. E. Langmuir 2000, 16, 4305. (b) Li, H. B.; Liu, B.; Zhang, X.; Gao, C. X.; Shen, J. C.; Zou, G. T. Langmuir 1999, 15, 2120. (30) Senden, T. J.; Ducker, W. A. Langmuir 1994, 10, 1003. (31) Fleer, G. J.; Cohen-Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: London, 1993. (32) Johner, A.; Joanny, J.-F. J. Chem. Phys. 1992, 96, 6257. (33) Tang, W. H.; Witten, T. A. Macromolecules 1996, 29, 4412.

Goodman et al.

Figure 2. Decompression curves of PNIPAM brushes obtained at various locations on a single sample surface as well as on different samples (same material). Curves are normalized by dividing by the maximum attractive force Fm in each measurement. Values are listed in the Supporting Information. Material and conditions are as in Figure 1b.

force does not return to the baseline until a separation of about 300 nm, a value well beyond the average contour length. The portion of the curve where L g Lm can be interpreted by considering the relationship between the fraction of adsorbed chains and the pulling force. Calculations using the Milner-Witten-Cates self-consistent mean-field model32 have found that the pulling force obeys a secondorder polynomial as a function of the fraction of chains in a polymer brush bridged to an adsorbing plate, N, according to -F ) a(bN - N2), where a and b are constants containing the reduced adsorption energy. While compression has been shown to favor the formation of bridges, pulling reduces the fraction of bridges and eventually leads to rupture. The pulling force at a given extension should be determined by the fraction of chains adsorbed to the tip at that particular length, assuming rupture occurs at an extension roughly equal to the contour length of the chain. The separation, Lm, at which the attractive force is a maximum, should therefore be close to the contour length of the highest fraction of adsorbed chains. If adsorption to the tip following compression occurs independently of chain length, Lm should approximate the average contour length determined by GPC. The observation of a decaying attractive force at separations much larger than the average contour length is attributed to the rupture of chains of above average contour length. This is reasonable due to the polydispersity of the sample (PDI ) 1.33), which indicates that numerous chains with contour length greater than the mean are present. To establish the extent to which above average length chains contribute to the force profile, we calculated the maximum number of chains of molecular weight Mi that can access an AFM tip located at separation L from the surface. Assuming a hemispherical tip of radius R ) 40 nm, the maximum number of accessible chains ηi,max was calculated from the grafting density σ and number fraction of chains Γi determined from GPC/MALLS according to

ηi,max ) Γiσπri2

(3)

where ri is the radius of a disk on the grafted layer defining the maximum area from which chains of contour length Lc,i can access the tip. We assume that fully extended

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Figure 3. Depiction of maximum accessible distance r for chain of contour length Lc and tip of radius R.

Figure 5. Decompression profile of PNIPAM brush with average contour length Lc ) 106 nm, Mn ) 47 700, PDI ) 1.33 (GPC results) showing direct relationship with cumulative weight fraction of grafted chains.

Figure 4. Maximum (left-hand ordinate) and minimum (righthand ordinate) number of chains η with molecular weight that have access to the AFM tip a separation L from the grafting surface. The calculation is based on the number fraction of chains in a PNIPAM brush of Mn,GPC ) 47 700 and σ ) 0.037 nm-2. The accessible number of chains is estimated from eq 3, see text for discussion of assumptions.

chains located along the circumference of the disk can access the tip from a normal distance equal to Lc,i, so that ri2 ) [(R + Lc,i)2 - (R + L)2] (Figure 3). The number of accessible chains is presented in Figure 4 for a brush of σ ) 0.037 nm-2 using the Γi data obtained by GPC for the PNIPAM brush of Mn,GPC ) 47 700. The calculation is performed for a tip located at the grafting surface, (L ) 0), a distance above the surface roughly equal to Lc, (L ) 100 nm), and a distance greater than Lc (L) 200 nm). The number of accessible chains having Mi > Mn is significant for all three tip positions. The calculation likely overestimates the number of accessible chains by assuming that all polymers located at a lateral distance between zero and ri from the center of the tip have equal access when chains located directly beneath the tip are more likely to adsorb. A calculation of the minimum number of accessible chains, ηi,min, where chains of any Mi located within a disk of radius r ) 40 nm were assumed accessible, predicts that a much smaller but still finite number of chains with Mi > Mn had access to the tip (Figure 4). We believe that it is therefore reasonable to attribute the attractive force detected at large separations to the rupture of long chains from the tip. The cumulative weight distribution, ξ, of chains determined from GPC/MALLS is found to be approximately parallel to the decaying region of the decompression profile, as demonstrated by the superposition of the two curves in the double Y-axis plot in Figure 5. The contour length along the x-axis is assumed to equal the separation, L, in the AFM trace, and is related to the individual molecular weights Mi determined by GPC according to

Lc,i Mi ) Mmon l

(4)

where Mmon is the molar mass of the monomer and l is the size of the monomer, 0.25 nm. We have defined ξ, a function of individual molecular weight Mi, as

ξ(Mi) ) ΣPi(Mi) i

(5)

where Pi is the weight fraction of chains having molecular weight Mi and i is counted down from the highest molecular weight chain. This is in contrast to the conventional representation of cumulative weight fraction, ξh, where ξh ) 1 corresponds to the highest molecular weight chain. The two notations are related by ξ ) 1 - ξh. The representation used here enables the fraction of chains adsorbed to the tip to be directly related to the cumulative fraction of chains present in the brush, since the smallest number of chains adsorbed on the tip occurs at high extension. The total number of chains adsorbed to the tip is reduced with increasing extension as the chains become fully stretched and sequentially detach from the tip. We have found the decaying attractive force profile to coincide with the cumulative weight fraction, as opposed to the cumulative number fraction of grafted chains, for high density brushes. We believe this observation indicates a slightly biased sampling of higher molecular weight chains by the AFM tip rather than a fundamental relationship between the pulling force and the cumulative weight fraction. In our earlier report14 where the molecular weights of PDMA brushes were measured by the more conventional single chain pulling technique, the Mn measured by AFM and GPC agreed but the Mw measured by AFM was overestimated. This was attributed to the ability of very long chains to adsorb onto the tip from locations outside the direct region of compression. The relationship between the force profile and the cumulative weight distribution is purely empirical and is not supported by theoretical calculations, however it enables the distribution of weight fractions to be estimated and the PDI to be determined. The weight fraction distribution, Pi(Lc,i), is obtained from the first derivative of the cumulative distribution, or in this case, the decompression profile. Before taking the derivative, we first fit the force profile for L g Lm to a monotonically increasing exponential function of separation L using a

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from the fit parameters according to34

Figure 6. Comparison of distribution of weight fraction of PNIPAM brush measured by GPC and calculated from analysis of decompression profile (see text). Very low weight fractions obtained at the low molecular weight end have been removed from the GPC distributions. Dashed and solid lines correspond to the log-normal distribution fits to the AFM data (S1 and S2 respectively). Brush properties obtained by GPC are for S1: (Mn ) 47 700, PDI ) 1.33, σ ) 0.037 nm-2) and for S2: (Mn ) 94 900, PDI ) 1.35, σ ) 0.067 nm-2).

least squares procedure:

F ) c(1 - e-k(L-d))

(6)

where c, d, and k are numerical fitting parameters. The use of eq 6 to fit the data was based on the empirically observed good fits, as demonstrated in Figure 1b. By obtaining Pi(Lc,i) from the derivative of the fitted curve rather than the actual force data, little weight was placed on variations between force profiles caused by discrete rupture events, such as the local maximum attractive force occurring at ∼225 nm in Figure 1b and at ∼275 nm in Figure 5. These discrete rupture events reflect the inhomogeneity in the lengths of the grafted chains attached to the tip. As can be seen from the superimposed curves in Figure 2, the data in the region between below 225 nm is relatively smooth, depicting a continuous rupturing of chains from the tip over the range of contour lengths. Discrete rupture events are observed predominantly in the high separation region, between 225 and 325 nm, where the fraction of chains is low and the probability of stretching a single chain is higher. The χ2 value for these fits ranged between 0.001 and 0.003, depending on the contribution of discrete rupture events. The weight fractions, Pi,AFM(Lc,i), were obtained by taking the derivative of the fitted force curve for separations L g Lm. Only the weight fractions of chains having Lc,i g Lm were obtained by this method. Weight fraction distributions calculated for the two PNIPAM samples are presented in Figure 6, along with the distributions measured by GPC. The individual weight fractions measured from the GPC data are such that ΣiPi,GPC ) 1. The weight fractions calculated from the force profile are normalized so that the peak value of Pi,AFM is equal to that of Pi,GPC. The weight fractions distributed as a function of molecular weight fit a log-normal distribution:

h )2/2s2] P ) Po + C(2πs2)-1/2 exp[-(ln M - ln M

(7)

where M h is the peak molecular weight on the log scale, s is the standard deviation, and Po and C are fitting constants. Values for Mn, Mw, and PDI can be obtained

Mn ) M h exp(- s2/2)

(8)

h exp(s2/2) Mw ) M

(9)

PDI ) exp(s2)

(10)

When both ln M h and s were free parameters in the fit, the value obtained for Mn,AFM by eq 8 was typically lower than Mn,GPC. Instead, we calculated Mn,AFM according to eq 4 by taking the separation Lm as an estimate for the average contour length. By setting the value of Mn,AFM and combining eqs 7 and 8, the distribution could be fit with only a single parameter, s. The χ2 values were on the order of 10-7. The PDIAFM was calculated from each of the fitted force curves according to eq 10. Values obtained for Mn,AFM and PDIAFM from individual force curves are tabulated in the Supporting Information. The average PDIAFM was 1.3 ( 0.1 for both PNIPAM samples, which agreed well with the PDIGPC values of 1.33 and 1.35. The Mn,AFM values for the two samples were (3.8 ( 0.5) × 104 and (9.4 ( 1.4) × 104, which compare to the corresponding values obtained from GPC/MALLS of 4.77 × 104 and 9.49 × 104. The Mw,AFM values for the two samples were (4.9 ( 0.7) × 104 and (12 ( 1.2) × 104, and the respective Mw,GPC values were 6.34 × 104 and 12.8 × 104. The values for the two samples were obtained by averaging over 9 and 12 curves, respectively. The values of Mn,AFM presented are the averages of individual measurements obtained on a given sample and the uncertainties are represented by the standard deviations. In addition, we consider here the contributions to the uncertainty in Mn,AFM based on a single estimate of Lm. Since the separation L is calculated from the difference between the change in piezo position and the change in deflection signal, the uncertainty in L depends on that of the optical lever sensitivity (( 5%). There is also an inherent error associated with defining L ) 0 nm, the constant compliance region on the polymer brush surface, due to the possible presence of an incompressible polymer layer between the tip and the latex. In our calculations we have assumed the thickness of this layer to be negligible14 based on the use of sharp tips and grafting densities which have been associated with small offset distances.12 To associate an error with the definition of zero separation however, we use a nominal offset distance of 5 nm and calculate an upper bound value for Lm where zero separation is L ) 0 ( 5 nm. The estimated uncertainties in the Lm values in this work were between 2% and 7%, providing values for the relative uncertainties in the individual Mn,AFM estimates. The individual values are given in the Supporting Information. A number of factors contribute to the uncertainty in the PDIAFM value obtained from a single force profile. The ability of the exponential function in eq 6 to fit the decompression profile depends on the “jaggedness” of the force data resulting from discrete rupture events. The uncertainty in the PDIAFM is also affected by the quality with which the Gaussian distribution fits the weight fraction data. We present the χ2 values above as a measure of evaluating the fits of the data but recognize that the good fits are misleading since the difference between the data and the fitted curve clearly does not represent the uncertainty in the PDI,AFM value. We believe the uncertainty associated with the estimation of PDI,AFM is best (34) Teraoka, I. Polymer Solutions: an Introduction to Physical Properties; John Wiley & Sons: New York, 2002.

Estimation of Adsorbing Polymer Brushes

indicated by the standard deviations obtained from multiple measurements. It should be noted that the lowest molecular weight data used in the fit of eq 7 was not Mn,AFM, but a data point slightly larger than Mn,AFM. This is due to the uncertainty associated with relating the decompression profile with the cumulative weight distribution in this region and with the availability of data on only the right side of the lognormal distribution. Typically, the peak in the weight fraction distribution corresponds to an inflection point in the cumulative distribution, as the two curves are related by the first derivative. Because data in the low molecular weight region is not obtained by this technique, the values of Pi,AFM do not plateau as M approaches Mn,AFM (for M > Mn,AFM), but steadily increase. The Pi,AFM in this region were therefore not included in the fits. The data presented in Figure 6 was used to obtain the fitted distributions. Although we have demonstrated that the probability of detecting chains larger than the average contour length is significant (Figure 4), there is another possible interpretation of the nonzero force at large separations. Spherical brushes that are adsorbed onto a flat substrate for analysis have a grafted layer between the particle and substrate as well as between the tip and particle. Compression of both layers could give rise to an equilibrium thickness twice as large as expected, while the stretching of both layers could result in rupture separations twice as long as the expected contour length. From our work with spherical PDMA brushes,14 however, we found that the average contour lengths measured by AFM agree with those estimated from the molecular weights obtained by GPC. We therefore concluded that contributions from the grafted layer adjacent to the glass substrate were minimal. The maximum attractive force Fm measured was -0.7 ( 0.2 nN and -1.2 ( 0.4 nN for the samples having Mn,GPC ) 47 700 and 94 900, respectively. Although the magnitude of the force varied between measurements, the values were always well below that required to break a typical C-C bond (∼5-6 nN).17 The quoted bond strength was calculated using the unperturbed potential well at zero loading rate, by solving for the derivative of the Morse potential V(x) at the separation xmax where the attractive force is a maximum. The above suggests that the measurements are associated with the dissociation of a collection of chains physisorbed to the tip. Nevertheless, we repeatedly checked the quality of the tip by obtaining force curves on clean glass after exposing the tip to the PNIPAM brush. As described in the Experimental Section, the force curves on glass did not have any distinguishing features that would suggest contamination of the tip for the results presented. It should be noted that contamination of the tip by the PNIPAM brushes occurred more frequently than when nonadsorbing polymer brushes such as PDMA were probed using the same sample preparation technique, presumably due to the larger attractive forces associated with PNIPAM. Two initial force measurements resulted in tip contamination but a more thorough rinsing procedure and more rigorously clean sample preparation allowed experiments to be successfully repeated without contamination. While reasonable estimates of the Mn and the PDI of the two samples were obtained by the method described here, it involves a number of approximations that contribute to uncertainties in the results. Equating the separation at maximum force with the average contour

Langmuir, Vol. 20, No. 8, 2004 3303

length gave Mn,AFM values that were underestimated as compared with GPC values. The inability of the method to account for chains with molecular weight lower than the average allows for only one side of the distribution of weight fractions to be fit. The fit parameters are therefore based on a symmetrical distribution of chain lengths on a log scale. Given that the weight fractions obtained by GPC are slightly positively skewed even after the logarithmic transformation (Figure 6), this assumption is inaccurate. The direct relationship between the decompression profile and the cumulative weight fraction, however, suggests that some of these sources of error cancel each other out. While high-molecular-weight chains that have a low probability of occurrence may not always be probed due to the small contact area between the tip and sample, the same chains may access the tip from locations beyond the direct region of compression and be over-represented. Conclusion We have investigated the stretching behavior of a dense PNIPAM brush that strongly adsorbs to a silicon nitride AFM tip. The shape of the decompression profile was closely related to the cumulative weight fraction of chains of each length determined by GPC. The separation at maximum attractive force, corresponding to the maximum fraction of adsorbed chains, approximated the average contour length. The weight fraction as a function of molecular weight was obtained from the derivative of the force profile and fit a log-normal distribution, enabling the PDI to be estimated. Characterization of the molecular weight and polydispersity of polymer brushes remains a challenge, particularly when the grafting surface is flat and the amount of grafted material is low. Even when the amount of material is sufficient for analysis by conventional methods such as GPC, cleaving the polymer from the grafting surface is not always possible. The method described here allows the molecular weight and polydispersity of polymer brushes to be estimated provided the adsorption of polymer to AFM tip is favorable. Although we investigated only the PNIPAM/silicon nitride system, the technique should be applicable to different polymers through modification of the tip surface chemistry to obtain the desired polymertip interaction. This technique is founded on a purely empirical relationship between the decompression profile and the cumulative weight fraction. The correlation between number average contour length and the separation at maximum attractive force is also empirical. On the basis of these observations, we have developed a practical application for the decompression profile of an adsorbing brush whereby the molecular weight and polydispersity of the polymer, under the appropriate conditions, can be determined. Acknowledgment. We thank the Canadian Institutes of Health Research, Canadian Blood Services, Natural Sciences and Engineering Research Council of Canada and Canada Foundation for Innovation for financial support. Supporting Information Available: Table of characteristics of PNIPAM brush. This information is available free of charge via the Internet at http://pubs.acs.org. LA036164L