# The ‘pop-in’ phenomenon¶

The nanoindentation (or instrumented or depth sensing indentation) is a variety of indentation hardness tests applied to small volumes. During nanoindentation, an indenter is brought into contact with a sample and mechanically loaded.

The following parts give a short overview of models existing in the literature used for the extraction of mechanical properties of homogeneous bulk materials from indentation experiments with conical indenters.

Parameters such as contact load $$F_\text{c}$$ and depth of penetration $$h$$ are recorded at a rapid rate (normally 10Hz) during loading and unloading steps of the indentation test. Usually, the depth resolution is around the fraction of $$\text{nm}$$-level and the resolution is around $$\text{nN}$$-level.

During the loading step of nanoindentation, a discontinuity in the measured depth is commonly referred to as a pop-in event.

## The pop-in event¶

A pop-in (event) is a sudden (load or displacement) burst during the loading of an indenter on a sample. If the nanoindentation experiment is load-controlled, an horizontal plateau is also observed on the load-displacement curve, when a pop-in occurs at the critical load $$F_\text{crit}$$ and critical displacement $$h_\text{crit}$$ (see Figure 1). In the case of a displacement-controlled nanoindentation experiments, a vertical drop of the load is observed on the load-displacement curve.

Figure 1 : Schematics of indentation load-displacement curve with a pop-in: a) load-controlled and b) displacement-controlled nanoindentation experiments.

$$F_\text{c,max}$$ is the maximum applied load, $$h_\text{t}$$ is the total indentation depth and $$h_\text{r}$$ is the residual indentation depth.

Find here the Matlab function to quantify pop-in by peak detection: peakdet.m.

Warning

In this toolbox, only load-controlled nanoindentation experiments are analyzed.

Many authors observed pop-in events on metals or metallic thin films, ceramics, semiconductors, hard brittle thin films deposited on a soft elastoplastic substrate...

The pop-in event is often explained by on of the following mechanisms in function of the indented specimen (ceramic, metal, semiconductor, coated or multilayer specimen...) and the experimental conditions (time, temperature, geometry of the indenter...) :

Some authors proposed to describe the statistics of the pop-in event with a cumulative Weibull-type distribution [4] or with a cumulative fraction function based on a rate equation, when a time or a temperature dependence of the pop-in is demonstrated.

## Weibull-type distribution¶

The cumulative Weibull distribution function with 2 parameters $$(\lambda, m)$$ is a continuous probability distribution and is often used in the description of particle size distribution and in survival or failure analysis [22].

(1)$V\left(x, \lambda, m\right) = exp\left(\left(\frac{x}{\lambda}\right)^m\right)$
(2)$W\left(x, \lambda, m\right) = 1 - exp\left(\left(\frac{x}{\lambda}\right)^m\right)$

With $$V$$ the survival probability function and $$W$$ the mortal probability function. $$m$$ is a dimensionless material-dependent constant, often named the Weibull modulus (from 0 to usually 50) [1]. If $$m=1$$, the rate of failure remains constant and there is random failure occurring.

In the case of several indentations performed on the same sample, $$x$$ can be the distribution of the critical loads $$F_\text{crit}$$ or the critical displacements $$h_\text{crit}$$, at which the pop-in events appear on the corresponding load-displacement curves. In this case, higher is $$m$$, more homogeneous is the distribution of the pop-in.

$$\lambda$$ is the scale parameter of the cumulative Weibull distribution. In the case of indentations, $$\lambda$$ is the mean critical load $$F^0_{crit}$$ or the mean critical displacement $$h^0_{crit}$$ at which the pop-in event appears for a given material.

Find here the Matlab function to calculate the cumulative mortal Weibull distribution: weibull_cdf.m.

Find here the Matlab function to calculate the complementary cumulative survival Weibull distribution: weibull_cdf_survival.m.

Chechenin et al. proposed to use a modified cumulative Weibull distribution function for the description of the statistics of the pop-in event [4]. This function is set to have a probability of 0.5, when $$F_{crit}$$ (the critical load) is equal to $$F^0_{crit}$$ (the mean critical load).

(3)$W\left(\frac{F_\text{crit}}{F^0_\text{crit}}\right) = 1 - exp\left(-ln2\left(\frac{F_\text{crit}}{F^0_\text{crit}}\right)^m\right)$

The cumulative Weibull distribution [22] and the modified cumulative Weibull distribution [4] are implemented in the PopIn toolbox.

Find here the Matlab function to calculate the cumulative modified mortal Weibull distribution: weibull_modified_cdf.m.

Find here the Matlab function to calculate the complementary cumulative modified survival Weibull distribution: weibull_modified_cdf_survival.m.

## Dislocations nucleation¶

According to many authors, pop-in corresponds to the nucleation of at least one dislocation, during indentation of crystalline materials (see Figure 2) [14], [15], [16], [9], [23], [12], [8], [13], [24] and [25]. Before the pop-in event, the indentation load-displacement obtained with a spherical indenter, can be fitted with the Hertz equation (see Figure 3) [6]:

(4)$F_\text{c} = (4/3) E^{*} \sqrt{R^{*}} F_\text{c}^{1.5}$

With $$E^{*}$$ the reduced Young’s modulus calculated from indentation test defined by (5) and $$R^{*}$$ the reduced radius of the spherical indenter defined by (6).

(5)$\frac{1}{E^{*}} = \frac{1}{E_\text{s}^{'}} + \frac{1}{E_\text{i}^{'}}$

With $$E_\text{s}^{'}$$ the reduced Young’s modulus of the sample and $$E_\text{i}^{'}$$ the reduced Young’s modulus of the indenter.

(6)$\frac{1}{R^{*}} = \frac{1}{R_\text{s}} + \frac{1}{R_\text{i}}$

With $$R_\text{s}$$ the radius of the sample (usually $$+\infty$$) and $$R_\text{i}$$ the radius of the indenter.

Find here the Matlab function to calculate the load with the Hertz equation: elasticLoad.m.

Find here the Matlab function to calculate the displacement with the Hertz equation: elasticDisp.m.

Find here the Matlab function to calculate the contact radius with the Hertz equation: elasticRadius.m.

Find here the Matlab function to calculate the reduced value of a variable: reducedValue.m.

Figure 2 : Schematics cross section of deformation profile of an elastic-plastic substrate under indentation : 1) elastic deformation, 2) elastoplastic deformation (nucleation of dislocation) and 3) transfer of dislocations across a grain boundary.

Figure 3 : Schematic of indentation load-displacement curve with two pop-in events (the 1st for the nucleation of dislocation and the 2nd for the strain transfer across a grain boundary).

The elastic-plastic transition is often studied statistically as a function of temperature $$T$$ and indentation rate $$F(t)$$. Schuh C.A. et al. observed a good consistency between experimental results and a thermally activated mechanism of incipient plasticity [14], [15], [16], [9]. They also proposed to use a statistical thermal activation model (see following equations) with a stress-biasing term. Based on this model, it is possible to extract the activation energy $$\epsilon-\sigma V$$, activation volume $$V$$, and attempt frequency for the rate-limiting event that controls yield $$W(F_\text{c})$$. The activation energy $$\epsilon-\sigma V$$ is separated into a term characterizing the activation enthalpy $$\epsilon$$, and a term giving the stress bias $$\sigma V$$, where $$\sigma$$ is the biasing stress over the activation volume $$V$$.

(7)$\dot{n} = \eta exp\left(-\frac{\epsilon-\sigma V}{kT}\right)$
(8)$\dot{N} = \eta exp\left(-\frac{\epsilon}{kT}\right)\cdot\int\int\limits_\Omega\int exp \left(\frac{\sigma V}{kT}\right)d\Omega$
(9)$\dot{F}(t) = \left[1-F(t)\right]{N}(t)$
(10)$F(t) = 1- exp\left(-\int_{0}^{t}{\dot{N}(t')dt'}\right)$
(11)$\tau_\text{max} = 0.31p_0 = 0.47p_\text{max} = \left(\frac{0.47}{\pi}\right)\left(\frac{4E^{*}}{3R^{*}}\right)F^{1/3}_{crit}$
(12)$\Omega \approx Ka^3 = K\left(\frac{3F_{crit}R^{*}}{4E^{*}}\right)$
(13)$F_\text{c} = \dot{F_\text{c}}\cdot t$

With $$\dot{n}$$ is the local rate at which the critical event occurs per unit volume of material, $$\eta$$ the pre-exponential frequency factor, $$k$$ the Boltzmann constant, $$\tau_\text{max}$$ the maximun shear stress obtained for the maximum pressure $$p_\text{max}$$, at a single point beneath the indenter given for an elastic Hertzian contact [6]. $$K$$ is a proportionality constant of order $$\pi$$.

Find here the Matlab function to calculate the maximum shear stress: maxShearStress.m.

Find here the Matlab function to calculate the maximum pressure: maxPressure.m.

Find here the Matlab function to calculate the mean pressure: meanPressure.m.

Finally, a statistical expression of the onset of plasticity can be formulated by combining previous equations, giving the cumulative fraction function $$W(F_\text{c})$$ :

(14)$W(F_\text{c}) = 1-exp\left(-\frac{9KR^{*}\eta}{4E^{*}\dot{F_\text{c}}\alpha^6}exp\left(-\frac{\epsilon}{kT}\right)\left(\beta(\alpha,F_\text{c})\right)\right)$
(15)$\alpha = \left(\frac{0.47}{\pi}\right)\left(\frac{4E^{*}}{3R^{*}}\right)^{2/3}\frac{V}{kT}$
(16)$\beta = 120exp(-F_\text{c}^{1/3}\alpha) + F_\text{c}^{5/3}\alpha^5 - 5F_\text{c}^{4/3}\alpha^4 + 20F_\text{c}\alpha^3-60F_\text{c}^{2/3}\alpha^2+120F_\text{c}^{1/3}\alpha-120$

Find here the Matlab function to calculate the cumulative survival distribution in function of the loadrate and the temperature of nanoindentation tests: mason_cdf.m.

Find here the Matlab function to calculate the $$\alpha$$ function: alphaMason.m.

Find here the Matlab function to calculate the $$\beta$$ function: betaMason.m.

Note

The surface mechanical state (presence of dislocations after polishing steps or surface free of dislocation) can modified the statistics of pop-in behavior [21].

Note

See this Github repository for the plot of stress distributions at the surface and along the axis of symmetry, caused by Hertz pressure acting on a circular area radius.

## Strain transfer across grain boundaries in metals¶

Some authors observed sometimes two pop-in on the load-displacement curve during indentation performed close to a grain boundary (see Figure 2 and Figure 3). The first pop-in is usually attributed to the nucleation of dislocation in a metallic material (see previous section of this documentation), and the second pop-in is related to the presence nearby of the grain boundary.

The occurrence of such a strain burst is found to be related to the slip activity (function of the phase material and the grain orientation), to the grain boundary resistance (function of the grain boundary misorientation [20], to the local chemistry [3] (impurities, embrittled hydrogen...), to the prior plastic deformation [3], to the distance between the indenter and the grain boundary [20], or other experimental parameters like the shape of the indenter and the grain boundary inclination...

Recently, the STABiX Matlab toolbox was developed to analyse in simple way slip transmission in a bicrystal [10], [18] and [17].

## Rupture of a hard brittle film on an elastic-plastic substrate¶

In the case of indentation made into a hard brittle film (e.g.: native or thermally/anodically grown oxide, ALD coatings ...) on an elastic-plastic (ductile) substrate, pop-in were observed experimentally and linked to the fracture of the brittle film (see Figures 4 and 5) [7], [2], [11] and [19].

Figure 4 : Schematic of indentation load-displacement curve with a pop-in event and the plot of the load carried by the elastic-plastic substrate.

Some authors explained that a circumferential crack appears at the location of the elastic–plastic boundary in the substrate [2] and [11]. The radius $$c$$ of this plastic zone in the ductile substrate is defined by the following equation :

(17)$c = \sqrt{\frac{3F_\text{crit,s}}{2\pi\sigma_\text{e}}}$

With $$F_\text{crit,s}$$ the load carried by the substrate at the critical indentation depth at which pop-in occurs, obtained usually with the power law relationship (18). $$\sigma_\text{e}$$ is the yield stress of the ductile substrate [5].

(18)$F_\text{crit,s} = Kh_\text{crit}^n$

With $$h_\text{crit}$$ the critical displacement at which the pop-in appears. The two constants $$K$$ and $$n$$ are obtained from a fitting procedure of the load-displacement curve obtained from nanoindentation tests performed on the substrate without the brittle film.

Figure 5 : Schematic cross section of deformation profile of a hard brittle film on an elastic-plastic substrate under indentation.

In the Figure 5, $$R$$ is the radius of the spherical indenter, $$t$$ is the thickness of the thin film, $$F_\text{c}$$ is the applied load, $$h$$ is the indentation displacement, $$c$$ the radius of the plastic zone in the substrate and $$a_\text{c}$$ the contact radius between the indenter and the thin film.

Find here the Matlab function to calculate the plastic radius $$c$$: plasticRadius.m.