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Two critical issues need to be considered when attempting nanoindentation measurements on soft materials:

(1) The first is the requirement that, in any force-displacement measurement platform, the stiffness of the machine ($k_{machine}$) must approximately match the stiffness of the sample ($k_{sample}$), at least in order of magnitude. If $k_{machine}$ is too high, then the indenter probe will simply run through the sample without being able to measure the force. On the other hand, if $k_{machine}$ is too low, then the probe simply will not indent into the sample, and no reading of the probe displacement can be made. For samples that are very soft, the first of these two possibilities is likely.

The stiffness of a sample is given by

$k_{sample}$≈$a$×$E_{sample}$

where $a$ is the size of the contact region between the indenter and the sample, and $E$ is the sample’s elastic modulus. Typical atomic-force microscopy (AFM) cantilevers have $k_{machine}$ in the range 0.05 to 50 N/m, and probe size in the range ~10 nm to 1 μm. Commercial nanoindenters are also similar. Therefore, if $k_{machine}$≈$k_{sample}$, then a typical AFM cantilever-tip or a commercial nanoindenter can only measure $E_{sample}$ in the ~kPa to GPa range. This range is wide enough to cover most synthetic materials including polymers, metals and ceramics, as well as a large variety of biological materials including tissues and adherent cells. However, there may be softer materials with moduli in the Pa range, such as floating cells, and these cannot be measured by an AFM or a commercial nanoindenter.

To measure $E_{sample}$ in the Pa range, “pico-indentation” using an optical tweezers system is suitable. Here, a laser beam is used to trap a translucent bead which is then brought into contact with the soft sample so as to indent it . The trap stiffness ($k_{machine}$) depends on the laser power and bead material, and a typical value is ~50 pN/μm. The probe size $a$ can be a micron or so. Then the optical trap can measure $E_{sample}$ (≈$k_{machine}$/$a$)in the Pa range.

(2) The second issue concerning soft samples is their viscoelasticity. Methods to handle viscoelasticity include the following.

(i) In the classical treatment of viscoelasticity, the load-displacement (P-h) response measured from the sample is fitted to predictions from an assumed constitutive model (e.g. the Maxwell model) of the material comprising spring and dashpot elements . Such an approach can be very time consuming, and cannot in general prove the assumed constitutive law in an unambiguous manner.

(ii) Dynamic indentation with an oscillatory load can be performed, and the viscoelastic behavior of the sample is presented in terms of the resultant storage and loss moduli, often as variations over the load frequency . However, the storage and loss moduli obtained this way are not intrinsic material constants, but depend on the oscillation frequency and the indenter probe geometry.

(iii) A “rate-jump” method can be used to return an intrinsic elastic modulus of the sample that is independent of the test conditions . In this method, a constitutive law comprising any network of (in general) non-linear dashpots and linear elastic springs is assumed to hold within a very short time window about the time instant tc at which a sudden step change in the loading rate is applied on the sample. Since the dashpots are described by relations of the form ${\dot {\epsilon }}$_ij=${\dot {\epsilon }}$_ij(${\dot {\sigma }}$_kl),the step change ∆${\dot {\sigma }}$_ij in the stress rate field ${\dot {\sigma }}$_kl at t_c does not result in any corresponding change in the strain rate field ${\dot {\epsilon }}$_ij across the dashpots, but because the linear elastic springs are described by relations of the form $\epsilon$_ij=S_ikjl$\sigma$_kl where S_ikjl are elastic compliances, a step change ∆${\dot {\epsilon }}$_ij across the springs will result according to

∆${\dot {\epsilon }}$_ij=S_ikjl∆${\dot {\sigma }}$_kl

The last equation indicates that the fields ∆$\sigma$_kl and ∆${\dot {\epsilon }}$_ij can be solved as a linear elastic problem with the elastic spring elements in the original viscoelastic network model while the dashpot elements are ignored. The solution for a given test geometry is a linear relation between the step changes in the load and displacement rates at tc, and the linking proportionality constant is a lumped value of the elastic constants in the original viscoelastic model. Fitting such a relation to experimental results allows this lumped value to be measured as an intrinsic elastic modulus of the material.

The rate-jump method

Specific equations from this rate-jump method have been developed for specific test platforms . For example, in depth-sensing nanoindentation, the elastic modulus and hardness are evaluated at the onset of an unloading stage following a load-hold stage. Such an onset point for unloading is a rate-jump point, and solving the equation $\epsilon$_ij=S_ikjl$\sigma$_kl across this leads to the Tang-Ngan method of viscoelastic correction

${\frac {1}{S_{e}}}$=${\frac {1}{2E_{r}a}}$=${\frac {\Delta {\dot {h}}}{\Delta {\dot {P}}}}$=${\frac {1}{S}}$-${\frac {{\dot {h}}_{h}}{{\dot {P}}_{u}}}$

where S = dP/dh is the apparent tip-sample contact stiffness at the onset of unload, ${\dot {h}}_{h}$ is the displacement rate just before the unload, ${\dot {P}}_{u}$ is the unloading rate, and $S_{e}$ is the true (i.e. viscosity-corrected) tip-sample contact stiffness which is related to the reduced modulus $E_{r}$ and the tip-sample contact size $a$ by the Sneddon relation . The contact size a can be estimated from a pre-calibrated shape function $f(h_{c})$=$\pi a^{2}$ of the tip, where the contact depth hc is obtainable using the Oliver-Pharr relation with the apparent contact stiffness $S$ replaced by the true stiffness $S_{e}$:

${\frac {1}{h_{c}}}$=$h_{max}$-${\frac {P_{max}}{S_{e}}}$=$h_{max}$-$P_{max}$$\left({\frac {1}{S_{e}}}-{\frac {{\dot {h}}_{h}}{{\dot {P}}_{u}}}\right)$