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Examination of an analytical phase-transformation model

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Abstract

A comparison between results of a recently published quasi-exact solution of the temperature integral used for the Avrami model of isochronal phase transformations and an analytical phase-transformation model in relation to exact solutions from numerical integration has been performed. The results for the transformed fraction from the quasiexact solution are more precise than the corresponding results of the analytical model, whereas the results for the transformation rate from both models are sufficiently precise for the nucleation mode of site saturation or continuous nucleation. It has been further shown that an analytical solution of the transformation rate cannot be obtained using a quasi-exact solution of the temperature integral in case of mixed nucleation, and that the results of the corresponding solution with the analytical model substantially alleviate the influence of the approximated temperature integral. By this method, an analytical approach of modeling, which can substantially alleviate the deviation (of model prediction) arising from approximations to the temperature integral, has been developed. The proposed approach has been successfully applied to experimental data of the crystallization of bulk amorphous Pd-Ni-P-Cu alloys.

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Abbreviations

a:

annealing time t for isothermal transformation or RT2/Φ for isochronal transformation

C c :

a constant defined by m, d, QN, and QG

D 0 :

pre-exponential term in expression for diffusion coefficient (m2/s)

d:

dimensionality of the growth

f:

transformed fraction

g:

particle-geometry factor

ΔH:

enthalpy change (J/mol)

ΔHpre:

enthalpy change for preannealing stage (J/mol)

ΔHtot:

enthalpy change for total transformation (J/mol)

K 0 :

pre-exponential factor for rate constant

m:

growth mode parameter

n:

growth exponent

N*:

number of pre-existing nuclei in mode of site saturation (m−3)

N 0 :

value of pre-exponential factor in mode of continuous nucleation (m−3s−1)

Q:

overall effective activation energy (kJ/mol)

Q G :

constant activation energy for growth (kJ/mol)

Q N :

constant activation energy for nucleation (kJ/mol)

r2/r1:

a ratio between the extended fraction conceived as due to pure continuous nucleation and that conceived as due to pure site saturation

R:

universal gas constant (J/K mol)

T:

temperature (K)

T 0 :

starting temperature for annealing (K)

T pre :

preannealing temperature (K)

t:

time for transformation (s)

τ:

time for nucleation (s)

V e :

extended volume (m3)

V:

real volume (m3)

v:

interface velocity (m/s)

v 0 :

pre-exponential term in expression for v (m/s)

Φ:

heating rate (K/s)

x e :

extended transformed fraction

Y:

volume at time t of a particle nucleated at time ı (m3)

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Authors and Affiliations

  1. State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, People’s Republic of China

    F. Liu, Y. Z. Ma, X. Hu & G. C. Yang

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  1. F. Liu
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  2. Y. Z. Ma
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  3. X. Hu
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  4. G. C. Yang
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Correspondence to F. Liu.

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Liu, F., Ma, Y.Z., Hu, X. et al. Examination of an analytical phase-transformation model. Journal of Materials Research 24, 1761–1770 (2009). https://doi.org/10.1557/jmr.2009.0194

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