Mapping Elastic Properties of Heterogeneous Materials in Liquid with Angstrom-Scale Resolution

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1 Supporting information Mapping Elastic Properties of Heterogeneous Materials in Liquid with Angstrom-Scale Resolution Carlos A. Amo, Alma. P. Perrino, Amir F. Payam, Ricardo Garcia * Materials Science Factory Instituto de Ciencia de Materiales de Madrid, CSIC c/ Sor Juana Inés de la Cruz, 8049 Madrid, Spain List of contents Numerical Simulator Probe calibration Data processing Purple membrane height Figure S1 Figure S Figure S3 Figure S4 Figure S5 Figure S6 Figure S7 Supporting references 1

2 Numerical Simulator To test the bimodal AM-FM theory and its range of applicability we have developed a numerical platform that simulates the operation of bimodal AM-FM. The numerical simulation software calculates the tip trajectory (time dependent deflection) z(t) = z 0 + z 1 (t) + z (t) (S1) which satisfies the following differential equation system given by both the dynamics of the cantilever and the feedbacks of the second mode for a given mean tip sample distance z c, effective Young Modulus and tip radius: + + = = (S) (S3) where z 0, z n, ω 0n, k n, Q n and F n are respectively the mean deflection (in our case is negligible), the deflection, the angular frequency, the stiffness, the quality factor and the driving force. The subscript n refers to the n th mode and m is the effective mass of the cantilever. Fn, ω and Fts are defined as: = / = + = + + = / = +, + 0 otherwise (S4) (S5) (S6) (S7) (S8) Here A n, A 0n and ϕ n are respectively the amplitude, the free amplitude and the phase shift between the deflection sinusoidal component and the excitation force of the n-th mode. A sp, ϕ sp and ω =π f are the amplitude set point, the phase set point and the frequency shift of the second mode; r and q are the proportional and integral gains of the amplitude gain controller (AGC) and K is the gain of the phase-locked loop (PLL) 1

3 (Figure S1). Regarding the force model, is the effective Young s modulus of the interaction (provided by eq 3), R is the tip radius, δ(t) is the time-dependent indentation and a 0 is the intermolecular distance. The AGC keeps the amplitude of the second mode fixed at A = A sp at the same time that the PLL keeps the phase shift between z (t) and the second mode driving force fixed to ϕ = ϕ sp. We perform the AM FM simulations by setting ϕ sp = π/ and solving eq S and S3 and with a fourth order Runge Kutta algorithm for a given, obtaining the observables distance curves (A (0) (0) 1 (z c ), ϕ 1 (z c ), A (0) (z c ), F (0) (z c ), ϕ (0) (z c ), f (0) (z c )), where the superscript denotes steady state. Using the A 1 (z c ) and f (0) (z c ) curves to obtain f (0) (A 1 ) the effective modulus and indentation are calculated respectively from eq 9 and eq 10. The sample Young s Modulus is obtained through the effective modulus by using eq 3. Figure S1 shows the block diagram of an AM-FM experiment modelled by the simulator. First, the total deflection is computed while the tip interacts with the sample. Then, the two components of the signal are treated separately. The amplitude of the first mode, A 1, is compared with the set point amplitude A sp1. The difference between both (error signal) is processed by a proportional-integral-derivative module (PID) to adjust the mean tip sample distance in order to get A 1 =A sp1. The phase shift ϕ 1 is also obtained but is not required for the eq 9 and eq 10. The bottom panel illustrates the main processing steps performed on the nd mode. The amplitude A is compared to its set point amplitude A sp and the resulting error signal is used to adjust the driving force of the second mode to achieve A =A sp. At the same time, the phase shift between the second eigenmode and its driving force is processed independently to keep its value fixed at π/. This is achieved by using a phase-locked-loop (PLL) that adjusts the frequency of the driving force to keep the oscillation at resonance. The changes in the frequency of the driving force of the second mode f are recorded. Then, for a tipsample force model the results given by eq 9 and eq 10 are compared with the numerical values produced by the simulator. Probe calibration To measure the Young s modulus requires the knowledge of the force constants and the quality factors of the cantilever. The optical lever sensitivity of the 1 st mode s 1 is determined by acquiring a force curve on a stiff surface (Muscovite mica). The optical 3

4 lever sensitivity of the second mode s is calculated from the relationship between the first and second mode optical lever sensitivities for a rectangular Euler-Bernoulli beam.,3 s = s (S9) The force constant of the 1 st mode and nd mode are determined by using the thermal noise method. 3,4 Figure S3 depicts the power spectral density (PSD) of the thermal motion of the AC- 40TS cantilever (Olympus, Japan). The PSD is fitted with the single harmonic oscillator (SHO) with white noise floor added, ω0 A0 A ( ) ω ω = + A White (S10) ω 0 ω 1 + ω ω0 Q where A 0 is the amplitude, ω 0 =πf 0 is the angular resonant frequency, ω the angular driving frequency, Q the quality factor and A white the white noise. The fittings applied to deduce the quality factor and the force constant of the cantilever (1 st and nd flexural mode) are showed in red. The value s extracted from eq S9 is locked in the SHO fit for the nd mode spring constant evaluation. For AC-40TS (Olympus) and the Arrow-UHF (Nanoworld AG) cantilevers, the force constant of the nd mode is calculated by using the thermal noise method and fitting the thermal curve with the theoretical optical lever sensitivity value of the nd mode as determined from eq S9. For the PPP-NCH cantilevers (Nanosensors), the force constant of the nd mode is determined by the theoretical relationship between the force constant of the 1 st mode, k 1, and the nd mode, 4 k, given by: 4

5 ζ f 0 k = k1 (S11) f 01 where f 01 and f 0 are the resonant frequencies of the 1 st mode and nd mode, respectively. We have used the corrected exponent ζ=.17 for this type of cantilevers. 5 Data processing Images analysis has been performed by using either the SPIP (Image Metrology) or the Gwyddion ( programs. Raw images have been plane-leveled and flattened by using the substrate as reference. In order to remove high frequency noise a Gaussian filter (Figure S6 a and b) or a Fourier transform (Figure 3 f and g) has been applied. The stiffness values showed in Figure 6 have been calculated by using the eq S11, k Hertz = E Rδ (S1) s where E s is the Young modulus as determined by eq 9, R is the tip radius used for the determination of the nanomechanical maps showed in Figure 5 d and e and δ is the indentation as obtained by eq 10. The cross-correlation averaging performed in Figure 6 has been performed as follows. First, one BR trimer was selected and cross-correlated with the topography image. From the cross-correlation each individual BR trimer was located. Second, the coordinates of each BR trimer were merged with the elastic modulus, deformation and dissipation channels. Finally, each unit cell is superimposed, averaged and three-fold symmetrized to obtain the corresponding insets (n=10). The code was written in Matlan. The structural model of the BR protein was obtained from the Protein Data Bank (code at9). The Chimera package 6 was used to represent the BR model. 5

6 Purple membrane height The cross-section (Figure S7 b) shows the height variation across different regions of the PM patches. Three different heights are obtained (from left to right) 7.0±0.1, 6.0±0.1 and 8.1±0.1 nm. Those height changes have been attributed to the imaging of different sides of the PM, cytoplasmic (CP) and extracellular (EC). 7 The lowest region as measured from the mica surface corresponds to a nonordered region of the PM. 8 The above height values exceed the structural thickness of the native PM ( nm). This height anomaly has been attributed to the effect of long-range electrostatic double-layer forces 7. 6

7 SUPPORTING INFORMATION FIGURES Supporting Figure 1. Scheme of the bimodal AM-FM feedbacks. The bimodal signal is separated in the 1st and nd components. The component corresponding to the 1 st mode is processed with a PID controller. The amplitude A is kept at a fixed value (A sp ) by changing the driving force of the second mode. The phase shift ϕ is processed independently to keep its value at π/. 7

8 Supporting Figure. Observables computed from simulations for compliant and stiff samples. (a), (b) and (c) shows the amplitudes of the first and second modes for the Hertz model characterized respectively by E s = 1MPa, E s = 1GPa and E s = 100GPa. The amplitude of the second mode is locked to A = A sp. (d), (e) and (f) depicts the corresponding phase shifts of the first and second mode and the frequency shift for the soft and stiff sample, respectively. The phase shift of the second mode is locked to ϕ = π/. All observables are calculated in the steady state (t >> f ). Simulation parameters: Soft material (E s = 1 MPa): A 01 = 50 nm, A 0 = 0.5 nm, k 1 = 0.01 Nm -1, k = Nm -1, Q 1 =.3, Q = 4.6, f 01 = 31.7 khz, f 0 = 63.5 khz, P = 0, I = 0.5 N/m Hz, K = 0.4 Hz. Stiff material (E s = 1 GPa): A 01 = 10 nm, A 0 = 1 nm, k 1 = 5 Nm -1, k = 40 Nm -1, Q 1 =.3, Q = 4.6, f 01 = 100 khz, f 0 = 68 khz, P = 0, I = 0.5 N/m Hz, K = 0.6 Hz. Stiff material (E s = 100 GPa): A 01 = 5 nm, A 0 = 0.1 nm, k 1 = 30 Nm -1, k = 1313 Nm -1, Q 1 =.3, Q = 4.6, f 01 = 19 khz, f 0 = 876 khz, r = 0, q = 0.1 N/m Hz, K = 0.5 Hz. In all cases, E t = 170 GPa, R = 5nm, ν t = ν s =

9 Supporting Figure 3. Power spectral density (PSD) of the thermal motion of the cantilever. (a) PSD (arbitrary units) showing the SHO fit to calibrate the 1 st and nd force constants of the cantilever. (b) PSD and SHO fit for the 1 st eigenmode frequency of the cantilever. The red line depicts the SHO fit done to calibrate k 1 with the optical lever sensitivity s 1 as one of the inputs in the fitting. (c) PSD and SHO fit for the nd eigenmode frequency of the cantilever. The red line depicts the SHO fit done to calibrate k with the optical lever sensitivity s as one of the inputs in the fitting. The data has been obtained with an AC-40TS cantilever (Olympus, Japan). 9

10 Supporting Figure 4. Nanomechanical maps of a polymer (PS-LDPE ) blend in air. (a) Topography map taken in AM-FM bimodal AFM. (b) map of the elastic modulus. Two regions are observed. The softer region in the dark circles corresponds to the LDPE while the stiffer region in the rest of the surface corresponds to the PS (see histogram in d. (c) Map of the indentation (see histogram in e). (d) histogram of the E s values. (e) histogram of the indentation. The data has been recorded in minutes. 10

11 Supporting Figure 5. Simulation results for the Young Modulus and indentation in the polymer blend sample. (a) shows the Young Modulus result (eq 3) from the simulation observables in the LDPE regions, characterized by a modulus of E s = 100 MPa. (b) indentation (eq 4) in the LDPE regions. (c) Young Modulus in the stiffer PS regions, characterized by a modulus of E s = GPa. (d) indentation in the PS regions. Simulation parameters: R = 6nm, ν t = ν s = 0.3, A 01 = 0 nm, A 0 = 0.5 nm, k 1 = 40 Nm -1, k = 1399 Nm -1, Q 1 = 300, Q = 500, f 01 = khz, f 0 =.073 MHz, P = 0, I = 0.05 N/m Hz, K = 0.06 Hz. 11

12 Supporting Figure 6. Angstrom-resolved maps on a mica surface. (a) topography channel taken at A 01 = 1.8 nm and A 0 =0.14 nm. (b) map of the elastic modulus. The mica structure (001) is overlaid; Oxygen (red), silicon or aluminum (yellow). (c) crosssection along the dashed line shown in (b). 1

13 Supporting Figure 7. Purple membrane (PM) observed in buffer solution using bimodal AM-FM. (a) topography of a patch of PM exhibiting the cytoplasmic and extracellular surface. (b) cross-section along the dashed line shown in (a). 13

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