Scanning Force Microscopy Based Rapid Force Curve Acquisition on Supported Lipid Bilayers: Experiments and Simulations Using Pulsed Force Mode

Scanning Force Microscopy Based Rapid ForceCurve Acquisition on Supported Lipid Bilayers:Experiments and Simulations Using PulsedForce ModeStephanie Kr¸ger, Daniel Kr¸ger, and Andreas Janshoff*[a]

Introduction

In situ imaging of surface confined biological and artificialmembranes by means of scanning force microscopy (SFM) haschanged the way we envisage membrane-associated process-es.[1±5] Besides delivering images of membrane topographydown to the submolecular level,[4] SFM offers the advantage ofmanipulating membranes and their constituents, such as trans-membrane proteins, and visualizing phase separation in artifi-cial bilayers by means of material contrast.[6, 7] The spatially re-solved detection of material properties, such as elasticity,breakthrough force, and surface charge density has greatimpact for the understanding of the role of certain lipids in bi-ological processes, such as receptor clustering and membranefusion and fission.[2,3,8±13] In particular, the detection of lipidrafts, a liquid-ordered lipid phase with lateral dimensions inthe nanometer range, which is thought to be responsible formany membrane associated processes, may only be achievedby imaging material properties such as stiffness, surface chargedensity, or adhesion.[6,13±16]

To date, material contrast of thin organic films has success-fully been investigated by means of phase imaging employingintermittent contact mode (tapping mode).[17±20] The theoreticalbasis of tapping mode, in which a cantilever is excited close toits resonance, has been thoroughly investigated.[17,21±24] For in-stance, depending on the contact area, it is possible to esti-mate differences in elasticity from phase-lag images as pointedout by Magonov et al.[17] However, the interpretation of phaseimages is still a matter of debate. It is generally accepted thatphase contrast can be attributed to energy dissipation differ-ences as discussed by Cleveland et al. , assuming harmonicmotion of the cantilever in the contact regime in order toobtain an analytical expression for the phase angle as a func-tion of the dissipated energy.[23] Noncontact and intermittentcontact mode SFM are well-suited to record high resolution

images in air and vacuum. However, in order to obtain reliableand interpretable information in terms of material propertiesof the sample, intermittent contact mode is not always thebest choice. Pulsed force mode (PFM)±SFM, however, althoughless sensitive than intermittent contact mode SFM due to oper-ation off resonance, allows reliable and interpretable data–interms of stiffness and adhesion–to be extracted.[6,25±36]

PFM±SFM combines the desired speed with high data rates,spatial resolution, and information from force curve analy-sis.[34,36] In PFM±SFM, the z-piezo is excited in a sinusoidal fash-ion, usually covering a frequency range of 100±2000 Hz. Fromthe raw data that contains the voltage provided by the posi-tion sensitive photodiode as a function of time, the PFM forcecurve can be calculated if the force constant of the cantileveris known (Figure 1). The difference between operating PFM±SFM in air and water is mainly governed by the higher qualityfactor in air leading to a characteristic ring down of the cantile-ver after successful detachment from the surface (Figure 1B).

Appropriate trigger settings enable one to extract local var-iations in stiffness and adhesion while scanning in the x±y di-rection. These images can be obtained simultaneously to thetopography data. The maximum force (Figure 1B) is used asthe setpoint/feedback signal for measuring the topography. Al-though the use of high-speed force curve sampling modessuch as PFM is straightforward with the cantilever operated ina gaseous environment, its operation in liquids requires adeeper understanding of the governing mechanisms. The criti-cal parameter that causes this difference between the two

[a] Dr. S. Kr¸ger, Dr. D. Kr¸ger, Prof. Dr. A. JanshoffJohannes Gutenberg Universit‰t MainzInstitut f¸r Physikalische Chemie, Jakob Welder Weg 1155128 Mainz (Germany)E-mail : [email protected]

In situ pulsed force mode scanning force microscopy (PFM±SFM)images of phase separated solid-supported lipid bilayers are dis-cussed with the help of computer simulations. Simultaneousimaging of material properties and topography in a liquid envi-ronment by means of PFM±SFM is severely hampered by hydro-dynamic damping of the cantilever. Stiffness and adhesionimages of solid-supported membranes consisting of cholesterol,sphingomyelin, and 1,2-dioleyl-phosphatidylcholine obtained in

aqueous solution exhibit contrast inversion of adhesion and stiff-ness images depending on parameters such as driving frequency,amplitude, and trigger setting. Simulations using a simple har-monic oscillator model explain experimental findings and give adeeper insight into the way PFM±SFM experiments have to beperformed in order to obtain interpretable results and hencepave the way for reliable material contrast imaging at highspeed.

ChemPhysChem 2004, 5, 989 ± 997 DOI: 10.1002/cphc.200301059 ¹ 2004 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim 989

media is the cantilever damping, which is drastically increasedin a liquid (e.g. , the dynamic viscosity of water is about fiftytimes higher than for dry air). At these magnitudes of damp-ing, a measurable phase lag between the driving force and themeasured force can appear; this has been shown to lead topossible misinterpretation of the adhesion data.[37] As ourscope is a quantitative analysis, we will initiate the investiga-tions with simulations of the PFM under liquid conditions. Theperformed simulations enable us to understand which systeminformation is extractable and to optimize the operating pa-rameters regarding speed and stability.

Results and Discussion

The method used for simulating the PFM mode is similar toone described earlier.[24] Figure 1A displays the essential geom-etry of the force probe and the sample surface. The modelsystem is a driven harmonic oscillator, Equation (1)

meff€zðtÞ þ a _zðtÞ þ kczðtÞ þ FTSðzÞ ¼ FPðtÞ ð1Þ

where meff, kc, and a are the effective mass, the force constantof the cantilever, and the damping constant, respectively.Equation (2) denotes the sinusoidal driving force used inpulsed force mode with a driving amplitude A0 (F0=A0kc) anda driving frequency fd.

FPðtÞ ¼ kcðA0sinð2pfdtÞ þ dstartÞ ð2Þ

FTS(z(t)) describes the tip-sample force at a given tip-sampledistance z(t), whereas dstart is the time-averaged position of thez-piezo (Figure 1A).

Commonly in SFM, the cantilever is piezo driven, thus thetip-sample force in Equation (1) does not directly reflect themeasured force, which is rather obtained from Equation (3)

FðtÞ ¼ kczðtÞ�FPðtÞ ð3Þ

Equation (1) was integrated numerically as described in theExperimental Section.[38] Even for a qualitative comparison ofthe simulation with the experimental force traces, it is necessa-ry to find good estimates for kc, a, meff, and the driving ampli-

tude A0. Here, kc has been calcu-lated from thermal noise spectraof the cantilever in air ;[39] meff

was estimated from the reso-nance curve in water. A0, whichis usually expressed in relativevalues, can be calculated fromthe PFM force curve amplitudewith the maximum force sethigh enough, and A0 set lowenough, to assure permanentcontact with the sample. a canbe obtained from the hydrody-namic dragging force maximumin the noncontact regime of the

oscillation. Noteably, more elaborate models are available thattake into account the exact geometry of the cantilever. Herein,we restrict ourselves to the simplified assumption that hydro-dynamic damping is represented by a constant a, which de-scribes our experimental data sufficiently well.[40,41]

The most challenging problem to be solved for simulatingforce±distance curves is, however, the assumption of a realisticinteraction potential. Conventionally, the SFM tip apex is de-scribed by a smooth parabolic half-space, or an elastic spherewith radius R. Assuming that the penetration depth is smallcompared to the extension of the elastically homogenousregion of the sample, the sample can be regarded as an ideallysmooth elastic half-space.

However, even the simple case of a purely linear elastic andfriction free contact between an elastic sphere and an infiniteelastic half-space lacks an exact analytic solution if attractiveforces are involved in the contact model. The analytical Hertzmodel describes only the case where attractive forces areabsent.

Approaches to include adhesion phenomena with certainsimplifications have been introduced by Derjaguin, Muller, andToporov (DMT)[42] and Johnson, Kendall, and Roberts (JKR).[43]

The models represent good analytical approximations for spe-cial cases of the tip-sample contact. The DMT model assumes aHertz-type shape of the contact and neglects the influence ofadhesion on the bodies geometry, whereas the JKR modeluses a thermodynamic approach neglecting the finite range ofthe attractive forces.[43] Both analytical models are inadequatefor describing the transition between contact and separationof two bodies–in particular at higher amplitudes. At the pointof contact, the force±distance relationship is not mathematical-ly differentiable in the DMT model, and is not steady in theJKR model.

The interaction between tip and surface is more completelydescribed by the MYD (Muller, Yushenko, Derjaguin) model,which treats contact without simplifications by considering thetype of the interaction force and its influence on elastic defor-mations.[44,45] Therefore, we calculated the interaction forceFTS(z) using the MYD model. Within this model, a general solu-tion of the contact problem allows us to model long-range ad-hesive forces by numerically solving the integral equation forthe shape xs(r,z) of the gap between tip and sample, a nonlin-

Figure 1. Geometry and parameters in pulsed force mode±scanning force microscopy. A) z(t) is the time-dependenttip-sample distance, A0sin(2pfdt)+dstart describes the movement of the z-piezo that is driven by a sinusoidal voltage,and zc(t) the cantilever deflection as detected by the photodiode. B) Typical PFM force curve of a cantilever operatingin air showing the ring down after detaching from the surface.

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A. Janshoff et al.

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ear Fredholm equation of second order with a singular kernel,Equation (4)[44]

xsðr,zÞ ¼r2

2R�z þ 16

3pK

Z1

0

p½xsðr0,zÞr0

r0 þ rI

�2

ffiffiffiffiffir0r

p

r0 þ r

�dr0 ð4Þ

where I is the complete elliptic integral of first order and p(xs)is a pressure function describing the surface forces betweenparts of the surfaces under consideration. In our case, we useda pressure function based on the Lennard±Jones potential andelectrochemical double layer (EDL) forces that arise due tocharged tip and sample surfaces, Equation (5)[44,46]

p½xsðr,zÞ ¼H

6pr30

��r0xs

�9

��

r0xs

�3�

þ 2e0er

�s2S þ s2

T þ sSsTðexskD þ e�xskDÞðexskD�e�xskDÞ2

� ð5Þ

where H is the Hamaker constant (~10�21 J) and r0 is the equili-brium distance.[47] sS and sT are the surface charge densities ofthe sample and the tip, respectively. kD is the inverse Debyelength, e0 is the permittivity of free space, and er is the dielec-tric constant.

The interaction force FTS(z) is then approximately determinedby simple integration of p[xs(r,z)] over the surfaces involved(the rapid pressure decay allows us to set the upper limit to in-finity), Equation (6)

FTSðzÞ ¼Z1

0

p½xsðr,zÞ2prdr ð6Þ

The shape xs(r,z) and consequently FTS(z) may change discon-tinuously with the distance z for very elastic bodies, as alsopredicted by the JKR model : strong attractive (adhesion) forcescan cause a mechanical instability of the surfaces resulting in ajump-to-contact of the two bodies.

As FTS(z) is calculated numerically, we will only obtain a dis-crete set of data FTS(zi) for the interaction potential and the cor-responding forces. To perform calculations in an acceptabletime, we interpolated FTS(z) by means of a cubic spline.

In Figure 2, numerical solutions for the pressure function[Eq. (5)] employing different Young moduli and surface chargedensities for the sample are shown. The simulated force curvesdemonstrate the impact of attractive and repulsive electrostat-ic forces and the strong indentation of soft materials by thetip. We assumed ES=20 MPa for a lipid bilayer in the liquidstate and 80 GPa for mica. The Young modulus of the tip wasvery high (ET=192 GPa for Si3N4).

In the following section, we will describe the impact of vari-ous parameters such as the force constant of the cantilever,the driving frequency of the z-piezo, the amplitude, the surfacecharge density of the sample and its Young modulus on theappearance of PFM force curves. Our objective is to give direc-tions for obtaining reliable material contrast images of soft ma-terials in aqueous solution at reasonably high scanning speeds.

Figure 3 shows PFM simulations of a typical situation as itoccurs during imaging of soft matter, such as solid-supportedlipid bilayers in aqueous solution. The forces between tip andsample FTS are modeled by the general MYD±EDL potentialunder constant charge conditions [Eq. (5)] . Figure 3A showsPFM force curves as a function of the driving frequency. It isapparent that higher frequencies result in stronger hydrody-namic damping since the velocity z« of the cantilever is in-creased [Eq. (1)] . Hydrodynamic damping is characterized by asecond force maximum directly before contact with thesample is established.

Even at these–for PFM operation–very low frequencies(100 Hz is still fast compared to conventional force±distancecurves) and small amplitudes (F0=400 pN), hydrodynamicdamping might be problematic and can obscure more subtleeffects arising, for instance, from double layer interactions. Onemajor consequence of strong hydrodynamic damping is thatneither the baseline nor the adhesion force can be determinedaccurately. Sometimes, as discussed later, stiffness informationmight also be obscured by hydrodynamic forces. Interestingly,hydrodynamic forces might exceed the maximum contactforce, which is used as the feedback signal in PFM imaging(Figure 3A, fd=3 kHz). In this case, not even the stiffness canbe extracted from the curve.

Figure 3B demonstrates the influence of the spring constantkc of the cantilever on the hydrodynamic damping. Cantileversexhibiting smaller force constants are more susceptible to hy-drodynamic damping than are stiff cantilevers. However, in-creasing the force constant also decreases the force resolution(DF/

ffiffiffiffikc

p). Depending on the situation, choosing a stiffer can-

tilever might easily avoid interference from hydrodynamicdamping. Figure 3C displays the influence of the surface

Figure 2. Numerical simulation of the interaction force between tip and sampleFTS(z) according to the MYD±EDL potential as described in the text. The impactof the surface charge density of the sample and its Young modulus are shown.The Young modulus of the tip (silicon nitride) was assumed to be 192 GPa. TheHamaker constant was set to 7î10�21 J, the radius of the tip R=20 nm, thePoisson ratio n=0.33, the equilibrium radius r0=0.3 nm, and the surfacecharge density of the tip sT was assumed to be �0.032 Cm�2. The Debyelength k�1

D was 0.75 nm.

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Rapid Force Curve Acquisition


charge density of the sample on the appearance of the PFMforce curve. Even at high-speed force recording (fd=300 Hz),electrostatic interaction might be mapped by setting the trig-ger appropriately. Quantitative data, however, are not accessi-ble due to interference from hydrodynamic damping. Adhe-sion forces are overestimated and the detection of stiffnessvariations becomes cumbersome if the difference betweencontact point, that is, when the tip touches the surface andthe maximum force is too small. Electrostatic interactions be-tween the tip and sample surface contribute to the slope, asshown in Figure 3C, and consequently might be responsiblefor an apparent stiffness contrast. Electrostatic repulsion re-duces the slope and thus might be falsely detected as a softermaterial. The benefit, on the other hand, is the possibility ofqualitatively mapping charge densities on the surface.[33,35, 36]

Figure 3D shows the impact of surface stiffness on the PFMcurve. Larger Young moduli result in a steeper slope and canreadily be detected if hydrodynamic damping is not toostrong.

To be consistent with the experimental procedure, the maxi-mum force as the feedback signal was held constant in thesimulations. In summary, the immense increase in viscosity ren-ders measurements in water severely influenced by hydrody-namic damping of the cantilever, whereas in air even free oscil-lation can be observed. Faster movement of the cantilever alsoleads to an increased damping, as apparent from the increasedhydrodynamic force maximum with increasing z-scanningspeed. Compensation of hydrodynamic damping might beachieved by using stiffer cantilevers, however, at the expenseof losing force resolution.

For the sake of simplicity, in the following simulations we as-sumed the interaction force FTS to be harmonic (sample stiff-ness ks) with the tip being in contact with the sample (FTS(z)=�ksz, z�0), and zero elsewhere (FTS(z)=0, z>0). This simplifi-cation does not influence the principle statements concludedfrom the simulations.

The comparison of experimental and simulated force curvesfor different driving amplitudes as shown in Figure 4 illustrates

Figure 3. A) Computer simulation of PFM force curves at various frequencies: 3000 Hz (c), 2000 Hz (b), 1500 Hz (g), 1000 Hz (±¥±), 500 Hz (±¥¥), 100 Hz(a), 10 Hz (d) by employing a general MYD±EDL potential to describe the tip-sample interaction force. The Young modulus of the sample ES=20 MPa andthat of the tip (silicon nitride) ET=192 GPa. The Hamaker constant H was 7î10�21 J, the radius of the tip R=20 nm, the Poisson ratio n=0.33, and the equilibriumradius r0=0.3 nm. The charge of the tip sT was assumed to be �0.032 Cm�2 and that of the sample sS=++0.003 Cm�2. The Debye length k�1

D was 0.75 nm. For theharmonic oscillator, the following system parameters were chosen: meff=36 ng, a=5260 nNsm�1, kc=0.1 Nm�1. The arrow indicates the occurrence of a secondmaximum due to hydrodynamic damping. B) Influence of the cantilever force constant kc on the hydrodynamic damping force, 0.01 Nm�1 (thick black line),0.03 Nm�1 (b), 0.05 Nm�1 (g), 0.07 Nm�1 (±¥¥), and 0.1 Nm�1 (c) at fd=300 Hz and assuming a Young modulus of ES=80 GPa for the sample. A higherforce constant ensures smaller hydrodynamic damping but reduces force resolution. The arrow indicates the occurrence of a second maximum due to hydrodynam-ic damping. C) Impact of surface charge density of the sample on the appearance of the PFM force curves. Using the same oscillator as in (A) (fd=300 Hz,ES=20 MPa) and assuming a negatively charged tip (sT=�0.032 Cm�2)). The surface charge density was varied from sS=�0.05 Cm�2 (b) over neutral (g)to +0.05 Cm�2 (c). D) Stiffness variation of the sample, (c) ES=80 GPa and (g) 20 MPa as typical values for mica and biological membranes, respectively.


A. Janshoff et al.


the relevant effects encountered in PFM under liquids as afunction of amplitude.

A typical effect is the (previously discussed) formation of asecond force maximum on the left-hand side of the force max-imum in the contact regime at about one half of the drivingperiod, which is caused by the large hydrodynamic dampingthat affects the SFM cantilever at the maximum driving veloci-ty. By choosing smaller setpoints for the maximum force trig-ger, and higher frequencies and amplitudes, the hydrodynamicforce maximum can easily become larger than the contactmaximum force (i.e. , Figure 3A). This is no major hindrance ina quantitative exploration of the PFM if the contact onsetpoint is still clearly identifiable.

However, the same mechanism is responsible for the hydro-dynamic force acting on the cantilever while retracting fromthe sample, which influences information on adhesion be-tween tip and sample (Figures 3A,C,D and 4). In Figure 4, thedetachment points, which are determined in the simulation as-suming zero adhesion force, are labeled by black triangles. It isevident that large hydrodynamic damping completely ob-scures the contact point, and consequently prevents reliabledetection of the adhesion maximum. Special post-processingroutines have been proposed to obtain corrected adhesionvalues.[37] However, this requires an exact knowledge of thedriving force, which might not be perfectly sinusoidal. In theexperiments presented herein, we followed an approach toreduce hydrodynamic damping to acceptable magnitudes bydecreasing both the driving amplitude and frequency. Decreas-ing the amplitude is limited by the need to overcome maxi-mum adhesion between tip and sample and by the need toexpress sufficient load for the observance of mechanical insta-bilities, that is, breakthrough events, of thin film systems. Inprinciple, the driving frequency can be reduced arbitrarily,however the intention is to keep the advantage of PFM as arapid acquisition mode. We found that amplitudes of 1±2 nN,together with driving frequencies fd of 100±300 Hz, satisfy thelatter needs and concomitantly reduce hydrodynamic dampingsufficiently to generate reliable adhesion data.

Analog PFM±SFM generally provides two-dimensionalimages of two main material properties±-adhesion and stiff-ness–the latter of which is more reliable, since adhesion datamight be flawed by hydrodynamic damping, the mechanicalstability of thin films, the electrolyte composition, and the sur-face charge of the tip and sample. The contact stiffness ksmight be approximately related to the Young modulus ES ofthe sample by Equation (7)

ks ¼2rcE

ð1�n2Þ ð7Þ

where rc is the contact radius and n is the Poisson ratio.[48] Stiff-ness imaging is a widely used application of PFM±SFM, andthus we will briefly try to explore to what extent qualitativeand quantitative stiffness analysis with the proposed standardprocedure is possible.[6, 25±33,35,36] Figure 5 shows simulations ofPFM force curves on samples with different harmonic stiffnessfor two driving frequencies, 100 Hz and 1 kHz. The slope SPFMof the force curve (Figure 1B) is plotted as calculated in thestandard PFM stiffness imaging from the force interval DF=F2�F1 and the time interval Dt= t2�t1, Equation (8)[36]

SPFM ¼ kskckc þ ks

¼ DFA0ð1�cosð2pfdDtÞÞ ð8Þ

As can be concluded from Figures 5B and 5D, PFM imag-ing–even with large damping effects–results in qualitativelycorrect images exploring a broad range of sample elasticity[using Eq. (1) and assuming R=20 nm; n=0.33; E ranged from1.12 MPa to 1.12 GPa] . The stiffness as obtained from theslope SPFM (Figures 5C,D) is not linearly dependent on the stiff-ness provided by the simulation parameters. Clearly, with thesoft cantilevers used here (kc=0.08 Nm�1), elasticity contraston stiff samples can only be poor and only the cantilever stiff-ness is displayed. Again, in the case of smaller hydrodynamicdamping, less care has to be taken with respect to possible ar-tifacts and the correct adjustment of the trigger points. Where-as the plateau regime of the PFM slope in Figure 5C is extend-ed both left and right from the force maximum (the stiffnesscontrast is independent of the chosen trigger point if no mate-rial failure occurs), the plateau on the left side of the forcemaximum in Figure 5D assuming a ten-fold increased frequen-cy is more narrow, and setting the trigger point on the right-hand side of the force maximum can even lead to a contrastinversion in the stiffness image. Contrast inversion can be ex-plained in terms of a significant phase lag between z-piezomovement and tip±sample distance, as shown in Figure 6 fortwo different frequencies employing a soft sample (ks=0.1 Nm�1). At 100 Hz, virtually no phase shift is observable,whereas at 1 kHz the phase lag is already considerable. Largehydrodynamic damping in combination with soft samples (Fig-ure 6B) is responsible for the fact that the maximum samplepenetration by the tip lags behind the maximum of the z-piezo movement in sample direction (see vertical lines in Fig-ure 6B). Consequently, the cantilever deflection, which is thedifference between the piezo movement and tip±sample dis-tance, is maximal before the piezo movement reaches its maxi-

Figure 4. Comparison of simulated PFM force curves (solid lines) and experi-mental force curves (dots) at different driving amplitudes: 3.48 (20.9%, A), 4.38(26.3%, B), 5.73 (34.4%, C), 8.51 (51.1%, D), and 15.04 nN (90.4%, E). Systemvalues used for the calculation were meff=36.6 ng, a=5260 nNsm�1,kc=0.09 Nm�1, and ks=10 Nm�1. The driving frequency was set to a typicalPFM value of fd=833 Hz. The solid black triangles indicate the detachmentpoints from the sample, as obtained from the simulation.




mum value. Thus, the sign of the force difference, which is re-sponsible for the stiffness determination [Eq. (8)] , mightchange when the stiffness of the sample changes due to thefixed trigger settings. Low frequencies, and therefore small hy-drodynamic dampings, however, abolish the phase shift andtherefore the trigger setting is no longer crucial for the correctdetermination of stiffness differences (Figures 5A,C).

Adhesion imaging may also be subject to misinterpretationdue to hydrodynamic damping, as demonstrated in Figure 7.Figures 7A and B show two PFM curves simulated at 100 Hzand 1 kHz, respectively. A strong electrostatic attraction be-tween the tip and sample was assumed, in order to modelstrong adhesion. Low frequency, and thus low hydrodynamicdamping, has a negligible effect on the adhesion force as de-termined by the difference between zero force–the base-line–and the local minimum that occurs as the tip detaches

from the surface. However, simu-lations reveal that even at theselow frequencies detachmentfrom the surface occurs earlierthan the force minimum implies.At higher frequencies, this effectis even more pronounced (Fig-ure 7B). The baseline cannot bedetermined accurately and thelocal minimum does not corre-spond to the adhesion force. De-tachment from the surface takesplace substantially before theminimum force is reached.

Contrast inversion in adhesion imaging is easily achieved ifthe two materials differ in their attractive potential and in theirYoung modulus. Assuming a soft material with a high adhesionbetween tip and sample coexisting with a stiff material exhibit-ing low adhesion, one obtains PFM curves as shown in Figur-es 7C and D. Adhesion was simulated by assuming strong elec-trostatic forces between the tip (�0.032 Cm�2) and sample(+0.05 Cm�2), as before. The curves displayed in Figure 7Cwere simulated at 100 Hz and show only a very narrow bandin which adhesion inversion might occur. However, the band isconsiderably increased at the higher driving frequency (Fig-ure 7D). It might happen that a trigger chosen at low frequen-cies provides correct adhesion contrast, whereas the same rela-tive position at higher frequencies shows inverted contrast.Sometimes switching within one image might occur due toslight drifts of the trigger position.

Figure 6. Phase lag between z-piezo movement zP(t) and tip±sample distance z(t). zc(t) is the cantilever deflectionfrom which the force F(t) was calculated according to Hook's law. Units are in nanometers. A) fd=100 Hz;B) fd=1000 Hz. System values used for the simulation were meff=36 ng, a=5260 nNsm�1, kc=0.1 Nm�1, F0=4 nN,and ks=0.1 Nm�1.

Figure 5. A,B) Simulations of PFM force traces for different sample stiffness together with C,D) the corresponding PFM stiffness values for various stiffness triggerpoints. System values correspond to those used in Figure 4 except for ks, fd, and A0. Figures A and C were obtained at a driving frequency fd=100 Hz; Figures Band D were simulated with a driving frequency fd=1000 Hz. In both cases, the driving amplitude F0 was set to 4 nN.


A. Janshoff et al.


The following paragraph is dedicated to the experimentalfindings upon imaging solid-supported lipid bilayers and isthought to provide evidence for the usefulness of the comput-er simulations. We concentrated on stiffness and adhesionimages, as these are the most frequently used material con-trasts in PFM±SFM. Figure 8 shows PFM±SFM images of phase-separated lipid bilayers spread from liposome suspension onfreshly cleaved mica, as described previously. The compositionof the bilayers (1,2-dioleoylphosphatidylcholine (DOPC):sphin-

gomyelin (SM):cholesterol=2:2:1) was expected to show stiffsphingomyelin-rich domainswithin a fluid matrix. Dependingon the cholesterol content, coex-istence of the so-called liquid-or-dered phase with the liquid-dis-ordered phase is expected.[6,13, 16]

The topography images (leftcolumn) show a typical phaseseparation, in which the highersphingomyelin-rich domains,which consist of SM and choles-terol (bright area), coexist withfluid, DOPC-rich areas (darkerarea). The height difference be-tween the two observed phasesis about 0.5±1 nm. From the top-ography measurement alone, itcan only be assumed that thehigher phase is enriched in thegel-state lipid SM due to the dif-ference in chain length andpacking geometry. Material con-trast imaging, however, providesproof for this assumption by

clearly distinguishing the phases in stiff and soft areas (imagesin the center of Figure8). Since DOPC is in the fluid state, andthus exhibits a smaller compressional modulus, it shows asmaller stiffness than the SM-enriched phase. From the stiff-ness mapping, it becomes evident that the higher domains arealso stiffer and can thus be identified as the SM-enrichedphase. Different driving frequency, amplitude, and trigger set-tings produce no significant change in topography, but maylead to contrast inversion of material properties, as expected

from our simulations. Choosing the right trigger-set-ting, at low amplitude and driving frequency, provid-ed the expected ™correct∫ contrast. The deeper(darker) DOPC-rich phase (left images, topography)shows higher adhesion (right images, light phases)and lower stiffness (center images, dark phases).Larger adhesion is probably a result of the larger con-tact area, since the tip penetrates deeper into thesoft material. However, contrast inversion can bereadily achieved by changing the driving amplitude,the frequency, or the trigger settings, in accordancewith our computer simulations. Figure 8 shows theimpact of increasing the driving frequency from263 Hz to 780 Hz on the adhesion images. Due tothe higher frequency, adhesion contrast is inverted.Setting the trigger to the right flank of the forcecurve (instead of the left one) resulted in stiffness in-version, as predicted from simulations (Figure 5D,dotted vertical line).

Besides artifacts in stiffness or adhesion imaging,the topography might also be subject to interferencefrom differences in material properties. For instance,

Figure 7. A,B) Determination of the lift-off point of the tip from the surface and comparison with the deflection signal.The dashed line indicates the contact with the surface at fd=100 Hz (A) and fd=1000 Hz (B). The lift-off point doesnot correspond to the local minimum. System values used for the simulation were: meff=36 ng, a=5260 nNsm�1,kc=0.1 Nm�1, F0=4 nN, d=++25 nm. A MYD±EDL potential was used as the interaction potential, with ES=80 GPaand sS=++0.05 Cm�2. The Debye length k�1

D was 0.75 nm. C,D) Possible scenario resulting in the observation of adhe-sion contrast inversion due to wrong trigger settings. One trigger is set to the baseline at zero force (zc(t)=0 nm); theother should ideally be set to the point of detachment of the tip from the surface. System values used for the simula-tion were: meff=36 ng, a=5260 nNsm�1, kc=0.1 Nm�1, and F0=4 nN. A MYD±EDL potential was used as the interac-tion potential, with ES=20 MPa and sS=++0.05 Cm�2 (solid line) and ES=80 GPa and a neutral sample surface(g). The Debye length k�1

D was 0.75 nm. The gray boxes indicate the region where adhesion contrast is inversed.

Figure 8. PFM±SFM images of solid-supported lipid bilayers (DOPC:SM:cholesterol=2:2:1)obtained in a buffer solution (kc=0.08 Nm�1). Left : topography (z-scale 10 nm), center : stiff-ness (z-scale 0.06 V), right : adhesion (z-scale 0.3 V). Images in (A) were obtained at 25.3%amplitude at fd=263 Hz. The stiffness and adhesion contrasts are as expected. The imageseries in (B) displays stiffness inversion due to setting the trigger at the right flank of thecurve (amplitude 15%). C) The adhesion contrast is inverted as compared to (A) due to thehigher driving frequency of 780 Hz.




a change in sample stiffness from 10 Nm�1 to 1 Nm�1 resultedin an apparent change of 1.8 nm in ™height∫ (F0=4 nN) due tothe fact that the maximum force is used as the feedbacksignal.

In conclusion, the presented simulation may be of greathelp for finding the best compromise between fast data ac-quisition and reliable contrast formation. Considering how hy-drodynamic damping changes imaging conditions, one maysafely assume that a correct stiffness and adhesion image canbe obtained with low frequency and amplitude parameters. Inthe future, digital PFM that allows one to record high resolu-tion PFM force curves during x±y scanning will help to produceimages offline with different trigger settings, avoiding artifactsfrom wrong settings, and to explore more subtle effects afterthe imaging procedure.

Experimental Section

Preparation of solid-supported lipid bilayers: Small, unilamellarvesicles of 1,2-dioleylphosphatidylcholine:sphingomyelin:cholester-ol (2:2:1) (Avanti Polar Lipids, Alabaster, Al, USA) were prepared bythe extrusion method using two stacked polycarbonate mem-branes with an average pore diameter of 100 nm. The vesicle sus-pension (200 mL; lipid concentration 0.2 mgmL�1) was incubatedovernight on freshly cleaved mica at 4 8C. Before heating for45 min at 60 8C, the samples were rinsed with buffer to removevesicles not adsorbed to the mica. The samples were cooled toroom temperature then rinsed with buffer again. All experimentswere performed in 2 mm CaCl2, 150 mm NaCl, 10 mm Tris/HClbuffer (Tris=Tris(hydroxymethyl)aminomethane) at pH 7.4.

PFM±SFM measurements: Experiments were carried out in aqueoussolution using a commercial scanning force microscope (Dimen-sion 3100 with Nanoscope IIIa+A/D controller, Veeco Digital In-struments, Santa Barbara, CA, USA) extended with a pulsed forcebox (WITec GmbH, Ulm, Germany) and silicon nitride cantilevers(OMCL-TR400PSA, Olympus, Japan).

Measurements of the thermal power spectrum densities yielded in-dividual spring constants of the cantilevers from 0.07 to0.09 Nm�1. The tip radii were estimated by deconvoluting theimages of a tip calibration grid TGT01 (Anfatec, Germany). A PCcard (NI 5911, National Instruments, USA) was used to digitalizethe PFM traces for the force curve mapping. SUBSTRACT BASELINEwas switched to ON at the PFM box for all measurements.

Computer Simulations: An iterative numerical algorithm modelingthe time evolution of the oscillating tip as it is approached or re-tracted from the sample surface was used. In this case, all types offorces FTS(z) acting on the SFM tip can be considered within thecalculations by integrating Equation (1). In the presence of a gener-al force FTS(z), Equation (1) can be solved using a velocity form ofthe Verlet algorithm that is based on a Taylor expansion of the po-sition z up to second order terms [Eq.s (9)±(11)][38]

zðt þ DtÞ ¼ zðtÞ þ _zDt þ Dt2

2€zðtÞ ) zðt þ DtÞ

¼ 2zðtÞ�zðt�DtÞ þ Dt2€zðtÞð9Þ

with

_zðtÞ ¼ zðt þ DtÞ�zðt�DtÞ2Dt

ð10Þ

and

€zðtÞ ¼ F0sinwt�kczðtÞ�a _zðtÞ�FTSðzðtÞÞmeff

ð11Þ

Equation (11) shows that the acceleration z» of the tip can be deter-mined if the system constants and FTS(z(t)) are known; kc can be es-timated from thermal noise experiments; and z is calculated by nu-merical iteration of Equation (9). Knowing the tip position z, allowsthe force F(z(t)) to be directly calculated at any time of the oscilla-tion. Time steps of 2�16 of an oscillation period were used toassure numerical stability. Substituting z» in Equation (9) providesEquation (12)[24]

zðt þ DtÞ ¼ C1½C2zðtÞ þ C3zðt�DtÞþC4½F0sinwt�FTSðzðtÞÞ

ð12Þ

with the constant coefficients given in Equations (13a±d)

C1 ¼�1þ Dt

a

2meff

��1

ð13aÞ

C2 ¼ 2�Dt2kcmeff

ð13bÞ

C3 ¼ Dta

2meff�1 ð13cÞ

C4 ¼Dt2

meff

ð13cÞ

Acknowledgements

This work was supported by the German Science Foundation (JA963/1±3).

Keywords: bilayers ¥ lipids ¥ membranes ¥ pulsed force mode ¥rafts ¥ scanning probe microscopy

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Received: November 17, 2003

Revised: April 1, 2004




Documents

Scanning Force Microscopy Based Rapid Force Curve Acquisition on Supported Lipid Bilayers: Experiments and Simulations Using Pulsed Force Mode