Grain size-dependent diffusion activation energy in nanomaterials

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Abstract

We report a unified model, free of any adjustable parameter, for size-dependence of intrinsic diffusion activation energy of elements in crystals. It is found that as the size of the nanocrystals decreases, the diffusion activation energy of atoms decreases and the corresponding diffusion coefficient strongly increases due to the Arrhenius relationship between them, which leads to evident diffusion at the room temperature. The model prediction is in agreement with the experimental diffusion results of N into bcc Fe and Ag into Au nanoparticles.

Introduction

It is well known that there are lower diffusion activation energy of atoms in nanocrystals or nanostructured materials and thus larger diffusion coefficient than the corresponding bulk counterpart due to the increase of surface (interface)/volume ratio of the nanocrystals or nanostructured materials [1], [2], [3]. The potential of this phenomenon for industrial application is an evident drop of temperature for any diffusion process. The latest example is that the nitrification in bcc nanostructured Fe can be made at 573 K while this operation usually is carried out at 773 K or higher [1]. Moreover, a rapid alloying between Ag shell and Au nanocores may occur in several days at the room temperature [2]. This kinetic property of materials in return is also meaningful to consider the thermal stability of the core-shell structure when the both sizes of the core and the shell are only several nanometers or less [2], [3]. The size and temperature dependent diffusion coefficient function is also an important parameter for any phase transition process through nucleation and growth where the size of the formed nucleus is 1–2 nanometers and inoculation time and growth time are certainly related to the kinetic properties of a material [4]. The understanding of this kind of scientific problem becomes more urgent due to recent development on the nanotechnology where the full size of the materials is in nanometer size range [5], [6].

Although the size-dependent properties of the kinetic parameters are so useful, a unique attempt to establish a quantitative model for describing this kind of size-dependence, based on simply the consideration on surface/volume ratio, cannot satisfactorily interpret the diffusion coefficient of Ag in Au nanoparticles [3]. Thus, it is necessary to model quantitatively the size and temperature dependence of the diffusion coefficient D(r,T) with r being the radius of nanoparticles or grains and T being the temperature.

In this contribution, a model for D(r,T) function without adjustable parameters is established through considering the size-dependent activation energy E(r). In light of the model, it is found that when r is several nanometers, in one side, at the same T, D(r,T)/D(∞,T) could even be more than 1010 due to the drop of E(r) where ∞ denotes the bulk size; in the other side, when a constant diffusion coefficient is needed, the diffusion temperature can decreases for several hundreds of degrees. The model predictions correspond to the experimental results of N diffusing into bcc nanostructured Fe and Ag shell diffusing into Au nanocores.

Section snippets

Model

To begin with, a well-known Arrhenius dependence for self-diffusion or intrinsic diffusion coefficient of interdiffusion D(r,T) is introduced [2],D(r,T)=D0(r)exp[−E(r)/(RT)]where D0(r) denotes a pre-exponential constant, R is the ideal gas constant.

To establish an E(r) function, D[r,Tm(r)]=D[∞,Tm(∞)] is assumed [2] where Tm(r) and Tm(∞) denote the size-dependent and the bulk melting temperature. Thus, D[r,Tm(r)]=D0(r)exp{−E(r)/[RTm(r)]}=D0(∞)exp{−E(∞)/[RTm(∞)]} in terms of Eq. (1). According to

Results and discussion

Fig. 2 shows a comparison between Eq. (9) and experimental results for D(r,T) function of N atoms diffusing into nanostructured bcc Fe [1]. Although only one experimental point shown in Fig. 2 is difficult to supply the consistency of D(r,T) function in the whole range of r with the experiment, this experimental point fits the model parameters. In Fig. 2, even T decreases from 773 to 573 K, the diffusion coefficient remains constant since the grain size of bcc Fe decreases to 6.5 nm.

Fig. 3

Conclusion

In summary, a model for D(r,T) and E(r) functions are established. D(r,T) increase of nanocrystals is induced by the decrease of E(r) due to the size effect. The model predictions for D(r,T) function of N diffusing into nanostructured bcc Fe and Ag diffusing into Au nanocores are in agreement with the experimental results.

Acknowledgements

The financial support from the NNSFC (Grant No. 50025101) is acknowledged.

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