ARTICLE pubs.acs.org/Langmuir
Nanoscale Electrowetting Effects Observed by Using Friction Force Microscopy Reynier Revilla,†,‡ Li Guan,† Xiao-Yang Zhu,† Yan-Lian Yang,*,† and Chen Wang*,† †
Key Laboratory of Standardization and Measurement for Nanotechnology, the Chinese Academy of Sciences, National Center for Nanoscience and Technology, Beijing 100190, China ‡ Centro de Estudios Avanzados de Cuba, La Habana 19370, Cuba
bS Supporting Information ABSTRACT:
We report the study of electrowetting (EW) effects under strong electric field on poly(methyl methacrylate) (PMMA) surface by using friction force microscopy (FFM). The friction force dependence on the electric field at nanometer scale can be closely related to electrowetting process based on the fact that at this scale frictional behavior is highly affected by capillary phenomena. By measuring the frictional signal between a conductive atomic force microscopy (AFM) tip and the PMMA surface, the ideal EW region (YoungLippmann equation) and the EW saturation were identified. The change in the interfacial contact between the tip and the PMMA surface with the electric field strength is closely associated with the transition from the ideal EW region to the EW saturation. In addition, a reduction of the friction coefficient was observed when increasing the applied electric field in the ideal EW region.
1. INTRODUCTION The influence of an external electric field across a solid/liquid interface on adhesion and friction at nanometer scale is keen to gain an insight into electrowetting (EW) processes. EW is the phenomenon in which the use of electric field improves the wetting of a dielectric surface. EW has several practical applications, such as in the operation of aqueous lens with variable focal length,13 controlled transportation of liquid volumes,46 mixing and delivering microliter or nanoliter volumes of liquids,7,8 among others. The EW effect is crucial for the electric power transmission lines to induce potential flashover, which is one of the reasons for power failure disasters. EW phenomenon at nanoscale is also a key issue to be considered in micro- and nanomechanical systems since the reduction of the scale can drastically increase local electric field values. The wide range of applications and the great number of open questions that still remain in the fundamental understanding of the EW phenomena prompt the needs of rigorous study at microscopic scale from both theoretical and experimental perspectives. r 2011 American Chemical Society
Atomic force microscopy (AFM) has shown to be a very important tool in the study of localized surface structures and properties. EW effects studied at nanoscale by using force spectroscopy method of AFM9 provided complementary information to that obtained by macroscopic techniques. The nanometer scale distance between tip and sample gives rise to extremely high electric field values, permitting the study of the EW phenomenon at a wider range of electric field strengths. In previous report,9 the adhesion force Fadh under the influence of the EW phenomena showed similar result as the macroscopic contact angle measurements at different electric field strengths. Fadh measured between a conductive AFM tip and a smooth dielectric film was expressed as a function of the external applied voltage (V)9 Fadh ¼ c1 þ c2 V 2
ð1Þ
Received: March 16, 2011 Revised: May 7, 2011 Published: May 23, 2011 7603
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Langmuir where c1 and c2 are constants, both of them are dependent on tip characteristics and meniscus geometry, and c2 also depends on dielectric properties or capacitance of the tipsample system.9 EW phenomenon following eq 1 is considered as ideal EW, which is an approximation based on the widely used Young Lippmann equation4,5,1013 and considering the tip with a spherical ending. In the force spectroscopy method the adhesion force is considered as the force required to break up the water meniscus formed between the tip and the sample. A disadvantage in using force spectroscopy method is that the adhesive behavior is highly affected by the relative humidity (RH). For relatively low or relatively high RH values, the adhesion force between the tip and the sample surface is very small, resulting in the increase of the relative errors in the adhesion measurements. Parallel to this, friction force microscopy (FFM) is another important tool for surface analysis. It provides nanoscale spatial resolution with rich information, such as surface chemical composition,14,15 molecular organization,1618 mechanical properties,19 etc. During the FFM measurement, the torsion or twisting of the cantilever is recorded while scanning along the direction perpendicular to the long axis of the rectangular cantilever. At environmental conditions a thin water film covers the sample surface; thus, the frictional signal also reflects the contribution of the adhesion force, which is highly affected by capillary phenomena. Therefore, it is also feasible to study the electrowetting phenomenon by using the FFM method. In this case the tip is always in contact with the sample surface, the intermolecular adhesive forces are commonly associated with an “internal” load.20 In this work we demonstrate the feasibility of using friction force microscopy method to study the electrowetting effect on the poly(methyl methacrylate) (PMMA) surface. Transition from ideal EW to EW saturation can be observed by studying the frictional behavior between the tip and the sample surface. This transition is related to changes in the interfacial contact between the tip and the PMMA surface. Transitions in the EW phenomenon can be clearly identified by observing the frictional force measurement errors. In addition, we can also obtain the change of the relative friction coefficient with the electric field in the ideal EW region.
2. EXPERIMENTAL SECTION A dielectric thin film of PMMA was prepared by spin-casting PMMA solution in chlorobenzene on Si(100) substrate doped with phosphorus (24 Ω 3 cm) (from Silicon Quest Int’l), followed by solvent evaporation at 90 °C for 5 h. FFM measurements were carried out by using a commercial AFM (D3100, Nanoscope II, Veeco Metrology) under environmental conditions. The experiments were repeatedly conducted over the course of several days to establish reproducibility, during which the RH values were ranged from 30% to 65%. A rectangular cantilever coated with PtIr with spring constant of 4.6 N/m and resonant frequency of 75.5 kHz was used to scan the surface in contact mode and as a top electrode. In all the FFM experiments, the PMMA sample surface was scanned within one scan area (2 μm 2 μm) with a fixed tip velocity (4 μm/s) in order to avoid systematic errors. In a data set the same tip is used to avoid tip-dependent variations. In order to calibrate normal forces two steps are carried out. First, photodetector sensitivity is calibrated by measuring a forcedistance curve on a very stiff sample. Mica is used as the substrate because the stiffness of the mica is large enough compared with that of the cantilever, and the surface is atomically flat to avoid topographic effects. Then the deflection during the force curve measurement is predominantly due to
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Figure 1. Tapping mode AFM image of PMMA surface. The scan size is 2 μm 2 μm. cantilever deflection. In the photodetector signal versus displacement plot, the photodetector sensitivity is the gradient at the repulsive region. The cantilever spring constant was measured from the thermal spectra of the cantilever response by considering this as a harmonic oscillator driven by thermal noise. During the FFM measurement, the scanning direction was fixed perpendicular to the long axis of the cantilever beam. The magnitude of the friction force is principally proportional to the degree of torsion or twisting of the cantilever, which gives rise to a differential signal in the photodetector given in unit of millivolts. The sample has been scanned for different values of applied voltage. Under the same potential condition 512 sample lines with trace and retrace per line are recorded for obtaining the averaged photodetector response signals (FFM signal, FS). Because of the fact that the photodetector may not be properly aligned with respect to the laser beam, FS for each sample line is calculated as the average between the absolute values of the trace and the retrace signal. The average friction force associated with a specific value of the applied voltage is determined by Gaussian fitting of the histogram of those 512 mean FS values. Applied load is obtained by multiplying the cantilever spring constant, the tip sensitivity in normal direction, and the difference between the deflection set point and the deflection signal when the tip is in its free state. By varying the normal load applied to the tip, it is possible to construct frictionload plots. The gradients of the frictionload plots were used to obtain relative friction coefficients. The coefficients of friction were normalized with respect to the highest friction coefficient value obtained. In order to obtain the topography of the PMMA sample, the same tip was used working at around its resonant frequency in tapping mode. The electric field strength is estimated as E = V/d (rough estimation as a parallel plate capacitor), where V is the external applied voltage and d the dielectric thickness. In any case, E is a magnitude proportional to the applied voltage.
3. RESULTS AND DISCUSSION The topographic image of the PMMA surface is shown in Figure 1. By using tapping mode AFM, the root-mean-square (rms) roughness of the PMMA surface was obtained at about 0.24 nm (scan size of 2 μm 2 μm), indicating the very good smoothness of the sample. Figure 2 shows, for different external electric field values, the FS as a function of the applied load for the sample shown in Figure 1 under environmental conditions. The nonzero FS at zero applied load is completely associated with 7604
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Table 1. Parameter Values Obtained from Fitting the FS Values versus Applied Loads under Different External Electric Field Strengths with eq 4
Figure 2. Photodetector response as a function of the applied load for PMMA surface under environmental conditions.
adhesive interactions. Adhesion between the tip and sample maintains the contact although, in some cases, there is a negative (tensile) load.20 Some of the data in Figure 2 are extended to substantial negative loads, reflecting maybe an augmentation of capillary force. It has been considered for several investigators that frictional force is proportional to the true contact area between surface asperities. Depending on the plastic or elastic deformation, the relationship between contact area and normal force will be different.21 The friction force will follow substantially nonlinear behavior with the applied load in case of elastic deformation,16,22 while linear relationship in case of plastic deformation.1417 The linear relationship can be modeled by using the modified version of Amonton’s law,23 which has a vertical offset based upon the adhesive force: Ff ¼ μðFL þ f0 Þ
ð2Þ
where Ff and FL are the friction force and the applied load respectively, μ is the friction coefficient, and f0 is associated with the total adhesive force. In the FFM measurements, FS is proportional to the friction force (Ff). The theories developed by Hertz,24 JohnsonKendall Roberts25 (JKR), and DerjaguinM€ullerToporov2628 (DMT) describe the interaction and contact area between a plane surface and a spherical tip when elastic deformations are involved. Equation 3 can be used to model the friction force between a sphere and a flat surface that are elastically deformed29 Ff ¼ kðR
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4=3 f0 þ FL þ f0 Þ
ð3Þ
where k depends on the material properties of the two interacting bodies and R varies from 0 to 1 (DMT and JKR limits, respectively), describing the intermediate regime between these two limits. Considering that most samples have a mixed behavior (elastic and plastic deformation) and that the tip ending has spherical shape, both of them, linear and nonlinear behaviors of the friction force as a function of the applied load, should be taken into account. Thus, we propose eq 4 with two terms by combining linear relationship23 (eq 2) with nonlinear relationship (eq 3)29
R
f0
adj R-square
0
0
105.165 99
0.995 63
0
0
157.294 72
0.997 84
0.156 28
0
0
298.7551
0.989 11
0.6
0.135 33
0.250 39
1
222.967 68
0.989 43
0.8 1.0
0.132 29 0.151 93
0.3188 0.244 23
1 1
227.165 31 268.613 92
0.995 25 0.997 45
1.2
0.151 19
0.225 65
1
298.033 46
0.996 68
1.4
0.106 78
0.520 58
1
221.525 31
0.997 13
E (MV/cm)
k1
0.0
0.186 45
0.2
0.170 27
0.4
k2
Ff ¼ k1 ðFL þ f0 Þ þ k2 ðR
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4=3 f0 þ FL þ f0 Þ
ð4Þ
where k1, k2, R, and f0 are parameters, and the parameter f0 is associated with the adhesion force between the tip and the sample. The data in Figure 2 are fitted by eq 4. During the fitting k1, k2, and f0 are forced to be greater than or equal to 0 and 1 g R g 0. Equation 4 contains the specific cases depending on the value of the parameters. Completely plastic deformation is considered at k2 = 0 and the elastic limit at k1 = 0. Under the elastic approximation, R = 1 and R = 0 correspond to the JKR and DMT limits, respectively. Equation 4 is in agreement with the general equation for fitting contact area proposed by Carpick et al.30 The fitting results are shown in Table 1. It can be noted that k2 = 0 for external electric field values lower than or equal to 0.4 MV/cm. This region coincides with the EW region identified in reference.9 The relationship between FS and the applied load is linear, which means that the second term of eq 4 can be neglected. This linear behavior is consistent with the modified version of Amonton’s law (eq 2). In Table 1 can be noted that for electric field values above 0.4 MV/cm the relationship between FS and the applied load has a nonlinear contribution. This is indicative of different interaction between the tip and PMMA surface for external applied electric fields below and above 0.4 MV/cm. These differences in the contact mechanism may be closely related with the adhesion force changes during the transition from the ideal EW region to the saturation region at 0.4 MV/cm.9 In this experiment, the applied electric field might cause local changes in the dielectric material, which can affect the dielectric properties of the system. Thus, variations observed in the contact mechanism around field strength of 0.4 MV/cm (transition from ideal EW to EW saturation) might be associated with an abrupt change in the dielectric properties of the polymer as mentioned above in eq 1. Adhesion force between the AFM tip and the PMMA surface as a function of the applied electric field is plotted in Figure S1 of the Supporting Information. Regions denoted with discontinuous lines in Figure S1 are consistent with regions observed in reference9 (from 0 to 0.4, 0.4 to 1.1, and greater than 1.1 MV/cm). Adhesion measurements were carried out considering a nominal spring constant of 0.3 N/m. Transitions between characteristic regions in the EW phenomena were clearly identified in the adhesive behavior.9 The adhesion force first of all increases until the external applied electric field reaches the value of ∼0.4 MV/cm and then remains approximately constant until 7605
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FS ¼ R1 þ R2 V 2
Figure 3. (a) Photodetector response versus the external applied electric field for an external load of 64 nN. The red line is the quadratic fitting of the first six points of the data. (b) Standard deviation of the photodetector response values shown in the left panel.
around 1 MV/cm, and a decrease is observed in the third region. The adhesion force decrease might be related to polarization of water molecules at the solidliquid interface.9 Hydrogen bonding between surface molecules can be affected by this polarization and therefore modify the magnitude of the interfacial surface tension.9 On the other hand, it is also possible that the application of extremely high electric field causes ionization in the liquid film and hence the breakdown of the capacitor formed by the liquid and the dielectric layer.9 Even though adhesion force values were relatively small, the general behavior for different electric field values is in good agreement with previous results.9 It has been previously observed16,22 that in systems where there is a lower adhesive interaction between the tip and the sample surface friction force principally varies linearly with the applied load, whereas the friction force varies in a nonlinear fashion with relatively high tipsample adhesive interaction. EW at nanoscale depends on water film thickness.3133 The adhesive behavior, which contributes to the friction force at this scale, is highly affected by the relative humidity, i.e., the water film thickness. For this reason, we have conducted our experiments repeatedly over the course of several days to establish reproducibility, during which the RH values were ranged from 30% to 65%. It could be concluded that the water film thickness in this RH region maintained approximately constant, or its variations did not affect the friction force behavior. The FS dependence on the applied electric field is shown in Figure 3a. Measurements were carried out at a constant load force of 64 nN. It can be observed that FS follows approximately a quadratic behavior when the electric field is lower than 0.4 MV/cm. This is in good agreement with the first region observed in previous results by using the force spectroscopy method,9 which can be called ideal EW region in accordance with the YoungLipmann equation. By increasing the applied external electric field, a marked deviation in the FS from the quadratic behavior can be observed above 0.4 MV/cm. FS still increases by increasing the external voltage but considerably lower than that predicted by the quadratic behavior. This deviation of the friction signal from the quadratic fashion can be associated with the saturation effect, which is consistent with our previous report by using force spectroscopy method9 and widely reported using contact angle measurements. As mentioned above, FS is proportional to the magnitude of friction force. Taking this proportionality into account and considering eqs 1 and 2, FS for a constant load force can be related, in the EW region, to the applied external voltage as
ð5Þ
where parameters R1 and R2 are dependent on tip and sample characteristics, photodetector sensitivity, and friction coefficient. Additionally, parameter R1 also depends on the applied load and R2 depends on the tipsample capacitance. By fitting a parabola (y = A þ Bx2) to the data points, FS can be described as FS = 24.8 þ 0.028V2 (FS given in millivolts and V in volts), with a regression coefficient of 0.975. Although adhesion force showed a decrease for extremely high electric field values (region 3 in Figure S1), this was not observed in the FS behavior. The reason for this may be associated with the fact that the FS dependence on the adhesive force for relatively high electric field values has a nonlinear contribution. Errors from the photodetector response measurement, indicated as the standard deviation, are shown in Figure 3b in further detail. Around 0.4 MV/cm a local maximum in the standard deviation can be identified. Larger standard deviation is generally associated with instability in the system, which can be considered as a typical behavior of transitional process. This local maximum in the standard deviation may be directly related to the transition from ideal EW region (region 1) to EW saturation region (region 2). For extremely high electric field values an increase in the measurement errors was observed. It is clear that at about 1.1 MV/cm the standard deviation drastically increase. This phenomenon along with the local maximum observed at ∼0.4 MV/cm suggests that the standard deviation in the friction force measurement can be a good indicator of the EW transition other than electronic noises originated from application of strong external electric field. Standard deviations remain high for almost all measurements in region 3 (electric field values above 1.1 MV/cm). At about 1.1 MV/cm coincidentally a dramatic decrease in the behavior of the adhesion force is observed in Figure S1. Thus, the augmentation of the standard deviation in region 3 may be directly associated with the adhesive force breakdown, previously reported.9 Although FS remains approximately constant in region 3, relatively high measurement errors indicate instability in the system with occurrence of dramatic changes at extremely high electric field values (region 3). Standard deviation of the FS measurements at the transition from EW saturation to EW breakdown (from region 2 to region 3) is much higher than the standard deviation at the region where the transition from ideal EW to EW saturation (regions 1 and 2) occurs, suggesting much higher instability, which might be caused by polarization of the water molecules in this second transition. This suggests that the two transitions previously reported9 in the EW phenomenon (from ideal EW to EW saturation and from EW saturation to EW breakdown) might be caused by different factors, such as dielectric breakdown and water molecules polarization, respectively. Figure 4 portrays FS for two different electric field values (0.09 and 1.09 MV/cm) as an example to compare the measurement errors. It shows the average of the absolute values of trace and retrace signals in one of the sample lines (Figure 4a). It can be noted that the standard deviation value of a sample line at 1.09 MV/cm is almost twice of the standard deviation value of a sample line at 0.09 MV/cm. This marked difference was also noted in the averaged FS from 512 sample lines (Figure 4b). Figure S2 in the Supporting Information shows the Gaussian fitting of the histogram of 512 FS values (one per sample line) 7606
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Figure 4. (a) Average lateral photodetector response between trace and retrace in a typical sample line for two electric field values (0.09 and 1.09 MV/cm). “Scan line” represent the scan size along which a trace and retrace cycle is made. (b) Lateral photodetector response, along the direction of the cantilever long axis (considering the average frictional signals of the sample lines) for two different electric field values (0.09 and 1.09 MV/cm). Solid horizontal lines represent the mean values, and the dashed lines represent the standard errors.
Figure 5. Normalized friction coefficients versus the external electric field (region 1). Friction coefficients were determined from the slopes of FSload plots for electric field values within the ideal EW region as shown in the inset.
for external applied voltage of 0.09 and 1.09 MV/cm. The wider distribution of FS at 1.09 MV/cm further confirms the high instability in the system in the EW breakdown region (region 3). Because of the linear dependence of the lateral friction signal on the applied load in region 1 (inset of Figure 5), the influence of the external electric field on the friction coefficient could be studied. Slopes of FSload plots were used to determine the relative friction coefficient values (Figure 5), normalized to the largest value. The result indicates a slight decrease of apparent friction coefficient with the increase of the applied external electric field within the ideal EW region. Friction coefficient at 0.4 MV/cm fell to 77% of the value at zero external applied electric field. Even if we cannot compare the friction coefficients obtained in this work to absolute values, we observed excellent reproducibility when comparing these coefficients among each other. In our case, these variations in surface friction due to the application of an external electric field may be closely connected with internal modifications of the material, since variations in the dielectric properties of the system have also been previously considered. AFM, in general, could be a very strong tool to consider in future studies of adhesive and frictional material properties under EW conditions.
4. CONCLUSIONS We demonstrated here the use of friction force microscopy method to study electrowetting phenomena. Lateral photodetector response has shown an approximately quadratic dependence on the external applied voltage for electric field values below 0.4 MV/cm, identifying this region as ideal electrowetting region. Differences in the contact mechanism between the AFM tip and the PMMA surface for electric field values below and above 0.4 MV/cm were observed by using FFM method, suggesting a close relation between this change and the transition from the ideal EW region to the EW saturation behavior. In the ideal EW region a reduction of the friction coefficient was observed when increasing the applied electric field. Errors in the friction force measurements are associated with transitional behavior in the electrowetting process and can be used as an indicator of the EW transitions. ’ ASSOCIATED CONTENT
bS
Supporting Information. Adhesion force versus electric field plots and histograms of the photodetector response per sample line. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*Tel: 86-10-82545559. Fax: 86-10-62656765. E-mail: yangyl@ nanoctr.cn (Y.L.Y.),
[email protected] (C.W.).
’ ACKNOWLEDGMENT Financial support from the National Natural Science Foundation of China (Grant No. 20873033) and National Key Project for Basic Research (Grant Nos. 2007CB936800 and 2011CB932800) is gratefully acknowledged. Reynier Revilla also acknowledges the financial support from the Ministry of Science and Technology of China (MOST). ’ REFERENCES (1) Hou, L.; Smith, N. R.; Heikenfeld, J. Appl. Phys. Lett. 2007, 90, 251114. 7607
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(2) Bergea, B.; Peseux, J. Eur. Phys. J. E 2000, 3, 159. (3) Yang, S.; Krupenkin, T. N.; Mach, P.; Chandross, A. Adv. Mater. 2003, 15, 940. (4) Blake, T. D.; Clarke, A.; Stattersfield, E. H. Langmuir 2000, 16, 2928. (5) Decamps, C.; De Coninck, J. Langmuir 2000, 16, 10150. (6) Kuo, J. S.; Spicar-Mihalic, P.; Rodriguez, I.; Daniel, T. C. Langmuir 2003, 19, 250. (7) Prins, M. W. J.; Welters, W. J. J.; Weekamp, J. W. Science 2001, 291, 277. (8) Lee, J.; Moon, H.; Fowler, J.; Schoellhammer, T.; Kim, C.-J. Sens. Actuators, A 2002, 95, 259. (9) Guan, L.; Qi, G.; Liu, S.; Zhang, H.; Zhang, Z.; Yang, Y. L.; Wang, C. J. Phys. Chem. C 2009, 113, 661. (10) Quinn, A.; Sedev, R.; Ralston, J. J. Phys. Chem. B 2003, 107, 1163. (11) Zhu, L.; Xu, J.; Xiu, Y.; Sun, Y.; Hess, D. W.; Wong, C.-P. J. Phys. Chem. B 2006, 110, 15945. (12) Wang, Z.; Ou, Y.; Lu, T.-M.; Koratkar, N. J. Phys. Chem. B 2007, 111, 4296. (13) Mugele, F.; Jean-Christophe Baret, C.-J. J. Phys.: Condens. Matter 2005, 17, R705. (14) Foster, T. T.; Alexander, M. R.; Leggett, G. J.; McAlpine, E. Langmuir 2006, 22, 9254. (15) Brewer, N. J.; Beake, B. D.; Leggett, G. J. Langmuir 2001, 17, 1970. (16) Whittle, T. J.; Leggett, G. J. Langmuir 2009, 25, 2217. (17) Brewer, N. J.; Leggett, G. J. Langmuir 2004, 20, 4109. (18) Chen, J.; Ratera, I.; Murphy, A.; Ogletree, D. F.; Frechet, J. M. J.; Salmeron, M. Surf. Sci. 2006, 600, 4008. (19) Dinelli, F.; Buenviaje, C.; Overney, R. M. J. Chem. Phys. 2000, 113, 2043. (20) Gao, J.; Luedtke, W. D.; Gourdon, D.; Ruths, M.; Israelachvili, J. N.; Landman, U. J. Phys. Chem. B 2004, 108, 3410. (21) Butt, H.-J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1. (22) Hurley, C. R.; Leggett, G. J. Langmuir 2006, 22, 4179. (23) Derjaguin, B. V. Z. Phys. 1934, 88, 661. (24) Hertz, H.; Angew, J. R. Math. 1882, 92, 156. (25) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, A 1971, 324, 301. (26) Derjaguin, B. V.; M€uller, V. M.; Toporov, Y. P. J. Colloid Interface Sci. 1975, 53, 314. (27) M€uller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1980, 77, 91. (28) M€uller, V. M.; Derjaguin, B. V.; Toporov, Y. P. Colloids Surf. 1983, 7, 251. (29) Schwarz, U. D. J. Colloid Interface Sci. 2003, 261, 99. (30) Carpick, R. W.; Ogletree, D. F.; Salmeron, M. J. Colloid Interface Sci. 1999, 211, 395. (31) Daub, C. D.; Bratko, D.; Leung, K.; Luzar, A. J. Phys. Chem. C 2007, 111, 505. (32) Bratko, D.; Daub, C. D.; Leung, K.; Luzar, A. J. Am. Chem. Soc. 2007, 129, 2504. (33) Bratko, D.; Daub, C. D.; Luzar, A.; Salmeron, M. Faraday Discuss. 2009, 141, 55.
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