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Interactions between Poly(ethylene oxide) Layers Adsorbed to Glass Surfaces Probed by Using a Modified Atomic Force Microscope G. J. C. Braithwaite,*,† A. Howe,‡ and P. F. Luckham† Department of Chemical Engineering, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, U.K., and Kodak European Research, Headstone Drive, Harrow HA1 4TY, U.K. Received February 20, 1996. In Final Form: June 12, 1996X We have investigated the adsorption of 56 000 molecular weight poly(ethylene oxide) in an aqueous system (good solvent) to glass using a development of the atomic force microscope technique. A glass particle is glued to a silicon cantilever to give a particle probe surface forces apparatus. The design of this custom built machine is discussed with reference to the particular problems inherent to the investigation. The data presented describe the evolution of the adsorbed polymer layer with time and the changes resulting from only allowing one surface to adsorb polymer. We also examine the change of the layer conformation with repeated compressions. Scans are carried out at close to Brownian collision rates and energies. The results are discussed in the light of previous surface force apparatus work. The development of the layer is clearly tracked from an initially thin coverage up to a stable equilibrium layer of some 90 nm. The “equilibrium” thickness is greater than those reported on the surface force apparatus. This is due to the increased resolution of the current apparatus, which enables energies as small as 0.5 µJ m-2 to be measured. At partial coverages of polymer on approach of the surfaces, a weak attraction is occasionally observed due to bridging of the polymer between the two surfaces. On separation a strong adhesion is noted. The lack of consistent strong attractions on approach of the surfaces is due to the relatively rapid rate of approach of the two surfaces, which does not allow sufficient time for the polymer to bridge between the surfaces and bring about an attraction. At full coverages of polymer, repulsive interactions at all surface separations are observed. However following many rapid approaches and separations at such coverages, attractive interactions may be observed, indicating that the structure of the adsorbed layer is changing and being disrupted with time. The results therefore demonstrate physically important interactions that would not be easily observed by any other force sensing technique.
Introduction Polymers have been used empirically to prevent particles from aggregating for many years. This method of stabilization, first termed “steric” by Heller and Pugh in the early 1950s,1 is now known as “steric stabilization” and is ubiquitous in the chemical and food industries. The first attempt to directly measure this steric interaction was made by Doroszkowski and Lambourne.2,3 In their experiment, sterically stabilized particles were floated on the surface of a Langmuir trough, and the surface pressure was determined. This gave an average force distance profile for the interactions between the particles. Later, osmotic pressure measurements were used to estimate similar properties for the interaction between particles bearing polymers in a three-dimensional pressure cell.4,5 However, the most accurate information to date for the interaction between polymer-coated surfaces has been obtained with the mica surface force apparatus (SFA). This machine was originally developed by Tabor and Winterton6 to study van der Waals interactions. It was then significantly improved upon by Israelachvili and * To whom correspondence may be addressed: e-mail,
[email protected]; telephone 0171 589-5111 (ext 55670); fax, 0171 5945604. † Imperial College of Science, Technology and Medicine. ‡ Kodak European Research. X Abstract published in Advance ACS Abstracts, August 1, 1996. (1) Heller W.; Pugh T. L. J. Chem. Phys. 1954, 22, 1773. (2) Doroszkowski, A.; Lambourne, R. J. Polym. Sci. 1971, C34, 253264. (3) Doroszkowski, A.; Lambourne, R. J. Colloid Interface Sci. 1973, 43, 97-104. (4) Cairns, R. J. R.; Ottewill, R. H.; Osmond, D. J. W.; Wagstaff, I. J. J. Colloid Interface Sci. 1976, 54, 45-51. (5) Homola, A.; Robertson, A. A. J. Colloid Interface Sci. 1976, 54, 286-297.
S0743-7463(96)00154-0 CCC: $12.00
Adams7 to determine the interaction between mica surfaces immersed in liquids. Using this technique, electrical double layer8 and solvent structuring effects9-11 were measured and found to significantly modify the interactions between the surfaces. The first systematic study for the interaction between polymer-coated surfaces on an SFA was by Klein12,13 in his study of the interaction between polystyrene-coated surfaces in a poor solvent (cyclohexane). An attraction between the surfaces was seen as a consequence of the polymer being in a poor solvent (we note that under these conditions particles would aggregate). Poly(ethylene oxide) (PEO) adsorbed to mica surfaces was the first case where the interaction between adsorbed polymers in a good solvent was studied.14,15 At full coverage of polymer the interaction was repulsive at all separations and could qualitatively be explained by scaling theory for the interactions between adsorbed polymers.16,17 It was found, for example, that the range of interaction scaled with the molecular weight of the adsorbed polymer and commenced at surface separations of some 5-6 radii of gyration (Rg) of the polymer in solution. The interaction between the adsorbed polymer layers at the onset of interaction could be (6) Tabor, D.;Winterton, R. H. W. S. Proc. R. Soc. London A 1969, 312, 435-450. (7) Israelachvili, J. N.; Adams, G. E. J. Chem Soc., Faraday Trans. 1 1978, 74, 975-1025. (8) Pashley, R. M. J. Colloid Interface Sci., 1981, 83, 531-545. (9) Horn, R. G.; Israelachvili, J. N. Chem. Phys. Lett. 1980, 71, 325. (10) Israelachvili, J. N.; Pashley, R. N. Nature 1983, 306, 249-250. (11) Christenson, H. K.; Horn, R. G.; Israelachvili, J. N. Nature 1982, 88, 79-88. (12) Klein, J. Nature 1980, 288, 248-249. (13) Klein, J. J. Chem. Soc., Faraday Trans. 1 1983, 79, 99-118. (14) Klein, J.; Luckham, P. F. Macromolecules 1984, 17, 1041-1054. (15) Klein, J.; Luckham, P. F. Nature 1982, 300, 429-431. (16) De Gennes, P. Macromolecules 1982, 15, 492-500. (17) De Gennes, P. Adv. Colloid Interface Sci. 1987, 27, 189-209.
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explained in terms of the increase in the osmotic pressure due to the increased polymer concentration in the gap between the surfaces. Later, the results could also be explained in broad terms by mean field theory.18 For very high molecular weight polymers, and at partial coverages of the surfaces by the polymer, an attraction was observed. This attraction was attributed to the polymer simultaneously adsorbing to both surfaces and giving rise to a bridging attraction. Recently, an alternative to the mica surface forces apparatus has been developed based on atomic force microscopy (AFM). In this method a small particle is attached to a very soft cantilever used in atomic force microscopy, and the particle is driven toward the surface. Initially, electrical double layer interactions were studied,19,20 and more recently interactions between adsorbed polyelectrolytes were probed.21,22 These experiments have all used commercially available atomic force microscopes, which have been adapted to enable force profiles to be obtained. In the work reported here the interactions between adsorbed poly(ethylene oxide) of a narrow molecular weight distribution have been revisited using the AFM approach. The polymer is now adsorbed to glass, rather than mica, surfaces. A custom built apparatus has been constructed which is considerably more flexible than a commercial AFM, making dynamic or time dependent experiments easier to perform. This apparatus is described in the next section. Experimental Section Equipment. Atomic force microscopy (AFM) has in recent years become a standard technique for the imaging of surface topographies in both industry and academia. However until recently the use of AFM as a force-sensing tool has been limited.18-21 The apparatus used here was designed and constructed with the specific aim of examining the surface forces at nanometer length scales and nanonewton forces in a variety of media. It is built around the principles of the AFM, although we have not incorporated any scanning capability. This greatly simplifies the machine and considerably reduces the cost. We shall therefore describe the apparatus in detail (see Figure 1). The fundamental approach used follows that of Binnig et al. with the original AFM.23 A small, soft cantilever is used to sense the forces at an interface. The lever/surface separation is controlled by a piezoelectric ceramic to nanometer resolution, and the deflection is determined using a detector system capable of such resolution. The most common approach used commercially for the detection of the nanometer deflections of the lever is that of optical beam deflection.24 Beam deflection is intrinsically vibration tolerant (due to its inherent gain) and is relatively immune to contaminants and the media used compared to other techniques. Beam deflection was therefore the approach adopted for our apparatus. In order to work in liquids, a “window” of glass was mounted on the back of the lever mount to provide a stable air/solvent interface (and hence a clear optical path) as shown in Figure 1. The beam deflection was monitored by a split, four-quadrant, position-sensitive detector (PSD). This allowed both the vertical and lateral spot motions to be resolved (i.e., the twisting as well as the bending of the lever). Both the PSD (SPOT-9D) and the signal amplifiers (301-DIV, 30kHz) were obtained from Optilas UK Ltd. The laser was a 3 mW diode laser (β-TX) with additional optics (Vector Technology Ltd, U.K.). (18) Fleer, G. J.; Scheutjens, J. M. H. M.; Cohen Stuart, M. A. Colloids Surf. 1988, 31, 1-29. (19) Ducker, W. A.; Senden T. J.; Pashley, R. M. Nature 1991, 353, 239-241. (20) Ducker, W. A.; Senden T. J.; Pashley, R. M. Langmuir 1992, 8, 1831-1836. (21) Biggs, S.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1994, 90, 3415-3421. (22) Biggs, S. Langmuir 1995, 11, 156-162. (23) Binnig, G.; Quate, C. F.; Gerber, Ch. Phys. Rev. Lett. 1986, 56, 930-933. (24) Meyer, G.; Amer, N. M. Appl. Phys. Lett. 1988, 53, 1045-1047.
Figure 1. Schematic of modified AFM for force sensing: (a) general construction of the apparatus without electrical connections; (b) a more detailed and pictorial representation of the sample section (not to scale). Data were captured into Snapshot commercial software running on an IBM compatible personal computer (PC) using a commercial data acquisition card (PCL812PG) obtained from Advantech, U.K. All processing thereafter was done off line on commercial spreadsheet software. An advantage of this system over the standard SFA is its increased force resolution. Since the particular setup used here has a resolution of (1 nm (noise limited), this implies a force resolution of 300 pN from a spring constant of 0.3 N m-1. The SFA used in the previous PEO work14,15 had a scatter of some 10-20 µN m-1 on its equivalent data. In terms of energy, where we divide by 2π according to the Derjaguin approximation,25 this gives our new apparatus a resolution which is an order of magnitude greater than was obtained in the SFA study of PEO adsorbed on mica. In many commercial AFMs the lever is driven by a piezoceramic. Although this allows any sample size to be used, it introduces alignment problems because the lever/laser/detector connection is variable. The laser spot can drift across the face of the detector. In constant force mode (the most common topographical imaging technique) this is not a problem, but if quantitative forces are required this is certainly not desirable. Additionally, by driving the surface, all mechanical motions are separated from the lever/detector system thereby increasing stability. Thus our choice was to drive the surface (sample) by a piezoelectric ceramic tube (Morgan Matroc, U.K.). A piezoceramic of the type PZT-5H was selected to allow a long working range for the range of voltages available but with stable and linear output. The piezo was driven from an amplified signal generator (Thandor TG501-RS Ltd, U.K.) giving a working voltage range of (250 V. This approach also allowed any desired signal to be easily applied to the driver. Typically a saw tooth voltage of ∼0.01 Hz was applied so that each compression and decompression experiment would last ∼100 s. However, faster experiments could be performed (up to 1 kHz) enabling dynamic measurements to be taken. The piezo was calibrated interfero(25) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155-164. Hunter, R. J. Foundations of Colloid Science; Clarendon Press, 1991; Vol. 1.
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Figure 2. Shows a topography of the glass surface obtained from a Topometrix Explorer SPM system in contact mode. The scan dimensions are roughly 1 µm × 1 µm. The figure demonstrates the roughness of the whole image and of a selected section of the image (partial image) within the box. metrically to yield a displacement constant of 2.5 nm/V, allowing an accessible range of 1300 nm. The errors due to hysteresis and nonlinearity were determined to be less than 5% and to only result in an offset error at the center of the piezo’s range. Coarse positioning was provided by a Burleigh Inchworm piezoelectric positioner. Probe particles were mounted on the end of commercially available silicon levers. The lever and detector were mounted on micropositioners (Speirs Robertson, U.K.) to allow easy adjustment and alignment but were not moved during the experiment once alignment was achieved. The sample holder mounted on top of the piezodriver was not fixed permanently, allowing the sample to be moved under the lever as the experiment progressed. The entire apparatus was mounted in a custom built aluminum frame. This was then supported on foam tiles on a commercial pneumatic antivibration (AV) table (Speirs Robertson, U.K.). By custom building our apparatus, we were able to sample at up to 32 kHz (limited by our choice of data acquisition card since in principle there is no limit), which is greater than commercial systems, and allows the study of faster phenomena. System. With the fundamentals of the apparatus dealt with, the experimental system will now be described in detail. The interactions were measured between glass surfaces. The substrate used was a readily available glass surface with no special treatment applied other than rigorous cleaning in a sonicator with a dilute RBS (surfactant) solution (Chemical Concentrates (RBS) Ltd, U.K.) and then thorough rinsing with Nanopure water. An AFM topographic image was obtained using a Topometrix Explorer (Topometrix) (Figure 2). The general surface roughness was below 2 nm, but rarely some large scale surface features (∼100 nm) were visible. The probe particles were typically 120 µm diameter glass spheres (BDH, U.K.). This particle was then mounted on a commercial silicon lever (Burleigh Instruments, U.K.) of quoted spring constant range 0.04-0.35 N m-1. The lever dimensions were approximately 400 µm long and 40 µm wide. This large size made beam reflection easy to achieve. Each lever came with an integral probe tip (15 µm long), which for our purposes was redundant and made the mounting of small particles more difficult. The probe was attached using Epikote 1001 epoxy resin (Shell Chemicals, U.K.). The spring constants of commercial levers are usually given in a range of values due to the uncertainty in the manufacturing process. This uncertainty is not crucial for topographic imaging, but for quantitative force work accurate values are required. For this reason, a technique was designed in our laboratories for determining the spring constant more accurately (similar to the approach used by Cleveland et al.26) using the resonance shift of a loaded lever (see Appendix). The probe particle (the load)
then became part of the calibration technique.27 The resonances are derived by monitoring the PSD output as the piezo driving voltage frequency is changed with the piezo close to, but not touching, the lever. The piezo acts as an acoustic oscillator, and the coupling through the air gap is good enough to excite the lever up to its resonant frequency of 16 kHz. The fundamental frequency is easily seen on a monitoring oscilloscope as a very sharp amplitude resonance peak. This apparatus is well suited to the study of any particle/ surface interactions. To date, however, no work on the interactions between adsorbed polymer-coated particles with an AFM have been reported. For this reason as an initial study we chose to examine the interactions of a well-characterized polymer, poly(ethylene oxide) (PEO). The molecular weight chosen was 56 000 (described as 56K from now on) although other polymers are also currently being investigated. The polymer was obtained from Polymer Laboratories, U.K. (batch number 20833-6), and has a quoted Mw/Mn ratio of 1.05 (where Mw is the weight average molecular weight and Mn is the number average molecular weight) and an expected radius of gyration in solution in a good solvent (Rg) of 7.4 nm. By choosing such a polymer, comparison could be made between this work and previous studies carried out by Klein and Luckham using the SFA.14,15 All data were obtained in aqueous solutions of approximately 0.25 M KNO3 (giving a predicted Debye length of less than 1 nm) with approximately 0.01% (by weight) of the polymer. The water used was filtered through a Nanopure system (resistivity 18 MΩ cm). All solutions were mixed at room temperature and then refrigerated for approximately 12 h before use to allow the polymer to fully dissolve.28 The working solution volume was 40 mL in all cases. The results were obtained at room temperature (usually 25 ( 2 °C) and pressure. The scan speed was of the order of 0.02 Hz or 25 nm/s unless otherwise stated. Which implies that a 100 nm layer is traversed in 4 s. Using the approach of Biggs,22 a 60 µm particle has a Brownian diffusion rate of 4 × 10-15 m2 s-1, which gives an interaction time scale of over 2 s for a 100 nm layer. (26) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 403-405. (27) Although the mass of glue used must be considered, it is clear from experiment that its influence on the resonance is considerably less than the resolution of the equipment. We estimate that with our current setup we can easily detect changes due to particle radii of 10 µm or less and that our derived values are accurate to better than 10%. (28) In some cases the polymer solution itself was filtered after the standing time through a 0.22 µm Millipore GS filter. Little difference was noted between filtered and unfiltered solutions, and for this reason later experiments were not generally performed with filtered solutions.
Interactions between PEO Layers Technique. Two approaches were used to gather data. In the first procedure both the probe and surface were allowed to adsorb the polymer from solution while in the rig. Initially salt solution was inserted into the system. The apparatus was then allowed to stabilize for up to an hour before the polymer was added.29 When stable PSD signals were obtained, a force profile was taken without any polymer added, then 10 mL of the 40 mL of salt solution was replaced with 10 mL of polymer solution. Once the polymer was added, the adsorption process was followed over time by repeatedly taking force profiles at different sites (to ensure the initial contact was on an undisturbed area) as the layer evolved. Note that although a fresh site on the surface was always chosen by moving the surface, the probe particle clearly could not be replaced every time. This may have resulted in the probe acquiring an atypical polymer layer around it. Due to the system geometry, adsorption probably proceeded faster for a given concentration in our experiments than it did for previous SFA work. The second procedure examined the interactions between a bare particle and a polymer layer. For this reason the approach was slightly different from the first procedure. Instead of adsorbing the polymer in the apparatus each time, a “snapshot” was obtained by adsorbing the solution onto an array of separate surfaces. The same solution mix was used as before, but the probe was never in the polymer solution apart from during the experiment. To terminate adsorption at a particular time, each dish was rinsed in three separate volumes of salt solution (at the same molar salt concentrations as the polymer solution). This gave an effective dilution of polymer concentration in solution of at least 104, which drastically reduced the adsorption rates. Each dish was then refilled with salt solution and replaced in the apparatus. The layer was then allowed to stabilize in the rig for at least half an hour before each data set was taken.31 Here extreme care was taken to ensure first contact was recorded.
Results and Discussion In the following section the results will be discussed and compared with respect to other work performed on similar systems. However all separation distances quoted here are relative to an assumed zero separation contact. This arises because calibration of the equipment is based on two assumptions: zero force is obtained when the lever is in equilibrium, well away from the surface and free from any interaction; zero separation is taken when the detector signal moves linearly with the drive voltage. This is known as constant compliance. Although there may have been polymer sandwiched between the probe and the surface, the layer apparently no longer changed thickness so the separation was now constant. There was no way to determine the absolute separation (and hence the layer thickness) or the contact area (unlike the SFA). There is therefore an offset error in any quoted distance value (of the order of 1-10 nm for two layers). These problems and assumptions are common to all AFM force sensing techniques. However, this does not alter the form of the interaction. It should also be noted that the surface energies (per unit area, E) reported here are in the form of E ) F/2πR (as required by the Derjaguin approximation,25 where F is the force between two crossed cylinders of radius R), whereas the data presented in previous work using the SFA7,10,14,15,32-34 were normalised as F/R and (29) Although low power, the laser does appreciable heat the system,30 specifically the lever and surrounding fluid. There must therefore be a period to reduce thermal drift after the laser was switched on (>30 min) before any data are collected. (30) Radmacher, M.; Cleveland, J. P.; Hansma P. K. Scanning 1995, 17, 117-121. (31) Although this may allow the layer to rearrange, it was felt that since adsorption must have stopped, this would be insignificant when compared to the errors due to the apparatus not having stabilized properly. (32) Israelachvili, J. N.; Tandon, R. K.; White L. R. Nature 1979, 277, 120-121. (33) Marra, J.; Hair, M. L. J. Colloid Interface Sci. 1988, 125, 552560.
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are therefore not true energies between flat plates, rather normalized forces. Raw Data. One of the advantages of building our own apparatus is the ability to define completely what is occurring during an experiment. Figure 3 is an example plot of raw data obtained from the PC after one run on glass in water. Only a section of the piezo range is displayed to highlight points of most interest. Clearly the times involved are much longer than those usually used by AFM,35 and this makes our system more prone to drift. The detector voltages are proportional to the deflection of the lever, and the piezo voltage is proportional to the surface motion. As the piezo voltage becomes more positive, the surface moves toward the lever. The surface is arranged such that it contacts the probe over a range corresponding to at least 100 nm applied motion. This is to ensure constant compliance and a long enough linear region to extract the detector calibration reliably. As is clear from the plot there is considerable drift in this particular set of data (indicated by the dashed line). This was not normally the case, but is included to illustrate the flexibility of our system. The drift could then be removed linearly by a simple subtraction (the data can be seen to follow the straight line very closely). All descriptions will therefore be given from the dotted line as a base line. Between 0 and 8 s there is no deviation from the equilibrium position. The detector output does not vary and no interaction is sensed. At about 8 s, the detector output goes slightly negative (corresponding to an attractive force) and is then rapidly ramped up through the equilibrium position and on up to greater than 1 V. Here the signal remains constant since the capture card amplifier has saturated (it was operated on a high gain to achieve a high signal to sampling noise ratio). Note that the piezo voltage is still ramping so the surface is still being compressed. This approach ensures that any hysteresis effects only occur as an offset, and not as a calibration error. At about 20 s the piezo drive voltage then reverses and the surface starts to retract. At 29 s the detector signal has reduced enough to come back into range and the lever can then been seen to follow the surface down linearly. At 30 s the detector output passes the equilibrium position (dotted line) and again goes negative, corresponding to adhesion. At 37 s the lever relaxes rapidly back to its equilibrium value. Also plotted on this graph is the output from the lateral detector quadrants (the filled symbols). Clearly there is no interaction until 8 s, where the vertical signal first sensed the surface. Beyond this there is a gradual, more or less linear, twisting of the lever until the turn-over point whereupon the twisting is sharply reversed. It then relaxes back toward its equilibrium value at 38 s. Pure Electrolyte. Initially it is necessary to examine a clean system. This provides a control for comparison with the experimental results. Before the polymer was added for each run, the apparatus was allowed to equilibrate for some time in the aqueous solution of 0.25 M KNO3. These data are presented in Figure 4 (the raw data for this plot are shown in Figure 3).36 On examination of Figure 4 there is no detectable interaction within experimental resolution until the probe approaches about 40 nm. Then there can be seen a gradually increasing (but very small) attractive force (negative interaction energy). This increases up to an energy of about -2 µJ m-2 at 20 nm. At 17 nm, the lever (34) Boils, D.; Hair, M. L. J. Colloid Interface Sci. 1993, 157, 19-23. (35) In the work of Biggs22 the standard scan rate is 10 µm s-1. In fact commercial AFM’s can run at much slower rates (for example the Topometrix Explorer is capable of 1 nm s-1 rates) but are rarely used in this way.
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Figure 3. An example plot of raw output data used to determine force curves. On the primary axis open circles (O) are the vertical PSD channel and closed circles (b) are the horizontal channel. The heavy dotted line describes the assumed linear drift that is removed. The triangular waveform is the applied piezo driver voltage plotted on the secondary axis.
then becomes unstable as the force gradient exceeds the spring constant resulting in it sharply jumping toward the surface over a period of 0.08 s. This jump is consistent with the van der Waals theory. Note however that the initial contact position is slightly offset from the zero separation line (by about 5 nm). This is probably due to the particle rolling under pressure. Plotted on top of the data in Figure 4 is the theoretical van der Waals plot using the Hamaker constant for fused quartz across water (A ) 0.83 × 10-20 J) in eq 1
WPLATE(D) )
F(D) 1 A )2πR 2π 6D2
(1)
Using a value of k ) 0.25 N m-1 (as obtained from the spring calibration) gives a jump-in distance of 9 nm, which implies ∼14 nm if offset by 5 nm as mentioned above. This compares well with the observed jump-in of 17 nm. Beyond this jump, the force rises rapidly and linearly along with the surface giving an apparent hard contact. On reversal, the probe stays attached firmly to the surface until a negative energy of -10 µJ m-2 is achieved, whereupon it is appears to move slightly away from hard contact (this is also probably due to the particle rolling on the surface under pressure). At -18 µJ m-2 the probe appears to be (36) It is extremely hard to obtain a van der Waals plot (where a clear attraction is seen at short ranges) on glass for a number of reasons. A rougher surface (see Figure 2) is less likely to exhibit a strong van der Waals interaction because the effective contact area is reduced, therefore a jump-in is unlikely. Also, not only is glass known to adsorb salts from solution quite strongly, but it also forms a “hydration layer” of adsorbed water molecules. These effects will result in a layer of adsorbates on the surfaces which will screen the attractive potential.
at a separation of 8 nm before it is released quickly (over 0.08 s) to its equilibrium position at about 44 nm. No further interaction can be seen. Particle roll is a particular problem if there are localised binding sites,37 and if this is the case, then the adhesion value shown here is not strictly correct. The larger the particle, the worse the effect will be. This complication must always be considered when examining force distance data generated by the AFM, particularly if a particle is used as the probe. Polymer Adsorbed to Both Surfaces. The following text describes Figures 5-7 for the case of polymer adsorbed from 0.25 M KNO3 solution onto both surfaces. This work is similar in principle to that of Klein and Luckham in their study of PEO adsorbed to mica.14,15 Note that the times quoted should be taken as a very rough guide, because the plots shown were seen to occur at different times for different runs (but similar setups). Clearly the adsorption rates are very sensitive to conditions not under direct control here. Figure 5 shows data gathered after 35 min of incubation of the glass surface in the 56K polymer solution. The lever was present in the solution at all times and the probe was never removed more than 25 µm away from the surface. Out at 200 nm the lever shows no deflection, and within experimental resolution it is subject to no interaction. This condition stays very stable until 45 nm is reached. At 45 nm there is a weak jump into some kind of contact at 35 nm (note that this corresponds to ∼5Rg). At 35 nm there is a negative interaction imposed on the probe of approximately -2 µJ m-2. This suggests that at this time there is only partial coverage of the surface by (37) Stuart, J. K.; Hlady, V. Langmuir 1995, 11, 1368-1374.
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Figure 4. Plot of processed data obtained from Figure 3. These data were taken in clean salt solution (0.25 M) with no polymer added. Inward (O) and outward (b) runs are plotted along with the theoretical van der Waals curve. The van der Waals curve is offset by 5 nm to account for the apparent nonzero inward contact position due to probe roll under pressure.
the polymer. Although there is a layer of polymer on the two surfaces, there is still also some free surface available for polymer to adsorb. For this reason, as the surfaces approach, there is a bridging effect where the “dangling” tails of the polymers on one surface can contact and adsorb to the other surface, thus exerting an attractive influence on it. This is much as would be expected, but it was rare to see the jump-in, and it was never of the magnitude described by Klein and Luckham.14,15 In their experiments, an attraction of 50 µJ m-2 was observed for PEO of molecular weight 1.1 × 106, although at the time it was not possible to observe any attraction for lower molecular weight PEO. Their results suggested that if the attraction was present, it would be less than 50 µJ m-2. Thus, our results are in broad agreement with this earlier study. The energy then rises rapidly and monotonically up to hard contact at 70 µJ m-2. Before this point is reached, the lever appears to be moving parallel to the surface from 50 to 60 µJ m-2 at a separation of about 5 nm. At 65 µJ m-2, the separation finally collapses to hard contact and no further interaction is seen. Although the effect is not all that clear, on this trace it is a common feature in other data not presented here and may well be due to the high surface pressures generated by the AFM. At even moderate loads, pressures of the order of 10 MPa can easily be achieved, because the contact area for this technique is so small. Biggs22 has also seen such a phenomenon in his study on polyelectrolytes. On reversal, the probe follows the surface without any apparent deviations down to the equilibrium position at 0 µJ m-2. Beyond this the lever parts slightly from the surface to 10 nm and an interaction energy of -20 µJ m-2, whereupon it is released. Note however that although this energy is the same as the adhesion in Figure 4, the release is not total this time.
The probe takes a considerable time to release. Initially it reaches 30 nm over a period of about 0.1 s, but then beyond 40 nm (although it now appears to have recovered it is still at a negative energy of -8 µJ m-2) it takes a further 3.5 s to recover gradually to its equilibrium value at 140 nm. This behavior suggests that up until -20 µJ m-2 the particle is stuck to the surface through van der Waals interactions, despite the screening layer of polymer (see Figure 8a(left). Beyond the van der Waals adhesive energy, the probe is released from contact with the surface, but is still “tethered” at a distance by the adsorbed bridging polymers (see Figure 8a(right). With time these polymers desorb (especially when partially stretched and hence in an unfavorable arrangement), eventually releasing the particle. This too is consistent with a low surface coverage on both surfaces. Figure 6 is a plot of data gathered after 25 min, but on a different run from the previous data. Judging from the evolution of the layer in each run, Figure 6 is later in adsorption terms than Figure 5. Beyond 90 nm there is no detectable interaction, but from this point (∼12Rg) the interaction energy rises monotonically up to about 10 µJ m-2 at a separation of 20 nm. Here the interaction changes slope and rises more quickly up to hard contact at 40 µJ m-2. No further deviation is seen. The comparatively large distance at which the interaction starts is somewhat surprising. We will discuss this in the next section. On reversal, little hysteresis is seen and the decompression curve follows the compression curve back to the equilibrium interaction at 90 nm. No attractive or adhesive interaction is seen, which implies that the system is at, or near, full coverage. The polymers cannot therefore attach to the approaching surface due to a lack of free sites. It is of interest that the second portion of the
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Figure 5. A plot of data taken after 35 min of adsorption of the polymer to both surfaces. Hard contact is assumed when the lever moves linearly with the surface. Inward (O) and outward (b) runs are plotted. Note the jump in at 40 nm and the nonzero jump out separation again due to probe roll. A semilog plot of the same data is inset.
interaction curve (between 20 nm and 0 nm) agrees very well with the similar section in Figure 5. As has been noted elsewhere,38 adsorbing polymers tend to form a thin stable layer for a long period before finally extending into solution. It is likely that what is seen here is a dilute regime of tails extending into solution (between 20 and 90 nm) followed by a still very dense area of loops and trains close to the surface (under 20 nm) (see Figure 8b), and hence this is not equilibrium conformation yet. Figure 7 represents the final equilibrium condition of the layer after over 24 h. Here there is no interaction out beyond 90 nm (12Rg); however as the particle approaches the surface at 90 nm, it begins to feel an interaction. A value of 12Rg (i.e., 6Rg per surface) is rather longer than has ever been detected using the SFA (typically 3Rg per surface is the norm). This we believe is principally due to the increased resolution of our apparatus over the SFA. The mica surface forces apparatus is sensitive to around 3 µJ m-2, while the current instrument is almost 10 times more sensitive because of the softer spring being used (approximately 3 orders of magnitude weaker). We note from Figure 7 that if we took the onset of interaction to correspond to the resolution of the mica SFA (i.e., 3 µJ m-2 and not our “0 µJ m-2”) then this would correspond to about 60 nm, which is close to that suggested by the interaction between mica surfaces bearing adsorbed PEO of a similar molecular weight.14 The respective techniques then yield similar results. As with Figure 6 there is no attractive component, but instead the interaction approaches hard contact exponentially at an energy of 60 µJ m-2. As in Figure 6 there is a slight change of slope at 20 nm, but this time the forces are much higher occurring (38) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman Hall: London, 1993.
at an energy of 25 µJ m-2. When the motion is reversed, the decompression curve follows that of the compression curve down to the point where the compression curve changed slope at 20 nm. From here, the decompression curve relaxes faster, returning to the zero interaction at 60 nm. Note that the ripple on the hard contact of the decompression curve is due to vibration. It is notable that the layer is now clearly denser than for Figure 6 with higher energies required to get to each separation. The semilogarithmic plot of the data, shown inset to Figure 7, has the same general form as the earlier data for PEO adsorbed to mica surfaces. A similar degree of hysterisis is also observed.14,15 Long Term Effects. Although the above descriptions agree reasonably well with those of other workers and with theory, one extra point should be noted. If the polymer is left to adsorb for a very long time (a matter of days) there should be no further adsorption after full coverage has been achieved. Figure 9 shows a plot that was obtained from a surface that had been allowed to adsorb long after an apparent equilibrium thickness had been obtained. No two plots are the same for this regime. However Figure 9 shows some of the features that can usually be seen. Note that this plot was obtained using the inchworm driver, so the calibration is less certain and the values are therefore less accurate. At distances beyond 1600 nm, there is no detectable interaction. As the probe approaches the surface, a slight repulsive interaction is seen beginning at 1600 nm. This is clearly a very large distance. The probe is repelled at a more or less constant energy (10 µJ m-2) between about 900 nm and 500 nm. At 400 nm, the probe experiences an increasing repulsion up to a point at about 250 nm where an almost hard wall repulsion (i.e., vertical on this plot) is observed beginning at 30 µJ m-2. Here, the force rises rapidly and linearly
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Figure 6. A plot of data taken after 25 min of adsorption, but determined to be later in adsorption terms than Figure 5. Note no hysterisis or adhesion. Inward (O) and outward (b) runs are plotted. A semilog plot of the same data is inset.
up to an interaction energy of 100 µJ m-2 at 200 nm separation. From here to hard contact, the force displays an oscillatory behavior with a period of around 50 nm, where the energy builds up almost linearly before collapsing rapidly. The original work on this system32 clearly showed a similar, high hysteresis, behavior, but further work by one of the present authors14,15 did not show this behavior. Israelachvili claimed that this effect was due to a diffuse polymer layer stretching out from the surface and that the sudden jumps were due to a partial collapse of the polymer’s “gellike” structure. However, as Klein and Luckham14 pointed out Israelachvili and co-workers used an extremely polydisperse sample and apparently did not filter their solution (we did not filter this particular sample either). This allows the possibility for partially undissolved polymer aggregates to enter the system. Although both these arguments are true, the situation was further confused by Marra and Hair.33 They used toluene as a solvent, and although this is different from the system used by the other workers, both it and water are generally considered good solvents for PEO at room temperature and pressure.34 Marra and Hair also observed long range repulsion with a progressive collapsing of the layer under pressure. They argued that over time the polymer was self-aggregating in solution and then adsorbing to the surface as bulk “macropolymers” rather than single molecules. For a long time the self-aggregation of PEO in good solvents34,39 was the accepted understanding of this system. Polik in particular obtained results from dynamic light scattering suggesting both high and low density aggregates existing alongside monomolecular PEO. He obtained aggregate sizes of at least 1 order of magnitude bigger than the Rg value for his polymer (20 000 (39) Polik, W. F.; Burchard, W. Macromolecules 1983, 16, 978-982.
Mw) at room temperature. However recently a more convincing argument has been introduced by Porsch and Sundelo¨f.40 Here they prove these “macropolymers” are in fact due to impurities sterically stabilized by PEO. After compression has been completed, at an energy of 130 µJ m-2, hard wall contact is finally achieved. On decompression, the probe follows the compression trace down to about 100 µJ m-2 where it jumps out by about 40 nm. It then relaxes gradually as the surface is retracted, crossing the equilibrium line at 300 nm and then moving into an adhesive regime. At 600 nm the adhesive force reaches its maximum value of -60 µJ m-2 before finally releasing. A further attraction is observed until at 1000 nm the probe appears to be finally free of the surface. In general, if compression is performed again, such a long range interaction is not observed. Figure 9 can therefore be pictured as the probe irreversibly forcing its way through a loosely adsorbed layer of aggregates due to impurities. For this reason long-term force curves are difficult to obtain reliably. Polymer Adsorbed to One Surface Only. Although attractive bridging was observed above in the symmetric system (polymer adsorbed on both surfaces), this was not the most usual situation, and when it did occur it was barely detectable. More often, no attraction was observed within the experimental resolution. This apparent lack of attraction is surprising, neither following previous work, nor as predicted by theory. Because of this anomaly it was decided to force the system into a situation where bridging should theoretically occur. This condition was obtained by only allowing the polymer to adsorb onto the surface, and not the probe. Figures 10 and 11 demonstrate (40) Porsch, B.; Sundelo¨f, L.-D. Macromolecules 1995, 28, 71657170.
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Figure 7. A plot of data taken after 24 h. This is the equilibrium condition. A slight hysterisis is evident consistent with a rearrangement of the layer under pressure but no adhesion is evident. Inward (O) and outward (b) runs are plotted. A semilog plot of the same data is inset.
b
Figure 8. Pictorial representation of the polymer layer at the surface. Part a shows a partial coverage situation of polymer on the surface. Here the particle can be stuck to the surface directly via van der Waal forces despite a screening layer of polymer (I) or, at greater separations, (II) adhered by a bridging interaction. Part b displays a possible conformation of polymer for the low coverage situation where the bulk of the polymer density is contained in loops and trains near to the surface.38
two force curves from such a system. The initial curves without polymer present (not shown) show little or no interaction other than the hard contact expected. From there the development in time is clearly evident. Another significant difference between these two curves and the symmetric, two surface, system is that here the polymer is allowed to approach the surface freely. In the previous graphs (Figures 5-7) the presence of the probe at roughly 25 µm from the surface must restrict access of the polymer to the surfaces. Therefore one would expect adsorption to be faster in this system than in the symmetric system
Figure 9. This plot shows data obtained long after the equilibrium condition of Figure 7 had been obtained. The Burleigh inchworm was used to obtain these data, therefore distance and force values have large errors (∼40%). Clearly there is still a very long range interaction with large hysterisis effects. This is attributed to aggregation of polymer in solution. Inward (O) and outward (b) runs are plotted.
(and consequently very much faster than adsorption in the SFA work). Figure 10 is a plot of data obtained by allowing the glass surface alone to incubate in the 56K polymer solution for 30 min. As the probe is brought in toward the surface, no interaction is observed until 60 nm (note that the slight
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Figure 10. These data were obtained after only the surface was incubated in the polymer solution for 30 min. Note the evidence of bridging attraction and adhesion. Inward (O) and outward (b) runs are plotted. A semilog plot of the same data is inset.
Figure 11. Data obtained after the surface was 3 h in solution. Note the adhesion has returned, but the attraction has not and the layer has stabilized at about 70 nm thick. Inward (O) and outward (b) runs are plotted. A semilog plot of the same data is inset.
apparent ripple on the signal is vibration noise). At 60 nm (8Rg) the probe is pulled in to 50 nm over about 0.1 s. From this point the interaction rises sharply up toward apparent hard contact at an interaction energy of 15 µJ m-2. As the load is increased still further, there is a final collapse to hard contact at a load of 20 µJ m-2. From here
there is no further change from a constant compliance behavior (hard contact). As for Figure 5, this plot represents a partial coverage condition. Free sites on each surface allow a bridging attraction to occur. However, the expectation would be that when Figure 10 is compared to Figure 5, there should be a much deeper attractive
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potential since one of the surfaces here is free of polymer and can thus be easily adsorbed and captured. This is not the case, but the lack of any strong bridging attraction on approach of the surfaces may be explained by the high scan rates since the surface moves roughly 0.5 nm between each sample point (25 nm/s). This rate of approach is simply too fast to pick up any strong attraction between the approaching particle and the surface because the polymer cannot rearrange fast enough. In the earlier experiments of Klein and Luckham, some 10-60 s elapsed between each point (this is typically the time scale of a whole force profile with the current set up). This probably allowed time for the very highest molecular weight polymer to bridge between the two surfaces and to pull them together. Indeed, by careful observation of the fringes of equal chromatic order (the interferometric separation determination method), Klein and Luckham could see the surfaces being “pulled” together before the forces stabilized. This process is therefore too slow to be observed in the current experiments. On retraction, the probe follows the surface down to almost its equilibrium interaction energy (0 µJ m-2), before it begins to diverge, until at -8 µJ m-2 it is 35 nm from hard contact. After this there is a gradual release over roughly 2 s to a separation of 80 nm and an interaction energy of -6 µJ m-2. The probe is stable here for a short spell of just under 1 s before releasing rapidly over 0.5 s to a separation of about 120 nm and a zero force interaction. No further interaction is seen within the experimental resolution. The protracted release, as for Figure 5 (and in contrast to Figure 4), is a clear indication of adhesive bridging where the polymers have been physically forced across the gap. The assumed layer thickness here (60 nm) is slightly greater than that obtained from Figure 5 (40 nm). However, the interaction for Figure 5 rises much faster to a considerably higher energy (roughly 60 µJ m-2 as compared to 20 µJ m-2). Both these plots show the final collapse from an apparent hard contact. The fact that the interaction energies for the two layers are larger is not unexpected, but the apparent layer thickness may be of concern even though these are taken early on in the adsorption process. Since the rate of approach of the surfaces is of the order of that observed in a Brownian collision, these results are likely to be of significance in the area of particle bridging flocculation. They suggest that adsorbed polymer plays a somewhat passive role in bridging flocculation. The particles having come together through a Brownian collision find themselves attached together by polymer “tethers” which are bridging between the particles, the polymer playing little or no role in bringing the particles together in the first place. Figure 11 shows a set of data obtained after the surface had been incubated for 3 h in the polymer solution. Here, out beyond 70 nm, there is no detectable interaction. At 70 nm the lever senses a monotonically rising force that achieves hard contact at about 27 µJ m-2. Again there appears to be a final collapse after the probe almost reaches the surface at a separation of about 5 nm. There is no attractive potential on approach. Adhesive bridging is therefore again present, and although the bridging is not strong it is important since it does operate over a long range. A single layer thickness of 70 nm (9Rg) is thick compared to theory and previous SFA experiments for this type of polymer system. Also it is roughly 0.7 of the value obtained for the two surface equilibrium case (Figure 7), where intuitively it might be expected to be closer to half. However Taunton et al.41 performed a similar comparison using end-grafted polymer chains (PS-PEO) (41) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Macromolecules 1990, 23, 571-580.
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on one and two surfaces. On one surface they obtained an equilibrium layer thickness that was 0.7 of the two layer thickness. Although the Taunton work was with an ideal brush, and is therefore not directly equivalent, it is certainly of interest that a similar effect has been seen here. As in the previous sets of data in this sequence, the energies are roughly half of those obtained from the two surface sequence. On reverse, the probe follows the surface back down to roughly 5 µJ m-2 whereupon it is gradually released so that at 0 µJ m-2 (this is not zero applied force) it is at a distance of 15 nm. Beyond this the lever sticks to the surface until at an interaction energy of about -3 µJ m-2 it is 30 nm from the surface. The interaction stays roughly constant until it is finally released from 100 to 110 nm. No further interaction is seen. Repeated Contact Effects. A long equilibration time in a saturated solution prevents bridging from occurring. This is because the polymer can adsorb to cover all the available surface leaving no available sites on which bridging can occur. Whereas the SFA is a pseudoequilibrium technique, and time is given for the system to stabilize before each sample, this is not true of our technique. This means that although, as Biggs21 points out, the scan rates are comparable to the impact rates in a real system (for example Brownian motion), there is a problem with the interpretation of the data, since all curves are now scan-rate dependent. The data must also be dependent on the past history of the sample; i.e., if the sample has undergone previous compressions, then the force profiles obtained will be different from the pseudoequilibrium of the SFA (which is generally at the first contact). This point is of concern and seems to be generally underplayed, because although the polymer would be expected to relax at rates much slower than our scan rate, it is conceivable that damage to the polymer layer occurs at the high contact pressures required to obtain the region of “constant compliance”. Figure 12 shows a series of four force profiles obtained from the same sequence of runs as Figure 11 (indeed Figure 12c is identical to Figure 11) after the surface has been incubated for 3 h in an aqueous solution of the 56K polymer. These four plots represent sequential compressions at a frequency of about 0.02 Hz. Figure12a is the first contact in the sequence. Here apparently the interaction does not start until 35 nm, although it must be pointed out that for this run only the form of the curve can be considered because the surface position was controlled by the Burleigh inchworm. Since this is not calibrated properly, absolute distances and forces must be treated with caution. On compression there is no attractive component. On reversal the lever is released slowly after the lever has passed the equilibrium position, indicating bridging adhesion. The adhesion is deep and quite localized close to the surface. Figure 12b represents the seventh contact. Here, the distance calibrations are reliable. The onset of interaction occurs at about 70 nm and rises almost linearly up to hard contact at 25 µJ m-2. On retraction there is again an adhesion, but it is less deep and more smeared out away from the surface than that shown in Figure 12a. The force curve detaches earlier from the surface than before. Figure 12c is a plot of the tenth contact. Again the interaction begins at about 70 nm, but now the curve is more rounded, exhibiting a more exponential than linear behavior. Hard contact is achieved at about 20 µJ m-2. Detachment is again a long, but relatively weak interaction, never exceeding -3 µJ m-2. Figure 12d shows data gathered after the 12th contact (and therefore some 10 min after the first contact). The trends described in the earlier plots have continued, with
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Figure 12. A sequence of four sets of data taken from the same set of runs: (a) the first contact (using an inchworm as the approach mechanism); (b) the seventh contact; (c) the tenth contact; (d) the twelfth contact. Note particularly how the interaction appears to become more defined in the later cases and that it develops an attractive component in the twelfth case. Inward (O) and outward (b) runs are plotted.
the start of the repulsive curve occurring at about 60 nm and proceeding up to 25 µJ m-2 before hard contact is achieved. However, at 90 nm there is a clear attraction before the repulsive force is experienced. On reversal, the probe detaches even earlier at a positive energy of 12 µJ m-2, but is again released slowly out to 100 nm at a very low adhesive energy. The series of curves clearly shows the interactions developing from an almost linear response to a more complicated, almost exponential response, with finally an attraction appearing. The adhesion also changes, becoming less localized and less intense as the contacts proceed. These two facts are evidence of the polymer layer being disturbed to the point where significant changes have occurred. It has been suggested that contact periods of at least 10 min are required to do this damage,42 but clearly here there is a short range attraction if the polymer layer is disturbed, even though our scan rates give a contact time of the order of only 30 s. This is possibly significant for sterically stabilized particles, since it implies that for a system that, on the face of it, should be purely repulsive (and hence stable) it is possible for the layer to generate an attractive interaction (hence causing the system to flocculate). However the contact energies (i.e., the level of compression) are well in excess of those available from Brownian motion alone. Therefore for this behavior to occur would require higher interaction energies (note that 10 µJ m-2 > 4kT). Examples with Mica as the Substrate. The original work of Klein and Luckham14,15 was made using the SFA. Their surfaces were mica and could therefore be expected (42) Granick, S.; Patel, S.; Tirrel, M. J. Chem. Phys. 1986, 85, 53705371.
to interact with the polymer differently from the system used here. For this reason, for some of the data sets described above, parallel data were captured using a piece of molecularly smooth (freshly cleaved) mica as the substrate. Figure 13 highlights the belief that there is no major difference between the two different substrates. This curve was taken after 9 h adsorption of the polymer onto a freshly cleaved piece of mica. Again, judging from the plots taken around it, this is equivalent to Figure 7. There is no interaction until the probe has approached up to 90 nm. Here it rises exponentially up to hard contact at about 60 µJ m-2. These values compare favorably to Figure 7, where the interaction starts at 90 nm and rises to 60 µJ m-2 before hard contact. This suggests that the increased ranges we see when compared to the SFA work are not due to surface roughness effects. With a mica substrate the effect of surface roughness should be considerably reduced and we would expect to see a pronounced shortening of the interaction range if there was a roughness problem. Clearly this has not happened to any significant degree, so we must conclude that the long interaction ranges observed are due to our instrument resolution. Conclusions A number of general points can be made from these data. The thickness of the equilibrium layer (12Rg) is of the correct order of magnitude and the development of the layer with time is much as would be expected when compared to previous work (6Rg). The general trends are well confirmed, with a thin polymer layer building up toward a stable conformation resembling an exponential repulsion. The lack of any strong attractive component
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Figure 13. A plot using a similar procedure to Figure 7 where both the probe and surface have polymer adsorbed onto them. However the surface used in this case was freshly cleaved mica. Clearly there is a very strong similarity between these data and those of Figure 7. Inward (O) and outward (b) runs are plotted. A semilog plot of the same data is inset.
in either system, especially early on in the adsorption process, is obviously of major significance. This must have repercussions when considering data in the light of colloid stability measurements. The complete lack of attraction on approach of the surfaces in the single surface case is due to the rate at which the surfaces approach. Although in our case it is slower than generally used for AFM, it is also considerably faster than that usually used in the SFA. This may suggest that the polymer does not have time to rearrange and hence “capture” the probe particle. However, since the scan rates here are of the same order as Brownian collisions, this would imply that flocculation of PEO-stabilized systems must proceed by a “hit-andhold” (rather than a “grab-and-hold”) process where the particles must collide to flocculate. Although the interactions are longer than those expected from the SFA results, this discrepancy can be accounted for by the increased sensitivity of our apparatus. This lets us see weaker interactions further away from the surface. Clearly, our apparatus is scanning at Brownian collision rates and can see Brownian collision energies. The SFA could never scan at these speeds, or achieve this time resolution. This is further evidence to support our belief that the current approach provides more directly applicable information for polymer-stabilized colloidal systems. Finally there is clear evidence of the layer becoming perturbed on repeated high-compression contacts. This in itself is not surprising (and has been noted on the SFA), the issue which is important is the development of adhesion. Clearly, if it is possible for an adhesive interaction to develop with repeated contacts at the rates expected by Brownian motion, then a system which may be expected to be stable could, in fact, flocculate. Of course the energies used here to achieve layer disruption are much higher than would be available from Brownian motion alone.
Acknowledgment. The authors of this paper are indebted to Professor Jonathan Colton of Georgia Institute of Technology for his suggestions and assistance in a tight spot. We thank Dr. Terry Blake at Kodak for many helpful comments. We are also grateful to the staff of Burleigh UK and Advantech UK for technical assistance. This work was partially funded by the EPSRC with assistance from Kodak European Research. Finally we thank the considered comments and suggestions of the reviewer of this paper. Appendix. Spring Constant Calibration If the mechanical properties of a lever are considered, it can be seen that the spring constant is a direct result of the dimensions and material properties of the lever. Using the general solution for a driven cantilever, a relationship for k and the material properties can easily be derived. Assuming a Hookean Spring (i.e. purely elastic)
F ) kx )
3EI x L3
(A1)
where E is the Young’s modulus, I is the moment of inertia, L is the spring length, k is the spring constant, x is the spring displacement at the tip, and F is the applied load. For a simple distributed homogeneous beam type cantilever the moment of inertia can be calculated as
I ) wt3/12
(A2)
which from eq A1 gives a spring constant of
k ) Ewt3/4L3
(A3)
Where w and t are the width and thickness, respectively.
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This allows an approximation to the spring constant from the lever’s assumed material properties and dimensions. It has been the most common method of determining spring constants until recently. This approach is however extremely sensitive to inaccuracies in the assumed beam properties. For this reason, a more reliable approach was chosen using the fundamental resonant frequency of a Hookean spring that is described by the solutions of the equation of motion of the lever43 4
mEFF ) 0.24mLEVER + mPROBE ) M + mPROBE
ωLOAD ) (A4)
which, when solved, yields the fundamental resonant frequency ω
x
k mEFF
mEFF ) 0.24mLEVER + mPROBE
(A6)
x
k M + mPROBE
M)
ωL2 ωN2 - ωL2
mPROBE
(A11)
where the subscripts denote the loaded and unloaded resonances. Equation A11 can then be placed back into eq A8 to remove any requirement on knowledge of the lever structure or properties
If no particle is mounted on the lever, then eq A6 reduces to
mEFF ) 0.24mLEVER ) M
(A10)
Now if eqs A8 and A10 are squared and rearranged, they can be equated in k to give an equation in M
(A5)
where mEFF is an effective mass depending on the lever. For a lumped system where a beam cantilever is loaded by a mass at one end (e.g., a particle),
(A9)
and the resonant frequency is now
2
∂x ∂x EI 4 + F 2 ) 0 ∂y ∂t
ω)
If there is a particle mounted then
k ) ωN2M )
(A7)
ωN2ωL2 ωN2 - ωL2
mPROBE
(A12)
and the resonant frequency from eq A5 is clearly
ωNOLOAD )
x
k M
(A8)
(43) Sarid, D. Scanning Force Microscopy; Oxford University Press: Oxford, 1991.
Therefore, if the fundamental resonances of a beam cantilever are determined before and after loading with a known mass, then the spring constant can be accurately obtained. LA960154L
