Systematic Multidimensional Quantification of Nanoscale Systems From. Bimodal Atomic Force Microscopy Data

Transcription

1 Systematic Multidimensional Quantification of Nanoscale Systems From Bimodal tomic Force Microscopy Data Chia-Yun Lai, Sergio Santos, Matteo Chiesa Laboratory for Energy and NanoScience (LENS), Institute Center for Future Energy (ifes), Masdar Institute of Science and Technology, bu Dhabi, UE

2 Contents Raw data and codes... Monitoring the tip radius... Derivation of the bimodal expressions for Hamaker and d min... 4 Derivation of main expressions in the text for the van der Waals forces... Raw bimodal SSS images for Fig Expressions derived from raw images and cantilever parameters... 5 Table of a, b, and c values for E* (Elastic modulus) in SSS... 7 Table II extended... 7 Supplementary Raw data and codes ll raw data and codes (produced via Matlab, Python, the R language/r studio and C) can be found at Readme.md file is also provided at the site with instructions of how to reproduce all the processed data and figures in this article from the raw data. Matlab, python and R should be installed in the system before running the scripts and python should be callable under Windows, Linux environments or other. Monitoring the tip radius

3 ll the experiments have been carried out with a Cypher FM from sylum Research and standard OLYMPUS cantilevers (C60TS -40 N/m, and C40TS N/m). Since it is wellknown that the tip radius R significantly affects the interaction force between the tip and the surface, R was constantly monitored in situ during the experiments. The initial value of R was assumed to agree with the nominal values provided by the manufacturer, i.e. R 0 nm. The experimental steps to take FM-PD curves are: ) sample was mounted for standard FM (Cypher FM from sylum Research) data acquisition. ) new FM cantilever (OLYMPUS C60TS with k=40n/m and Q factor 500 or OLYMPUS C40TS with k= N/m and Q factor 00) was mounted on the FM cantilever holder. 3) The value of R was monitored by acquiring standard PD curves and these were used to compute the critical amplitude 3 c value in raw Volt units. The initial value of R was assumed to agree with the nominal values provided by the manufacturer, i.e. R 0 nm. Provided c did not change for a given sample during the experiments, we assumed that R remained constant. 4) pproximately PDs were acquired immediately after computing the value of R for each tip. 5) The raw PD curves were then converted into F ts versus distance d profiles as those presented in Figs. Sa for HOPG and Sb for PFD. In Fig. S the raw data is shown in red dots and the

4 Hamaker fits in blue lines. Each PD produces an F ts versus d profile. We defined d=0 at minima in F ts, i.e. when F ts coincides with the force of adhesion F D. Derivation of the bimodal expressions for Hamaker and d min General Virial expression for the van der Waals force This data is reproduced from Ref. 4 4 The bimodal theory 5-9 states that the virial V for modes and can be obtained directly from observables via V m ( dmin) km0 m ) F z dt cos m( d ) T ts m = φ (S) Q m( dmin min m By combining (S) and (3), where (3) is the model for the force employed in this work in the attractive regime as described in the main text, it can be shown 0 that RH RH dmin+ V ( H, dmin) = z dt 6 6 (S) T d 3/ Combining (S) and (S) one equation with unknowns, i.e. d min and H, results. second equation can be derived from the second mode Virial. On the other hand, the second virial should be expressed in terms of the tip position of the first mode z in order to make the calculations tractable. This simplification was proposed by Kawai et al. and the approximations discussed in detail by ksoy and talar 8 and others 7,. ssuming that during a full first mode

5 cycle the derivative is an even function of position, the simplifications proposed in the literature, assuming the model in (3), are equivalent to V ( H, d min ) = T RH 6d RH zdt 6π du [ d + ( + u) ] 3 min u (S3) The above results from combining T F f, 3 ts dt k (S4) z f0 and 6-7 V f k (S5) f0 from which the second mode virial can be rewritten as V T F z ts dt (S6) The expression in (S5) results from inserting (3) into (S6) and assuming harmonic motion as usual, i.e. z cos(ω t-ϕ ) III.B Solution of second mode virial for the van der Waals force The objective is to solve the integral in (S7) that can be written as I( a, b) du = 3 [ a + b( + u) ] u (S7)

6 where a=d min and b=. nalytical solution of (S9) fter a few changes of variable the solution of (S9) can be found in close form as rctan Hereafter equation (S8) is considered the analytical solution of (S7). Since (S8) is cumbersome, an approximation is sought. (S8) pproximate solution of (S0) Considering that rctan[..57 (S9) Furthermore, from dimensional analysis, and neglecting the second term of (S8), (S0)

7 Furthermore, whenever b>>a, it follows that (S) Or I( d min, ) 0.83 d 5 min (S) Hereafter (S), or equivalently, (S), is considered the approximate solution to (S7) and (S8) and it is simple enough for the purpose of this work. Comparison between the numerical integration and the analytic solution (S8) and the approximate solution (S) s shown in the example in Fig. S, the numerical solution coincides with the analytical solution in (S8) with errors smaller than % as compared to numerical integration of (S7). In the figure, the x axis is b= and a=d min has been set to nm. The physical implication is that for the range of set point amplitudes explored here, i.e. - 0 nm, the analytical solution coincides with the numerical integration of (S7). These results have practical use since in attractive bimodal FM the set point amplitude or sp lies in such range 3-6.

8 Fig. S. Comparison between numerical integration of (S7) and the analytical solution in (S8) and as a function of b, with a=nm. The next two figures (Fig. S and Fig. S) show the results of the analytic solution in (S8) that, again, coincides with numerical integration of (S7) with errors smaller than % and the approximate solution in (S). In Fig. S3 b =0 nm and a d min is varied from 0.5 nm to nm. The approximation in (S) produces errors smaller than %. n extreme case of operation is presented in Fig. (S6) for which b =0.5 nm and a d min varies from 0.5 to nm. In this extreme case errors are still predicted to be smaller than 0%.

9 Fig. S- nalytical-numerical solution compared with the approximation in (S3) for b=0nm. Fig. S3 nalytical-numerical solution compared with the approximated solution given by equation in (S3) with b=0.5nm.

10 Figs.S3 and S4 show the comparison between the numerical solution as a function of b when is = 0.nm and a= nm respectively. Fig. S4 nalytical-numerical solution compared with the approximated solution given by equation in (S) with a=0. nm.

11 Fig. S5nalytical-numerical solution compared with the approximated solution given by equation in (S) with a= nm. In summary, the error of the approximation in (S) is always lower than 5% except in cases where a>b. In such cases the maximum error reaches % the range of parameters explored in the figures above. In principle, provided a>b the error from (S) tends to zero. The practical result here is that one can employ the approximation in (S) provided the oscillation amplitude = sp, i.e. b, is larger than the minimum distance of approach d min, i.e. a. In any case, this is true in all our examples since, as corroborated from the force profiles, the decay length of the van der Waals forces is always in the order of 0.5- nm while oscillation amplitudes employed in this work for sp ar4e always larger than nm.

12 Derivation of main expressions in the text for the van der Waals forces Combining (S3) and (S) V ( H, d min ) RH 6π 0.83 d 5 min (S3) Expression (S3) is the second equation necessary with the two unknowns required, i.e. d min and H, that combined with (S) results in a system of two equations in two unknowns. Solving for d min the solution can be expressed as d / 3 min + bdmin + c= 0 (S4) in accordance with (4) in the main text. The coefficients b and c are found to be 3πk 0Q cosφ b= 0.83k 0Q cosφ /3 ( ) / 3 (S5) c= (S6)

13 The zeros of (S4) were found here in Matlab with the help of the standard fzero function and initial values were assumed to lie in the nm range. Provided dmin was found for a given pixel in the image, the solution for H in terms of V was trivial ( sp = ) H = k0 cosφ d min + RQ 3/ (S7) and in terms of V H 3πk cosφ 0 5 = dmin (S8) 0.83RQ When no solution for (S4) were found H was not computed. Errors resulted in some pixels for which dmin was not found. Images for which 0% of pixels produced errors were discarded. In our case, this resulted mostly when working outside resonance, i.e. this formalism is based on the FM being operated at the two modal resonances.

14 Raw bimodal SSS images for Fig.

15 Fig. S6 We note that and are in the sub-nm range and, in particular, lies in the sub- ngstrom range as discussed in the main text. Expressions derived from raw images and cantilever parameters The energy transfer from mode to mode can be shown to be written as 9 nπk ( m) ( m) 0( m) ET ( m) ( dmin) Ftsdz( m) = sinφ( m) ( dmin) T Q( m) ( m) 0( m) (S9) where n= for the mode or m= and n approximately 6 for mode or m=. The kinetic energy for each mode m is T( m) ( dmin) k( m) ( m) (S0) then E (m) follows as E Q E + πn ( m) ( m) ( m) ( dmin) T( m) (S) The energy dissipated E dis is

16 Edis (dmin ) ET () + ET ( ) (S) The modal virial is V( m ) (d min ) ( m ) (d min )k( m ) 0( m ) Fts z( m) dt = cos φ( m ) (d min ) T Q( m ) Examples of these maps for the four model systems in the main text are given in Fig. S7. Fig. S7. (S3)

17 Table of a, b, and c values for E* (Elastic modulus) in SSS a b c Table II extended

18 E * [GPa] T () [ ] T () [ ] V () [ev] V () [ev] E T() [ev] E T() [ev] E () [ev] E () [ev] Reference. Santos, S.; Guang, L.; Souier, T.; Gadelrab, K.; Chiesa, M.; Thomson, N. H., Method to Provide Rapid in Situ Determination of Tip Radius in Dynamic tomic Force Microscopy. Rev. Sci. Instrum 0, Ziegler, K. J.; Lyons, D. M.; Holmes, J. D.; Erts, D.; Polyakov, B.; Olin, H.; Svensson, K.; Olsson, E., Bistable Nanoelectromechanical Devices. ppl. Phys. Lett. 004, Santos, S.; Guang, L.; Souier, T.; Gadelrab, K. R.; Chiesa, M.; Thomson, N. H., Method to Provide Rapid in Situ Determination of Tip Radius in Dynamic tomic Force Microscopy. Rev. Sci. Instrum 0, Lai, C.-Y.; Perri, S.; Santos, S.; Garcia, R.; Chiesa, M., Rapid Quantitative Chemical Mapping of Surfaces with Sub-nm Resolution. Nanoscale 06.

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