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Download PDF File Short Communication Real and extended volumes in simultaneous transformations T. Kasuya, K. Ichikawa, M. Fuji, and H. K. D. H. Bhadeshia T he relationship between real and extended volumes is expanded to an indefinite number of simultaneous transformation processes. T he assumption used in the present study is that the growth and nucleation of each product phase occurs randomly throughout the transforming region. T he real volume of each phase is given in the form of an integral, using the extended volumes. Once the extended volumes of diVerent product phases are related, the real volume of each product phase can be analytically or numerically calculated, and analytical solutions for cases where the extended volumes are related linearly or parabolically are given. MST /4226 Dr Kasuya, Mr Ichikawa, and Mr Fuji are in the Welding and Joining Research Center, Nippon Steel Corp., 20-1 Shintomi, Futtsu-City, Chiba, 293–8511 Japan. Dr Bhadeshia is in the Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK. Manuscript received 4 October 1998; in final form 24 November 1998. ` 1999 IoM Communications L td. two equations Introduction V +V dV = 1− 1 2 dV e .........(3) A model for a single transformation begins with the 1 A V B 1 calculation of the nucleation and growth rates using 1–6 V +V classical theory, but an estimation of the volume fraction dV = 1− 1 2 dV e .........(4) requires impingement between particles to be taken into 2 A V B 2 account. This is generally achieved using the extended volume concept of Kolmogorov, Johnson and Mehl, and which in general must be solved numerically although an Avrami (details are given in Ref. 1) as illustrated in Fig. 1. analytical solution which gives both V1 and V2 has been proposed for the case where the ratio of the volume Suppose that two particles exist at time t: a small interval 11 dt later, new regions marked a, b, c, and d are formed fractions of the two phases is constant. The method assuming that they are able to grow unrestricted in can in principle be extended to incorporate an indefinite extended space whether or not the region into which number of reactions occurring together. The purpose of they grow is already transformed. However, only those the present work is to generalise the previous work and components of a, b, c, and d which lie in previously to present analytical expressions for cases other than the untransformed matrix can contribute to a change in constant ratio described above. the real volume of the product phase (identified by the subscript 1) Method V = − 1 e dV1 A1 B dV 1 ...........(1) V To obtain the general relationship between real and extended volumes in simultaneous transformations, the where it is assumed that the microstructure develops variables are first redefined as follows randomly. The superscript e refers to extended volume, V is the volume of phase 1, and V is the total volume. V V e n n 1 v = i ,ve= i ,z=∑v , and ze=∑ve (5) Multiplying the change in extended volume by the i V i V j j probability of finding untransformed regions has the effect j=1 j=1 = of excluding regions such as b, which clearly cannot where Vi (i 1 to n) is the real volume of the product e contribute to the real change in volume of the product. phase i, and V i is its extended volume. This equation can readily be integrated to obtain the real From the same analysis as described above with the volume fraction assumption of random microstructure development = − e V V e dvi (1 z) dvi .............(6) 1 =1−exp − 1 ...........(2) V A V B Summing equation (6) from i=1 to n and using equa- tion (5) gives In practice, there are many instances where several transformations occur together. The different reactions dz=(1−z) dze and z=1−exp(−ze) ....(7) interfere with each other in a manner which is funda- Equations (6) and (7) yield mental to the development of many steel microstructures. Therefore, considerable effort has recently been devoted to dve dve dve dv = i dz= i (1−z) dze= i exp(−ze) dze ..(8) the development of an Avrami model for simultaneous i dze dze dze reactions.7–14 Most of these works calculated the total volume of transformed product, but did not calculate the Hence real volume of each transformed phase. dve A simple method of accounting for two precipitates v = i exp(−ze) dze ..........(9) (1 and 2) is that the above equation becomes a set of i P dze ISSN 0267–0836 Materials Science and Technology April 1999 Vol. 15 471 472 Kasuya et al. Real and extended volumes in simultaneous transformations where B is a positive constant and C is zero or a positive e constant. Since v2 cannot be negative, equation (10) implies that ve (0∏ve ∏C/B) ze= 1 1 .....(11) G + e − e > (B 1)v1 C(v1 C/B) ∏ e ∏ Equation (9) can then be integrated for 0 v1 C/B as = − − e = v1 1 exp( v1) and v2 0 .......(12) e > and for v1 C/B as B 1 v =1− exp(−C/B)− exp[−(B+1)ve +C] 1 B+1 B+1 1 (a) .........(13) B v = {exp(−C/B)−exp[−(B+1)ve +C]} (14) 2 B+1 1 e The new parameters (v1)s and (v1)s are defined here as e v1 and v1 respectively, obtained during the simultan- eous transformation. The parameter (v1)s is calculated by subtracting [1−exp(−C/B)] from equation (13), hence 1 (v ) = {exp(−C/B)−exp[−(B+1)ve +C]} 1 s B+1 1 1 = exp(−C/B){1−exp[−(B+1)(ve −C/B)]} B+1 1 v = 2 ...............(15) B The term exp(−C/B) occurs because [1−exp(−C/B)] of (b) phase 1 exists before commencement of the simultaneous e − e reaction, and (v1 C/B) is (v1)s. Equation (15) shows that two precipitate particles have nucleated together and grown to finite there is a simple linear relation between (v1)s and v2. An size in time t: new regions c and d are formed as original particles example of an actual simultaneous reaction is a precipitation grow, but a and b are new particles, of which b has formed in region phenomenon in ferritic steel. Suppose that the isotropic which is already transformed = growth rate of phase i(i 1, 2) is Gi, and that all particles 1 Illustration of concept of extended volume of phase i start growth at time t=0 from a fixed number of sites Ni per unit volume, then In equations (8) and (9), ze is treated as the only e N 4p N G3 independent variable, and vi is considered to be a function ve = 1 G3t3 and ve =Bve B= 2 2 (16) e = 1 V 3 1 2 1 A N G3B of z . The calculations of vi (i 1 to n) require the relations 1 1 between the various ve values (i=1 to n) throughout the i It is emphasised that the specific assumptions made to reaction paths. To explain this necessary condition simply, express ve can be selected at will, for example, to include two reaction paths, I and II, are considered, where the 1 a nucleation rate. Details can be found in Ref. 1. Equa- relations between the ve values are different for the two i tion (16) is the special case where C in equation (10) is paths. In this case, even if each ve value is equal for i zero, which has been used by Robson and Bhadeshia14 to reactions I and II at a certain moment, the v values are, i examine the precipitation phenomena in power plant steel. at that moment, different because of the difference of the A negative value of C would require a rearrangement of reaction paths. Determination of the relationships between e equation (10) with corresponding changes to the subsequent the various vi values throughout the reaction path is hence e derivations. necessary to calculate vi. Once all the vi values are related e = e to one another, only one vi (i 1 to n), or z , can be an independent variable. Equation (9) indicates that vi can be PARABOLIC calculated from only ve (i=1 to n) and gives the general e e i The case is now discussed where v1 and v2 are related relation between real and extended volumes. parabolically e = e 2+ e − > Á v2 A(v1) Bv1 C(A 0, B, C 0) ...(17) e ¬ − + Solution Equation (17) implies that when v1 is less than R( [ B 2+ 1/2 e e < (B 4AC) ]/2A), v2 is zero. Hence, for v1 R, equation Analytical solutions of equation (9) exist for special cases (9) becomes ff e where the extended volumes of the di erent phases (or vi v =1−exp(−ve ) and v =0 .......(18) for i=1 to n) can be related. Hereafter, the case of n=2 is 1 1 2 e and for ve ÁR considered and, for mathematical convenience, v1 is treated 1 as an independent variable. (B+1)2 1 v =1−exp(−R)+exp +C (p/A)1/2 1 C 4A D 2 LINEAR e e ×[erf( )−erf( )] ..........(19) Suppose that v1 and v2 are related linearly according to t t0 e = e − = − − − e 2− + e + v2 Bv1 C .............(10) v2 1 v1 exp[ A(v1) (B 1)v1 C] ...(20) Materials Science and Technology April 1999 Vol.
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