Phase transformation sequence of Ti-6Al-4V as a function of the cooling rate

. The growth kinetics of allotriomorphic α along the prior β grain boundaries and of globular primary α in Ti-6Al-4V during continuous cooling is described. A physical model is developed based on classical nucleation and growth of platelets for the allotriomorphic α. The growth of the primary α is modelled based on the growth of spherical particle immerged on a supersaturated β-matrix. Continuous cooling tests at two different holding temperatures in the α+β field, 930°C and 960°C, and five different cooling rates, 10, 30, 40, 100 and 300°C/min, are conducted to validate the proposed models and elucidate the growth sequence of those α morphologies. Additionally, interrupted tests at different temperatures are conducted to determine the progress of growth of primary α and formation allotriomorphic α during cooling. The size of primary α increases while its size distribution broadens with a decrease in cooling rate. Area fractions of primary α decrease with increasing cooling rate and increasing holding temperature. Moreover, the lower the cooling rate, the thicker the plates of allotriomorphic α. At the beginning of the cooling, growth of primary α, as well as formation of allotriomorphic α plates is observed. The experimental and modelled results show good agreement.


Introduction
The mechanical proper es of Ti-6Al-4V, especially fa gue resistance, toughness and duc lity, are strictly correlated with its microstructure formed during the industrial thermo-mechanical treatments [1]. Different microstructures can be achieved, i.e. martensi c, lamellar, equiaxed or bimodal, and they are mainly related to the cooling rate during thermo-mechanical treatment. A bimodal microstructure consis ng of lamellar α and globular primary α (α p ) is normally desired owing to the combina on of high duc lity and high toughness [1]. The globular α p phase grows during cooling, and the allotriomorphic α phase is formed along the grain boundaries.
The growth of α p phase is a diffusion-controlled process. For very slow cooling rates, the β phase transformes mainly into α p . Thus, the resul ng microstructure consists almost en rely of large equiaxed αp with small amounts of retained β phase [2]. The amount of α p decreases for increasing cooling rates and other morphologies of α phase are formed [3]. Semia n et al. [4] observed that α p exhibits epitaxial growth for Ti-6Al-4V, and that it is controlled by the diffusion of vanadium. The presence of a rim-α phase in a near α Ti60 [3] surrounding the α p during cooling with the same crystallographic orienta on, evidences the epitaxial growth of α p phase. The nuclea on and growth of allotriomorphic α (α GB ) is dependent on the cooling rate and follows a platelet-like forma on [2], [4]. Extensions from α p with same crystallographic orienta on were observed preferen ally along the β/β boundary sugges ng symbio c growth from the α p phase [3].
Semia n et al. [4] proposed a model for the growth of αp that uses exact solu ons of the diffusion equa on and takes into considera on: a) the diffusion coefficients with a thermodynamic correc on for the specific composi on of the material, and b) the large super-satura on formed during cooling [4]. Meng et al. [5] complemented the model proposed in [4] by considering the effect of the thermal history on the diffusion field of growing par cles, and the overlap of these diffusion fields [6].
The development of robust, accurate but computa onal simple models to simulate the β→α transforma on are of great importance to predict and control the evolu on of the microstructure in complex shape and variable cross sec ons components during thermomechanical processing. A coupled model for the growth of primary α and forma on of allotriomorphic α phases is here proposed. The results are compared with the measured data acquired for different con nuous cooling experiments.

Material
A cogged Ti-6Al-4V in the β and α+β fields with further annealing at 730°C for 1 hour followed by air cooling was used for this inves ga on. The es mated β transus is ~1020°C [7]. Cylindrical samples with a diameter of 5.5 mm and a length 10 mm were u lized for the heat treatments in a dilatometer.

Heat treatments
A dilatometer DIL 805A/D (TA Instruments, Hüllhorst, Germany) was used to perform con nuous cooling heat treatments. The tests were carried out in a protec ve atmosphere of argon. A�er hea ng with a rate of 30°C/min, the samples were held for 1 h at two different holding temperatures in the α+β field, 930 and 960°C, followed by con nuous cooling un l room temperature. The cooling was conducted using five different cooling rates: 10, 30, 40, 100 and 300°C/min. Interrupted con nuous cooling heat treatments were carried out in order to elucidate the mechanism/s governing the transforma on during cooling. A�er hea ng with 30°C/min, the samples were held for 60 min at a constant temperature of 930°C or 960°C. The subsequent cooling was performed with 10 and 100°C/min. The samples were quenched using Argon at four different temperatures, 900, 875, 850 and 800°C.

Metallography and microstructure investigation
The samples were polished using OP-S (oxide polishing suspension) a�er a conven onal grinding procedure from grit 500 un l 2000. The samples were etched with Kroll's reagent, 91 ml water, 6 ml HNO 3 and 3 ml HF. The metallographically prepared and etched samples were inves gated using light op cal (LOM) and scanning electron (SEM) microscopy. The SEM analysis were conducted using a Tescan Mira3 microscope using an accelera on voltage of 10 kV and working distance of 12 mm. A minimum of five representa ve micrographs were analysed for each cooling rate and holding temperature for the quan fica on of globular α p . The globular phases were marked using the so�ware GIMP (GNU Image Manipula on Program) and analysed with ImageJ so�ware.

Modelling strategy
The growth of α p during cooling in heat treatments conducted below the β-transus temperature was modelled based on [4]. The growth of allotriomorphic α was modelled based on a classical model of nuclea on and diffusion equa on for growth of platelet. The microstructure is modelled as consis ng of three major components: α p , α SEC and α GB , as schema cally shown in Figure 1. The nuclea on and growth of α SEC is not considered in the present model.

Figure 1: a) Schema c representa on of the different morphologies of the alpha phase: primary (αp), secondary (α SEC ) and allotriomorphic (α GB ), formed during cooling of a typical Ti-6Al-4V alloy; b) growth of a spherical par cle of radius R and composi on C P immersed in a matrix of composi on C M , in which C I is the chemical composi on at the interface.
For Ti-6Al-4V Semia n et al. [4] showed that the growth rate of globular αp phase is limited only by V diffusion. The intrinsic diffusion coefficient of V in beta tanium (D) is considered as obtained from Zwicker [8] and also adopted in the model and simula ons of Semia n et al. [4].
The growth of a spherical par cle immersed in a matrix of composi on C M can be given according to Equa on 1.

Equa on 1
Where R is the radius of the par cle, D is the diffusion coefficient and λ is a parameter that can be calculated according to Equa on 2.
Equa on 2 The parameter Ω denotes supersatura on, and can be calculated according to Equa on 3.

Equa on 3
Where C I is the composi on of the matrix-par cle interface, C P is the composi on of the α p phase. C M is considered as the equilibrium phase composi on of the alloy at the actual temperature, and C P is considered as the equilibrium phase composi on of the α phase. The equilibrium phase frac on as well as the chemical composi on were calculated using the so�ware JMatPro ® v. 10. In order to account for the so� impingement on the "far-field" matrix composi on, C M is calculated using a usual mass balance between the frac on of α (f α ) and β phases, as given by Equa on 4.

Equa on 4
Where C 0 is the nominal concentra on of V in the material and C α is the concentra on of V in the α phase. Similar to the growth of α p , the par al enrichment of V along the formed α/β phase boundary is observed. However, this phenomenon stabilizes locally the β phase. Thus, a plate-like morphology of α phase is obtained along the β/β grain boundaries. When the nuclea on me is neglected, the rate of nuclea on of precipitates can be given according to Equa on 5.

Equa on 5
Where N 0 is a pre-exponent term and considered as 4.5x10 4 for a prior β grain size of 200 µm, Q is the ac va on energy for diffusion, is the energy barrier for heterogeneous nuclea on, R the molar gas constant and T the temperature. f αp and f αGB are the volume frac on of α p and allotriomorphic α formed along the prior beta grain boundaries, respec vely. The energy barrier for heterogeneous nuclea on can be calculated according to Equa on 6.

Equa on 6
Where A* is a parameter obtained according to the nuclea on at grain boundary, or sympathe c nuclea on [9], or a sum of contribu ons of both (adopted in the current model).
The cri cal thickness (B crit ) for a disk-like α GB to nucleate can be calculated according to Equa on 7.

Equa on 7
Where γ αβ is the interface energy between α and β phases, ∆G V is the chemical free energy of phase transforma on obtained using Equa on 8 [9].

Equa on 8
Where is the concentra on of V in the α GB , considered to be equal to the V concentra on in the α p ( = C P ).
The nucleus of α GB will grow by diffusion process, and it is considered to be similar to the growth of the platelets, i.e. thickening of a planar disordered boundary via ledge growth mechanism. In this case, the lengthening of a platelet is significantly faster than the thickening due to high anisotropy in interfacial energy. Therefore, the volume frac on of α GB is only dependent on the number density and its thickness. The thickening of the α GB can be then modelled according to Equa on 9.
Equa on 9 Where B is the thickness of the α GB , m is a ledge coefficient to account for the planar disordered growth, D is the diffusivity of V in the β matrix and λ GB is parameter that can be calculated according to Equa on 10.

Equa on 10
Where is a dimensionless supersatura on parameter, and considered equal to Ω because . Similar to the growth of precipitates, the mean thickness of the platelets ( ) is calculated according to Equa on 11 [10].

Equa on 11
The first term corresponds to the growth of the exis ng platelets of allotriomorphic α, while the second represents the contribu on of new nuclei of cri cal size calculated according to Equa on 7.
A Matlab ® rou ne was developed to implement the model for the growth of the αp previously described. Figure 2 shows the microstructure of the Ti-6Al-4V argon quenched a�er 1 h holding at 960°C . Sparsely and nearly separated αp par cles are found in a matrix of martensite (α'), originally a matrix of β phase.

Figure 2: Representa ve op cal micrograph of the inves gated Ti-6Al-4V a�er 1 h at 960°C followed by argon quenching.
From the microstructure shown in Figure 2, different frac ons and sizes of α phase are formed during cooling, as shown in Figure 3. The cooling rate of 10°C/min (Figure 4a) lead to a nearly fully equiaxed microstructure, thus mostly growth of α p is exhibited. It is difficult to dis nguish the lamellas of α SEC from the globular α p for the cooling rate of 10°C/min. The increase in cooling rate leads to forma on of α SEC , as well as α GB phase. For 30°C/min and 40°C/min (Figure 3b and Figure 3c, respec vely), α GB exhibits irregulari es in the interface surface, as well as the α p phase. Higher cooling rates lead to less pronounced growth of α p and α GB (Figure 3d and Figure 3e). Moreover, regular plate-like morphology of α GB is observed for 100 and 300°C/min. Figure 4 exhibits the typical micrographs a�er interrupted con nuous cooling heat treatments. The α GB is highlighted in red do�ed line, and the α SEC in do�ed green circles. The forma on of α GB ini ates from the globular α p , i.e. symbio c growth. The 2D interconnec vity of the plates of α GB along the grain boundaries is very low for 900°C (Figure 4(a,b)). At 850°C the grain boundaries are nearly decorated with α GB . Irregular growth of α GB seems to occur at ~800°C, and is more pronounced for 10°C/min cooling rate (Figure 4g). The nuclea on and growth of α GB do not seem to be significant at temperatures higher than 900°C for the two inves gated cooling rates, leading to comparable evolu on behaviour of the α GB a�er holding at temperatures of 930°C and 960°C (Figure 4a). Forma on of α SEC is not pronounced un l ~850°C. Although Figure 4e and Figure 4g show the presence of α SEC for 10°C/min cooling rate, its area frac on is notable small. Similar conclusions can be obtained comparing Figure 3a with Figure 3d.

Figure 4: Representa ve SE-SEM micrographs of the interrupted heat treatments a�er holding at 960°C for 1 h followed by con nuous cooling at 10°C/min (a,c,e,g) and 100°C/min (b,d,f,h). The tests were interrupted at: a,b) 900°C; c,d) 875°C; e,f) 850°C; g,h) 800°C.
The mean α p diameter, area frac on of α p and the mean thickness of α GB is compared for the simulated and measured results in Figure 5. The diameter of α p increases with decrease of cooling rate and the model predicts with good accuracy this behaviour. The increase in area frac on of α p is pronounced for decreasing cooling rates. The developed model describes well the area frac on evolu on apart from 30 and 40°C/min for 6 MATEC Web of Conferences 321, 12038 (2020) https://doi.org/10.1051/matecconf/202032112038 The 14 th World Conference on Titanium holding temperature of 960°C. The thickness of the α GB was in the range of 1-4 µm for the inves gated condi ons and it increased with decreasing cooling rates. The differences between 930°C and 960°C are not significant.

Figure 5: Measured and simulated microstructural features for the inves gated cooling rates for the holding temperatures of 930°C and 960°C: a) mean α P phase diameter, and b) area frac on of α P ; c) measured and simulated thickness of the α GB
To illustrate the results of the developed model, the evolu on of the mean diameter of αp, mean thickness of α GB and vanadium supersatura on is exhibited in Figure 6 for the holding temperature of 960°C. A significant growth of α p as well as α GB is predicted down to 800°C, which was also observed in Figure 4. The supersatura on of V in β phase matrix is predicted to increase sharply un l ~850°C, when the α SEC is observed to be formed more pronounced (Figure 4).

Summary and conclusions
The sequence of growth of the α phases (primary, secondary and allotriomorphic) during cooling from an isothermal temperature below the βtransus is clarified and modelled for Ti-6Al-4V. Nuclea on of α GB starts at the globular α p , as well as in other regions. A coupled physical model for the growth of α p phase and the growth of the α GB is developed and the following conclusions can be drawn: Growth of α p is more pronounced for low cooling rates and occurs more notable un l ~800°C.
Regular (planar) interface shape of α GB is observed decora ng the prior β grain boundaries for high cooling rates. For low cooling rates irregular shapes are observed. α GB nucleates preferen ally from exis ng α p phase. For higher cooling rates nuclei in triple points and other regions of the prior β/ β grain boundary are observed. The higher nuclea on rate for those cooling rates can explain the different behaviour. A sharp increase of V supersatura on in the β matrix is observed un l ~850°C, which is a�ributed to contribute to the forma on of α SEC , especially for cooling rates higher than 10°C/min.