Z. Phys. B 96, 79-86 (1994) ZEITSCHRIFT FOR PHYSIK B 9 Springer-Verlag 1994 Antiphase domain growth in BCC metallic alloys via vacancies Carlos Frontera, Eduard Vires, Antoni Planes Departament d'Estructura i Constituents de la Mat~ria, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain (Tel.: 34 34021181, Fax: 34 34 021174, e-mail: carlos at ebubecm 1.bitnet) Received: 11 May 1994 Abstract. We study by Monte Carlo simulation the influ- ence of vacancies on the growth of antiphase domains after a quench of a BCC binary alloy. We find a power- law regime R(t)~t x with x depending on temperature and larger than the Allen-Cahn value 0.5. We have veri- fied the scaling of the structure factor and performed a finite size analysis of the excess energy evolution and va- cancy diffusion. PACS: 64.60.Cn; 81.30.Hd; 61.70.Bv I. Introduction The development of long range order in metallic alloys after fast quenches from the disordered phase to tem- peratures below the order-disorder transition (To), rep- resents an important topic in material science as well as in statistical mechanics. From a fundamental point of view this problem is used as a prototype in the study of the kinetics of phase transitions [1 ]. Immediately after the quench, order develops by the decay of the initial disordered state to the corresponding equilibrium state at the quenching temperature (Tq). For shallow quenches (T~-- Tq~ T~) evolution usually initiates by nucleation and proceeds by growth of the droplets with favorable sizes. For deep quenches (Tc- Tq~ T~), the disordered state de- cays through an spinodal ordering process. Indepen- dently of the initial ordering mechanism, in both cases the late-time emerging structure will be an interpenetrat- ing array of antiphase boundaries (APB) separating or- dered regions in the different, but energetically equiva- lent, ground states. The APB contain the energy excess of the system which is reduced by growing larger and larger domains. This is called the antiphase domain growth or coarsening. Two striking features observed in these late stages are: first, the existence of a paramount time-dependent length - the mean domain size R (t) - and, second, the scaling behaviour of the structure factor with R (t) [2]. For most systems the domain size shows a power law growth R (t) ,-~t x [ 1]. A lot of work in this field has been focused on the determination of the growth exponent x. Theo- retical arguments and numerical simulations on simple models, have revealed that the exponent x is quite uni- versal and depends only, on very few factors. The most important one is whether or not the order parameter governing the transition is a conserved quantity. For an alloy undergoing an order-disorder transition, the order parameter (concentration in one sublattice) is a non con- served quantity. In this case it is expected that, at least for pure systems, x = 1/2. This is the so called Allen- Cahn law [3, 4]. In real systems, ordering is always affected by a num- ber of uncontrolled parameters. Impurities [5, 6], struc- tural defects, vacancies, lack of stoichiometry [7,8], etc ..... can influence the value of the growth exponent x. In this context, numerical simulations are a very suitable tool to discriminate the effects associated to different factors that in the experiments come up all together [9-11]. In previous papers [12-15] we have analysed from Monte Carlo simulations results, the role of a small frac- tion of vacancies in the ordering kinetics of a 2d binary alloy on a square lattice. Taking into account that dif- fusion in alloys occurs through the vacancy mechanism, only atom-vacancy exchanges were allowed in the sim- ulations. Our results proved that the equilibrium prop- erties of the system were practically not affected by the existence of an small amount of vacancies. Nevertheless, the coarsening kinetics was clearly modified by their pre- sence. If only exchanges between atoms and nearest neighbour (n.n.) vacancies were permitted, a logarithmic growth law was obtained. However if next nearest neigh- bour (n.n.n.) vancancy jumps were possible (even with a quite small probability) the exponent x appeared to be greater than 1/2, decreasing towards this value when Tq approaches the ordering temperature T c. This behaviour was attributed to the non-homogeneity of the excitations