Substructure imaging of heterogeneous nanomaterials with enhanced refractive index contrast by using a functionalized tip in photoinduced force microscopy

The opto-mechanical force response from light-illuminated nanoscale materials has been exploited in many tip-based imaging applications to characterize various heterogeneous nanostructures. Such a force can have two origins: thermal expansion and induced dipoles. The thermal expansion reflects the absorption of the material, which enables one to chemically characterize a material at the absorption resonance. The induced dipole interaction reflects the local refractive indices of the material underneath the tip, which is useful to characterize a material in the spectral region where no absorption resonance occurs, as in the infrared (IR)-inactive region. Unfortunately, the dipole force is relatively small, and the contrast is rarely discernible for most organic materials and biomaterials, which only show a small difference in refractive indices for their components. In this letter, we demonstrate that refractive index contrast can be greatly enhanced with the assistance of a functionalized tip. With the enhanced contrast, we can visualize the substructure of heterogeneous biomaterials, such as a polyacrylonitrile-nanocrystalline cellulose (PAN-NCC) nanofiber. From substructural visualization, we address the issue of the tensile strength of PAN-NCC fibers fabricated by several different mixing methods. Our understanding from the present study will open up a new opportunity to provide enhanced sensitivity for substructure mapping of nanobiomaterials, as well as local field mapping of photonic devices, such as surface polaritons on semiconductors, metals and van der Waals materials.

Schematic diagram of the image charges with finite dipole method. (a) Dipole-image dipole interaction based on the ellipsoidal finite dipole model. The induced charge on the tip due to the incident field is Q0 which is positioned at Z0. The induced charge due to the multiple reflection process with the sample is Q1 positioned at Z1. The image charge of them is positioned at X0 and X1 in the layered sample. The tip is modeled as an ellipsoid with length of 2L. The plane wave light is illuminated to the sample with angle of . The H is the gap distance from the tip end to the sample surface. (b) Potential responses of a charge Q0 in the distance z0 above a flat layered sample.
In this model, the potential response U of the sample to the tip can be calculated by considering the two induced charges on the tip and the other two image charges in the sample. The induced charge on the tip due to the incident field is Q0 which is positioned at Z0. The induced charge due to the multiple reflection process with the sample is Q1 positioned at Z1. The image charge of them is positioned at X0 and X1 in the layered sample. By considering the boundary conditions, the potential above the sample is given as 3 The potential inside the sample is given as The potential below the sample (inside substrate) is given as where i=0,1 with 0 = 4 0 1− 21 23 ⅇ −2 d . The is the electrostatic reflection factor for the layered system, given as = − + for n, m=1,2,3. The g is the empirical geometric factor due to the tip shape. For typical PiFM (or s-SNOM) tip geometries, |g| = 0.7±0.1 4 . The total potential response is the sum of the potentials given as Un=Un(Q0, z0) +Un(Q1, z1) where n = 1,2,3, and the electric field is given by differentiating the potential with respect to the z-axis, = − , in the regions. Note that this ellipsoidal geometry doesn't show any resonant antenna effect in the mid-IR, which is related to the tip's geometry 5 .
The electrostatic potential and the normalized field distribution inside and outside of the 10 nm PS film on Si substrate and on are plotted in Fig. S2a and S2b where the gap distance H is fixed by 2 nm, by implementing the library value of the bulk polystyrene 6 , gold 7 and Si substrate 8 . The simulation parameters are R = 30 nm, L = 450 nm, p = 30 ns, H = 2 nm,  = 30 degree and the 0 = 1268 cm -1 . Let's assume the incident electric field as E0 = 10 6 V m -1 which covers the experimental parameters of the I0 = 5 mW (incident power), A = (10 m) 2 (focal area), f = 1.6 MHz (repetition rate) and p = 30 ns (pulse width). The tip end is located at z = 2 nm.

Figure S2
Electric potential and field distribution on the polystyrene film over Si substrate. (a) Calculated electrostatic potential and (b) normalized field distribution by E0 on the PS film over Si substrate (red solid line) and ZnSe substrate (blue solid line) from tip end to substrate at the resonance of the PDMS (1268 cm -1 ). The sample surface is located at z = 0 and the tip end is located at z = 2 nm.

Waals interaction in PiFM
Let us next calculate the tip-enhanced photo-thermal expansion force of the contaminated PDMS on the tip over the PS film on the Si substrate. The photo-thermal expansion force is the result of several causal processes: First, there is an energy exchange with the light field, which scales with the optical absorption coefficient and results in a temperature rise (ΔT ~ Pabs). Second, the accumulated heat diffuses to deform the sample to induce a thermal expansion (ΔL ~ ΔT). Third, the thermal expansion changes the tip-sample distance, which introduces a modulation of the tip-sample interaction force (ΔF ~ ΔL). Then the gradient of the modulated force is coupled to the PiFM.
When we assume the PDMS layer is 1 nm 9 , it may be ignorable for about the field distribution. In this case, the electric field is approximately calculated for the clean Au tip case. The absorbed power is obtained by integrating |E| 2 from the tip end toward the substrate by 1 nm in Fig. S2b. After calculating the electric field distribution, the PDMS will be revisited in the force calculation. By integrating the |E| 2 at the tip end and substituting it into = ∫ , then one can obtain the absorption coefficient of the PDMS to 91.2 x 10 4 m -1 at the vibrational resonance of 1097 cm -1 . If we assume that the thermal expansion is proportional to the temperature change, the calculation of the thermal expansion is much simplified by regarding the maximum temperature change, given as 11,12 : where ℎⅇ is the heated volume related to the thermal wavelength 13 , given as ≈ √ where where  is the linear thermal expansion coefficient. The relaxation time is given as ⅇ ≈ 2 ⁄ . The linear thermal expansion coefficient, density, heat capacity and thermal conductivity of the PDMS are 907×10 -6 K -1 , 965 kg/m 3 , 1460 J/kg.K and 0.15 W/m.K, respectively. The eff is the effective thermal conductivity which serially connects the sample material to the thermal bath given as 1 ≈ 1 + ℎ where the h is the interfacial thermal resistance between the sample and the thermal bath. For the thin films of low thermal conductivity such as PDMS is the interfacial thermal resistance between PDMS and the gold coating become the dominant factor for the heat transfer. Unluckily the interfacial thermal resistance is often unknown which make modelling of the thermalization dynamic difficult, but has the general effect of increasing the relaxation time with more prominent effect the thinner the sample. According to Ref. [ 14 ] we assume this effect decreases the k of the sample around by 1/10 for the 1 nm PDMS so that eff ≈ We assume the ℎⅇ ≈ ℎⅇ d and ℎⅇ ≈ for the PDMS contamination. For the PDMS contaminated tip on the Si substrate with H = 2 nm, the electric field is plotted as in Figure S2b. Because the PDMS thickness is regarded as 1 nm, by integrating the electric field from the tip end by 1 nm in Figure S2b and substituting it into Eq. (S13), then one can obtain the thermal expansion of ~24 pm at the PDMS vibrational resonance of 1268 cm -1 . Finally, the photo-thermal expansion force in tapping mode PiFM is mediated by the modulated tip-sample interaction force which constitutes the attactive van der Waals force and the repulsive DMT contact force. Because the modulated DMT contact force 15 due to the oscillotary thermal expansion is under the noise level (sub pN) due to the extremly small elasticity of the PDMS (0.6 MPa), the modulated van der Waals force and its force gradient contribute to the PiFM signal, given as:

S4. Laser focal spot images with respect to different tip materials on ZnSe
The measured focal spot images are obtained by different tip materials on ZnSe in Figure S4. The scan size is 33 m x 33 m and the measured focal spot is optimized with the size of 2.5  for long axis and 1.5  for short axis where  = 9.1 m (1100 cm -1 ). After mapping the focal spot, we put the tip into the center of the spot (red circle dot) to obtain the spectrum in Figure 2b. We keep the same laser power of ~10 mW with pulse width of 30 ns and tried to maintain the same focus for the three different tips by scanning the shape of the focal spot with the piezo electric parabolic mirror.

S5. Refractive index of polymers in mid-IR
The complex index of refraction shows the wavenumber dependent behavior which is generally the dispersive line shape near the molecular resonance. The refractive indices of NCC 18 , PAN 17 , PDMS 19 and PS 20 in mid-IR range are plotted with respect to wavenumber from the library in Figure S5.