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Direct Comparison of Atomic Force Microscopic and Total Internal Reflection Microscopic Measurements in the Presence of Nonadsorbing Polyelectrolytes Simon Biggs,*,† Dennis C. Prieve,‡ and Raymond R. Dagastine§ Institute of Particle Science and Engineering, School of Process, Environmental and Materials Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom, Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, and Department of Chemical and Biomolecular Engineering, University of Melbourne, Victoria 3010, Australia Received January 6, 2005. In Final Form: March 22, 2005 We have investigated the structural and depletion forces between silica glass surfaces in aqueous, salt-free solutions of sodium poly(styrene sulfonate). The interaction forces were investigated by two techniques: total internal reflectance microscopy (TIRM) and colloid probe atomic force microscopy (AFM). The TIRM technique measures the potential energy of interaction directly, while the AFM is a force balance. Comparison between the data sets was used to independently calibrate the AFM data since the separation distances cannot be unequivocally determined by this technique. Oscillatory structural forces are excellent for this work since they give multiple reference points against which to analyze. Comparison of the data from the two techniques highlighted significant uncertainties in the AFM data. At low polymer concentrations, a significant uncertainty in the apparent zero separation distance was seen as a result of the AFM cantilever reaching an apparent constant compliance region prior to any real contact between the surfaces. Further complications arising from the number and position of the measured minima were also seen in the dilute polymer concentration regime as a result of hydrodynamic drainage between the approaching surfaces in the AFM perturbing the delicate structural components in the fluid.
Introduction Depletion flocculation has been observed in many colloidal systems. The fundamental requirements for depletion are the presence of two particulate phases that are mutually repulsive (i.e., noninteracting), and a significant size difference between those particles.1 The magnitude of the attraction is related to the size ratio of the particles and the concentration of each species. Asakura and Oosawa were the first to describe this force and its origins in 1954.2 Essentially, the force arises from the increased entropy of the smaller particles after flocculation as a result of the extra volume available to them. In the simplest case of two large particles approaching one another, it is clear that when the distance of surface-surface separation is less than the characteristic diameter of the smaller particles, then the smaller particles must be excluded from this gap. Under these conditions, there will be a solvent-rich region between the larger particles that is surrounded by a bulk fluid of the smaller particles. This particle density gradient leads to an osmotic pressure that must then push the larger particles together.3 The release of this excluded volume after flocculation increases the entropy of the smaller particles. Simple depletion theories, of this type, assume a constant bulk density for the smaller particle system. In these cases, an attractive force is predicted only when the distance of large particle separation is less than the size of the smaller colloids. The main features of this * Corresponding author. † University of Leeds. ‡ Carnegie-Mellon University. § University of Melbourne. (1) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1983. (2) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (3) Bechinger, C.; Rudhardt, D.; Leiderer, P.; Roth, R.; Dietrich, S. Phys. Rev. Lett. 1999, 83, 3960.
attractive force have now been described by many authors and several excellent reviews are available.4,5 The depletion attraction has also been confirmed experimentally for systems of particles dispersed in neutral polymer solutions (good solvent conditions) both directly6,7 and indirectly.8 More recently, a number of authors have reported experimental data that illustrate the role that structural correlations between the smaller colloids can have on the interactions between the larger particles.3,9-16 As the concentration of any colloid increases, it will exhibit liquid-structural correlations that can have a profound effect on interparticle interactions. Typically, these structural effects give rise to an oscillatory distribution of the smaller particles between two surfaces. As the larger particles approach, this distribution of the smaller particles will then give rise to an oscillatory interaction force. For simple particle depletants, the distribution function will be related to the characteristic size of the particles; in aqueous systems this will include a contribution from (4) Jenkins, P.; Snowden, M. Adv. Colloid Interface Sci. 1996, 68, 57. (5) Seebergh, J. E.; Berg, J. C. Langmuir 1994, 10, 454. (6) Milling, A.; Biggs, S. J. Colloid Interface Sci. 1995, 170, 604. (7) Kuhl, T.; Guo, Y. Q.; Alderfer, J. L.; Berman, A. D.; Leckband, D.; Israelachvili, J.; Hui, S. W. Langmuir 1996, 12, 3003. (8) Vincent, B.; Edwards, J.; Emmett, S.; Croot, R. Colloids Surf. 1988, 31, 267. (9) Dinsmore, A. D.; Yodh, A. G.; Pine, D. J. Nature 1996, 383, 239. (10) Ohshima, Y. N.; Sakagami, H.; Okumoto, K.; Tokoyoda, A.; Igarashi, T.; Shintaku, K. B.; Toride, S.; Sekino, H.; Kabuto, K.; Nishio, I. Phys. Rev. Lett. 1997, 78, 3963. (11) Crocker, J. C.; Matteo, J. A.; Dinsmore, A. D.; Yodh, A. G. Phys. Rev. Lett. 1999, 82, 4352. (12) Richetti, P.; Kekicheff, P. Phys. Rev. Lett. 1992, 68, 1951. (13) McNamee, C. E.; Tsujii, Y.; Matsumoto, M. Langmuir 2004, 20, 1791. (14) Piech, M.; Walz, J. Y. J. Colloid Interface Sci. 2000, 232, 86. (15) Piech, M.; Walz, J. Y. J. Phys. Chem. B 2004, 108, 9177. (16) Biggs, S.; Dagastine, R. R.; Prieve, D. C. J. Phys. Chem. B 2002, 106, 11557.
10.1021/la050041e CCC: $30.25 © 2005 American Chemical Society Published on Web 05/06/2005
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the electrical double layer around the particles as well as their physical volume. For polymer systems, the presence of structural oscillations is further complicated by the ability of polymer chains to interpenetrate at concentrations above the chain overlap concentration, c*. For polymer systems below c*, as well as for particle and micelle systems, the periodicity of the oscillations seen has been shown to correlate directly with the radial distribution of the depletant species in the bulk.15,16 Across a wide concentration range the periodicity scales as c-1/3, as expected for a close-packed system of spheres. Above c*, for polymer systems, the periodicity correlation changes to c-1/2, as predicted from scaling theory.17 Under these conditions, the characteristic length of interest is the mesh size, ξ. In many practical cases where depletion interactions are of interest, the depletant molecules are polyelectrolytes. This adds significant further complications to the analysis and understanding of the interaction.18 In the case of polyelectrolytes, the osmotic pressure is dominated by the counterion concentration and so a significant pressure, and hence depletion force, can be developed at relatively low concentrations.19 However, the residual double layer interaction between the particle surfaces inside the depletion layer will act against this interaction and a full understanding will require knowledge of the interstitial ionic structure. Direct measurement of depletion and structural forces between surfaces in the presence of polyelectrolytes have been reported in both dilute and semidilute concentration regimes. The data reported indicate good correlation with predictions from scaling theory in both dilute and semidilute regimes.15,18,20-22 The magnitude of both the depletion and structural terms in the force profile between the larger particles will depend on factors such as the particle type(s), their concentrations, and the relative sizes. Such structural/ depletion interactions have now been demonstrated for small colloidal particles such as Ludox silica,15 surfactant micelles,12,13 polymer-surfactant complexes,23-26 and polyelectrolytes.16,18,21,27 In general, both the depletion and structural forces are weak and direct measurements have only recently become possible with the improved sensitivity afforded by techniques such as atomic force microscopy (AFM)6,15,18,21,28-30 and total internal reflectance microscopy (TIRM).3,11,16,31-33 Related phenomena have also been (17) Degennes, P. G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. Phys. 1976, 37, 1461. (18) Milling, A. J. J. Phys. Chem. 1996, 100, 8986. (19) Dautzenberg, H.; Jaeger, W.; Ko¨tz, J.; Philipp, B.; Seidel, C.; Stscherbina, D. Polyelectrolytes; Hanser: Munich, Germany, 1994. (20) Biggs, S.; Walker, L. M.; Kline, S. R. Nano Lett. 2002, 2, 1409. (21) Milling, A. J.; Kendall, K. Langmuir 2000, 16, 5106. (22) Qu, D.; Baigl, D.; Williams, C. E.; Mohwald, H.; Fery, A. Macromolecules 2003, 36, 6878. (23) Kolaric, B.; Jaeger, W.; von Klitzing, R. J. Phys. Chem. B 2000, 104, 5096. (24) Kolaric, B.; Jaeger, W.; Hedicke, G.; von Klitzing, R. J. Phys. Chem. B 2003, 107, 8152. (25) Klitzing, R. V.; Kolaric, B. Abstr. Pap. Am. Chem. Soc. 2000, 219, 187. (26) Theodoly, O.; Tan, J. S.; Ober, R.; Williams, C. E.; Bergeron, V. Langmuir 2001, 17, 4910. (27) Asnacios, A.; Espert, A.; Colin, A.; Langevin, D. Phys. Rev. Lett. 1997, 78, 4974. (28) Milling, A. J.; Vincent, B. J. Chem. Soc., Faraday Trans. 1997, 93, 3179. (29) Biggs, S.; Burns, J. L.; Yan, Y. D.; Jameson, G. J.; Jenkins, P. Langmuir 2000, 16, 9242. (30) Burns, J. L.; Yan, Y. D.; Jameson, G. J.; Biggs, S. Colloid Surf. A: Physicochem. Eng. Asp. 2000, 162, 265. (31) Odiachi, P. C.; Prieve, D. C. Ind. Eng. Chem. Res. 2002, 41, 478. (32) Sharma, A.; Tan, S. N.; Walz, J. Y. J. Colloid Interface Sci. 1997, 191, 236. (33) Sharma, A.; Walz, J. Y. J. Chem. Soc., Faraday Trans. 1996, 92, 4997.
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reported from investigations of the stepwise reduction of thin-film lamellae in the presence of surfactants and/or polymers.23-25,27,34 In this work, we report investigations performed by both TIRM and colloid probe AFM to investigate structural and depletion interactions. The combination of these two techniques allows interaction measurement across a large concentration regime. The sensitivity of TIRM means that it is well suited to very weak interactions. However, strongly attractive interactions, of the type seen at higher polymer concentrations in structural interaction measurements, are often problematic. Conversely, AFM does not have good sensitivity for extremely weak interactions but can be used to measure both strong attractions and repulsions between surfaces. The regimes of validity for these two techniques are not readily apparent from the available literature. However, it is worth noting that the limited AFM data available that were collected in the dilute polymer regime did not show the expected correlation of c-1/3 that has been seen with TIRM. These two techniques are highly complementary, therefore, and measurements using both might offer insight into the accuracy of earlier data from both approaches. The purpose of this work is therefore 2-fold: to confirm the validity of measurements by AFM, where the surfacesurface separation distance can only be determined indirectly and is therefore subject to uncertainty, and to probe a wide concentration range for depletion/structural interactions between surfaces in the presence of polyelectrolytes. Experimental Section Materials. All water used was of Millipore Milli-Q quality. The spheres used were 5 µm borosilicate glass particles (Duke Scientific). Spheres were used as supplied with no further treatment. Silica microscope slides (Escoproducts, Oak Ridge, NJ) were cleaned by soaking for 30 min in chromic acid, 30 min in concentrated hydrochloric acid, and 1 h in 100 mM sodium hydroxide solution. The polyelectrolyte, sodium poly(styrene sulfonate), was obtained from Polysciences Inc. (Warrington, PA). The sample used here had a charge density of 100% and a nominal weight-average molecular weight of 35 300 and a polydispersity index, Mw/Mn, of 1.01. Methods: Energy Measurements by Total Internal Reflection Microscopy. Interaction energy profiles between a single silica sphere and a microscope slide were determined by TIRM. This technique has been described in detail by a number of authors35-37 and only relevant details will be given here. An evanescent wave is generated at the solid/liquid interface by a total internally reflected laser light source within the glass slide. Any particle located close to the surface of the glass slide will scatter the incident evanescent wave. The scattering intensity is related to the distance between the sphere and the surface of the glass slide via
I(h) ) I0 exp(-βh)
(1)
where β-1 is the decay length of the evanescent wave and I0 is the “stuck” particle intensity, where h ) 0. Measuring the scattering intensity over time provides a histogram of the separation distances. The probability of finding a particle at any given separation distance, F(h), is related to the total potential energy at that point through a Boltzmann relationship:
(
p(h) ) A exp -
)
φ(h) kT
(2)
(34) Bergeron, V.; Langevin, D.; Asnacios, A. Langmuir 1996, 12, 1550. (35) Prieve, D. C. Adv. Colloid Interface Sci. 1999, 82, 93.
Direct Comparison of AFM and TIRM Measurements The potential energy profile is obtained by inverting this distribution. Full details are presented elsewhere.35 In a given experiment, the TIRM cell was initially filled with water at pH 10. Glass spheres were then introduced into the flow cell. The evanescent wave with a decay length of 113.7 nm was produced by use of a 35 mW HeNe laser operating at 632 nm. A glass sphere of average brightness and size was selected and held in place with optical tweezers while the rest of the particles were flushed from the cell with pH 10 water. The optical tweezers are generated by use of an argon ion laser (150 mW, λ ) 488 nm) with the beam delivered from both above and below the sphere. The radiation pressure from the optical tweezers is also used to bias the height distribution of the particle of the slide, making the apparent weight decrease or increase depending on direction. Once the excess particles have been pushed through the flow cell, the interaction energy of the glass sphere in pH 10 was recorded as a function of radiation pressure from below (decreasing the apparent weight of the sphere). Intensity measurements (200 000-300 000) were collected at 5 ms spacing between measurements through single photon counting by use of a photomultiplier tube (PMT). The apparent weight of the particle is then determined as a function of radiation pressure and extrapolated to zero radiation pressure according to the method of Walz and Prieve.38 The actual weight of the particle is then used to determine the diameter of the sphere. The solution of interest was introduced in the flow cell while the particle was held in place with the optical tweezers. A minimum of 7-10 mL of fluid was used to change solutions (changing the volume of the fluid cell at least 5 times). Radiation pressure was then employed from either above, below, or both to ensure that the sphere sampled the largest number of elevations above the plate. The data collection procedure was the same as above. The solution can be changed in this manner for increasing concentrations of polymer. The last portion of the experiment is the distance calibration of the height data where the polymer solution is replaced with pH 10 water again and then a 10-15 mM sodium chloride solution is introduced to screen the repulsion levitating the particle. The particle will then come into contact with the plate (becoming “stuck”) and the scattering intensity is recorded for 100 000 points. A region where there are no particles in the microscope field is then selected to determine the background scattering intensity. The difference between the average of the “stuck” particle intensity and the average of the background intensity is taken to be I0 in eq 5. Force Measurements by AFM. The use of an AFM for direct force measurements on colloidal systems is an established technique.39-41 Two types of probe were used here for these measurements. One set of data were collected with commercially available silicon cantilevers with an integral tip. These probes are assumed to have an inherent oxide layer. The second set of data was collected with probes prepared by attaching one of the borosilicate glass spheres (used in the TIRM work) to a commercially available silicon nitride AFM cantilever (Veeco Instruments) by use of a small amount of adhesive. Forcedistance data were collected between either of these probes and a flat silica microscope slide. In the case of the colloid probe, the use of a similar particle, a similar substrate, and the same solutions was designed to allow direct comparison to the TIRM work. Force-distance data were collected on a Nanoscope III AFM (Veeco Instruments). In force mode, the standard X-Y raster scan is suspended and the flat substrate is moved toward and away from the sensing cantilever probe in the Z-direction. This is achieved by the application of a triangular voltage ramp to the piezoelectric scanner. In a typical AFM experiment, the probe of choice was mounted into the AFM liquid cell, which was then mounted into the AFM (36) Bevan, M. A.; Prieve, D. C. Langmuir 1999, 15, 7925. (37) Odiachi, P. C.; Prieve, D. C. Colloid Surf. A: Physicochem. Eng. Asp. 1999, 146, 315. (38) Walz, J. Y.; Prieve, D. C. Langmuir 1992, 8, 3073. (39) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831. (40) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (41) Butt, H. J. Biophys. J. 1991, 60, 1438.
Langmuir, Vol. 21, No. 12, 2005 5423 head unit before being affixed to the scanner. All of these operations were performed in a clean environment. After the two surfaces were manually brought into close proximity, the solution of interest was introduced into the AFM liquid cell. Force data were then collected after a short equilibration time (typically 15-30 min). Whenever the solution was changed, at least 30 min was allowed to elapse before measurements were taken to ensure equilibrium was attained. Conversion of the raw data followed the standard protocol described previously by many authors.39 Quantitative force measurements require an accurate knowledge of the spring constants used. The spring constants were determined according to the method of Hutter and Bechhoefer.42 The values ranged from 0.009 to 0.015 N/m for the cantilevers used. All measurements were performed with scan sizes of (250 nm and at the slowest practical scan rate (usually 0.05-0.5 Hz) where significant drift in the data was not seen.
Results and Discussion All the experiments reported here involved measurements of the interaction between two (silica) glass surfaces in various solutions of a nonadsorbing polyelectrolyte, sodium poly(styrene sulfonate) [NaPSS]. To ensure that there was no possibility of adsorption occurring, the experiments were performed at a solution pH of 10, where both the surfaces and the polymer are highly negatively charged.30 Previously, we have shown that, from available scaling theory, for the polymer used here a value for the rigid rod length (L) of 9.8 nm can be used.16 Also, the chain overlap concentration, c*, was determined to be approximately 6000 ppm. This value is in good agreement with data from neutron scattering experiments on similar polymers.43 Data of the energy of interaction versus surface separation collected by TIRM are shown in Figure 1. The measured potential energy profiles between a single 5 µm glass sphere and a glass plate in the presence of poly(styrene sulfonate) [Mw ∼ 35 000] are shown at a range of concentrations between 50 and 1000 ppm. At polymer concentrations of 100 ppm and below, the data presented accurately define both the positions and the relative energies of all minima in a given data set. For concentrations of 200 ppm and above, the presented data were collected in a set of consecutive experiments since when the particle entered one of the inner minima, at small separation distances, the height of the barrier for escape was too large. As discussed in our earlier publication, the consequences of this are that the positions of the energy wells are accurate but the relative energies are unknown.16 A full discussion of these data and their comparison to available scaling theory has been previously reported. Only those details of relevance to the AFM measurements will be discussed here, where appropriate in comparison between the data sets. In Figure 2, equivalent force-distance data collected by AFM are presented. These data were collected with the same polymer solutions and surfaces as used in the TIRM, excepting the AFM tip of course. The data shown were collected at a range of concentrations between 100 and 10 000 ppm. No reasonable data were successfully collected in this work below a concentration of 100 ppm. Initial examination of the data in Figures 1 and 2 indicate broadly similar behavior. As the polymer concentration is increased, the number of observable oscillations increases. Clearly, the resolution available from TIRM is greater than that of AFM. Furthermore, the data collected by use of the tip when compared to the probe, for the AFM work, show a higher level of background noise. (42) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 1868. (43) Kassapidou, K.; Jesse, W.; Kuil, M. E.; Lapp, A.; Egelhaaf, S.; vanderMaarel, J. R. C. Macromolecules 1997, 30, 2671.
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Figure 1. TIRM data of potential energy (kT) as a function of surface-surface separation distance for a 5 µm silica sphere interacting with a flat silica substrate in salt-free NaPSS solutions. The concentrations of NaPSS used are indicated in the figure. All data were collected at pH 10 and at 25 °C.
This is expected since the relative size of the interaction force is directly related to the probe size and so, for the tip, the contribution of the thermal noise to the total signal will be greater. The data for the tip have been normalized by use of a nominal radius of 100 nm; the F/R values are therefore unlikely to be accurate but are consistent in a relative sense. It is also worth noting that the scan rate was maintained as low as possible during these experiments. Analysis of data collected at a range of scan rates,
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for a fixed scan size, suggested that there was a progressive degradation in the quality of the oscillatory force data as a function of increasing scan rate. This was attributed to the hydrodynamic drainage between the macroscopic surfaces during approach that is most likely sufficient to disrupt the relatively weak polyelectrolyte structure in solution. Interestingly, the inner minimum in the force data was always seen as relatively insensitive to the approach rate of the surfaces. Initial reports concerning the measurement of structural forces between surfaces in polyelectrolyte solutions by use of an AFM focused on the semidilute regime.18 Milling and Kendall21 reported a limited amount of data in the dilute regime, but no correlation with the expected scaling theory results was observed. In contrast, in our later TIRM study a complete investigation of the dilute regime was undertaken and the data were shown to correlate well with the expected theory. The correlation length was directly related to the length of the rigid rod plus 2-3 Debye lengths, and each rod sweeps out a spherical volume in the bulk solution. In addition, detailed analysis of the TIRM data (Figure 1) indicated the presence of an inner layer of rods that are aligned parallel to the surface, suggesting that the presence of a surface disrupts the bulk arrangement of the polymers. Such correlations are not readily apparent from the published AFM data. The choice of concentration ranges for the TIRM and AFM allow investigation of the data from the dilute through to the semidilute regime. Furthermore, there is a significant regime of overlap between the two data sets that facilitates direct comparison between the two measurement approaches. In Figures 3-5, data collected by colloid probe AFM and TIRM at 100, 500, and 1000 ppm are shown, respectively. Direct comparisons between force and energy data for the same interaction, while not simple, should be quite feasible. A schematic of an idealized interaction between two surfaces across a structured fluid is shown in Figure 6. Clearly, the simplest point to compare in data such as these should be the location of the innermost energy minimum where the force has a zero value. If the distance data are correct in both the force and energy,
Figure 2. AFM data of reduced force (F/R) as a function of apparent surface separation distance for (a) a 5 µm silica sphere and (b) an integral silica tip, interacting with a flat silica substrate in salt-free NaPSS solutions. The concentrations of NaPSS used are indicated in the figure. All data were collected at pH 10 and at 25 °C.
Direct Comparison of AFM and TIRM Measurements
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Figure 3. Comparison of force and energy data collected at 100 ppm polymer concentration by AFM and TIRM, respectively. Points A and B represent the position of zero force (AFM) and the inner energy minimum (TIRM), respectively. If the separation distance for each data set is correct, these points should be aligned (in distance). Figure 6. Schematic representation of the ordering of polymer molecules (represented as spheres) between two surfaces and the resultant force and energy profiles as these surfaces are brought together. The region close to each surface, represented by a dashed line, corresponds to the electrical double layer. The energy is a minimum when a full layer of molecules (or multiple thereof) can exist between the surfaces. Table 1. Position of the Measured Zero Force by AFM and the Innermost Potential Energy Minimum by TIRM
Figure 4. Comparison of force and energy data collected at 500 ppm polymer concentration by AFM and TIRM, respectively. Points A and B represent the position of zero force (AFM) and the inner energy minimum (TIRM), respectively. If the separation distance for each data set is correct, these points should be aligned (in distance).
Figure 5. Comparison of force and energy data collected at 1000 ppm polymer concentration by AFM and TIRM, respectively. The arrow indicates the near-perfect alignment of the zero force (AFM) and inner energy minimum (TIRM) positions.
then this point will provide a fixed reference. Comparison of the data in Figures 3-5 shows that only the data sets at 1000 ppm are aligned in this way. Before these data are analyzed further, in particular the number and location of all the minima, it is pertinent to examine the poor correlation of this inner minimum position at lower concentrations somewhat further. The distance between the position of force zero and the energy minimum is shown in Table 1. Although the data
polyelectrolyte concn (ppm)
PE minimum [TIRM] (nm)
force zero [AFM] (nm)
difference (nm)
100 200 500 1000
59 52 31 16
36 28 16 16
23 24 15 0
set is limited, it seems that the offset is directly related to the concentration of polyelectrolyte, and hence electrolyte. In a typical AFM experiment, distance is not independently determined as it is in TIRM. Therefore, it seems likely that the TIRM energy minima are correctly located, and hence the error is most likely in the AFM data. In the work reported here, the spring constants chosen for the AFM probes were relatively weak. The location of zero separation in an AFM experiment, and from this all separation distance, is based upon the assumption that the constant compliance regime in the raw data is caused by the direct coupling between surfaces in hard contact. Such an assumption is known to be flawed when surface adsorbates such as polymers are present, and in these cases, the separations quoted are only apparent distances from the point of closest approach possible.44,45 However, it is normally assumed that for clean bare oxide surfaces constant compliance always equates to surface-surface contact. The data reported here suggest that this may not always be true. A simple analysis of double layer forces shows that it may be possible for an apparent constant compliance regime to be reached before the surfaces come into contact and while they are still interacting through the double layer. The electrostatic double layer force was used to simulate AFM force curves in Figures 7 and 8. Plots of the electrostatic double layer force versus surface separation and details of their calculation are given in Appendix A. (44) Biggs, S.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1994, 90, 3415. (45) Biggs, S. Abstr. Pap. Am. Chem. Soc. 1996, 212, 199.
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Figure 7. Calculated values for force and separation distance as a function of piezo travel distance in a standard AFM experiment. Solid lines correspond to the force data, and dashed lines indicate the separation distance. Data are shown at two typical cantilever spring constants, as indicated in the figure.
The AFM force curves are generated through the use of a simple distance balance for the AFM:
∆l ) ∆d - ∆D
(3)
where no deformation to the surfaces is assumed, ∆D is the change in separation, ∆l is the change in piezo motion, and ∆d is the deflection, related to the force via
F ) Kc∆d
(4)
where Kc is the cantilever spring constant. The force curves in Figure 7 show the calculated effect of changing the spring constant from 0.01 to 0.1 N/m and the corresponding separation distance as a function of piezo motion. The curve with the lower spring constant is, as expected, more sensitive to a weak force, but this results in a requirement for very large piezo travel to reach constant compliance. Furthermore, depending on the maximum of the force scale, the curve may appear linear if one examines the weaker force range as shown in the inset. In contrast, the force curve simulated with the larger spring constant is far less likely to have a pseudolinear region and the true constant compliance is more likely to be achieved in the weaker force range scale shown in the inset. The force scale and piezo travel required to reach constant compliance for the cantilever with the lower spring constant can easily pass beyond the piezo travel or photodiode detector limit of many commercial instruments, resulting in observing an apparent compliance region for interaction force between bare oxide surfaces. The above example displays how the cantilever spring constant can affect the distance at which an apparent constant compliance can occur. However, other factors such as the magnitude and extent of the double layer may also affect the separation distance at which this occurs. This is demonstrated in Figure 8 for two separate ionic strengths corresponding to the 100 and 1000 ppm concentrations of PSS. As the electrostatic double layer is screened, the separation distance reaches a smaller separation before the onset of an apparent constant compliance. The above examples are obviously only semiquantitative illustrations, since they do not include the depletion and structural effects caused by the polymer. However, they clearly demonstrate the possibility of an apparent constant compliance region that is well away from the real contact as a result of simple double layer surface force interactions in the absence of any adsorbate.
Figure 8. Calculated values for force and separation distance as a function of piezo travel distance in a standard AFM experiment. Solid lines correspond to the force data, and dashed lines indicate the separation distance. Data are shown at two values of the Debye length, as indicated in the figure.
It is important to note that, while this is the first study to quantify the distance offset in an AFM depletion measurement by an independent experimental method, evidence for the apparent constant compliant behavior arising from an electrostatic double layer repulsion has been seen in a recent colloidal probe AFM study by Peich and Walz15 examining the depletion forces from highly charged spherical depletant. Evidence of this behavior has also been seen in two separate AFM studies measuring repulsive van der Waals forces.46,47 The estimated distance offsets in these earlier studies were on the order of several nanometers. The implication for other colloidal probe AFM studies with strong repulsive forces is that, even with bare surfaces in the absence of compliant adsorbates, there may be a distance error if a cantilever with very low spring constant is used. For electrostatic double layer interactions, it is unlikely that the Debye length determinations will be affected if the distance offset is a constant with piezo travel, but the error in the surface potential from regression of the AFM data could be increased significantly. The consequence for depletion measurements is that the spacing between minima may be correct, but the absolute positions in the force-distance profile may have a distance offset that is a function of ionic strength and/or depletant concentration. A closer inspection of the data in Figure 5, at 1000 ppm, shows that while the inner energy minimum and the initial force zero position are well aligned, there is significant discrepancy between the number and position of the minima seen by AFM and TIRM. This is highlighted further by directly comparing the AFM data with the derivative of the TIRM data. Such a comparison is shown in Figures 9 and 10 for the 500 and 1000 ppm data, respectively. Note that the 500 ppm AFM data set has been offset such that the inner energy minimum and force zero are aligned. In both cases, the data show good correlation at large separation distances, but as the amplitude of the oscillations increases at shorter separation distances, TIRM appears to indicate the presence of a greater number of oscillations. The most likely explanation of this is that the AFM measurements are highly (46) Milling, A.; Mulvaney, P.; Larson, I. J. Colloid Interface Sci. 1996, 180, 460. (47) Lee, S.; Sigmund, W. M. J. Colloid Interface Sci. 2001, 243, 365.
Direct Comparison of AFM and TIRM Measurements
Figure 9. Comparison of AFM force data with the differential of TIRM energy data, collected at 500 ppm polymer concentration. Note that the position of the AFM data has been artificially adjusted such that the innermost repulsive interaction (due to double layer interactions) is aligned.
Figure 10. Comparison of AFM force data with the differential of TIRM energy data, collected at 1000 ppm polymer concentration.
dynamic, and in the presence of a strongly attractive interaction the cantilever becomes unstable and will “snap” inward. Such motions may result in significant perturbations of the fluid between the surfaces. Given that these structural interactions are very weak, these motions may disrupt the structure and prevent accurate measurement of the interactions. Notwithstanding these difficulties, it is apparent that for concentrations of polymer >1000 ppm the separation distances measured by the AFM are essentially accurate. Also, for measured force oscillations at larger separation distances the correlation with TIRM is excellent, suggesting that these data can be used to assess the structural correlations in the intervening fluid as the surfaces are brought together. In our earlier work, the peak-to-peak spacing between adjacent minima in the TIRM data (Figure 1) were analyzed at each concentration. A similar analysis of the AFM data between 500 and 10 000 ppm has been performed here, on the furthermost visible oscillations in the force data. Comparison between the AFM and TIRM data is shown in Figure 11. The correlation between the two sets of data is excellent up to 5000 ppm and indicates a scaling law relationship of c-1/3, as expected for a dilute polymer solution. At 10 000 ppm polymer, the data appear to show a deviation from this relationship in the direction expected for the semidilute regime (c-1/2). However, in the absence of a more complete set of concentration data, it is difficult to make any real conclusions. The transition in behavior between the dilute and semidilute concentration regimes for potassium poly-
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Figure 11. Scaling law relationship between oscillatory period of the force or energy data and polymer concentration. Solid and open symbols represent calculated periodicity from TIRM and AFM data, respectively. The line of best fit shown was calculated by use of only the TIRM data, which are exclusively in the dilute concentration regime. The scaling law relationship calculated for these data is ζ ) 372c-0.35.
(acrylic acid) (KPAA) has recently been investigated by Piech and Walz15 using AFM. The reported scaling behavior for this polymer between silica and titania surfaces followed the expected c-1/3 and c-1/2 relationships for dilute and semidilute regimes, respectively. It is worth noting here, however, that the oscillatory force data reported were averages of up to 60 curves and so some detailed information may have been lost. It is also unclear exactly where the periodicity for the scaling behavior was taken from these force data; that is, whether it was from the furthermost or innermost oscillations. However, the authors did note that the instability seen at the inner oscillation rendered these data problematic for analysis. In addition, it was noted that the data did not reveal the transition from c-1/3 to c-1/2 behavior for outer and inner minima within a single force curve in the dilute regime. Given the uncertainties apparent in the inner oscillations for the data reported here, this is perhaps not surprising. The data reported here also cast some doubt on the previous results and analysis of Milling and co-workers.18,28 In general, although the data are of very high quality, no evidence for a c-1/3 relationship was found in the dilute regime for a variety of different polyelectrolytes, including NaPSS as used here. In light of the TIRM data and the work of Piech and Walz,15 this is surprising. However, examination of this work indicates that the innermost minima of the AFM data were used to calculate the scaling relationships. Given the uncertainties in the number and position of the minima highlighted here, it seems likely that the analysis of the scaling relationship may have been flawed. Conclusions A comparison between TIRM and AFM data collected for similar surfaces and the same NaPSS polymer solutions indicates a number of important features that are of general relevance to all AFM force measurements as well as direct relevance to structural and depletion measurements. Possibly the most significant finding is the uncertainty in zero distance caused by the attainment of a false constant compliance region. This is of particular importance when weak cantilever springs are used to measure strongly repulsive forces; importantly, it may occur for measurements of simple double layer forces between hard surfaces in the absence of adsorbates.
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Comparison of the structural and depletion data from AFM and TIRM indicated a broad agreement between the data collected by either route; for example, the number of observed oscillations and their magnitudes increased with increasing polymer concentration in both cases. However, detailed examination of the data at each concentration indicated significant differences that may complicate the interpretation of such data collected by AFM. The weakness of these interactions suggests the need to use highly sensitive springs. However, at low electrolyte concentrations this leads to uncertainty in the apparent region of constant compliance that results in a significant error in the separation distances. Further caution is needed when the number and period of the oscillations are examined, since the data here indicate that some detail, apparent in the more sensitive TIRM data, may be lost as a result of the relative surface motion and the related hydrodynamic perturbance in the system. It is worth noting, however, that at higher concentrations (>1000 ppm here) where the double layer is sufficiently compressed, the inner energy minimum is well predicted by the AFM data; this corresponds to the depletion force
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minimum for the system and so, under these conditions, AFM can be used to analyze depletion interactions. Appendix A The electrostatic double layer force for a silica sphere of 2.5 µm in radius and a silica plate was calculated by solution to the full numerical solution to the PoissonBoltzmann equation according to the method of Chan et al.48 with a constant surface charge boundary condition as shown in Figures 7 and 8 at a series of ionic strengths relevant to the PSS concentration in these studies. The surface charge at pH 10 for silica at a series of ionic strengths was used from the study by Harltey et al.,49 which measured both electrokinetic methods and surface force measurements. The ionic strengths are based on the monomer contraction and a pH of 10. LA050041E (48) Chan, D. Y. C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283. (49) Hartley, P. G.; Larson, I.; Scales, P. J. Langmuir 1997, 13, 2207.
