Thickness-Dependent Strain Rate Sensitivity of Nanolayers via the Nanoindentation Technique

Abstract

The strain rate sensitivity (SRS) and dislocation activation volume are two inter-related material properties for understanding thermally-activated plastic deformation, such as creep. For face-centered-cubic metals, SRS normally increases with decreasing grain size, whereas the opposite holds for body-center-cubic metals. However, these findings are applicable to metals with average grain sizes greater than tens of nanometers. Recent studies on mechanical behaviors presented distinct deformation mechanisms in multilayers with individual layer thickness of 20 nanometers or less. It is necessary to estimate the SRS and plastic deformation mechanisms in this regime. Here, we review a new nanoindentation test method that renders reliable hardness measurement insensitive to thermal drift, and its application on SRS of Cu/amorphous-CuNb nanolayers. The new technique is applied to Cu films and returns expected SRS values when compared to conventional tensile test results. The SRS of Cu/amorphous-CuNb nanolayers demonstrates two distinct deformation mechanisms depending on layer thickness: dislocation pileup-dominated and interface-mediated deformation mechanisms.

1. Introduction

For most metallic materials, the plastic deformation can be regarded as a thermally-activated process which can be quantitatively characterized by the values of the strain rate sensitivity (SRS), m, and the dislocation activation volume, V* [1,2]. SRS is an important indicator of the plasticity of metallic materials, which can be estimated by the amount of activation volume required for dislocation motions. These two inter-related material properties are important for understanding thermally-activated plastic deformation, such as creep.

In general, for crystalline metals, higher values of SRS are often accompanied by excellent ductility and deformability. The values of SRS are strongly size-dependent: for face-centered-cubic (fcc) metals, SRS normally increases with decreasing grain sizes, whereas the opposite holds for body-center-cubic (bcc) structures [3]. Metallic glasses (MGs) exhibit high strength, excellent abrasion and corrosion resistance [4], but they are generally brittle due to the shear band (SB) controlled-deformation mechanisms therefore the SRS is close to zero [5]. Prior studies have shown that adding a crystalline phase to the amorphous matrix can increase the toughness/plasticity of ZrTi-based MGs [6] and crystalline/amorphous (C/A) multilayers [7,8,9,10,11,12,13,14]. For the C/A multilayer composites, their thermally-activated plastic deformation mechanisms can also be revealed by quantifying their SRS. Thus, various methods have been developed for determining SRS in the past few decades [15,16,17,18,19,20,21,22].

The method for determining SRS can be roughly classified as macroscopic testing and “localized” testing, usually nanoindentation. Macroscopically, the SRS is often measured by uniaxial tension [15,16,17,18,19,20] and compression tests [21,22] on bulk specimens, whereas nanoindentation is widely adopted to measure SRS of small specimens, such as thin films. Figure 1 illustrates the differences between SRS measurement under tension/compression and nanoindentation. In addition to the different scope of applications, the key difference between uniaxial tension/compression and nanoindentation measurements lies in the determination of strain rate ($\dot{\epsilon }$). In the tensile/compression testing method, $\dot{\epsilon }$ is calculated based on the original length of sample ($l_o$), and its increment ($\Delta l$) under displacement-control mode. In comparison, the nanoindentation SRS measurements achieve a constant indentation strain rate $\dot{h}/h$ through maintaining $\dot{P}/P$ based on their relationship (h is the indentation depth, $\dot{h}$ is the displacement rate, $P$ is the loading force, and $\dot{P}$ is the real-time loading rate) [23]. Although the nanoindentation method determines SRS on a local scale, many nanoindentation experiments are typically performed to ensure the results are representative of their overall mechanical properties.

In 1999, the $\dot{P}/P$ technique for SRS measurement was first presented by Lucas and Oliver, and they applied this technique for measuring the hardness of high-purity indium under various strain rates [23]. This method shows good applicability during the high strain rate experiment, however, when it comes to low strain rates, the displacement of the indenter will be significantly affected by thermal drift. In 2010, the continuous stiffness measurement (CSM) that could decrease the influence of thermal drift was proposed by Hay et al. [24]. In 2011, an indentation strain-rate-jump test method for measuring SRS proposed by Maier et al. showed a significant mitigation of the thermal drift [25]. Unlike the previous methods, this special nanoindentation creep method alleviates thermal drift by first applying the highest strain rates and then the lowest strain rates. However, during the strain-rate-jump test, the indentation depths must be greater than 500 nm for reliable determination of SRS.

Recent studies on mechanical behaviors of multilayers focused on deformation mechanisms when the individual layer thickness is 20 nm or less where the majority of multilayer systems reach high strength [26,27,28]. However, there are limited studies on the size-dependent SRS in multilayers using the nanoindentation test, except some atomistic simulations on the deformation mechanisms of nano-sized thin films [29,30,31,32]. Here, we review a new nanoindentation test method that renders hardness nearly insensitive to thermal drift. Such a technique permits reliable determination of SRS for Cu films. The SRS measured using this technique reveals size-dependent variation of deformation mechanisms in Cu/a-CuNb (crystalline Cu/amorphous CuNb) multilayers.

2. Experimental Methods

The 1.5 μm Cu film, 1 μm thick a-CuNb single layer and the Cu/a-CuNb nanolayers with different individual layer thickness (h, ranges from 5 to 150 nm; and total thickness ranges from 1 to 2.4 μm) were deposited on HF-etched Si wafers by DC (direct current) magnetron sputtering. Pure Cu (99.995%) and CuNb alloy (Cu 50 at%-Nb 50 at%) targets were used to prepare these nanostructured materials. The microstructure was characterized by FEI Tecnai G2 F20 TEM (Thermo Fisher Scientific, Hillsboro, OR, USA). All the nanoindentation tests were conducted on Agilent G200 (Agilent Technologies, Santa Clara, CA, USA) and Hysitron TI950 TriboIndenter (Bruker, Billerica, MA, USA) in the continuous stiffness measurement (CSM) mode. For a-CuNb and Cu/a-CuNb multilayers, at least 10 tests were conducted for each sample at each specific strain rates of 0.2, 0.05, and 0.01 s⁻¹, respectively. For the Cu single-layer, the specific testing strain rates are 0.05, 0.01, and 0.002 s⁻¹.

This new modified testing method developed by Liu et al. [33] involves directly measuring the contact stiffness and calculating the contact area from the measured stiffness and modulus. Figure 2 depicts the differences between the conventional and new modified method. It is easy to find that both the conventional and modified method are based on the relationship performed by Sneddon’s stiffness equation [34]:

$$E_r=\frac{\sqrt{\pi }}{2} \frac{S}{\sqrt{A}}$$

(1)

where E_r is the reduced elastic modulus, A and S represent the contact area and contact stiffness, respectively [35,36]. The primary difference between these two methods is reflected in the different calculation process of A. As for the conventional method, h (indentation displacement), P (load), and S at different $\dot{\epsilon }$ can be directly measured, the h_c (contact depth) can be calculated by using the equation:

$$h_c=h-\frac{0.75P}{S}$$

(2)

where $0.75P/S$ is a calculated value which represents $h_s$ (the vertical distance from the contact point of the sample and the indenter to the sample surface) [35], then A could be derived by:

$$A=m_0{h}_c^2+m_1{h}_c$$

(3)

where $m_0$ and $m_1$ are determined by the standard calibration process [35]. At last, E_r and H (hardness) could be derived by those parameters. At high strain rates, the measurement error of h is negligible. However, at lower strain rate, the measurement of h will be significantly affected by thermal drift which makes the later calculations become unreliable. Unlike the conventional method, the modified method calculates the contact area from the measured stiffness and modulus, and the process can be summarized as follows: Firstly, measured E_r at high $\dot{\epsilon }$, then E_r as a known parameter can be used at lower $\dot{\epsilon }$ later. Secondly, at low $\dot{\epsilon }$ condition, A can be derived by Equation (1) (thus, H is calculated), then h_c is calculated from Equation (3), and at last, h can be calculated by using the Equation (2). The calculated h instead of directly measured can reduce the thermal-drift effects significantly and control the real-time indentation displacement [33].

3. Results and Discussion

The hardnesses of single-layer Cu and a-CuNb films were measured by using this modified method. Figure 3 shows the comparison of nanoindentation results obtained from conventional (blue) and modified (red) methods using the same sets of nanoindentation data at various strain rates. Both the methods show good stability during the high strain rate experiment, and lead to similar results. However, at the lower strain rate, for a-CuNb multilayers (0.01 s⁻¹), the scattering data presented by the conventional method is not acceptable, while the convergent and consistent hardness values were obtained from the modified method. For Cu films (0.002 s⁻¹), both methods returned similar results, except the hardness obtained by the conventional method is slightly larger than the modified method, and the convergence of the conventional method is slightly better than the modified method. This is because under the condition of a small load or small indentation depth, the determination of the contact area from only contact stiffness and load is less accurate than from contact stiffness, load, and indentation depth, despite the thermal drift.

The SRS values of Cu film were calculated by using the hardness obtained at a depth of 300 nm from the two methods under different strain rates. As shown in Figure 4a, the SRS value (m = 0.048 ± 0.006) produced from the modified method (half-filled red circles) are less scattered than he conventional method. In contrast, the conventional method (half-filled black squares) leads to relatively erroneous and unacceptable SRS values (m = 0.112 ± 0.020), which are more than twice that of the modified method.

In order to further verify the reliability of the new method, we compare the measured SRS value of Cu film with the results that are obtained from bulk Cu samples or Cu films recorded in the previous literature, such as uniaxial tension, compression, and indentation jump tests [17,18,19,20,37,38,39,40,41], and plot the results in Figure 4b. For fcc metals, SRS normally increases with decreasing grain sizes. Here, the SRS value of Cu film with average grain size of ~70 nm (confirmed by TEM shown in Figure 5a) obtained from the modified technique is consistent with the general trend reported before, shown in Figure 4b. However, the conventional method shows very different results due to the influence of thermal drift at lower strain rates.

After the new technique is validated through Cu films and returns expected SRS values, the SRS of C/A multilayers can be easily determined. Thus, in this section, the relationship between individual thickness (h), the plastic deformation mechanism, and the SRS of Cu/a-CuNb multilayers with different individual layer thickness (h, ranging from 5 to 150 nm) are systematically discussed. The image in Figure 5b is the cross-sectional microstructure of Cu 100 nm/a-CuNb 100 nm multilayers (referred as Cu/a-CuNb 100 nm) characterized by TEM. The layered structures can be clearly distinguished, and the selected area diffraction (SAD) pattern in the lower right corner shows diffuse diffraction halo and diffraction spots which represent the featureless amorphous layer and polycrystalline Cu layers, respectively.

Before conducting the nanoindentation experiments, we can briefly discuss the relationship between the SRS and individual layer thickness in crystalline (fcc metals)/amorphous multilayers. First, we assume that the C/A multilayers are under an isobaric stress condition during the nanoindentation experiment (i.e., the stress applied to crystalline layers and amorphous layers are equal). In this case, the total plastic deformation can be divided into two parts: the plastic deformation (strain) from the amorphous phase and from the crystalline phase. Moreover, the plastic deformation of these two phases is different partly due to the hardness of the amorphous phase being much higher than the crystalline phase. Thus, when the individual thickness of the amorphous phase and the crystalline phase are designed to the same value, the strain of C/A multilayers can be expressed as:

$$\epsilon =\frac{1}{2}\left({\epsilon }_a+{\epsilon }_c\right)$$

(4)

where $\epsilon$ is the total strain (indenter displacement divided by the film thickness: $\Delta L/L$), ${\epsilon }_a$ and ${\epsilon }_c$ are the strain of the amorphous layers and crystalline layers. Then, the strain rate can be shown as:

$$\dot{\epsilon }=\frac{1}{2}\left(\dot{{\epsilon }_a}+\dot{{\epsilon }_c}\right)$$

(5)

Finally, according to [42], the SRS of C/A multilayers can be calculated as:

$$\frac{1}{m}=\frac{\partial \ln \dot{\epsilon }}{\partial \ln \mathsf{\sigma }}= \frac{\Delta L_a}{\Delta L_a + \Delta L_c} \times \frac{1}{m_a} + \frac{\Delta L_c}{\Delta L_a + \Delta L_c} \times \frac{1}{m_c}$$

(6)

where $\Delta L_a$, $m_a$, $\Delta L_c$, and $m_c$ represent the displacement and SRS of the amorphous phase and the crystalline phase, respectively. Here, it is important to note that we should figure out the proportion of ${\epsilon }_a$ and ${\epsilon }_c$ in the total plastic deformation. That means the key in this study is to determine how the displacement is distributed. In Equation (6), both the SRS and the percentage of plastic deformation accommodated by the crystalline ($\frac{\Delta L_c}{\Delta L_a+\Delta L_c}$) and amorphous ($\frac{\Delta L_a}{\Delta L_a+\Delta L_c}$) layers should change with the change of h.

For the SRS of Cu/a-CuNb multilayers, we take the previously measured SRS of the 1.5 μm Cu film (m = 0.048, assuming that m does not change in this regime) into Equation (6) and treat m_a as a value close to zero, then we can qualitatively discuss the relationship between m and h. As shown in Figure 6a, the modeled curve (dashed line) indicates the evolution of co-deformation of both Cu and a-CuNb. In addition, the SRS of a-CuNb, Cu, their average value (dotted lines) and Cu/a-CuNb are also plotted on the graph.

The experimental data are consistent with the model discussed previously. At lager h, the plastic deformation is mainly generated by the crystalline phase, the value of $\frac{\Delta L_c}{\Delta L_a+\Delta L_c}$ will be close to 1 and $\frac{\Delta L_a}{\Delta L_a+\Delta L_c}$ close to zero. Therefore, the SRS should be similar to the single layer Cu. However, when h is smaller, the $\frac{\Delta L_c}{\Delta L_a+\Delta L_c}$ and $\frac{\Delta L_a}{\Delta L_a+\Delta L_c}$ would be close to 0.5 due to the co-deformation of crystalline phase and amorphous phase (the two phases share the deformation equally) [8,13], and the SRS would be closer to the single-layer amorphous film based on Equation (6).

Figure 6b is a schematic showing the different deformation mechanisms at different h. When h > 100 nm, Cu layers deform plastically due to its low strength and accommodate most of the strain, and thus Cu/a-CuNb multilayers have an apparent SRS value similar to that of the single layer Cu film. The mechanism of plastic deformation in this regime is dominated by dislocation activities inside the Cu layers.

When h < 50 nm, the SRS of Cu/a-CuNb multilayers decreases with decreasing h. The yield strength of Cu film increases with decreasing h due to the Hall-Petch effect (although there may occasionally be an inverse Hall-Petch effect when the grain size decreases to a few nanometers, here we leave this possibility aside). At the same time, the density of dislocations in the Cu grains decreases and the activation volume increases. The generation and propagation of dislocation is very difficult, and the stress concentration caused by the dislocations at the interface is not enough to transmit the dislocations across the interface. The limited dislocations will be confined within the Cu layers and slide along the interface [43]. On the other hand, for the a-CuNb layers, the deformation mechanism is the coalescence of shear transformation zones instead of major shear banding, and at this layer thickness, Cu and a-CuNb layers can co-deform.

4. Conclusions

A new nanoindentation test method that enables reliable determination of SRS is reviewed. The new method is applied in two model systems (single-layer Cu film and Cu/amorphous-CuNb multilayers) to reduce the thermal drift effect and yields reliable results. The method yields correct SRS of single-layer Cu film compared with the conventional method, and the layer thickness-dependent SRS of Cu/a-CuNb nanolayers under the iso-stress condition is systematically discussed: when h > 100 nm, the plastic deformation is mainly accommodated by the Cu layers through dislocation pile-up, and Cu/a-CuNb multilayers have an apparent SRS value similar to that of the single layer Cu film; when h < 50 nm, the crystalline and amorphous layers co-deform, and the SRS of Cu/a-CuNb multilayers decreases with decreasing h. The main deformation mechanism is interface-mediated. This new method is beneficial for the measurement of hardness and the SRS of nano-scale materials without the thermal drift error at low strain rate under nanoindentation.