High-resolution high-speed dynamic mechanical spectroscopy of cells and other soft materials with the help of atomic force microscopy

Dynamic mechanical spectroscopy (DMS), which allows measuring frequency-dependent viscoelastic properties, is important to study soft materials, tissues, biomaterials, polymers. However, the existing DMS techniques (nanoindentation) have limited resolution when used on soft materials, preventing them from being used to study mechanics at the nanoscale. The nanoindenters are not capable of measuring cells, nanointerfaces of composite materials. Here we present a highly accurate DMS modality, which is a combination of three different methods: quantitative nanoindentation (nanoDMA), gentle force and fast response of atomic force microscopy (AFM), and Fourier transform (FT) spectroscopy. This new spectroscopy (which we suggest to call FT-nanoDMA) is fast and sensitive enough to allow DMS imaging of nanointerfaces, single cells, while attaining about 100x improvements on polymers in both spatial (to 10–70 nm) and temporal resolution (to 0.7s/pixel) compared to the current art. Multiple frequencies are measured simultaneously. The use of 10 frequencies are demonstrated here (up to 300 Hz which is a rather relevant range for biological materials and polymers, in both ambient conditions and liquid). The method is quantitatively verified on known polymers and demonstrated on cells and polymers blends. Analysis shows that FT-nanoDMA is highly quantitative. The FT-nanoDMA spectroscopy can easily be implemented in the existing AFMs.


Verifications of the FT-nanoDMA method a. Comparison of the moduli obtained with FT-nanoDMA method and well-established techniques
The accuracy of the method was verified on well-characterized homogeneous polymer, PDMS. Here we assumed that the values of the macroscopic storage and loss moduli (measured with independent methods, DMA instrument, nanoDMA mode of nanoindenter) should be equal to the ones obtained at the nanoscale with FT-nanoDMS. PDMS was chosen as it is one of the most homogeneous polymers known. Fig.S2 shows the storage, loss, and static (Young's) moduli measured on the same PDMS sample using all three techniques mentioned above. One can see that all moduli are indeed quite similar.

b. The absence of cross-talk between frequencies
A multi-frequency method on the base of atomic force microscopy (AFM) and FPGA electronics was implemented. We demonstrated feasibility of using 10 different frequencies simultaneously to measure the storage and loss moduli of soft polymers (PDMS, polyurethane, and polystyrene samples; static elastic modulus was in MPa -GPa range). The correct work of the implementation was verified by comparing the obtained results with those when measured using the same frequency range sequentially. Figure S3 shows an example of such verification for a sample of PDMS resin. Each point shows the average and one standard deviation for measurements done at ten different points of the surface, and with 3 different amplitudes (the results should be amplitudeindependent). Dynamic calibration of the AFM setup was done for the whole frequency range of 3-300Hz. A sinusoidal -signal of 2.5--10 nm amplitude was sent to the piezo-scanner in two different modes: One frequency in a time and 10 frequencies all at once. The dull probe used in this experiment had the geometry close to an ideal spherical shape with a radius of curvature of 1000± 120 nm. The cantilever and scanner positions were recorded with the sampling rate of 250kHz. One can see an excellent agreement between measurements when 10 frequencies were measured sequentially and when measured simultaneously.

c. Linearity and the minimum radius of the probe
The resolution of FT-nanoDMA is defined by the diameter of the contact between the indenting probe and sample. The diameter of contact is defined by the static load and adhesion force between the sample and probe. At first glance, such a diameter can be minimized by using a sharp probe. However, as was demonstrated 1,2 , the use of an excessively sharp probe can easily result in overstretching the material (non-linear stressstrain response). To minimize the stress, one needs either to increase the area of contact between the sharper indenter and sample or to use a duller indenter. As a result, the diameter of contact for the linear stress -strain regime is about the same for both dull and sharp indenters. The optimal radius R* of the indenting probe that gives the best resolution is given by the following formula 1 : where F Lmax is the maximum load force allowed for the linear stress-strain relation, E* is the reduced Young's modulus of the material,  max is the maximum vertical compressive stress, adh w is the adhesion energy (per unit area) of interaction of two flat interfaces made of the materials of the AFM probe and sample. As a matter of rule, the lateral resolution is higher on more rigid samples.
An example of full analysis of polyurethane was done in 3 . It was found that the maximum stress corresponding to the linearity limit was ~60MPa (with the help of macro DMA instrument), adh w 0.035 J/m 2 . Taking the maximum load force or 50 nN, one can find that the AFM probe radius should be not less than 210 nm be able to work in the linear stressstrain regime.
The probes used in this work have radii in the range of 550-1000nm. This is a bit excessive, but gives additional guarantee that we do not reach the linearity limit. This brings another important question, how to find the linearity limit in the case of heterogeneous materials at the nanoscale, and in the case of materials which cannot be scaled up to macroscopic size (for example, biological cell). The study of this question is beyond the scope of this work; it will be studied in the future. Two examples are shown in figure S4.
PDMS resin is tested in air and water. One can see perfect linearity within the error of measurements. The only noticeable deviation is seen for PDMS in water for the largest amplitude of 50nm.

d. Possible contribution of liquid medium to damping
Starting from equation (3), we neglected a possible contribution of damping of the fluid environment. There is a relatively simple way to test it. Measuring the viscous parameters the (loss modulus or tangent of loss angle (loss tangent)) for a homogeneous isotropic polymer, one should see no difference in the values of those viscous parameters when changing the oscillation amplitude Z i (as soon as the amplitude is much less than static indentation). If such dependence is seen starting from some large amplitude, it presumably means that our approximation of equation 3 was not correct, and one has to take into account damping due to the liquid environment. Figure S5 shows the measurements of loss tangent ( tan  ) for a homogeneous and isotropic polymer (PDMS) as the function of amplitude Z i (three different frequencies are shown). One can see that tan  is independent of the amplitude up to 15nm (and higher in air). Some changes in tan  is seen in water for higher amplitudes. These may be considered as the limit up to which we can neglect damping of the medium. It should be noted that 15 nm is quite large amplitude, and there is no any particular need to work in such regime. As to the error coming from the creep effect of the sample, it was found that the polymer creep could cause overestimation of the elastic modulus up to ~30% 4,9 . As to the FT-nanoDMA method, the creep effect can be neglected because it gives less uncertainty to the contact area then the one coming from the error in definition of the probe shape (creep can be measured directly).

f. Influence of non-ideally vertical indentation of AFM probe.
This phenomenon is a known potential problem when using the AFM probes indentation because the probes are attached to a tilted cantilever. This actually was one of the reasons of designing nanoindenters in which the indenting probes move vertically downward. This issue is schematically shown in Figure S6 (left). If the consequent lateral shift is excessively large, it can lead to interference with the vertical response of the material. If the lateral shift is not too large, the lateral shift is not as serious of a problem as it might look. The lateral and vertical deformations can still be considered independently while working within the proportionality elastic limit. However, if the lateral shift becomes too large, the measurements can be out of the proportionality elastic limit. Moreover, the optical system of detection of the cantilever deflection can become nonlinear. The former effect can be detected by searching nonlinearity as previously described. The later effect can be taken into account by not exceeding a maximum allowable deflection limit of the cantilever (such a limit is clearly seen in the calibrating plots against a rigid surface, Fig.S6 (right)). In either case, it is possible to choose the appropriate AFM cantilever to minimize these effects. For example, a long (but still stiff) cantilever will decrease the described problems. Finally, when using spherical probes with indentation depths much smaller than the probe radius, it obviously decreases the described effect due to probably sliding between the probe and sample (in particular, when oscillating the probe via applied additional sinusoidal signal).
As we demonstrated, 1,2 AFM indentation can be used to derive macroscopic values of homogeneous polymers with rather high accuracy. Moreover, the modulus values were found to be independent of the indentation depth, which means a lateral shift. These results mean that the effect of non-ideally vertical indentation of AFM probe is negligible when the appropriate cantilever and probe are used.

Comparison of FT-nanoDMA technique with existing top-of-the-line nanoindenter (Hysitron TI950)
Here we compare of the FT-nanoDMA method with the existing top-of-the-line nanoindenter  Table 1. The measurements using the FT- Right: The deflection of the cantilever may become nonlinear for the excessively large load force. An example of such deviation from linearity is clearly seen in the calibration plot (cantilever deflection versus Z displacement of the scanner).
nanoDMA method are described in the main text. The measurements done with Hysitron nanoindenter are described below.

i. Lateral resolution
The lateral resolution of the contact canning probe methods is limited by the diameter of the indentation contact. When doing indentation on PDMS resin, the nanoindenter has a contact diameter of ~20 µm when its indentation reaches stable values of the elastic modulus, and 33 µm when the values get close to the microscopic value (and the stress under the indenter is within the proportionality limit of the strain-stress ratio). The AFM setup described has a contact diameter of ~160 and 230 nm under similar conditions. In the case of polyurethane, these numbers are 9 or 15µm and 140 or 170nm for the nanoindenter and AFM, respectively. Thus, the lateral resolution of the proposed AFM setup is higher than that for nanoindenter by 150x for PDMS and 64-88x for polyurethane.
It should be mentioned that this increase in resolution is rather weekly dependent on the details of definition of the contact diameter. Specifically, we defined the lateral resolution to be the diameter of the contact when indentation reaches values of the elastic modulus close to the macroscopic value. This makes sense because of the high homogeneity of PDMS.
However, some skeptics may reason that such a definition of lateral resolution is an excessively conservative definition. In principle, one can reach the stable values of the modulus well before approaching the macroscopic values. It is a well-known issue that the elastic modulus typically decreases with indentation and reaches its bulk value when a substantial indentation depth is reached (see, 10,11 , 12,13 , 14 and references therein). We demonstrated 2 that the higher values of the elastic modulus near the surface is frequently a result of nonlinearity (and sometimes because of not taking adhesion into account).
Nevertheless, if we ignore those results, because they are not generally accepted as of yet, we may correlate the lateral resolution with the diameter of the contact when indentation reaches stable values of the elastic modulus ("stable reading" definition).   15 . These methods allow for measuring some sort of elastic modules, say at 512x512 surface points, as fast as 30-90 min. Both techniques use a rather high operational (and single) frequency (tens and hundreds of kHz for HarmoniX and ~KHz for the other modes). Besides using just a single frequency, this frequency is too high to be interesting for soft materials. Finally, the modulus derived with those modes cannot be directly translated into the storage and loss modulus. The value of the elastic modulus in these modes is approximated by static modulus calculation (using typically the DMT model) despite the force curves are collected in a highly dynamic way. It makes this "modulus" dependent, for example, on the load force. Although this modulus can be close to the expected values of the static modulus for materials weakly dependent on the indentation speed (see, e.g., ref. 3 ), these methods can be treated as relative imaging rather than truly quantitative.