# Modeling of the Atomic Diffusion Coefficient in Nanostructured Materials

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## Abstract

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## 1. Introduction

_{gb}) is less than the surface energy (γ

_{sv}) [7]. NSs are widely used, but receive more attention in magnetic data storage, ultra-large-scale integration, thermoelectric power generation, and structural engineering [8,9,10]. In order to provide the critical information for the above applications, the atomic diffusion property of NSs should be taken into consideration regarding diffusion processing [2,11].

_{T}(L,T) of NSs increased as L declined [11,14], where T denotes the temperature. The D

_{T}(L,T) of nanostructured Cu is about 10~16 orders of magnitude higher than the bulk Cu when L is approximately 13 nm [15], while an increase in D

_{T}(L,T) from 0.33 × 10

^{−8}m

^{2}/s to 1.0 × 10

^{−8}m

^{2}/s was observed for nanostructured Ni when L decreased from 9 nm to 4 nm [16]. Owing to this, NSs are usually used in diffusion processing, including coating [16,17], nitriding [18], and carburizing [19], significantly improving the processing efficiency [20]. Since NSs consist of nanoscaled grains with a solid–solid interface or grain boundary between them [21], the increase in the diffusion shortcuts should have had a significant impact on raising the diffusivity. Considering that the coordination imperfection exists at the interface, the diffusion might also have been influenced. Compared to NPs with free surfaces, the coordination imperfection should have been weak at the interface, influencing the diffusion of NSs. However, a theoretical way to elucidate this has still remained unavailable.

_{T}(∞,T) can be expressed as D

_{T}(∞,T) = ${D}_{T}^{0}(\infty )\mathrm{exp}\left[-{E}_{a}(\infty )/\left(RT\right)\right]$, where ∞ denotes the bulk size, ${D}_{T}^{0}$ is the pre-exponential constant, the activation energy is denoted by E

_{a}, and the ideal gas constant is denoted by R. Upon the atomic diffusion in NPs, the surface effect on the diffusion coefficient of atoms, denoted as ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$, has been considered and modeled [14]. According to this work, extending the above D

_{T}(∞,T) expression into the nanometer regime, ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$ can be given as [22],

_{m}being the melting point. Moreover, based on Lindemann’s criterion and Mott’s equation of vibrational entropy, the size-dependent T

_{m}of NPs denoted as ${T}_{m}^{\mathrm{NP}}(L)$ is formulated as [23]

_{NP}= 2S

_{vib}(∞)/3R + 1 denotes the surface effect factor, S

_{vib}is the vibrational contribution of the overall melting entropy of the bulk crystals. L

_{0}is the critical diameter of a nanoparticle, where almost all atoms are located on the surface, where L

_{0}= 2(3 − d)h, d represents the dimension of the nanoparticle, with d = 0 for nanoparticles, d = 1 for nanowires, d = 2 for thin films, and h is the atomic diameter [24]. In light of this, ${T}_{m}^{\mathrm{NP}}(L)$ decreased as L was lowered [23]. An important reason for this change is relevant to the decrease in atomic cohesive energy associated with the atomic coordination imperfection at the surfaces. Based on Equation (1), and with the help of Equation (2), as L decreases, the diffusion coefficient ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$ is increased by several orders of magnitude [14,25]. When L is as small as several nanometers, ${D}_{T}^{\mathrm{NP}}\left(L,T\right)/{D}_{T}\left(\infty ,T\right)$ could be greater than 10

^{10}due to the drop of ${E}_{a}^{\mathrm{NP}}(L)$. Moreover, the diffusion temperature could be lowered by several hundred degrees when a constant diffusion coefficient is required [14]. This increase is believed to be attributed to the surface energy, which is associated with the coordination imperfection at the surface.

## 2. Model

_{T}[∞, T

_{m}(∞)] is assumed [22], and one sees ${D}_{T}^{\mathrm{NS}}[L,{T}_{m}^{\mathrm{NS}}(L)]$ = ${D}_{T}^{0}(L)\mathrm{exp}\{-{E}_{a}^{\mathrm{NS}}(L)/[R{T}_{m}^{\mathrm{NS}}(L)]\}$ = ${D}_{T}^{0}(\infty )\mathrm{exp}\{-{E}_{a}(\infty )/[R{T}_{m}(\infty )]\}$. In terms of the point defect mechanism [14], ${E}_{a}^{\mathrm{NS}}(L)$ in this expression means the activation enthalpy with ΔH = ${E}_{a}^{\mathrm{NS}}(L)$, and ${D}_{T}^{0}$ is an amount proportional to λ

^{2}, Z and exp(ΔS

_{NS}/R), where λ is the interplanar crystal spacing, Z is the nearest neighbor gap number and ΔS

_{NS}is the activation entropy. The sizes of λ, Z, and ΔS

_{NS}do not affect the thermal vibration energy. According to the thermodynamic knowledge [14], T [∂ΔS

_{NS}(L)/∂T]

_{P}= ${[\partial {E}_{a}^{\mathrm{NS}}(L)/\partial T]}_{P}$, where P = 4f/L is the internal pressure of the sphere particles under a specific size. Hence, ΔS

_{NS}(L) changes with ${E}_{a}^{\mathrm{NS}}(L)$. However, regarding the activation process, the change in ΔS

_{NS}(L) caused by the change of vibrational frequency is less than 5%, which is quite small even when L is changed from the bulk size to 4–6 nm [26]. ${E}_{a}^{\mathrm{NS}}(L)$ is therefore temperature independent. ${D}_{T}^{0}(L)$ is a weak function of L, but the exponential effect $\mathrm{exp}\left\{-{E}_{a}^{\mathrm{NS}}\left(L\right)/\left[R{T}_{m}^{\mathrm{NS}}\left(L\right)\right]\right\}$ on ${D}_{T}^{0}(L)$ is very strong. As a first order approximation, assuming ${D}_{T}^{0}(L)\approx \text{}{D}_{T}^{0}(\infty )$ [14], we thus have,

_{sv}(∞)/γ

_{gb}(∞) − 1]α

_{NP}} where δ is an additional parameter showing the role of grain boundaries relative to free surfaces. Thus, by substituting Equation (5) into Equation (3) with the help of Equation (4), we get,

## 3. Results and Discussion

_{a}(∞). A significant decrease in ${E}_{a}^{\mathrm{NS}}(L)$ occurred at about L ≈ 5 nm, although the decrease of ${E}_{a}^{\mathrm{NP}}(L)$ was observed at around L ≈ 10 nm. When L > 10 nm for NSs or L > 20 nm for NPs, E

_{a}(L) → E

_{a}(∞). Compared to the bulk case, the decrease in ${E}_{a}(L)$ should be relevant to large thermal vibrational energies of atoms at the surfaces or grain boundaries. The observation that ${E}_{a}^{\mathrm{NS}}(L)$ > ${E}_{a}^{\mathrm{NP}}(L)$ was attributed to the thermal vibration energy of atoms at the grain boundary being lower than that at the surface. The model prediction agrees roughly with the experimental results.

_{0}, where ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$ > ${D}_{T}^{\mathrm{NS}}\left(L,T\right)$ > D

_{T}(∞,T). An obvious increase of ${D}_{T}^{\mathrm{NS}}\left(L,T\right)$ occurred at about L ≈ 4 nm, although the increase of ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$ happened at around L ≈ 6 nm. When L > 10 nm for NSs or L > 20 nm for NPs, the values of ${D}_{T}^{\mathrm{NS}}\left(L,T\right)$ and ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$ approached D

_{T}(∞, T). The increase in ${D}_{T}^{\mathrm{NS}}\left(L,T\right)$ is related to the decrease of ${E}_{a}^{\mathrm{NS}}(L)$ because of the coordination imperfection at grain boundaries. It can also be seen that ${D}_{T}^{\mathrm{NS}}\left(L,T\right)$ < ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$, which is attributed to the fact that the thermal vibration of atoms at the grain boundary is weaker than at the surface. The model prediction is consistent with the experimental results.

_{gb}(∞)/γ

_{sv}(∞) ratio. To show how ${E}_{a}^{\mathrm{NS}}\left(L\right)$ will vary with γ

_{gb}(∞)/γ

_{sv}(∞), Figure 3 shows a plot of $\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)/\mathsf{\Delta}{E}_{a}^{\mathrm{NP}}(L)$ as the function of γ

_{gb}(∞)/γ

_{sv}(∞) at L = 4 nm using Equations (4) and (5) with $\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)/\mathsf{\Delta}{E}_{a}^{\mathrm{NP}}(L)$ = $[{E}_{a}^{\mathrm{NS}}(L)-{E}_{a}(\infty )\left]\text{}/\text{}\right[{E}_{a}^{\mathrm{NP}}(L)-{E}_{a}(\infty )]$. It can be seen that $\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)/\mathsf{\Delta}{E}_{a}^{\mathrm{NP}}(L)$ increases almost linearly as γ

_{gb}(∞)/γ

_{sv}(∞) rises. Since 0 < $\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)/\mathsf{\Delta}{E}_{a}^{\mathrm{NP}}(L)$ < 1 exists in the range 0 < γ

_{gb}(∞)/γ

_{sv}(∞) < 1, $\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)$ < $\mathsf{\Delta}{E}_{a}^{\mathrm{NP}}(L)$ is available in the whole γ

_{gb}(∞)/γ

_{sv}(∞) range, while $\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)$ approaches $\mathsf{\Delta}{E}_{a}^{\mathrm{NP}}(L)$ as γ

_{gb}(∞)/γ

_{sv}(∞) tends to be in unity. Thus, the weakening of the $\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)$ function can be scaled by the γ

_{gb}(∞) size.

_{m}(∞), and a significant decrease occurred at about L ≈ 5 nm for ${T}_{m}^{\mathrm{NS}}\left(L\right)$ but at around L ≈ 10 nm for ${T}_{m}^{\mathrm{NP}}\left(L\right)$. The T

_{m}(L) value was closer to T

_{m}(∞) when L > 10 nm for the NSs or L > 20 nm for the NPs. The decrease in T

_{m}(L) is related to the coordination imperfection at the interface and the surface. Regarding ${T}_{m}^{\mathrm{NP}}\left(L\right)$ < ${T}_{m}^{\mathrm{NS}}\left(L\right)$, this could be due to the fact that γ

_{gb}(∞) < γ

_{sv}(∞), since the coordination imperfection at the grain boundary is weak relative to that at the surface. The validity of Equation (5) can be confirmed by the available experiments and computer simulation results, showing that the L-dependences of ${E}_{a}^{\mathrm{NS}}\left(L\right)$ and ${D}_{T}^{\mathrm{NS}}\left(L,T\right)$ can be influenced by the grain boundary energy effect.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**${E}_{a}^{\mathrm{NS}}(L)$ as the function of L (solid) in terms of Equations (4) and (5) for (

**a**) Au, (

**b**) Bi in Cu, (

**c**) Cu, and (

**d**) Fe, where Bi in Cu in (

**b**) means the diffusion of Bi in nanostructured Cu and Au in (

**a**), Cu in (

**c**), and Fe in (

**d**) give the self-diffusion data. ${E}_{a}^{\mathrm{NP}}\left(L\right)$ functions (dashed) are also plotted with Jiang’s prediction [14] for comparison. The symbols show experimental results: (

**a**) ■ [22] for Au nanostructured materials (NSs), (

**b**) ♦ [15] for Bi in Cu NSs, (

**c**) ● [27] and ▲ [28] for Cu nanoparticles (NSs), and (

**d**) ○ [4] for Fe NPs. The parameters for the calculation are shown in Table 1.

**Figure 2.**${D}_{T}^{\mathrm{NS}}\left(L,T\right)$ as a function of L (solid) in terms of Equation (6) for (

**a**) Cu and (

**b**) Ni; ${D}_{T}^{\mathrm{NP}}\left(L,T\right)$ functions (dashed) are also plotted with Equations (1) and (2) for comparison. The symbols denote the experimental results with ▼ [27] for Cu NSs and ▲ [16] for Ni NSs. The parameters for the calculation are shown in Table 1.

**Figure 3.**$\mathsf{\Delta}{E}_{a}^{\mathrm{NS}}(L)/\mathsf{\Delta}{E}_{a}^{\mathrm{NP}}(L)$ as the function of γ

_{gb}(∞)/γ

_{sv}(∞), with L = 4 nm in terms of Equations (4) and (5) for seven elements. The averaged values of S

_{vib}and L

_{0}among these seven elements are used for the calculation with S

_{vib}= 7.251 Jmol

^{−1·}K

^{−1}and L

_{0}= 1.578 nm. Other parameters are shown in Table 1.

**Figure 4.**${T}_{m}^{\mathrm{NS}}\left(L\right)$ as the function of L (solid) in terms of Equation (5) for (

**a**) Ag, (

**b**) Sn, and (

**c**) Pb, where the case of ${T}_{m}^{\mathrm{NP}}\left(L\right)$ (dashed) is also given for comparison with Equation (2). The symbols show experimental or simulation results with (

**a**) ▼ [7] for Ag NSs and ▽ [32] for Ag NPs, (

**b**) ◆ [33] and ● [34] for Sn NSs, and (

**c**) ■ [35] for Pb NPs and ★ [33] for Pb NSs. The parameters necessary for the calculation are shown in Table 1.

h [29] (nm) | T_{m} (∞) [30] (k) | S_{vib} (∞) [29] (Jmol^{−1·}k^{−1}) | γ_{sv} (∞) [24] (Jm^{−2}) | γ_{gb} (∞) [24] (Jm^{−2}) | ${\mathit{D}}_{\mathit{T}}^{\mathbf{0}}({\mathbf{m}}^{2}\xb7{\mathbf{s}}^{-\mathbf{1}})$ | E_{a} (∞) (kJ·mol^{−1}) | |
---|---|---|---|---|---|---|---|

Ag | 0.289 | 1234 | 7.82 | 1.250 | 0.392 | - | - |

Pb | 0.350 | 600.61 | 6.65 | 0.600 | 0.111 | - | - |

Sn | 0.281 | 505.08 | 9.22 | 0.649 | 0.179 | - | 56.93 [31] |

Fe | 0.248 | 6.82 | 2.420 | 0.528 | - | 79.11 [4] | |

Au | 0.288 | 7.62 | 1.500 | 0.400 | - | 169.81 [22] | |

Cu | 0.256 | 7.85 | 1.790 | 0.601 | - | 95.52 [15] | |

2 × 10^{−18} [27] | 66.57 [27] | ||||||

Ni | 0.249 | 8.11 | 2.380 | 0.866 | 1.77 × 10^{−7} [16] | 43.65 [16] |

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**MDPI and ACS Style**

Hu, Z.; Li, Z.; Tang, K.; Wen, Z.; Zhu, Y.
Modeling of the Atomic Diffusion Coefficient in Nanostructured Materials. *Entropy* **2018**, *20*, 252.
https://doi.org/10.3390/e20040252

**AMA Style**

Hu Z, Li Z, Tang K, Wen Z, Zhu Y.
Modeling of the Atomic Diffusion Coefficient in Nanostructured Materials. *Entropy*. 2018; 20(4):252.
https://doi.org/10.3390/e20040252

**Chicago/Turabian Style**

Hu, Zhiqing, Zhuo Li, Kai Tang, Zi Wen, and Yongfu Zhu.
2018. "Modeling of the Atomic Diffusion Coefficient in Nanostructured Materials" *Entropy* 20, no. 4: 252.
https://doi.org/10.3390/e20040252