Modeling grain growth kinetics of binary substitutional alloys by the thermodynamic extremal principle

Abstract

The thermodynamics and kinetics fundaments of grain growth in binary substitutional alloys were analyzed using the thermodynamic extremal principle. Applying the regular solution approximation, a new equation for solute segregation at steady-state diffusion is proposed, which suggests reduced solute segregation as the grain boundary (GB) solute concentration increases, differently from previous models [Acta Mater 2009;57(5):1466, Acta Mater 2012;60:4833, Scripta Mater 2010;63:989] that adopt constant segregation enthalpy. Furthermore, a self-consistent consideration has been carried out to account for the coupled changes in GB energy and GB mobility as a result of solute segregation. On this basis, the quantitative relation is evaluated between the thermodynamic and kinetic effects of solute segregation to determine the dominant role in retarding and even suppressing grain growth, by comparison of the dimensionless GB energy (i.e., the GB energy of alloy over that of pure solvent) and the dimensionless effective GB mobility (i.e., the effective GB mobility over that of pure solvent): the kinetic effect prevails if the dimensionless effective GB mobility is smaller than the dimensionless GB energy, and vice versa. The present model is adopted to describe well the experimental results for Fe–P alloys, and nanocrystalline Ni–P and Pd–Zr alloys.

Appendices

Appendix 1

In Hillert’s derivation [39, 47], as the equilibrium state is corresponding to the minimum of total Gibbs energy in the system, the change of total Gibbs energy must remain zero when tiny solvent atoms with amount of dn _A and solute atoms with amount of dn _B are transferred from the bulk phase to the GB one, i.e., [47]

$$ {\text{d}}G = \left( {\mu_{\text{A}}^{\text{GB}} - \mu_{\text{A}}^{\text{bulk}} } \right){\text{d}}n_{\text{A}} + \left( {\mu_{\text{B}}^{\text{GB}} - \mu_{\text{B}}^{\text{bulk}} } \right){\text{d}}n_{\text{B}} = 0 $$

(32)

In general, the solution of Eq. (32) is $\mu_{\text{A}}^{\text{GB}} = \mu_{\text{A}}^{\text{bulk}}$ and $\mu_{\text{B}}^{\text{GB}} - \mu_{\text{B}}^{\text{bulk}}$ due to the arbitrary dn _A and dn _B. However, under the assumption of constant GB width yielding dn _A + dn _B = 0, the precondition of Eq. (32) is $\mu_{\text{B}}^{\text{GB}} - \mu_{\text{B}}^{\text{bulk}}$ = $\mu_{\text{B}}^{\text{GB}} - \mu_{\text{B}}^{\text{bulk}}$ identical to Eq. (15). That is to say, the steady-state diffusion of components implies the equilibrium between the bulk and GB phases.

Appendix 2

The change of total Gibbs free energy in the system can be given by,

$$ {\text{d}}G = \sum\limits_{\text{i}} {\mu_{\text{i}}^{\text{bulk}} {\text{d}}n_{\text{i}}^{\text{bulk}} + n_{\text{i}}^{\text{bulk}} {\text{d}}\mu_{\text{i}}^{\text{bulk}} + \mu_{\text{i}}^{\text{GB}} {\text{d}}n_{\text{i}}^{\text{GB}} { + }n_{\text{i}}^{\text{GB}} {\text{d}}\mu_{\text{i}}^{\text{GB}} } $$

(33)

According to Gibbs–Duhem relation, Eq. (33) is rewritten as,

$$ {\text{d}}G = \sum\limits_{\text{i}} {\mu_{\text{i}}^{\text{bulk}} {\text{d}}n_{\text{i}}^{\text{bulk}} + \mu_{\text{i}}^{\text{GB}} {\text{d}}n_{\text{i}}^{\text{GB}} { = }\mu_{\text{A}}^{\text{bulk}} {\text{d}}n_{\text{A}}^{\text{bulk}} } { + }\mu_{\text{B}}^{\text{bulk}} {\text{d}}n_{\text{B}}^{\text{bulk}} + \mu_{\text{A}}^{\text{GB}} {\text{d}}n_{\text{A}}^{\text{GB}} + \mu_{\text{B}}^{\text{GB}} {\text{d}}n_{\text{B}}^{\text{GB}} $$

(34)

Then, with mass conservation equations dn ^GB_A  + dn ^bulk_A  = 0 and dn ^GB_B  + dn ^bulk_B  = 0, the last equation is further simplified as,

$$ {\text{d}}G = \left( {\mu_{\text{A}}^{\text{GB}} - \mu_{\text{A}}^{\text{bulk}} } \right){\text{d}}n_{\text{A}}^{\text{GB}} + \left( {\mu_{\text{B}}^{\text{GB}} - \mu_{\text{B}}^{\text{bulk}} } \right){\text{d}}n_{\text{B}}^{\text{GB}} $$

(35)

Thereafter, combining Eq. (35) with Eq. (15) yields

$$ {\text{d}}G = \left( {\mu_{\text{A}}^{\text{GB}} - \mu_{\text{A}}^{\text{bulk}} } \right){\text{d}}n^{\text{GB}} $$

(36)

With ${\text{d}}n^{\text{GB}}$ = δdS/V _m, Eq. (36) is changed into

$$ {\text{d}}G = \frac{\delta }{{V_{\text{m}} }}\left( {\mu_{\text{A}}^{\text{GB}} - \mu_{\text{A}}^{\text{bulk}} } \right){\text{d}}S $$

(37)

Further incorporating Eq. (18) into Eq. (37) leads to

$$ {\text{d}}G = \gamma_{\text{gb}} {\text{d}}S $$

(38)

On this basis, the driving force P of grain growth can be presented according to its definition (i.e., P = dG/dV) as,

$$ P = \frac{{{\text{d}}G}}{{{\text{d}}V}} = \gamma_{\text{gb}} \frac{{{\text{d}}S}}{{{\text{d}}V}} $$

(39)

By means of S = 4πR ²/2 (1/2 implying a GB shared by two adjacent grains) and V = 4/3πR ³, Eq. (39) is thus simplified as,

$$ P = \frac{{\gamma_{\text{gb}} }}{R} $$

(40)

Definitely, under steady-state diffusion, the GB energy can be regarded as the driving force of the grain growth affected by solute segregation.