*2.1. Model Development*

Intrinsically speaking, plastic deformation of pure single-crystal materials is mainly determined by the mobile ability of dislocations when there exist sufficient initial dislocations. At low temperatures, the slipping of existing dislocations, especially for BCC metals, is dominated by short-range barriers like the Peierls potential. With the increase of *T*, long-range barriers induced by the network dislocation interaction become the dominant role in determining materials hardness [**? ?** ]. Therefore, the temperature dependent critical resolved shear stress *τ*CRSS(*T*), indicating the impediment of slipping dislocations, can be expressed as

$$
\pi\_{\rm CRSS}(T) = \pi\_{\rm n}(T) + \pi\_{\rm f}(T), \tag{1}
$$

where *τ*n(*T*) and *τ*f(*T*), respectively, denote the dislocation hardening term and lattice friction. Therefore, *τ*n(*T*) is influenced by both materials properties and dislocation microstructures at a given temperature. For the latter, statistically stored dislocations (SSDs) and GNDs should be simultaneously addressed when performing nano-indentation on pure single crystals. Following the Taylor law, the general form of *τ*n(*T*) with temperature effect yields as

$$
\pi\_{\mathbf{n}}(T) = b\mu(T)a(T)[\rho\_{\mathbf{G}}(T) + \rho\_{\mathbf{S}}(T)]^{m(T)},\tag{2}
$$

where *b* denotes the magnitude of Burgers vector. *μ*(*T*) and *α*(*T*) are, respectively, the shear modulus and dislocation strength coefficient, which both decrease with the increase of *T*. *m*(*T*) is the hardening coefficient that is generally around 0.5 at room temperature but also decreases with increasing temperature [**?** ]. Moreover, *ρ*S(*T*) and *ρ*G(*T*) denote the density of SSDs and GNDs, respectively. Following the theory of Nix and Gao [**?** ], GNDs are considered to generate beneath the indenter tip in order to coordinate with the gradient plastic deformation, and be stored within the plasticity affected region which is assumed to be a hemisphere, as illustrated in Figure **??**. Therefore, *ρ*G(*T*) can

be deduced by dividing the length *λ* of GNDs by the volume *V*(*T*) of the plasticity affected region, that is,

**Figure 1.** (Color online) Schematic of the nano-indentation of single crystal materials at two different temperatures, that is, *T*<sup>2</sup> > *T*1. The plasticity affected region is assumed as a hemisphere with radius of *R*, and *R*(*T*2) > *R*(*T*1).

$$\rho\_{\mathcal{G}}(T) = \frac{\lambda}{V(T)} = \frac{3}{2b \text{tan}\theta M^3(T)} \frac{1}{h'} \tag{3}$$

where *λ* = *πh*2/(*b*tan*θ*) and *V*(*T*) = 2/3*πR*3(*T*). Here, *θ* is the angle between the surface of the indenter and sample, and *R*(*T*) denotes the radius of the plasticity affected region that proportionally scales with the indentation depth *h* with a proportional coefficient *M*(*T*), that is, *R*(*T*) = *M*(*T*)*h*. At elevated temperatures, both lattice friction and dislocation impediment strength get weakened, which dramatically facilitate the expansion of the plastic region, and lead to the increase of *M*(*T*) and *R*(*T*) with *T* [**?** ], as presented in Figure **??**. Moreover, *ρ*S(*T*) is expressed as

$$\rho\_{\mathbb{S}}(T) = \frac{3}{2b\tan\theta} \frac{1}{h^\*(T)},\tag{4}$$

where *h*∗(*T*) represents a characteristic length that is connected to the bulk hardness [**?** ]. According to Ashby's definition [**?** ], SSDs are formed and accumulated in pure crystals during straining. Therefore, with higher internal strain stored in the materials at higher temperatures, more dislocations will be formed within the crystal that result in the higher density of SSDs.

Concerning the lattice friction, it is well known that *τ*f(*T*) for most face-centered cubic (FCC) materials is negligible when compared with the dislocation hardening term, therefore, the contribution of *τ*f(*T*) is generally ignored when addressing the temperature effect on materials' hardening [**???** ]. Whereas, for BCC materials, the stress required to move a dislocation over the Peierls potential is a thermally activated event, and takes a dominate role in determining the materials strength at low temperatures [**???** ]. Following the work of [**?** ], the expression of *τ*f(*T*) for BCC metals follows as

$$\tau\_{\rm f}(T) = \begin{cases} \tau\_{\rm f0} [1 - \sqrt{\frac{k\_{\rm f} T}{2I\hbar\_{\rm k}} \ln \left( \frac{\dot{\gamma}\_{\rm f0}}{\ell} \right)}] & (T \le T\_0) \\ \tau\_{\rm f0} [1 - \frac{k\_{\rm f} T}{2I\hbar\_{\rm k}} \ln \left( \frac{\dot{\gamma}\_{\rm f0}}{\ell} \right)]^2 & (T > T\_0) \end{cases} \tag{5}$$

where *τ*p0 and *τ*f0 are, respectively, the reference stress for the screw dislocations when *T* is below and above the critical temperature *T*0, which divides the deformation region into the elastic interaction and line tension regimes. *k*<sup>B</sup> indicates the Boltzmann constant and 2*H*<sup>k</sup> is the formation enthalpy of the kink pair on a screw dislocation. *γ*˙ p0 and *ε*˙ are the reference strain rate and loading strain rate, respectively.

By further considering the Mises flow rule [**?** ] and Tabor's factor [**?** ], one can connect the temperature dependent hardness *H*(*T*) with *τ*CRSS(*T*) as

$$H(T) = 3\sqrt{3}\tau\_{\text{CRSS}}(T) = H\_{\text{f}}(T) + H\_{\text{n}}(T),\tag{6}$$

where *H*f(*T*) = 3 <sup>√</sup>3*τ*f(*T*) is the hardness component induced by lattice friction, and *<sup>H</sup>*n(*T*) denotes the dislocation hardening component deduced by submitting Eqsuations (**??**) and (**??**) into Equation (**??**), that is,

$$H\_{\mathbf{h}}(T) = H\_0(T)[1 + \frac{\bar{h}^\*(T)}{h}]^{m(T)} \tag{7}$$

where

*H*0(*T*) = 3 √ 3*bμ*(*T*)*α*(*T*)*ρ m*(*T*) <sup>S</sup> (*T*), (8)

and

$$\bar{h}^\*(T) = h^\*(T) / M^3(T) \quad \text{with} \quad h^\*(T) = \frac{3[3\sqrt{3}b\mu(T)a(T)]^{\frac{1}{m(T)}}}{2b\tan\theta H\_0^{\frac{1}{m(T)}}(T)}.\tag{9}$$

Further derivation of Equation (**??**) indicates that [*H*n(*T*)/*H*0(*T*)]1/*m*(*T*) scales linearly with 1/*h*, and the slope ¯ *h*∗(*T*) is determined by both the characteristic length *h*∗(*T*) and proportional coefficient *M*(*T*). On the one hand, it shows that *ρ*S(*T*) tends to increase with *T*, which results in the decrease of *h*∗(*T*) at elevated temperatures [**?** ]. On the other hand, as the impediment of slipping dislocations gets weakened at high temperatures, the expansion of the plasticity affected region becomes comparatively easy, which results in the increase of *M*(*T*) with *T* [**?** ]. Therefore, ¯ *h*∗(*T*) tends to decrease at high temperatures that results in the weakened indentation size effect at elevated temperatures. Furthermore, increasing temperature not only leads to the decrease of *H*0(*T*) and *m*(*T*), but also weakens the lattice friction, thus, it becomes rational to experimentally observe that *H*(*T*) decreases with the increase of *T* for most crystalline materials [**???** ].

One may also note that Equation (**??**) offers a general law characterizing the hardness-depth relationships of pure single crystals at various temperatures. When ignoring the temperature effect, Equation (**??**) can be degraded into the hardness model involving lattice friction effect at room temperature. Once the hardening contribution of lattice friction is further ignored, the model is ultimately reduced to the classical Nix-Gao model [**?** ]. This simplification is reasonable and rational for materials with small lattice friction. Whereas, when the lattice friction is relatively comparable with the dislocation hardening term, the ignoring of the former will result in the overestimation of *H*<sup>0</sup> and *ρ*S.
