Image Anal Stereol 2012;31:149-162 Original Research Paper ON MODELLING RECRYSTALLIZATION PROCESSES WITH RANDOM GROWTH VELOCITIES OF THE GRAINS IN MATERIALS SCIENCE Elena Villab>1 and Paulo R Rios2 !Dept. of Mathematics, University of Milan, Via Saldini 50, 20133, Milano, Italy; 2Universidade Federal Fluminense, Escola de Engenharia Industrial Metalurgica de Volta Redonda, Av. dos Trabalhadores 420, 27255-125, Volta Redonda, RJ, Brasil e-mail: elena.villa@unimi.it, prrios@id.uff.br (Received April 24, 2012; revised July 4, 2012; accepted July 16, 2012) ABSTRACT Heterogeneous transformations (or reactions) may be defined as those transformations in which there is a sharp moving boundary between the transformed and untransformed region. In materials science such transformations are normally called nucleation and growth transformations, whereas birth-and-growth processes is the preferred denomination in mathematics. Recently, the present authors in a series of papers have derived new analytical expressions for nucleation and growth transformations with the help of stochastic geometry methods. Those papers focused mainly on the role of nuclei location in space, described by point processes, on transformation kinetics. In this work we focus on the effect that a random velocity of the moving boundaries of the grains has in the overall kinetics. One example of a practical situation in which such a model may be useful is that of recrystallization. Juul Jensen and Godiksen reviewed recent 3D experimental results on recrystallization kinetics and concluded that there is compelling evidence that every grain has its own distinct growth rate. Motivated by this practical application we present here new general kinetics expressions for various situations of practical interest, in which a random distribution of growth velocities is assumed. In order to do this, we make use of tools from stochastic geometry and geometric measure theory. Previously known results follow here as particular cases. Although the motivation for this paper was recrystallization, the expressions derived here may be applied to nucleation and growth reactions in general. Keywords: birth-and-growth process, casual cone, formal kinetics, phase transformations, Poisson process, random set, recrysrallization. INTRODUCTION Heterogeneous transformations or reactions may be defined as those transformations in which there is a sharp moving boundary between the transformed and untransformed region. This definition aims at chemical reactions in general. Specifically, it is applied to nucleation and growth transformations in materials science. Notwithstanding its specific application to materials science, the geometrical idea pertaining to the definition of heterogeneous transformations is quite general. Indeed in mathematics this geometrical idea is called a birth and growth process. In treating heterogeneous transformations one normally envisages that these transformations may be decomposed in two stages. In the first stage, the nucleation or birth stage, new transformed regions originate at certain "nucleation sites". In the second stage, the growth stage, those nuclei, which are very small transformed regions, grow consuming the parent untransformed matrix. one might see the nucleation/birth as the stage in which the interfaces between the transformed and untransformed regions are "created". As expected, such a general geometrical idea finds a wide range of application in diverse fields of knowledge (Tomellini and Fanfoni, 2008), such as the phase separations in multicomponent alloys (Starink, 2004), the film growth on solid substrates (Fanfoni and Tomellini, 2005), the kinetics of Ising lattice-gas model (Ramos etal., 1999), and the DNA replication (Jun and Bechhoefer, 2005). To these we may add a recent extensive work by Aquilano et al. (2009) on crystallization processes. In the present paper, the practical application that we often bear in mind is the modeling of phase transformations in general and recrystallization in particular. For the development of mathematical models, nucleation and growth or birth and growth may be considered to be operational concepts. In other words, no specific physical mechanism is associated with nucleation and growth. They are wholly defined by their stochastic geometric properties. These ideas constitute the basis of a methodology to analyse transformation kinetics, which is often called "formal kinetics". Formal kinetics had its inception in the early work by Kolmogorov (1937), Johnson and Mehl (1939) and Avrami (1939; 1940; 1941), henceforward referred to as "JMAK". It is worthy of note that JMAK papers were originally motivated by phase transformations. Indeed, examples of formal kinetics modeling can be found associated with a variety of transformations, such as, austenite to perlite transformation (Johnson and Mehl, 1939), recrystallization (Vandermeer and Juul Jensen, 2001), abnormal grain growth in BaTi03 (Kondo etal., 1998), martensite "spread "(Rios and Guimaraes, 2007; 2008) and polymer crystallization (Capasso etal., 2003; Burger etal., 2002). The early work quoted above considered that nucleation sites were uniform randomly located in space, or more precisely, that nuclei were located in space according to a homogeneous Poisson point process. They also considered that the velocity of the moving boundaries was constant and was the same at every point of the moving boundaries. Based on these assumptions, two exact analytical expressions could be obtained. In both cases, one assumed that the new phase grows with a spherical shape and constant velocity, G, but with different nucleation conditions. In the first of these, the nucleation is supposed to be site- saturated, which means that all possible nucleation sites are exhausted at the very beginning of the reaction. One often describes this by saying that all nuclei are already present at t 0. If the number of nuclei per unit of volume is Ny, then the volume fraction transformed, Vy, is Vv(t) = l-zxp(-^NvG3t3^ whereas for a constant nucleation rate per unit of volume, Iy, FF(i) = l-exp(-|/FG3i4). Subsequent work generalized both the distribution of the nuclei in space (Rios and Villa, 2009; Villa and Rios, 2009; 2010; 2011), and the time-dependence of the growth velocity (Vandermeer etal., 1991). Still, the key assumption about the velocity made in the work of JMAK remained the same; namely, that the velocity of the moving boundary at a particular point in time was the same at all points of the moving boundaries. This condition is slightly relaxed when the growth model is assumed to be shape-preserving, for instance the growing regions are ellipsoids of invariant shape (Vandermeer etal., 1991; Rios and Villa, 2009). More general growth models admitting different velocities for different boundary points can be obtained by assuming space-and-time dependent velocity G = G(t,x) (e.g., see Capasso etal., 2003; Burger etal., 2002; Capasso and Villa, 2007b; Villa, 2008 and references therein). In all the mentioned models the growth velocity field is assumed to be deterministic; such an assumption is possibly a good approximation for certain practical cases, whereas for others the boundary velocity may not reasonably be thought to be neither deterministic nor to be the same for each grain. As an example, suppose that spherical regions nucleate at uniform randomly located nuclei. Suppose further that their growth is controlled by the diffusion of an alloying element in the untransformed region. One concrete case would be the nucleation and growth of ferrite from austenite in an iron-carbon alloy. Ferrite has virtually no carbon and, as it grows, carbon must diffuse away from the moving ferrite -austenite interface. Eventually, carbon diffusion fields in the austenite around adjacent a growing ferrite region must overlap. This overlapping will obviously change the velocity of the moving boundaries. Because interparticle distance is random it is reasonable to infer that when overlapping starts to take place each growing ferrite region should have it is own distinct velocity. In the situation of growing ferrite just described, it is qualitatively reasonable and intuitive to expect a deviation from JMAK's velocity assumption. Unfortunately, it is not so easy to assess whether or not such an effect might be quantitatively important. One of the strong reasons for this is that measuring the boundary velocity of individually growing grains presents a significant challenge. Juul Jensen and her group at RIS0 laboratories, e.g., Lauridsen et al. (2003), developed a methodology, which can non-destructively measure the growth kinetics of individual bulk grains in situ by 3-dimensional synchrotron X-ray diffraction (3DXRD) microscopy. Juul Jensen and coworkers applied those techniques mainly to determine the growth velocity of individual grains during the recrystallization of pure aluminum. In a recent paper, Juul Jensen and Godiksen (2008) reviewed their experimental measures of growth velocities of individual grains obtained by neutron and 3-dimensional synchrotron X-ray methods. They concluded that there is compelling evidence to support that "every single grain has its own kinetics different from the other grains" (Juul Jensen and Godiksen, 2008). In order to model such a situation, Juul Jensen and Godiksen emphasize that it is necessary to assume that there is a continuous distribution of growth velocities (Juul Jensen et al., 2002; Godiksen etal., 2007). Nonetheless, in spite of this experimental evidence very few papers deal with this problem theoretically. Godiksen et al. (2007) approached this problem using computer simulation. Juul Jensen et al. (2002) derived an analytical expression for the volume fraction transformed for the case in which nucleation was site-saturated. In this paper, we obtain general analytical solutions to the nucleation and growth model considering that there is a probability distribution of growth velocities of the grains. We published a preliminary version of this work (Rios and Villa, 2011), in which we focused on particular cases of practical interest. In this paper we present a more detailed theoretical mathematical treatment covering several and more general cases. Namely, we consider the cases in which the velocity of each grain is - random but constant in time - random and time-dependent - random dependent on the spatial location of the corresponding nucleus, both in the case of site-saturation and in the case of time dependent nucleation. It is worthy of note that even though we have been motivated by the observations of recrystallization behavior (Juul Jensen and Godiksen, 2008) the present results may be applied quite generally to birth-and-growth processes. Also for this reason, throughout the paper we provide explicit expressions for arbitrarily integer dimension d, so that the 2D and 3D case usually considered in materials science applications follow as particular cases. MATHEMATICAL BACKGROUND AND BASIC NOTATION Detailed mathematical background may be found in previous work by the authors Rios and Villa (2009) and Capasso and Villa (2007a;b); in particular we refer to our recent paper (Villa and Rios, 2011) for basic definitions. Here only some essential definitions and some useful relationships will be presented to make this paper more self contained and easier to read. For a brief description of homogeneous and inhomogeneous Poisson point process, the reader is referred to Rios and Villa (2009) or for a more detailed presentation to specific texts on stochastic geometry (Stoyan etal., 1995). MODELLING BIRTH-AND-GROWTH PROCESSES A birth-and-growth (stochastic) process is a dynamic germ-grain model (Stoyan etaJ., 1995), used to model situations in which nuclei (germs) are born in time and are located in space randomly, and each nucleus generates a grain evolving in time according with a given growth law. Since, in general, nucleation and growth are random in time and space, then the transformed region 0' at any time t > 0 is a random set (Stoyan etal., 1995) in Rrf, that is a measurable map from a probability space (Q,#,P) to the space of closed subsets in Rrf; namely where F denotes the class of the closed subsets in Rrf, and Op is the cr-algebra generated by the so called Fell topology, or hit-or-miss topology, that is the topology generated by the set system where CS and Z are the system of the open and compact subsets of Rrf, respectively (e.g., see Matheron, 1975; Schneider and Weil, 2008). The family {©'}, is called birth-and-growth process. Birth-and-growth and nucleation and growth will be used as synonyms in this paper. Of course, different kinds of nucleation and growth models gives rise to different kinds of processes {©'}/. As mentioned in the Introduction, we shall consider two kinds of nucleation of interest in recrystallization; the first one is that of sitesaturation, in which all nucleation sites are exhausted at the beginning of the reaction, the second one is that of time-dependent nucleation, in which nuclei can be born randomly both in space and time. Site-saturated nucleation processes and space-time dependent nucleation processes can be modeled by point processes and marked point processes, respectively. We give here some basic concepts and definitions useful for the sequel. (See also, for instance, Rios and Villa, 2009, Sec. 2.3.) Throughout the paper ¿^ r denotes the Borel cr-algebra of any space 3t . We remind that a point process in Rrf is an almost surely locally finite sequence of points, say N = {Xi}i, randomly located in Rrf, according with a given probability law. Equivalently, it can be seen also as a counting measure as well, so defined N(A) := number of the X/s, which belong to A. for any A 0 of the nucleus Xj, the transformed region 0' at time t > 0 is e' = U ©'(*/)• j Given a complete separable metric space K, a marked point process in R+ x K is a sequence TV := {(TJ, K,) },• of points in R+ x K such that the sequence { T,-},- is a point process in R+, while each K, e K is said to be the mark associated to the point TJ. K is called mark space. Then, a space-time nucleation process can be modeled by a marked point process, identifying 7} as the time of birth of the /-th nucleus, and X, as its spatial location in Rrf. Analogously to the site-saturation case, a counting process TVon R+ x Rrf can be defined as N([s, t\ x A!) := number of nuclei, which are born in A! during the time interval [5, . By denoting ®'T.(Xj) the grain obtained as the evolution up to time t> 7) of the nucleus born at time Tj in Xj, then the transformed region 0' at time t > 0 is given by = (J ®7>(*/) * Of course a site saturated process may be seen as a particular case of the time-dependent one by assuming Tj = 0 for any j. Given a point process TV on Rrf, the measure ma) := E|TVM)| for all a e -/>.,1 is called intensity measure of TV; whereas, if TVis a marked point process process in R+ x K, then its intensity measure A is the measure on R+ x K defined as MA) := E|TVM)| for all a 0 , having denoted by vd the ¿/-dimensional Lebesgue measure in Rrf. In particular, it is of interest to consider the expected volume measure E[vrf(0'n •)] and its density (i.e., its Radon-Nikodym derivative), called mean volume density of®' and denoted by Vy, provided it exists: E[vd(®'nA)}= [ Vv(t,x)dx, \/Aes§Wd. (1) J a It is well known and easy to prove that Vv(t,x) =F(xe®') , forvrf-a.e. xG (2) Whenever A is the region of the physical sample under observation, the ratio VvM) := fAVy(t,x) dx vd(A) is also called volume fraction. Let us notice that whenever Vy is independent of x (e.g., under assumptions of homogeneous nucleation and growth), then Vy is independent of A and Vv(t) = Vy(t) (Rios and Villa, 2009; Stoyan etal., 1995, p. 342). We also mention that other quantities of interest in real applications are the so-called mean extended volume density al time denoted by V/f t.-), defined as the density of the mean extended volume measure at time t,E[^](•) :=-E\Lf.TJ) € Rrf x R+, such that a grain born at point y that grows with velocity ¿j covers the point x at time t; it is worthy of note that in contrast with the causal cone shown in Fig. 1, the visualization of such a causal cone is difficult. The results stated in the next theorem can be easily derived by the well known theory for inhomogeneous Boolean models with spherical grains (Fallert, 1996; Schneider and Weil, 2008); we give here a sketch of the proof for a non-expert reader's convenience. Theorem 2 Under the above assumptions, Eqs. 4 and 6 hold with ^(t,x) = {(y^)GRdxR+:yGB,t(x)}. (9) Sketch of the proof. By definition K(t,x) := {(.y, & € Rrf x R+ : x€ b.t(y)} = Eq. 9, and so, being Na Poisson process, it follows that Vv{t,x) = l-zxv{-A{tf{t,x))} = 1 - exp { - J^ J^ ^ X (j) dyQ(dt;)} . (10) By the definition (Eq. 3) of Ve, and by the application of the Campbell's formula (Baddeley etal, 2007), / vE(t,x)dx= f f vd(b,t(y)na))X(y)dyQ(d^) J A J r+ md for any a ■> :,/. By applying Fubini's theorem and a suitable change of variable, we get f VE(t,x)dx J A Eq=10 / A(r€(t,x))dx J A for any a 0 in the particular case in which Gis deterministic, according with Eq. 5. We recall that a function /: Rrf —> R is said to be harmonic if it is twice continuously differentiable and it satisfies the Laplace's equation X^i d1fjd£i= o. Note that, in particular, if the nucleation process is homogeneous in space, then A (x) = A > 0 is harmonic. Corollary 4 Under the above assumptions, if X is a non-negative harmonic function in the spatial region where the nucleation takes place, then the Eqs. 4 and 11 simplify as follows: VE{t,x)=X{x)bdtdE[Gd], (12) SE(t,x) =X{x)dbdtd-lnGd'1} , (13) where b,i is the volume of the unit ball in Rrf. Moreover, = ■ (14) Proof In order to obtain Eqs. 12 and 13, it is sufficient to remind that for any harmonic function / in Rrf it holds )Br(x) f(y)dy= f(x)vd(Bß(0)) = fix)/^1. Vß> 0, and [ f(y)J?d-\dy) = fix)dbdR> \ \/R> 0, JdBR(x) and then to apply this to the expressions for VE and Sy given in the theorems above. Eq. 14 directly follows by Eqs. 12 and 13. □ For sake of simplicity, we assume in the next theorem that the intensity A of the point process driving the spatial location of the nuclei is bounded and continuous; such an assumption is usually satisfied in real applications. We point out that the validity of Eq. 7 stated in the theorem below, may be derived by the result proved in Schneider and Weil (2008, Theorem 11.1.3); nevertheless, the proof provided there requires notions {e.g., specific intrinsic volumes and mixed measures) which might be difficult to understand for a non-expert reader. So we give here a sketch of an alternative proof, which might be appear a bit more intuitive. Theorem 5 Under the above assumptions, if the intensity X is bounded and continuous, Eq. 7 still holds; moreover ifX is harmonic in the spatial region where the nucleation takes place, then it holds j-Vv(t,x) E^] E [Gd-1 Sv(t,x). (15) Sketch of the proof By the general results on the mean density of inhomogeneous Boolean models proved in Villa (2010a) (in particular by applying here Theorem 3.9, observing that the grains of the transformed region 0' are balls for any / > 0, and so the ,jf'd~l-measure of the topological boundary of 0' equals the ,jf'd~l-measure of its the essential boundary), we can claim that Sv{t,x) lim 40 (16) where :={a-g set of 0' at distance r. Let us notice that : dist(x, 0') < /•} is the parallel 0' U BgMZ) (X¡,G¡)eN and denote by '/r the subset of Rrf x R+ so defined: 5V := Rrf x R+ : B,t+r{y)} . We may observe that = ¥{{N{S>r) > 0} n {N{tf{t,x)) = 0}) = F(N({(y^):yGB,t+r(x)\B,t(x)})>0) ■F(N(tf(t,x)) =0) 1-exp \-f f X(y)dyQ(d^)V JR+ J B.t+r(x)\B.,(x) > > ■(1 -Vv(t,x)), (17) where the last two equations follow by the assumption the N is a Poisson point process. By applying Villa (2010a, Theorem 3.5b) to the measure p{dx) := X (x) dx, we get that lim- / X (y) dy 40 rJB,t+r(x)\B,t(x) f X(yWd-\dy), (18) JdB^ix) X{y)dy 0 (i.e., Q{dt,) = -ct^lcdE,), then Vv(t, x) = 1 -exp{-A(x) b^^dl} - if G is a discrete random variable with P( G = Gi) = Pi, Pi € [0,1], Y.iPi = 1, then Vv(t.x) = 1 - exp { -AUjV'T G!p> Explicit expressions for the other mean densities Sy, VE and SE can be obtained similarly. RANDOM TIME-DEPENDENT VELOCITY Assumptions, each grain develops with random time-dependent velocity during the reaction, of the type: G(t) = Gog(t,a), (21) where Gq is a non-negative random variable and g is a non-negative function depending on time and on a random vector parameter a in R". We assume that Gq and a are independent on the spatial location of the nucleus of the associated grain, with joint probability distribution Q(d(£, a)) on R+ x R". Even in this case, different grains may have different velocity, and we may model such a birth-and-growth process by a marked Poisson point process N {Xj. (Gj, (Xi)} i in Rrf with independent marking in R+ x R", with mark distribution Q. Then, the intensity measure A of the nucleation process TVis of the type A(d(y^a))=X(y)dyQ(d(^a)) (22) while the transformed region 0' at time / is given by ©'= U (Xj,(Gj,case: VE{t,x) = A-TzX{x)nGlW[l-a)\, so that we get Vv(t,x) = l-txp{-X(x)^KE[Gl] jy^dzj , (24) when a ~ U[0,1], and Vv(t,x) = 1 -exp{ -X(x)^kE[GI\ r' \ d-h g{ T,a)dr /0 H^'Bm^1-^}' (25) when a ~ Beta(a, b). (See also Rios and Villa, 2011 for a more exhaustive discussion about this.) RANDOM CONSTANT POSITION-DEPENDENT VELOCITY Assumptions, each grain develops with random velocity constant in time during the reaction, whose probability distribution depends on the spatial position of the corresponding nucleus. We may model the nucleation process by a position dependent marked Poisson point process with intensity A(d(x^)) = X(x)dxQ(x,d^). Fig. 2. The straight line without any symbols on it corresponds to the KJMA kinetics for a constant, non-random velocity The symbols correspond to Godiksen et al. simulated data and the solid lines correspond to the theoretical result and were calculated using a uniform distribution, Eq. 24: (•) and using a Beta distribution, Eq. 25: (□), for different values of a and b. There is, as expected, excellent agreement between simulation and analytical solution. Note that now Q{y dg) represents the probability distribution of G given that the associated nucleus is born in j. Of course, the case discussed in the previous section "Random constant velocity" can be seen now as a particular case. By proceeding along the same lines as before, Eqs. 6, 4 and 7 still hold now with the causal cone given by ^(t,x):{(y^)eRdxR+:xeB,t(y)} = {(y^)e Rrf X R+ : ^ > dist(y,x)/t} , and so VE(t,x)=A(V(t,x)) = f (f Q{yd^)X(y)dy. (26) md ^ JAisi(y,x)/1 ' As we can see, such a case could be more difficult to handle because the evaluation of the integral above might be quite hard. Example Let us consider the case Qiy. dc,) := ¿l[0,7l](^, i.e., G(y) ~ £/[0^]; then 1 - exp \ - ( d.,, n-^x),tXiyW). Particular case: position-dependent deterministic velocity Let us assume that each grain grows with constant velocity, depending on the spatial location of the corresponding nucleus, i.e., the velocity Gj associated to a grain with nucleus in X, dist(j;x)} , Eq. 26 simplifies as follows: VE{t,x)=[ X{y)dy. JA{x,t) Since the nucleation process is Poissonian we still have Vv(t,x) = l-exp{-VE(t,x)}. Note that, under homogeneous nucleation with X(x) = X, we have Vv(t,x) = 1 - exp{-Avrf(^)}, so that only the volume of the set A(x. t) has to be evaluated; explicitly, for practical purposes in the 3D case A = {(ji,j2,J3) € R3 : f(j)t > Vijl - XI )2 + (j2 - X2)2 + (J3 - )2 } • RANDOM VELOCITY IN THE TIME-DEPENDENT CASE In this section we consider the case in which the nucleation takes place randomly both in time and space, and the growth velocity of the grains is random. In the site-saturated case we modelled the random velocity of a certain grain as a mark of the corresponding nucleus, and so by modelling the nucleation process as a marked Poisson point process. Analogously, we shall add a further mark to the nucleation process {T^Xj}, by marking the point process {T], in R+ of the birth-times of the nuclei with marks representing the spatial location of the nucleus and the velocity of the corresponding grains. It is intuitive that arguments and results are similar to those of the previous section; difficulties in the time-dependent case might be arise in evaluation of the involved integrals. We shall discuss the three cases of interest studied in the site-saturated case. We assume that the point process { T,-},- describing the birth time of the nuclei is an inhomogeneous Poisson point process in R+ with intensity A (5). We also assume independent marking of { 7}}j, that is the intensity measure of the marked point process is of the type A(d(s,k)) = X(s)dsQ(dk), where Q is the mark distribution, i.e., a probability measure on the mark space. RANDOM CONSTANT POSITION-DEPENDENT VELOCITY Assumptions: each grain develops with random velocity, constant in time during the reaction. The velocity Gj associated to the grain born at time 7} at point Xi is in general dependent on the spatial location X,. We denote by Q(d(y. c,)) the joint probability law oi(Xj,Gj). Therefore we may consider the spatial location Xi and the velocity G, associated to the birth-time 7} as marks of 7J, and model such a birth-and-growth process by a Poisson point process N= {(T,. (Xi, Gj))} in R+ with marks in Rrf x R+, having intensity measure A(ds,d(y^)=X(s)dsQ(d(y^)). Note that Q(d(y. q)) represents the probability that any nucleus is born in the infinitesimal region dy and the growth velocity of the associated grain is in the infinitesimal interval dg. So, by giving a suitable joint probability Q we can easily handle the case of position-dependent velocity. It follows that the transformed region 0' at time / is given by ®'= U BGi(t_Ti](Xi) , (UXifi^eN so that the causal cone of any point x € Rrf at time t e R+ is now the subset of R+ x Rrf x R+ so defined Again we have that Vv(t,x) = l-cxp{-VE(t,x))}, with VE(t,x))=A(^(t,x)) X(s) M <2(d(j^)) )ds. Remark 8 (Position-independent velocity) In case Gi and Xi are independent, then the mark distribution factorizes as Q(d(y. c,)) = Qx(dx)Qc(dt,), having denoted by Qx and Qc the probability distribution of the random location X of the nuclei and of the velocity G, respectively, and so the intensity measure A of the process is of the type A(ds,d(j^)) = X(s)dsQx(dy)Qc(d^). In some applications it is of interest to evaluate the mean volume density Vy in the centre of the specimen; note that in the particular case in which the nucleation is homogenous in time (i.e., X(s) = X> 0), the nuclei are uniformly located in a compact window \—M. M\d and G is bounded, say G< K € R+, then by Eq. 27 we get: vem Xbdtd+l 2dMd(d+l) , ViG [0,Af/Ä] . (28) Example Under the above assumptions, we get the following explicit expressions for the mean volume density V\/(t,x) in some cases of interests for applications: - if the velocity G and the spatial location X of the nuclei are independent, with U(0,K) and X ~ Uil-M.M]''), then and so 1 2 dMdK [0,Ä] Vv(t,x) = 1 - exp \ - A(s) j\d(B,(t_s](x) n \-M,M\d)dt;)ds}. If in particular X(x) = X > 0, then the volume density at the origin is given by 1,, ™ 1 f XbdKdtd+l 1 F^0) = l-exp{-2(/MV+i)2), Vt£ [0 ,M/K\ . - if the velocity G and the spatial location X of the nuclei are independent, with X ~ U([—M,M\d) and G discrete random variable with P( G = Gj) = Pi, Pi € [0,1], 1,/J/ = 1, then 1 (27) <2(d(j^)) = and so Vv(t,x) = l-cW{-^f\(s) Y,vd{BGi(t_s){x)C\[-M,M\d)p)ds VE{t,x) = [ A(s)([[ Q(d(y^a)))ds. J0 K-JSR{s[)(x)x R+xR" Sometimes it is needed to model nucleations such that the birth-time and the spatial location of the nuclei are dependent; a particular case in which they are independent of the growth velocity of the grains can be model by choosing an intensity measure of the type A(d(s,y^a))=X(s,y)dsdyQG,a(d(^,a) , (29) where Qc.u is the joint distribution of Go and a. By proceeding along the same lines of Theorem 3 and of Theorem 5, it is not difficult to prove that, if A is of the type 29 with A bounded and continuous, then Eq. 7 holds and SeM If in particular A (5) = A >0, and if sup,- G, = K, then the volume density at the origin is given by Vy(t,0) = 1 -exp{ -A^ f^GHt-s/pAs} W € [0, M/K\, in accordance with Eq. 28. RANDOM TIME-DEPENDENT VELOCITY Assumptions', each grain develops with random time-dependent velocity during the reaction of the type Eq. 21. Thus we can model the nucleation process by a marked Poisson point process N {(T,-. (X,-, G,. ay))} in R+ with marks in Rrf x R+ x R", with intensity measure A(d(s,y^a))=X(s)dsQ(d(y^a)), where (?is the joint probability law of (X, Go, a), and, in general, with G dependent on the spatial location X of the corresponding nucleus. The growing region 0' at time t is given by ©'= U B^X,), (Ti,(Xi, Gj, ai))eN-.T,:Tir := %, a) € [0, t] x Rrf x R+ x R" : je BR^+r(x)}. If moreover A(5,-) is harmonic for any s \j){l-*){d-j) + la i - It is clear that some cases of interest in applications (e.g., Gb or a deterministic), follow now as particular cases. Let us consider the very simple case in which nucleation takes place homogeneously both in time and space, and G(t) = Gb(l — a)ra with Gb and a independent each other. Denoted by A the mean number of nuclei per unit of time and unit of volume, it follows that Vv{t,x) = 1 -exp{-XbdE[G$]j\tl-a-i-a)dds} , if Gb is random and a is deterministic, whereas Vv{t,x) = 1 -exp{-kbdGf{£E[(ti-a-J-a)d]ds} , if Gb is deterministic and a is random. SUMMARY AND CONCLUSIONS - General expressions were derived for the mean volume density and the mean surface density of birth-and-growth processes with Poisson nucleation process and spherical grains with distinct random velocities, modelling recrystallization processes in materials science, both in case of site-saturation and in case of time-dependent nucleation. To this aim, tools from Stochastic Geometry and recent results from Geometric Measure Theory (namely in the proof of Theorem 5), together with the notion of causal cone have been used. The general case of random growth velocity of the grains with distribution dependent on the spatial location of the corresponding nucleus has been also considered. Some particular cases of practical relevance in applications have been discussed as examples throughout the paper, also in connection with previous literature. - Specifically with regard to materials science, these results may be used to model and/or assess the importance of a random velocity distribution in transformations kinetics, as opposed to assume a non-random velocity. It is particularly useful in the cases where the velocity is known to be different for each grain such as in recrystallization as experimentally demonstrated by luul lensen and coworkers. - This paper provides the mathematical basis for the development of more general expressions to be used in practical applications. Although the motivation was recrystallization the expressions derived here may be applied and extended to nucleation and growth reactions in general. 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