Mg segregations at and near deformation-distorted grain boundaries in ultrafine-grained Al–Mg alloys

I. A. Ovid’ko; A. G. Sheinerman; R. Z. Valiev

doi:10.1007/s10853-014-8351-8

Journal of Materials Science

October 2014, Volume 49, Issue 19, pp 6682–6688 | Cite as

Mg segregations at and near deformation-distorted grain boundaries in ultrafine-grained Al–Mg alloys

Authors
Authors and affiliations

I. A. Ovid’ko
A. G. Sheinerman
R. Z. Valiev

Ultrafinegrained Materials

First Online: 13 June 2014

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6 Citations

Abstract

The formation of Mg segregations at and near deformation-distorted grain boundaries (GBs) in ultrafine-grained Al–Mg alloys is theoretically described as a process enhanced by stress fields of extrinsic dislocations existing at such GBs. The equilibrium Mg concentration profiles near low-angle and high-angle GBs containing extrinsic dislocations are calculated. The results of the calculations explain the experimental observations (reported in the scientific literature) of spatially inhomogeneous Mg segregations characterized by high Mg concentrations at and near GBs in ultrafine-grained Al–Mg alloys processed by severe plastic deformation.

Keywords

Severe Plastic Deformation Edge Dislocation Misorientation Angle Atomic Probe Tomography Tilt Boundary

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Notes

Acknowledgements

The work was supported, in part, (for I.A.O. and R.Z.V.) by the Russian Ministry of Education and Science (Grant 14.B25.31.0017) and (for A.G.S.) by St. Petersburg State University research grant 6.37.671.2013. The authors would also like to thank Dr. N.A. Enikeev for fruitful discussions.

Appendix

Equilibrium concentration of Mg in an Al–Mg alloy under an applied stress

In this Appendix, we consider a binary Al–Mg alloy with concentration (defined as ratio of the number of atoms of one kind to the total number of atoms) $ c $ of Mg and concentration $ 1 - c $ of Al. Let an elastic field $ \sigma_{ij} $ act in the Al–Mg alloy and lead to its decomposition. As a result of the decomposition, the distribution of the Mg concentration $ c $ approaches the equilibrium distribution corresponding to the minimum of the free energy that specifies the Al–Mg alloy.

Let us calculate the equilibrium Mg concentration c in the presence of the stress field $ \sigma_{ij} $. We assume that the initial Mg concentration $ c_{0} $(prior to the action of the stress field $ \sigma_{ij} $) is spatially uniform. In order to calculate the equilibrium Mg concentration $ c $, we calculate the free energy $ F $ of the Al–Mg alloy. The free energy can be expressed as $ F = \int\limits_{V} {f(c(x,y,z)\,){\text{d}}V'} $, where $ V $ is the volume of the solid, $ V' $ is the integration parameter, ($ x $,$ y $,$ z $) are the coordinates of a point in the solid, and $ f(c) $ is the free energy density (the free energy per unit volume).

In a first approximation, the free energy density $ f $ of the solid solution can be written as the sum of the chemical energy density $ f^{\text{ch}} $ and strain energy density $ w^{\text{el}} $:

$$ f = f^{\text{ch}} + w^{\text{el}} . $$

(3)

The chemical energy density $ f^{\text{ch}} $ can be presented as the sum of the chemical energy density $ f_{0}^{\text{ch}} $ at the temperature $ T = 0 $ K and the configurational entropy term $ f_{\text{ent}}^{\text{ch}} $ ($ f^{\text{ch}} = f_{0}^{\text{ch}} + f_{\text{ent}}^{\text{ch}} $). In order to calculate the energy density $ f_{0}^{\text{ch}} $, we assume that the concentration $ c $ is typically small enough, so that the Al–Mg alloy has the fcc crystal lattice characteristic of pure Al. Also, we approximate the chemical energy density $ f_{0}^{\text{ch}} $ as the chemical energy density of a homogeneous solid solution. Following [24], we present the chemical energy density $ f_{0}^{\text{ch}} $ as the sum of the total energy (per unit volume) that specifies the interaction of atoms with the electron density and the term associated with the pair interaction between atoms. Also, we adopt the approximation of nearest neighbors assuming that each atom interacts only with $ z $ nearest neighbors (where $ z = 12 $ for the fcc crystal lattice). Then, the energy density $ f_{0}^{\text{ch}} $ can be presented as

$$ f_{0}^{\text{ch}} = (1/\bar{V}_{\text{at}} )\,\left( {cE_{\text{Mg}} + (1 - c)E_{\text{Al}} } \right), $$

(4)

where $ \bar{V}_{at} $ is the average volume per atom, $ E_{\text{Mg}} $ and $ E_{\text{Al}} $ are the atomic energies. These are given as follows:

$$ E_{\text{Mg}} = E_{\text{Mg}}^{\text{el}} + (cz/2)E_{\text{Mg - Mg}} + ((1 - c)z/2)E_{\text{Mg - Al}} , $$

(5)

$$ E_{\text{Al}} = E_{\text{Al}}^{\text{el}} + (cz/2)E_{\text{Mg - Al}} + ((1 - c)z/2)E_{\text{Al - Al}} , $$

(6)

where $ E_{\text{Mg - Mg}} $, $ E_{\text{Al - Al}} $ and $ E_{\text{Mg - Al}} $ are the energies of the pair interaction between corresponding neighboring atoms, $ E_{\text{Mg}}^{\text{el}} $ ($ E_{\text{Al}}^{\text{el}} $, respectively) is the energy that characterizes the interaction of a Mg (Al, respectively) atom with the electron density.

The energy density $ f_{\text{ch}}^{\text{ent}} $ is as follows (e.g., [25]):

$$ f_{\text{ch}}^{\text{ent}} = k_{\text{B}} T[c\;\ln \;c + (1 - c)\ln (1 - c)], $$

(7)

where $ k_{\text{B}} $ is the Boltzmann constant and $ T $ denotes the absolute temperature.

From formulas (3)–(7), we obtain the following expression for the chemical energy density $ f_{ch}^{{}} $:

$$ \begin{gathered} f^{\text{ch}} = \frac{1}{{2\bar{V}_{\text{at}} }}\left\{ {z\left( {c^{2} E_{\text{Mg - Mg}} + (1 - c)^{2} E_{\text{Al - Al}} + 2c(1 - c)E_{\text{Al - Mg}} } \right)} \right. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { + 2cE_{\text{Mg}}^{\text{el}} + 2(1 - c)E_{\text{Al}}^{\text{el}} + 2k_{\text{B}} T[c\;\ln \;c + (1 - c)\ln (1 - c)]} \right\}\;. \hfill \\ \end{gathered} $$

(8)

The strain energy density $ w^{\text{el}} $ can be written in the isotropic approximation in spirit of [26, 27, 28] as

$$ w^{\text{el}} = \eta^{2} [K_{\text{Al}} \,c(1 - c_{0} )^{2} + K_{\text{Mg}} c_{0}^{2} (1 - c)] - 3\sigma {\kern 1pt} \eta (c - c_{0} ), $$

(9)

where $ K_{\text{Al}} = 2G_{\text{Al}} (1 - \nu_{\text{Al}} )/(1 + \nu_{\text{Al}} ) $, $ K_{\text{Mg}} = 2G_{\text{Mg}} (1 - \nu_{\text{Mg}} )/(1 + \nu_{\text{Mg}} ) $, $ G_{\text{Al}} $ and $ G_{\text{Mg}} $ are the shear moduli of pure Al and pure Mg, respectively, $ \nu_{\text{Al}} $ and $ \nu_{\text{Mg}} $ are Poisson’s ratios of pure Al and pure Mg, respectively, $ \sigma = (\sigma_{xx} + \sigma_{yy} + \sigma_{zz} )/3 $, $ \sigma_{xx} $, $ \sigma_{yy} $ and $ \sigma_{zz} $ are the components of the stress field $ \sigma_{ij} $, $ \eta = \frac{1}{{\bar{a}}}\frac{\partial a}{\partial c} $, $ \bar{a} = a(c_{0} ) $ is the average lattice constant of the Al–Mg alloy, and $ a $ is the alloy lattice constant that depends on the Mg concentration $ c $. The first two terms on the right-hand side of formula (9) represent the proper strain energy density of the solid solution, while the last term is the energy density that specifies the interaction of the solid-solution induced stress with the stress $ \sigma_{ij} $.

We now consider the situation where the Mg concentration $ c $ is the equilibrium one. In this situation, the chemical potential of Mg atoms should be constant [22], that is, $ \partial f/\partial c = {\text{const}} $. Let the decomposition of the solid solution occurs due to the presence of defects, whereas the solid solution in the absence of defects is characterized by the spatially homogeneous concentration $ c_{0} $. Let us assume that defects are localized in a finite volume, so that their stresses tend to zero at a remote boundary of the solid. With this assumption, the characteristics of the solid at such a boundary are as follows: $ c = c_{0} $ and $ \sigma = \sigma_{\infty } $ (where $ \sigma_{\infty } $ is the stress associated with the applied load). Then, we obtain $ \partial f/\partial c = (\partial f/\partial c)|_{\begin{subarray}{l} c = c_{0} , \\ \sigma = \sigma_{\infty } \end{subarray} } $.

Substitution of formulas (8) and (9) as well as the equality $ f = f^{\text{ch}} + w^{\text{el}} $ to the latter equation yields:

$$ \frac{1}{{\bar{V}_{\text{at}} }}\left\{ {2z(c - c_{0} )\varOmega + k_{\text{B}} T\left( {\ln \frac{c}{1 - c} - \ln \frac{{c_{0} }}{{1 - c_{0} }}} \right)} \right\} - 3(\sigma - \sigma_{\infty } )\eta = 0, $$

(10)

where $ \varOmega = (E_{\text{Mg - Mg}} + E_{\text{Al - Al}} )/2 - E_{\text{Mg - Al}} $. Formula (10) coincides with formula (1) of the main text in the case of $ \sigma_{\infty } = 0 $.

In order to roughly estimate $ \varOmega $ for the Al–Mg alloy, we use the value of the energy $ E_{f}^{\text{sub}} $ of a Mg impurity atom (substitutional point defect) in pure Al. This energy was calculated in Ref. [29] and defined [29] as

$$ E_{f}^{\text{sub}} = E_{\text{tot}} [{\text{Mg}}_{\text{Al}} ] - E_{\text{tot}} [{\text{Al}}_{\text{Al}} ] - E_{\text{Mg}}^{c} + E_{\text{Al}}^{c} , $$

(11)

where $ E_{\text{tot}} [{\text{Mg}}_{\text{Al}} ] $ is the energy of an Al specimen with one atom of Mg substituting an Al atom, $ E_{\text{tot}} [{\text{Al}}_{\text{Al}} ] $ is the energy of a similar Al specimen without substitution of an Al atom by a Mg atom, $ E_{\text{Mg}}^{c} $ and $ E_{\text{Al}}^{c} $ are the cohesive (atomic) energy of Mg and Al in pure Mg and pure Al, respectively.

The energies appearing in formula (11) can be presented as

$$ \begin{gathered} E^{\text{tot}} [{\text{Mg}}_{\text{Al}} ] = E_{\text{Mg}}^{\text{el}} + zE_{\text{Mg - Al}} ,\;\;\;\;E^{\text{tot}} [{\text{Al}}_{\text{Al}} ] = E_{\text{Al}}^{\text{el}} + zE_{\text{Al - Al}} , \hfill \\ E_{\text{Mg}}^{c} = E_{\text{Mg}}^{\text{el}} + (z/2)E_{\text{Mg - Mg}} ,\;\;\;\;\;E_{\text{Al}}^{c} = E_{\text{Al}}^{\text{el}} + (z/2)E_{\text{Al - Al}} . \hfill \\ \end{gathered} $$

(12)

From formulas (11) and (12) and the definition of $ \varOmega $, we obtain $ z\varOmega = - E_{f}^{\text{sub}} $, where, following Ref. [29], $ E_{f}^{\text{sub}} = 0.05 $ eV.

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Authors and Affiliations

I. A. Ovid’ko
- 1
- 2
- 3
A. G. Sheinerman
- 1
- 2
- 3
Email author
R. Z. Valiev
- 1
- 4

1.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
2.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia
3.St. Petersburg State Polytechnical UniversitySt. PetersburgRussia
4.Ufa State Aviation Technical UniversityUfaRussia

Mg segregations at and near deformation-distorted grain boundaries in ultrafine-grained Al–Mg alloys

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