Modeling the kinetics of consecutive phase transitions in the solid state

Abstract

A theoretical approach for describing the kinetics of consecutive phase transformations ruled by nucleation and growth is reported. In the considered system, the mother phase (M) transforms to an intermediate phase (α) which, in turn, transforms to the final product (β). The classical Kolmogorov–Johnson–Mehl–Avrami theory is generalized to deal with a finite-size phase with moving boundary. To this end, the statistical method based on the differential critical region has been employed. The exact solution of the kinetics is computed in closed form for the transformation of a spherical α-nucleus growing into the mother phase. By resorting to an approximate expression for the probability function entering the differential critical region method, the consecutive transformation is studied in the case of nucleation and growth of the α-phase. The time dependence of the β/α volume fraction is found to be in very good agreement with the stretched exponential kinetics, and the dependence of Avrami’s exponent on both nucleation and growth rates of the two phases is investigated. Modeling of the non-isothermal kinetics at constant heating rate has also been performed which provides an insight into the shape of the differential scanning calorimetry curves for consecutive phase transitions.

Introduction

Phase transformations are very important in Materials Science since their occurrence in materials processing may affect, significantly, the properties of the final product. This topic has been extensively studied from both experimental and theoretical perspectives, in the case of isothermal and non-isothermal conditions, paying particular attention to the kinetic aspects of the process [1–7].

With reference to first-order phase transformations, which occur by energy activated nucleation and growth, the kinetics is usually described by means of the Kolmogorov–Johnson–Mehl–Avrami (KJMA) model which assumes transitions to take place in infinite space [8–12]. This theory has been improved in order to deal with various effects, not considered in the original formulation of the theory, related to the spatial distribution of nuclei [13–17], growth law [18–20], nucleus shape [21–25], and both finite size and metric of the system where the transition takes place [26, 27]. The kinetic theory of phase transformation of small particles has been also developed in [28] where the effect of particle boundary has to be considered in the formulation of the model. Besides, phase transitions taking place in finite domain can be modeled by means of the time-cone method presented in Ref. [29]. All these approaches refer to the formation of a single phase in either infinite or finite environment.

Transformation kinetics of multi-phase formation has been discussed in Ref. [30]. In this case, difficulties in modeling the transition arise from the possible overgrowth of the phantom nuclei of the phase with the highest rate of growth. To overcome this problem, the theory has to be formulated in terms of the actual nucleation rate leading to a more involved expression of the solution that is now given by an integral equation [30]. In this modeling, the same mother phase transforms in the various new phases. However, the growths of the phases are not independent to each other since they compete for transforming the available space occupied by the mother phase. This aspect is made evident by the stochastic approach presented in Ref. [30].

Another topic linked to transitions involving more than one phase is represented by the consecutive phase transformations. Differently to the above, in this case, the new phase becomes the mother phase of a successive transition. In the case of two-phase system, the process can be thought as a two-step reaction where the reagent (initial mother phase) transforms to the transient species (second phase) which eventually transforms to the final product (third phase). Two cases can be distinguished depending on whether the two-phase transformations take place simultaneously or not. On one hand, in non-isothermal transformations, the two transitions can occur at quite different temperatures and the kinetics are representative, in fact, of two independent processes. In this context, kinetic studies performed using the differential thermal analysis (DTA) are enlightening. For instance, well-separated DTA peaks are obtained for consecutive phase transitions in systems which are important for technological application such as, Bi–Zn–B glass [31] and BIHFVOX (formulated as Bi₄Hf _x V_2−x O_{11−x/2−δ} ) ionic conductors [32]. Non-isothermal methods based on the KJMA rate equation [33] have been employed in [31] for describing the kinetics and for determining activation energy and Avrami’s exponent. On the other hand, in some systems, the DTA peaks of the consecutive transitions overlap, markedly, and this implies the kinetics of the phase transitions to overlap as well. The kinetics cannot be considered as independent, anymore. This situation has been put in evidence by the study reported in Ref. [34] on the structural transformations of Fe₇₅Ni₂Si₈B₁₃C₂ amorphous alloy (metallic glasses). In particular, the differential scanning calorimetry (DSC) peak has been ascribed to successive transformations of α-Fe(Si) to α ₁-Fe(Si) and to Fe₂B phases, where Fe(Si) denotes the solid solution. At the kinetic level, as also maintained by the authors, the overlap of the kinetic steps makes the KJMA rate equation not fully appropriate for modeling the kinetics [34]. Metastable intermediate phases have been characterized during structural transformation of the γ-Al₂O₃ cubic phase. It is found that the γ → α transition, at temperature of ≈1200 °C, proceeds via the formation of the intermediate θ-phase (monoclinic) [35, 36], while in the presence of Ar through the metastable δ-phase (orthorhombic) [37]. Consecutive phase transformations are also expected to occur, at constant temperature after cooling, in Fe-alloys which form the basis of the steel industry. This is the case of the peritectic Fe–Co alloys. On the iron-rich side of the system, δ-ferrite transforms to γ-austenite which in turn transforms to α-ferrite [38].

Intermediate phases can be obtained during chemical reactions at the solid state driven by nucleation and growth [7]. In this case, the situation is more involved than either the phase separation in solid solutions or allotropic transformations. In this context, it is worth mentioning, for example, the phase transformations in Cu–Al–O system which undergoes the oxidation reaction 4CuAlO₂ + 1/2O₂ → 2CuAl₂O₄ + Cu₂O, where the metastable spinel decomposes according to CuAl₂O₄ → Al₂O₃ + CuO [39]. Similarly, the carburization reaction of scheelite CaWO₄ + C → 1/3Ca₃WO₆ + 2CO + 2/3WC gives the intermediate solid phase Ca₃WO₆ which is finally carburized to produce CO, CaO and WC [40].

On the theoretical side, the kinetics of consecutive phase transitions ruled by nucleation and growth has not been so far comprehensively studied, apart from the case of uncoupled transitions. In this context, the present work is aimed at modeling consecutive transitions which take place simultaneously. In the following, a model kinetics is developed which considers nucleation and growth of an intermediate phase and its transformation, through nucleation and growth, to the final phase. The solution of the model kinetics is given in closed form and can be used as starting point for future development of the theory aimed at describing more complex transformations such as those occurring in successive chemical reactions, as mentioned above.

The paper is organized as follows. The first two sections are devoted to the basic equations of the model and to the definition of the probability function that is the key quantity of the approach. In the third section, the computation is developed for a single particle of metastable phase. In the same section, an approximate probability function is proposed and its validity analyzed on the basis of the numerical computations reported in the fourth section. The last sections deal with the modeling of the transformation by considering nucleation and growth of the metastable phase as well. Both isothermal and non-isothermal consecutive phase transitions are discussed.

Results and discussion

Formulation of the problem and basic equation

In this work, we tackle the problem of describing phase transitions taking place by the formation of an intermediate phase which, eventually, transforms to the final stable phase. It can be schematized as M → α → β, where M denotes the mother phase, and α and β the two phases. This process can be thought as a consecutive reaction at the solid state. The formation of α and β is considered to occur by nucleation and growth processes of spherical nuclei. The kinetic quantity we want to determine is the conversion degree of the β phase or, alternatively, the time dependence of the β/α volume fraction. To derive the basic equation let us first consider the simple case of a single-phase formation: M → α. From now on, the actual time of the kinetics will be indicated as t _f ; this is in order to better distinguish it from other time variables required by the theoretical approach presented in the next section. By denoting with V ₀ the volume of the system, the fraction of the α phase at time t _f , $\frac{{V_{\alpha } (t_{f} )}}{{V_{0} }}$, is equal to the probability a generic point within V ₀ is transformed at t _f . This probability can equally be expressed in terms of the probability the point is untransformed at t _f , P(t _f ), according to

$$ \frac{{V_{\alpha } (t_{f} )}}{{V_{0} }} = 1 - P(t_{f} ) . $$

(1)

In the case of a random distribution of nuclei, the computation of this probability leads to the celebrated KJMA kinetics [8–12]. In the case of two consecutive transitions, the first phase is now the mother phase of the second one. An expression similar to Eq. 1 is expected to hold for this case as well, although now the situation is more involved owing to the time dependence of the volume of the mother phase (α-phase). In fact, in the present case Eq. 1 is satisfied but in differential form according to

$$ \frac{{{\text{d}}V_{\beta } (t_{f} ,t_{1} )}}{{{\text{d}}V_{\alpha } (t_{1} )}} = 1 - P(t_{f} ,t_{1} ) , $$

(2)

where dV _α (t ₁) designates an element of volume of the α-phase formed in the time interval dt ₁ at t ₁ and dV _β (t _f , t ₁) is the part of it which is transformed at later time t _f  > t ₁ (Fig. 1). In Eq. 2, P(t _f , t ₁) is the probability a generic point within dV _α (t ₁) has not been transformed by the β-phase up to time t _f . The kinetics of the β-phase is eventually obtained by integration of Eq. 2:

Fig. 1

Pictorial 2D view of the α-particle at time a t ₁ + dt ₁ and b t _f with t _f  > t ₁. In panel a the annulus represents a just-formed shell of α-phase and Q a generic point of this region. The internal part of the sphere is made up of both α and β-phases. At time t _f , the shell generated at t ₁ < t _f has been partially transformed in the β-phase (panel b)

$$ V_{\beta } (t_{f} ) = \int\limits_{0}^{{t_{f} }} {{\text{d}}t_{1} } \left[ {1 - P(t_{f} ,t_{1} )} \right]\frac{{{\text{d}}V_{\alpha } (t_{1} )}}{{{\text{d}}t_{1} }} . $$

(3)

Evaluation of Eq. 3 requires the knowledge of both the kinetics of the transformation M → α and the empty-probability function, P(t _f , t ₁). For instance, in the case of a random distribution of nuclei, the former can be modeled through the KJMA approach. On the other hand, the computation of the empty-probability, P(t _f , t ₁), is a more involved problem for which an approximate estimation has been proposed, for 2D transitions, by exploiting the KJMA theory [41]. In particular, as discussed in more detail below, in this approach, the probability is assumed to be of the form P(t _f  − t ₁). To the best of author’s knowledge, so far the exact expression of P(t _f , t ₁) has not been reported in the literature, even for the model case of the growth of a single spherical α-particle. Aim of the following section is to determine this probability in closed form, for random nucleation and growth of the β-phase, and to provide an assessment on the validity of the approximate expression previously proposed.

Rate equation for the P(t _f , t ₁) probability

To determine the empty-probability, we employ the differential critical region method which is shown to be particularly suitable to deal with finite systems with moving boundaries, as considered here. The differential critical region approach has been originally introduced by Alekseechkin [16, 20, 30] and it is based on the formulation of a rate equation for the probability that a generic point of the space is transformed by the new phase. A comparative analysis of the various methods developed for describing phase transformation kinetics can be found in Ref. [42].

To determine the rate equation for the P(t _f , t ₁) probability, let us consider the quantity dP _c (t, t ₁, t″) defined as follows: It is the probability the generic point Q, lying within the spherical shell dV _α (t ₁) (generated between t ₁ and t ₁ + dt ₁), is transformed in the β-phase, between time t and t + dt, by a β-nucleus born between t″ and t″ + dt″, where t ₁ < t and t″ < t. By denoting with R _β (t, t″), the growth law of the β-nucleus and with I _β (t″) the nucleation rate, we get

$$ {\text{d}}P_{c} (t,t_{1} ,t^{\prime\prime}) = P(t,t_{1} )I_{\beta } (t^{\prime\prime}){\text{d}}t^{\prime\prime}{\text{d}}t\int\limits_{\varDelta \varOmega } {{\text{d}}\varOmega R_{\beta }^{2} (t,t^{\prime\prime})} \partial_{t} R_{\beta } (t,t^{\prime\prime}) , $$

(4)

where the solid angle, Ω, spans the differential critical region (ΔΩ) and the notation $\partial_{t} \equiv \frac{\partial }{\partial t}$ was used. Eq. 4 has the following meaning: the probability the point is transformed is given by the product of (i) the probability a nucleus appears at t″ within dt″ at distance R _β (t, t″) from Q in the volume elements dυ = R ² _β (t,t″)∂ _t R _β (t, t″)dtdΩ and (ii) the probability the point Q is untransformed at time t. In turn, the former probability is just equal to $I_{\beta } (t^{\prime\prime}){\text{d}}t^{\prime\prime}\int\limits_{\varDelta \varOmega } {{\text{d}}\upsilon }$. Eq. 4 can be rewritten as

$$ {\text{d}}P_{c} (t,t_{1} ,t^{\prime\prime}) = P(t,t_{1} )I_{\beta } (t^{\prime\prime})\partial_{t} \varpi (t,t_{1} ,t^{\prime\prime}){\text{d}}t^{\prime\prime}{\text{d}}t , $$

(5)

where ϖ is the volume of the critical region for the point Q.¹ Integration of Eq. 5 leads to,

$$ {\text{d}}P_{c} (t,t_{1} ) = P(t,t_{1} ){\text{d}}t\int\limits_{0}^{t} {{\text{d}}t^{\prime\prime}I_{\beta } (t^{\prime\prime})\partial_{t} \varpi (t,t_{1} ,t^{\prime\prime})} . $$

(6)

Since dP _c (t, t ₁) = −dP(t, t ₁), we eventually get,

$$ \begin{aligned} P(t_{f} ,t_{1} ) &= \exp \left[ { - \int\limits_{{t_{1} }}^{{t_{f} }} {{\text{d}}t} \int\limits_{0}^{t} {{\text{d}}t^{\prime\prime}I_{\beta } (t^{\prime\prime})\partial_{t} \varpi (t,t_{1} ,t^{\prime\prime})} } \right] \hfill \\ & = \exp \left[ { - \int\limits_{{t_{1} }}^{{t_{f} }} {{\text{d}}t} J(t,t_{1} )} \right], \hfill \\ \end{aligned} $$

(7)

where t _f is the actual time and $J\left( {t, t_{ 1} } \right) = \int\limits_{0}^{t} {{\text{d}}t^{\prime\prime}I_{\beta } \left( {t^{\prime\prime}} \right) \partial _{t} \varpi \left( {t, t_{ 1} , t^{\prime\prime}} \right)}$.

The P(t _f , t ₁) probability for a spherical α-nucleus

As reported in the previous section, the determination of the empty-probability requires the knowledge of the critical region. Here, we consider a single α-nucleus with growth law R _α (t), where $\dot{R}_{\alpha } > \dot{R}_{\beta }$. This constraint is dictated by the consecutive process we are dealing with, where the overgrowth of the intermediate phase is precluded. The volumes of α- and β-nuclei are $\upsilon_{\alpha } = \tfrac{4\pi }{3}R_{\alpha }^{3}$ and $\upsilon_{\beta } = \tfrac{4\pi }{3}R_{\beta }^{3}$, respectively, and the growth law is linear according to R _α (t) = v _α t and R _β (t) = v _β t. Such a time dependence not only simplifies the presentation of the modeling but also allows a comparison to be made with the KJMA kinetics.

In order to determine the critical region, we distinguish the following two cases: (i) 0 < t″ < t ₁ < t < t _f and (ii) 0 < t ₁ < t″ < t < t _f . As schematized in Fig. 2, in the first case at time t″, when the nucleation event takes place, the available volume for nucleation is a sphere entirely contained within the volume of the α-nucleus at time t ₁. The reverse situation occurs in case (ii). In both cases and in dependence on the ratios $\frac{{t_{1} }}{t}$ and $\rho = \frac{{v_{\beta } }}{{v_{\alpha } }} < 1$, one considers three characteristic times, T ₀, T ₁, and T ₂, which enter the definition of the ∂ _t ϖ(t, t ₁, t″) versus t″ function. These times have to satisfy the inequalities 0 < T _i  < t ₁ (case i) and t ₁ < T _i  < t (case ii). In particular, the time $T_{0} = \frac{{\rho t - t_{1} }}{1 + \rho }$ is defined by the equation R _β (t, T ₀) = R _α (t ₁) + R _α (T ₀), i.e., for t″ ≤ T ₀, ϖ(t″) = υ _α (t″) (for both cases i and ii); the time $T_{1} = \frac{{t_{1} - \rho t}}{1 - \rho }$ satisfies the relationship R _α (T ₁) + R _β (t, T ₁) = R _α (t ₁) implying, at t″ < T ₁, ϖ = 0 (case i); and the time $T_{2} = \frac{{t_{1} + \rho t}}{1 + \rho }$ which satisfies the equation R _α (T ₂) = R _α (t ₁) + R _β (t, T ₂) and implies, at t″ > T ₂, ϖ(t, t″) = υ _β (t − t″) (case ii). For the sake of completeness, the details of the computation are reported in Appendix 1 and the results are summarized in the graphical representation illustrated in Fig. 3. Notably, the inequalities derived in Fig. 3 also hold at ρ = 1. However, for the case (i), the condition ϖ = 0 is never satisfied at ρ = 1, and the time T ₀ is given by $T_{0} = \frac{{t - t_{1} }}{2}$.

Fig. 2

Schematic view of the differential critical regions of the point Q (Fig. 1). A β-nucleus born between t″ and t″ + dt″ will transform the point Q between t and t + dt provided it is located within the differential critical region (thick segments of circumference). The nucleation center of the β-nucleus has to lie within the volume of the α-particle at time t″, when its radius equals R _α (t″). Panels a and b refer to the cases t″ < t ₁ < t < t _f and t ₁ < t″ < t < t _f , respectively, being t _f the actual time

Fig. 3

The volume of the critical region, ϖ(t, t ₁, t″), is reported as a function of the birth time of the nucleus t″. Panels a and b refer to the cases t″ < t ₁ < t < t _f and t ₁ < t″ < t < t _f , respectively. The allowed range of variation for t″ is evidenced by thick line; for completeness, the t and t _f values are also reported on the time axis. In the panels, ρ = v _β /v _α , is the velocity ratio. See the main text for the meaning of υ _α , υ _β , and ω functions and Appendix 1 for the details of the computation of ϖ(t, t ₁, t″)

In Eq. 7, the integral over t″ splits into the contributions of “case (i)” and “case (ii)” above, as

$$ J(t,t_{1} ) = \int\limits_{0}^{{t_{1} }} {{\text{d}}t^{\prime\prime}I_{\beta } (t^{\prime\prime})\partial_{t} \varpi (t,t_{1} ,t^{\prime\prime})} + \int\limits_{{t_{1} }}^{t} {{\text{d}}t^{\prime\prime}I_{\beta } (t^{\prime\prime})\partial_{t} \varpi (t,t_{1} ,t^{\prime\prime})} . $$

(8)

Using the time dependence of ∂ _t ϖ reported in Fig. 3, we eventually get

$$ \int\limits_{0}^{{t_{1} }} {{\text{d}}t^{\prime\prime}I_{\beta } (t^{\prime\prime})\partial_{t} \varpi (t,t_{1} ,t^{\prime\prime})} \equiv \chi_{{\frac{\rho }{2 + \rho },\rho }} (\tfrac{{t_{1} }}{t})\int\limits_{{T_{0} }}^{{t_{1} }} {f(t,t_{1} ,t^{\prime\prime}){\text{d}}t^{\prime\prime}} + \chi_{\rho ,1} (\tfrac{{t_{1} }}{t})\int\limits_{{T_{1} }}^{{t_{1} }} {f(t,t_{1} ,t''){\text{d}}t^{\prime\prime}} $$

(9a)

$$ \begin{aligned} \int\limits_{{t_{1} }}^{t} {{\text{d}}t^{\prime\prime}I_{\beta } (t^{\prime\prime})\partial_{t} \varpi (t,t_{1} ,t^{\prime\prime})} \equiv \chi_{0,1} (\tfrac{{t_{1} }}{t})\int\limits_{{T_{2} }}^{t} {g(t,t^{\prime\prime}){\text{d}}t^{\prime\prime}} \hfill \\ \, + \chi_{{0,\frac{\rho }{2 + \rho }}} (\tfrac{{t_{1} }}{t})\int\limits_{{T_{0} }}^{{T_{2} }} {f(t,t_{1} ,t^{\prime\prime}){\text{d}}t^{\prime\prime}} + \chi_{{\frac{\rho }{2 + \rho },1}} (\tfrac{{t_{1} }}{t})\int\limits_{{t_{1} }}^{{T_{2} }} {f(t,t_{1} ,t^{\prime\prime}){\text{d}}t^{\prime\prime}} { ,} \hfill \\ \end{aligned} $$

(9b)

where χ _a,b (x) is the characteristic function (χ _a,b (x) = 1 for a < x < b, χ _a,b (x) = 0 otherwise), f(t, t ₁, t″) = I _β (t″) ∂ _t ω(t, t ₁, t″), g(t, t″) = I _β (t″) ∂ _t υ _β (t − t″) and T ₀, T ₁, and T ₂, are functions of t and t ₁.

Before proceeding with the numerical computation of Eqs. 7–9, we briefly consider the approximation proposed in the literature to model the empty-probability [41]. In the framework of the KJMA theory, Eq. 7 takes the form

$$ P_{a} (t_{f} ,t_{1} ) = \exp \left[ { - \int\limits_{{t_{1} }}^{{t_{f} }} {{\text{d}}t} J_{a} (t,t_{f} )} \right] $$

(10a)

with

$$ J_{a} (t,t_{f} ) = I_{\beta } (t)\upsilon_{\beta } (t_{f} - t) $$

(10b)

which entails, for constant nucleation rate, P _a (t ₁,t _f ) ≡ P _a (t _f  − t ₁). Eq. 10b implies that only nucleation events occurring at time greater than the birth time of the point Q, t ₁ (generation of the α-phase), are capable of transforming Q. According to the discussion above this is not strictly true, as nucleation events taking place before time t ₁ can contribute to the phase transition as well (see also Fig. 2). In fact, the exponential argument in Eq. 10a is just given by the last integral in Eq. 8 by setting ϖ(t, t ₁, t″) = υ _β (t − t″) and changing the order of integration. In the following, we investigate the validity of the approximate expression Eq. 10b and show that according to Eq. 7–9 the functional form of the probability is $P(t_{1} ,t_{f} ) \equiv P(t_{1} ,\tfrac{{t_{1} }}{{t_{f} }})$.

Numerical computations of Eqs. 8 and 9 have been performed for constant nucleation rate, I _β . The probability P(t, t _f ) and the volume of the β-phase are then computed through Eqs. 7 and 3, respectively. The expressions of the volumes of the critical regions are $\upsilon_{\beta } (t - t^{\prime\prime}) = \frac{4}{3}\pi v_{\beta }^{3} (t - t^{\prime\prime})^{3}$ and ω(t, t ₁, t″) = ω[v _α t″, v _β (t − t″); v _α t ₁], where ω[r ₁, r ₂; z] is the overlap volume of two spheres of radii r ₁ and r ₂ located at relative distance z. The time derivatives of the volumes of the critical regions are (see also Appendix 2 for details)

$$ \partial_{t} \upsilon_{\beta } (t - t'') = 4\pi v_{\alpha }^{3} \rho^{3} (t - t'')^{2} $$

(11a)

$$ \partial _{t}\omega \left( {t,t_{1} ,t^{\prime\prime}} \right) = \pi \nu_{\alpha }^{3} \left[ {2\rho^{3} \left( {t - t^{\prime\prime}} \right)^{2} - \rho^{2} t_{1} \left( {t - t^{\prime\prime}} \right) + \frac{{\rho^{2} }}{{t_{1} }}\left( {t - t^{\prime\prime}} \right)\left[ {t^{{\prime\prime}2} - \rho^{2} \left( {t - t^{\prime\prime}} \right)^{2} } \right]} \right]. $$

(11b)

The dimensional analysis of Eq. 11 shows that ∂ _t ϖ(t, t ₁, t) scales as t ^k t ^m₁ (t″)^s with k + m + s = 2 (k, m, s relative integers). The integral Eq. 9 is therefore given by the linear combination of terms t ^k t ^m₁ , with k + m = 3. Consequently, $J(t_{1} ,t) = I_{\beta } v_{\alpha }^{3} t_{1}^{3} \tilde{J}(\tfrac{{t_{1} }}{t}) = I_{\beta } v_{\alpha }^{3} t_{1}^{3} \tilde{J}(x)$ with $x = \frac{{t_{1} }}{t}$. By changing the integration variable, the last integral of Eq. 7 becomes, $\int\limits_{{t_{1} }}^{{t_{f} }} {{\text{d}}t} J(t,t_{1} ) = I_{\beta } v_{\alpha }^{3} t_{1}^{4} \int\limits_{{\tfrac{{t_{1} }}{{t_{f} }}}}^{1} {{\text{d}}x\tilde{J}(x)} x^{ - 2}$. Therefore, in terms of dimensionless variables, the probability function takes the form

$$ \begin{aligned} P(\tau_{1} ,\tau_{f} ) &= \exp \left[ { - \tau_{1}^{4} \int\limits_{{\tfrac{{\tau_{1} }}{{\tau_{f} }}}}^{1} {{\text{d}}x\tilde{J}(x)} x^{ - 2} } \right] \hfill \\ \, &= \exp \left[ { - \tau_{1}^{4} \eta (\tfrac{{\tau_{1} }}{{\tau_{f} }})} \right] \, , \hfill \\ \end{aligned} $$

(12)

where τ ₁ = (I _β v ³ _α )^1/4 t ₁ and τ _f  = (I _β v ³ _α )^1/4 t _f are the normalized times and $\eta (x) = \int\limits_{x}^{1} {{\text{d}}z\tilde{J}(z)z^{ - 2} }$. As anticipated above, Eq. 12 implies $P(t_{1} ,t_{f} ) \equiv P(t_{1} ,\tfrac{{t_{1} }}{{t_{f} }})$.

The transformed volume fraction of the β-phase is estimated using Eq. 3, with V _α  = υ _α , since we are considering a α-nucleus growing in isolation:

$$ X_{\beta } (t_{f} ) = \frac{3}{{t_{f}^{3} }}\int\limits_{0}^{{t_{f} }} {{\text{d}}t_{1} } \left[ {1 - P(t_{f} ,t_{1} )} \right] \, t_{1}^{2} , $$

(13)

with X _β  = V _β /V _α . In terms of normalized times, the volume fraction becomes

$$ X_{\beta } (\tau_{f} ) = \frac{3}{{\tau_{f}^{3} }}\int\limits_{0}^{{\tau_{f} }} {{\text{d}}\tau_{1} } \left[ {1 - e^{{ - \tau_{1}^{4} \eta (\tfrac{{\tau_{{_{1} }} }}{{\tau_{f} }})}} } \right]\tau_{1}^{2} , $$

(14)

where Eq. 12 was employed. Finally, we point out that the approximation Eq. 10 implies considering in Eq. 14 the function

$$ \eta (\tfrac{{\tau_{1} }}{{\tau_{f} }}) = \frac{\pi }{3}\rho^{3} \left( {\frac{{\tau_{f} }}{{\tau_{1} }} - 1} \right)^{4} . $$

(15)

Numerical results

In this section, numerical outputs of the theoretical approach presented in “The P(t _f , t ₁) probability for a spherical α-nucleus” section are reported and discussed. Particular attention is devoted to the validity of the approximate Eqs. 12 and 15. This is an important issue since Eq. 15 allows to treat, in a quite manageable fashion, the complex process of nucleation and growth of α-nuclei in the mother phase and their successive transformation in the β-phase.

Typical behaviors of the empty-probability, P(τ ₁, τ _f ), are displayed in Fig. 4 as a function of normalized times for several values of ρ. The approximate empty-probability has also been computed using Eq. 15 and the deviation from the exact computation reported in Fig. 5 for the same ρ’s as in Fig. 4. Specifically, the plots refer to the quantity (P(τ ₁, τ _f ) − P _a (τ ₁, τ _f )) with P _a (τ ₁, τ _f ) being the approximate function. As far as the probability function is concerned, it results that the approximate Eq. 15 works better the lower the velocity ratio ρ. The transformed volume fraction of the single nucleus, X _β (τ _f ), (Eq. 14) is shown in Fig. 6 for several ρ values. Besides, the kinetics satisfies the asymptotic behavior $\left. {X_{\beta } (\tau_{f} )} \right|_{{\tau_{f} \to \infty }} = 1$, as it follows from Eq. 14 by using z = τ ₁/τ _f as integration variable. It is worth noticing that the stretched exponential function, widely employed in connection to the KJMA approach, does fail in describing the kinetics of Fig. 6. In fact, the fit of the function $X(\tau_{f} ) = 1 - e^{{ - K(\tau_{f} )^{n} }}$ (with K temperature-dependent constant and n Avrami’s exponent) to the exact kinetics is quite poor (not shown in the figure).

Fig. 4

3D plot of the probability P(τ ₁, τ _f ) as a function of normalized times. It is the probability a point of the α-phase, generated at τ ₁, is untransformed up to time τ _f . a ρ = 0.2, b ρ = 0.5, c ρ = 0.9

Fig. 5

3D plot of the difference between the probability P(τ ₁, τ _f ) and its approximate expression P _a (τ ₁, τ _f ) (Eqs. 12, 15), as a function of normalized times. P(τ ₁, τ _f ) is the probability a point of the α-phase, generated at τ ₁, is untransformed up to time τ _f . a ρ = 0.2, b ρ = 0.5, c ρ = 0.9

Fig. 6

Kinetics of the V _β /V _α volume fraction for a single α-particle as a function of normalized time. Parameter values are ρ = 0.05, ρ = 0.2, ρ = 0.4, ρ = 0.6, ρ = 0.8, ρ = 1, where the duration of the kinetics decreases with increasing ρ

To check the validity of the approximation which led to Eq. 10, the computation of X _β (τ _f ) has been performed by employing Eqs. 15 in 14. The result of this computation is displayed in Fig. 7 for the difference between the exact and the approximate kinetics, the former being greater than the latter. The curves of Fig. 7 show that the approximation proposed in Ref. [41] is indeed quite good for ρ < 1. Specifically, it is obtained that the uncertainty increases with ρ being lower than 15 % for ρ ≤ 0.8. This result is important since, once checked the reliability of Eq. 15, this approximation can be exploited to deal with the more involved process of the nucleation of the α-phase.

Fig. 7

The difference between the V _β /V _α volume fraction for a single α-particle computed using the P(τ ₁, τ _f ) probability and the approximate expression P _a (τ ₁, τ _f ). Parameter values are Panel a ρ = 0.2, ρ = 0.4 (left scale) and ρ = 0.05 (right scale). Panel b: ρ = 0.6, ρ = 0.8, ρ = 0.9

Consecutive transformation with nucleation and growth of the α-phase

Isothermal kinetics

On the basis of the results of the previous sections, we are now in the position to study the effect of the nucleation of the α-phase on the kinetics. The determination of the exact solution of this kinetics is a formidable task, since it requires considering the morphology of the α-phase ruled by impingement. Nevertheless, this task can be accomplished by using the approximate expression Eq. 15 in Eq. 12 which allows us to perform the computation, analytically. In fact, considering P _a (t ₁, t _f ) as obtained from Eqs. 7, 10, means that the maximum extension of the critical region for the volume element of the α-phase, generated at t ₁ within dt ₁, is just equal to υ _β (t _f  − t ₁). This domain is entirely contained within the α-phase, whatever its morphology. In other words, the shape of the α/M boundary does not enter the computation of the critical region.

The M → α transformation kinetics is given by the KJMA kinetics according to

$$ \frac{{V_{\alpha } (\tau_{f} )}}{{V_{0} }} = 1 - e^{{ - \, \frac{\pi }{3}\lambda \tau_{f}^{4} }} , $$

(16)

where V ₀ is the total volume of the system, λ = I _α /I _β and τ _f the normalized time, defined in Eq. 12. The volume of the β-phase is given by Eqs. 3, 12, and 15 as

$$ V_{\beta } (\tau_{f} ) = V_{0} \frac{4\pi }{3}\lambda \int\limits_{0}^{{\tau_{f} }} {{\text{d}}\tau_{1} \left[ {1 - e^{{ - \, \frac{\pi }{3}\rho^{3} \tau_{f}^{4} \left( {1 - \frac{{\tau_{1} }}{{\tau_{f} }}} \right)^{4} }} } \right]e^{{ - \, \frac{\pi }{3}\lambda \tau_{1}^{4} }} \tau_{1}^{3} } . $$

(17)

The transformed volume fraction is eventually obtained from Eqs. 16 and 17 according to

$$ \frac{{V_{\beta } (\tau_{f} )}}{{V_{\alpha } (\tau_{f} )}} = \frac{4\pi }{3}\frac{\lambda }{{1 - e^{{ - \frac{\pi }{3}\lambda \tau_{f}^{4} }} }}\int\limits_{0}^{{\tau_{f} }} {{\text{d}}\tau_{1} } \left[ {1 - e^{{ - \frac{\pi }{3}\rho^{3} \tau_{f}^{4} \left( {1 - \frac{{\tau_{1} }}{{\tau_{f} }}} \right)^{4} }} } \right]e^{{ - \frac{\pi }{3}\lambda \tau_{1}^{4} }} \tau_{1}^{3} . $$

(18)

The behavior of the volume fraction, as a function of actual time, is reported in Fig. 8 for several values of λ = I _α /I _β and ρ = v _β /v _α . At odds with the results of “Numerical results” section, these kinetics are in very good agreement with the stretched exponential function above reported ($X(\tau_{f} ) = 1 - e^{{ - K(\tau_{f} )^{n} }}$). The fits of this function to the numerical outputs of Eq. 18 are shown in Fig. 8 as dashed lines. The fitting function provides a very good description of the theoretical results in the range of λ and ρ values considered here. The Avrami exponents obtained by these fits are displayed in Fig. 9 as a function of λ and ρ. Therefore, as far as the β/α fraction is concerned, these results provide a theoretical basis for using Avrami’s equation in modeling consecutive phase transformations. It is worth noticing that Avrami’s exponent for the β/α kinetics reported in Fig. 9 is greater than n = 4 namely, the value predicted by the classical KJMA kinetics for liner growth of 3D nuclei and constant nucleation rate. These results suggest that, whenever experimentally accessible, the $\tfrac{{V_{\beta } }}{{V_{\alpha } }}$ kinetics should be measured being suitable to be described by Avrami’s equation.

Fig. 8

Kinetics of the V _β /V _α volume fraction in the case of nucleation and growth of the α-phase (solid symbols). Computation is reported for ρ = 0.2 (panel a) and ρ = 0.4 (panel b). The various values of λ = I _α /I _β are also indicated on the curves. The dashed lines are the best fit of the stretched exponential function to the computed curves

Fig. 9

Behavior of Avrami’s exponent as a function of λ = I _α /I _β for the kinetics displayed in Fig. 8. Parameter values are ρ = 0.2 open diamonds; ρ = 0.4 solid circles; ρ = 0.5 solid squares

Non-isothermal kinetics

The fact that the time dependence of $\tfrac{{V_{\beta } }}{{V_{\alpha } }}$ is well described by Avrami’s kinetics makes it possible to model non-isothermal consecutive reactions as commonly studied by differential scanning calorimetry technique. As outlined in the introduction section, the KJMA rate equation, which is obtained by the derivative of the stretched exponential function [33, 43–45], is not fully consistent with kinetic data of consecutive phase transformations. This finding can be justified on the basis of the present approach. In fact, according to the present modeling, the volume fraction $\frac{{V_{\beta } }}{{V_{0} }} = \frac{{V_{\beta } }}{{V_{\alpha } }}\frac{{V_{\alpha } }}{{V_{0} }}$ behaves as the product of two stretched exponential functions. On the ground of this result in the following, we study the effect of this kinetics on the shape of the DSC peak. In Ref. [33], it is shown that for the formation of a single phase, under linear temperature variation T(t) = T ₀ + Φt with Φ constant heating rate, the volume fraction is given by

$$ X(T) = 1 - \exp \left\{ { - [A\tfrac{E}{{k_{B} }}p\left( {\tfrac{E}{{k_{B} T}}} \right)]^{n} } \right\} , $$

(19)

where T is the actual temperature, k _B Boltzmann’s constant, E the apparent activation energy, A a constant, and $p(x) = \int\limits_{x}^{\infty } {\frac{{e^{ - u} }}{{u^{2} }}{\text{d}}u}$ the exponential integral. This formulation applies to the fraction $\tfrac{{V_{\alpha } }}{{V_{0} }}$ with n = 4. As regards the kinetics $\tfrac{{V_{\beta } }}{{V_{\alpha } }}$, Eq. 19 has to be considered an approximation owing to the dependence of n on both λ = I _α /I _β and ρ = v _β /v _α , i.e., on temperature (Fig. 9). Therefore, in the following calculation, which is aimed at providing information on the shape of the peak, a mean value $\bar{n} = 5$ will be adopted. Usually, in several experiments, E/k _B T ≫ 1 and the series expansion p(x) ≈ e ^−x/x ² is exploited in Eq. 19 [33, 43, 44]. Provided the fitting parameter K is energy activated, the non-isothermal kinetics of α and β phases can be written according to

$$ \frac{{V_{\alpha } (T)}}{{V_{0} }} = Y_{1} (T) $$

(20a)

$$ \frac{{V_{\beta } (T)}}{{V_{0} }} = Y_{1} (T)Y_{2} (T) , $$

(20b)

where $Y_{1} (T) = \left( {1 - \exp \left[ { - \left( {A_{1} \frac{{T^{2} }}{{q_{1} }}e^{{ - q_{1} /T}} } \right)^{4} } \right]} \right)$, $Y_{2} (T) = \left( {1 - \exp \left[ { - \left( {A_{2} \frac{{T^{2} }}{{q_{2} }}e^{{ - q_{2} /T}} } \right)^{5} } \right]} \right)$ and q _i  = E _i /k _B . The rate of enthalpy change of the transformation is proportional to

$$ \frac{{{\text{d}}H}}{{{\text{d}}T}} = \varDelta h_{M,\alpha } \frac{{{\text{d}}V_{\alpha } }}{{{\text{d}}T}} + \varDelta h_{\alpha ,\beta } \frac{{{\text{d}}V_{\beta } }}{{{\text{d}}T}} = \varDelta h_{\alpha ,\beta } \frac{\text{d}}{{{\text{d}}T}}(aV_{\alpha } + V_{\beta } ) , $$

(21)

with Δh _i,j being the specific enthalpy change for the process i → j and $a = \frac{{\varDelta h_{M,\alpha } }}{{\varDelta h_{\alpha ,\beta } }}$.² In terms of the dimensionless variables ξ = T/q ₁, κ ₁ = A ₁ q ₁, κ ₂ = A ₂ q ²₁ /q ₂, and w = q ₂/q ₁, the functions Y _1,2 become $Y_{ 1} \left( \xi \right) = \left( { 1 - { \exp }\left[ { - \left( {\kappa_{ 1} \xi^{ 2} e^{ - 1/\xi } } \right)^{ 4} } \right]} \right)$ and $Y_{ 2} \left( \xi \right) = \left( { 1 - { \exp } \left[ { - \left( {\kappa_{ 2} \xi^{ 2} e^{ - w/\xi } } \right)^{ 5} } \right]} \right)$. From Eq. 21, one eventually obtains

$$ \frac{{{\text{d}}H}}{{{\text{d}}\xi }} \propto \frac{\text{d}}{{{\text{d}}\xi }}\left[ {Y_{1} (\xi )\left( {a + Y_{2} (\xi )} \right)} \right] . $$

(22)

The trend of Eq. 22 as a function of the normalized temperature (ξ) has been displayed in Fig. 10 for several values of a and w (at a > 0 and κ ₁ = κ ₂). The computation is aimed at illustrating how the shape of the peak changes with parameters governing the kinetics and, in particular, whether two separate features may be resolved in the spectrum, or not. As Fig. 10 does show, the answer to this question is in the affirmative. In principle, the presence of consecutive transformations could manifest itself either with an asymmetry of the peak or with two separate peaks depending on materials parameters. Specifically, at w > 1, increasing the w value the DSC curve splits into two separate peaks, the first of which being related to the M → α transformation (peak at lower ξ value). In fact, in the limit of large w the product Y ₁(T)Y ₂(T) becomes Y ₁(T)Y ₂(T) ≈ Y ₂(T) and the heat flux, Eq. 22, is just equal to the linear superposition of the two kinetics with weighting parameter a: $\frac{{{\text{d}}H}}{{{\text{d}}\xi }} \propto \frac{\text{d}}{{{\text{d}}\xi }}\left[ {aY_{1} (\xi ) + Y_{2} (\xi )} \right]$. This is due to the fact that the Y ₁(T) kinetics is faster than the Y ₂(T) one, i.e., Y ₁(T) ≥ Y ₂(T)∀T. On the contrary, in the case w < 1, still at κ ₁ = κ ₂, the Y ₂(T) kinetics is faster than Y ₁(T) and the condition Y ₁(T)Y ₂(T) ≈ Y ₁(T) is now fulfilled. Under these circumstances, the DSC peak is asymmetric and does not separate in two distinct features as obtained in the previous case. It is worth noticing the difference between the structure of Eq. 22 and that describing non-isothermal kinetics of non-competing multi-phase transformations, previously modeled in Ref. [46], i.e., $\frac{{{\text{d}}H}}{{{\text{d}}\xi }} \propto \frac{\text{d}}{{{\text{d}}\xi }}\left[ {bY_{\alpha } (\xi ) + Y_{\beta } (\xi )} \right]$. In this equation, Y _α and Y _β refer to the kinetics of the transformations M → α and M → β, respectively, with $b = \frac{{x_{\alpha } }}{{x_{\beta } }}\frac{{\varDelta h_{M \to \alpha } }}{{\varDelta h_{M \to \beta } }}$ and x _i volume fraction of the phase at the end of the transformation.

Fig. 10

Computation of the heat flow, Y(ξ′) ∝ dH/dξ′, in non-isothermal consecutive transformations at constant heating rate (Eq. 22). The dimensionless variable ($\xi ' = \sqrt {\kappa_{1} } \xi$) is proportional to the temperature, i.e., to the actual time. Calculations have been performed for several values of w = q ₂/q ₁ and $a = \frac{{\varDelta h_{M,\alpha } }}{{\varDelta h_{\alpha ,\beta } }}$, at κ ₁ = κ ₂ = 5 × 10³ (Eqs. 21, 22). Panel a: a = 0.2; panel b a = 0.5; panel c a = 0.8. The values of w are indicated on the curves

The present study provides a rationale for investigating consecutive phase transitions as detected by thermal analysis. The DSC curve may, therefore, result in either an asymmetric peak due to the non-linear coupling of Avrami’s kinetics or in two nearly separate peaks when linear coupling of the kinetics is operative. All this depending on materials parameters, for instance, on the w ratio at κ ₁ = κ ₂ as considered in the evaluation here reported (Fig. 10).

Before concluding, we comment on the possibility of analyzing the non-isothermal kinetics using isoconversional methods [4, 47–49]. These approaches rely on the factorization of the reaction rate in the product of volume fraction and temperature-dependent functions according to $\dot{X} = f_{1} (X)f_{2} (T)$, where $f_{2} (T) = Ae^{{ - E_{\text{a}} /k_{\text{B}} T}}$ with E _a apparent activation energy and A temperature-independent pre-exponential factor. The plot $\ln \left[ {\left( {\dot{X}} \right)_{X} } \right]$ against $\frac{1}{T}$, at constant X and varying the temperature program (i.e., Φ in the case of linear temperature profile), leads to the determination of E _a [4]. For the KJMA rate equation $f_{1} (X) = n(1 - X)\left[ { - \ln (1 - X)} \right]^{{\frac{n - 1}{n}}}$.

The present computation shows that for the consecutive transitions here investigated, the differential isoconversional analysis is based on approximations. As far as the kinetics, X _β  = V _β /V _α is concerned, the results of “Isothermal kinetics” section indicate that it matches a stretched exponential with a temperature-dependent Avrami’s exponent, n. In other words, the factorization above reported does not hold, strictly, for the rate $\dot{X}_{\beta }$. In addition, in terms of the actual time the kinetics reads, $X_{\beta } (t_{f} ) = 1 - e^{{ - \tilde{K}(t_{f} )^{n} }}$ where $\tilde{K} = K[I_{\beta } v_{\alpha }^{3} ]^{n/4}$. At constant X _β , it follows that $\ln \left[ {(\dot{X}_{\beta } )_{{X_{\beta } }} } \right] \propto \ln (n) + \frac{n - 1}{n}\ln [ - \ln (1 - X_{\beta } )] + \frac{1}{n}\ln \tilde{K}$ and the isoconversional method does not provide a straightforward estimate of the apparent activation energy. However, owing to the weak dependence of n on λ (Fig. 9), a mean n value can be used as previously done in studying the DSC curve:

$$ \ln \left[ {(\dot{X}_{\beta } )_{{X_{\beta } }} } \right] \propto \frac{1}{{\bar{n}}}\ln \tilde{K} = \frac{1}{{\bar{n}}}\ln K + \frac{1}{4}\ln I_{\beta } + \frac{3}{4}\ln v_{\alpha } , $$

(23)

where the last two terms are linked to the activation energies for nucleation and growth of β and α phases, respectively. Next, we consider the temperature dependence of the fitting parameter K, a function of both λ and ρ. Taylor’s expansion of K in terms of lnλ provides $K\left( {\lambda , \rho } \right) = \sum\limits_{m} {C_{m} \left( \rho \right)\left[ { { \ln }\lambda } \right]^{m} }$ where, it is worth recalling, $\lambda \propto e^{{ - (E_{{{\text{N}},\alpha }} - E_{{{\text{N}},\beta }} )/k_{\text{B}} T}}$ and $\rho \propto e^{{ - (E_{{{\text{G}},\beta }} - E_{{{\text{G}},\alpha }} )/k_{\text{B}} T}}$ with E _N (E _G) actual activation energy for nucleation (growth). Numerical computations of the kinetics in the range 0.05 < ρ < 0.8 and 0.005 < λ < 1000 show that K(λ, ρ) is well approximated by retaining in the series terms up to the second order. In addition, the behavior of C _m (ρ) is in excellent agreement with the power law $C_{m} (\rho ) \propto \rho^{{p_{m} }}$ with p ₀ = 2.2, p ₁ = 2.7, and p ₂ = 3.2. The logarithmic term in the series expansion brings about a quadratic dependence of K on 1/T which can be neglected when compared to the exponential terms. As a first approximation, lnλ can be considered constant and equal to its mean value in the explored interval of temperatures. It is obtained that the polynomial matches the power law $K(\overline {{ \ln }\lambda} ,\rho ) \propto \rho^{{p_{\lambda } }}$ with high accuracy (specifically, 1.7 < p _λ  < 2.7 for $- 2 < \overline{{ { \ln }\lambda }} < 6$). The apparent activation energy is eventually obtained from Eq. 23 according to $E_{\text{a}} = \frac{{p_{\lambda } }}{{\bar{n}}}(E_{{{\text{G}},\beta }} - E_{{{\text{G}},\alpha }} ) + \frac{{E_{{{\text{N}},\beta }} + 3E_{{{\text{G}},\alpha }} }}{4}$. Under these circumstances, one identifies E ₂ with E _a in Eq. 20b. On the experimental side, this analysis is based on the possibility of measuring the kinetics of both M → α and α → β phase transitions.³ Finally, when the isokinetic conditions E _G,β  = E _G,α and E _N,β  = E _N,α are fulfilled, the apparent activation energy is equal to $E_{\text{a}} = \frac{{E_{{{\text{N}},\beta }} + 3E_{{{\text{G}},\beta }} }}{4}$.

Conclusions

In this work, the kinetics of consecutive phase transformations ruled by nucleation and growth has been studied. To this purpose, the differential critical region method has been employed which is shown to be particularly suitable to deal with the stochastic problem related to the process under investigation. In fact, in the consecutive transformations, the finite size and the moving boundary of the metastable phase have to be taken into account in the formulation of the theory.

The exact solution of the kinetics is derived for the transformation of a single α-nucleus in the case of constant nucleation rate and linear growth. Key point of the model is the determination of the empty-probability P(t _f , t ₁), that is the probability a point of the metastable phase (α), formed at time t ₁, is not transformed in the β-phase within time t _f . It is found that the behavior of the kinetics of β/α volume fraction does not match Avrami’s kinetics. Also, an approximate expression of the P(t _f , t ₁) function, P _a (t _f , t ₁), is derived by means of the KJMA model and its validity checked by comparison with the exact solution. The P _a (t _f , t ₁) function is useful to tackle the very involved problem linked to the nucleation and growth of the α-phase during the consecutive transformations. For nucleation and growth of the α-phase, the β/α volume fraction is found to be in excellent agreement with a stretched exponential with Avrami’s exponent in the range 4 < n < 5.6, in dependence on the ratios between nucleation and growth rates of the two phases.

The kinetics of the consecutive transformations has also been studied under non-isothermal conditions at constant heating rate. It is found that consecutive transformations give rise to DSC spectrum with either an asymmetric peak or two separate peaks, in dependence on materials parameters. At the mathematical level, these findings are described by linear and non-linear superpositions of two Avrami’s kinetics. This output is in qualitative accord with the experimental results quoted in the introduction section.

Finally, the present modeling can also be applied for describing consecutive transitions ruled by diffusional-type growth. For this growth mode, Eq. 7 is not the exact solution of the kinetics owing to the overgrowth of the phantom nuclei. Moreover, the computation of the volumes of the critical regions is more involved because the times T _i are now the solutions of non-linear equations.

Appendices

Appendix 1

In this Appendix, the computation of the volume of the critical region, ϖ(t, t ₁, t″), is reported. According to Fig. 2, the following cases are considered:

Case (i) 0 < t″ < t ₁ < t < t _f

For R _α (t″) + R _β (t, t″) < R _α (t ₁), the volume of the critical region is nil, ϖ = 0. For linear growth, this inequality implies,

$$ 0 < (1 - \rho )t^{\prime\prime} < t_{1} - \rho t , $$

(24)

where $\rho = \frac{{v_{\beta } }}{{v_{\alpha } }} < 1$. Therefore, for $\, \rho < \, \frac{{t_{1} }}{t} < 1 { }$ one defines the time $T_{1} = \frac{{t_{1} - \rho t}}{1 - \rho }$ and Eq. 24 provides

(a) $\, \rho < \, \frac{{t_{1} }}{t} < 1 { }$

$$ \varpi = 0 \, \leftrightarrow \, t'' < T_{1} < t_{1} $$

(25)

$$ \varpi = \omega (t,t_{1} ,t'') \, \leftrightarrow \, T_{1} < t'' < t_{1} $$

(26)

where ω(t, t ₁, t″) = ω[R _α (t″), R _β (t, t″); R _α (t ₁)], with ω[r ₁, r ₂; z] overlap volume of two spheres of radii r ₁ and r ₂ located at relative distance z.

For R _β (t, t″) > R _α (t ₁) + R _α (t″), the volume of the critical region is equal to ϖ(t″) = υ _α (t″) and its derivative, in Eq. 7, is nil. This condition implies

$$ (1 + \rho )t'' < \rho t - t_{1} . $$

(27)

For $\, \frac{{t_{1} }}{t} < \rho \,$ one defines the time $T_{0} = \frac{{\rho t - t_{1} }}{1 + \rho }$. Accordingly, for $\frac{\rho }{2 + \rho } < \frac{{t_{1} }}{t} < \rho,$ the inequalities 0 < T ₀ < t ₁ hold and the volume of the critical region becomes

b) $\frac{\rho }{2 + \rho } < \frac{{t_{1} }}{t} < \rho$

$$ \varpi = \upsilon_{\alpha } (t'') \leftrightarrow \, t'' < T_{0} < t_{1} $$

(28)

$$ \varpi = \omega (t,t_{1} ,t'') \, \leftrightarrow \, T_{0} < t'' < t_{1} . $$

(29)

On the other hand, when $\frac{{t_{1} }}{t}$ is lower than $\frac{\rho }{2 + \rho },$ the volume of the critical region is

c) $0 < \frac{{t_{1} }}{t} < \frac{\rho }{2 + \rho }$

$$ \varpi = \upsilon_{\alpha } \left( {t^{\prime\prime}} \right) \leftrightarrow t^{\prime\prime} < t_{ 1} $$

Case (ii) 0 < t ₁ < t″ < t < t _f

In this case, the volume of the critical region is always different from zero (Fig. 2).

For R _α (t″) > R _α (t ₁) + R _β (t, t″), the volume of the critical region is equal to ϖ(t, t″) = υ _β (t − t″). This inequality implies (1 + ρ)t″ > ρt + t ₁, namely $t'' > T_{2} = \frac{{t_{1} + \rho t}}{1 + \rho }$, where t ₁ < T ₂ < t. Furthermore, for R _β (t, t″) > R _α (t ₁) + R _α (t″), the volume of the critical region is equal to ϖ(t″) = υ _α (t″) and the derivative in Eq. 7 is nil. This condition leads to the inequality Eq. 27, (1 + ρ)t″ < ρt - t ₁ and, as above, for $\, \frac{{t_{1} }}{t} < \rho \,$ the time $T_{0} = \frac{{\rho t - t_{1} }}{1 + \rho }$ is defined. Consequently, for 0 < t″ < T ₀, the volume of the critical region is ϖ(t″) = υ _α (t″). Also, for $0 < \frac{{t_{1} }}{t} < \frac{\rho }{2 + \rho }$, t ₁ < T ₀ < T ₂ < t, while, for $\frac{\rho }{2 + \rho } < \frac{{t_{1} }}{t} < 1$, it follows that T ₀ < t ₁. All these conditions are summarized according to

a) $0 < \frac{{t_{1} }}{t} < \frac{\rho }{2 + \rho }$

$$ \varpi = \upsilon_{\alpha } (t'') \, \leftrightarrow \, t_{1} < t'' < T_{0} $$

$$ \varpi = \omega (t,t_{1} ,t'') \, \leftrightarrow \, T_{0} < \, t'' < T_{2} $$

$$ \varpi = \upsilon_{\beta } (t - t'') \, \leftrightarrow \, T_{2} < \, t'' < t $$

b) $\frac{\rho }{2 + \rho } < \frac{{t_{1} }}{t} < 1$

$$ \varpi = \omega (t,t_{1} ,t'') \, \leftrightarrow \, t_{1} < \, t'' < T_{2} $$

$$ \varpi = \upsilon_{\beta } (t - t^{\prime\prime} ) \leftrightarrow T_{ 2} < t^{\prime\prime} < t. $$

A graphical representation of the results here obtained for both cases (i) and (ii) is reported in Fig. 3.

Appendix 2

The overlap volume of two spheres of radii r ₁ and r ₂ whose centers are located at relative distance z is

$$ \omega (r_{1} ,r_{2} ;z) = \pi \left[ {\frac{2}{3}(r_{1}^{3} + r_{2}^{3} ) + \frac{1}{12}z^{3} - \frac{1}{2}z(r_{1}^{2} + r_{2}^{2} ) - \frac{1}{4z}[(r_{1}^{2} - r_{2}^{2} )^{2} ]} \right] , $$

(30)

which is further computed for r ₁ ≡ R _α (t″), r ₂ ≡ R _β (t, t″) and z ≡ R _α (t ₁). The derivative of Eq. 30 eventually gives

$$ \partial_{t} \omega (t,t_{1} ,t^{\prime\prime}) = \pi \left[ {2v_{\beta }^{3} (t - t^{\prime\prime})^{2} - v_{\beta }^{2} v_{\alpha } t_{1} (t - t^{\prime\prime}) + \frac{{v_{\beta }^{2} }}{{v_{\alpha } t_{1} }}(t - t^{\prime\prime})[v_{\alpha }^{2} t^{{\prime\prime}2} - v_{\beta }^{2} (t - t^{\prime\prime})^{2} ]} \right] $$

(31)

that is Eq. 11b.