Kinetics of the allotropic hcp–fcc phase transformation in cobalt

The allotropic, martensitic phase transformation (hcp → fcc) in cobalt was investigated by differential scanning calorimetry (DSC) upon isochronal annealing at heating rates in the range from 10 K min−1 to 40 K min−1. The microstructural evolution was traced by optical microscopy and X-ray diffractometry. The kinetics of the phase transformation from hcp to fcc Co upon isochronal annealing was described on the basis of a modular phase transformation model. Appropriate model descriptions for athermal nucleation and thermally activated, anisotropic interface controlled growth tailored to the martensitic phase transformation of Co were implemented into the modular model. Fitting of this model of phase transformation kinetics to simultaneously all isochronal DSC runs yielded values for the energy of the interface separating the hcp and fcc Co phase and the activation energy for growth.


Introduction
Pure cobalt exhibits an allotropic phase transformation at the equilibrium temperature T 0 (at constant pressure) with the hcp modification as low temperature phase and the fcc modification as high temperature phase. This allotropic transformation shows characteristics of a martensitic transformation [1,2]: the transformation needs no diffusion, and composition change has a distinct athermal nucleation nature leading to spontaneous initiation of the reaction upon reaching the 'martensite' start temperature M S upon cooling or the 'austenite' start temperature A S upon heating.
It appears that depending on the initial microstructure and experimental conditions (as heating/cooling rate) a significant number of hcp$fcc transformation cycles must be passed through in order to establish reproducible characteristics of the hcp $ fcc transformation (see [1,2,5,10,15,19] and, in particular, results of this study presented in Section 4). This may contribute to the discrepancies apparent from the literature cited above.
A full, quantitative, description of the kinetics of the allotropic transformation in Co has not been presented until now. The present paper for the first time provides such a model description of the allotropic, martensitic transformation kinetics of Co, departing from a general modular model of phase-transformation kinetics composed of separate modes of nucleation, growth and impingement [20,21], thereby incorporating an athermal nucleation mode, as proposed in [19], interface-controlled growth and anisotropic impingement. This approach was recently applied successfully by our group to the polymorphic transformation of Laves phases [22].
The present paper focusses on the hcp ! fcc phase transformation using 'stabilised' specimens (i.e. after a number of preceding transformation cycles). Isochronal annealing experiments (i.e. experiments using a constant heating/cooling rate) in a fixed temperature range were performed using differential scanning calorimetry (DSC). The resulting enthalpy changes as function of time and heating rate were interpreted quantitatively using the modular phase transformation model.

Theoretical background of transformation kinetics
Solid state phase transformations can take place as soon as the hitherto existing phase is not stable anymore; i.e. a thermodynamic driving force can be indicated. Such a phase transformation can be realised in different ways. In general, a phase transformation can be subdivided into three (overlapping) steps: nucleation, growth and impingement. This type of modular approach has been described in [20,23] (see, especially, the review in [21]) and has been applied successfully to a variety of phase transformations: crystallisation of amorphous metal alloys [24][25][26][27][28][29], the austenite-ferrite transformation in Fe-based alloys [30][31][32] and the polytypic transformations of Laves phases [22].
Assuming, hypothetically, that each individual product particle, emanating from a successful nucleation process, grows into an infinitely large parent phase, in the absence of other product particles, the so-called extended volume, V e , given by the sum of the volumes of all these (hypothetical) particles, can be calculated. In a second step, the extended transformed fraction, x e (¼V e /V S ; with V S as the volume of the specimen), has to be corrected for (hard) impingement to obtain the real transformed fraction, f, by adopting a certain impingement mode. In the following, after discussing the hcp $ fcc transformation mechanism, nucleation, growth and impingement modes relevant for the hcp $ fcc transformation are indicated briefly.

The hcp $ fcc transformation mechanism
The dislocation (line) energy of a so-called perfect dislocation can be reduced by dissociation into two Shockley partial dislocations inducing a stacking fault (SF) in-between both partials, with the stacking fault energy (SFE) . The width of the dissociated partial dislocations is given by the balance of the elastic repulsion force, forcing the dissociation, and the stacking fault energy, , opposing the dissociation: dissociated (perfect) dislocations are a basic component of the microstructure [33].
The transformation of an fcc (ABCABC . . . stacking sequence) into an hcp (ABABAB . . . stacking sequence) crystal structure, and vice versa, can be realised by the motion of Shockley partial (SP) dislocations, with Burgers vectors, of type 1=6 11 " 2 , on every second closest packed plane [6,19]. This process can be called 'ordered glide' as an ordered array of Shockley partials is required for the phase transformation. So this ordered array of x partials transforms a region of thickness 2x closest packed layers. The transformation fcc ! hcp occurs by dissociation of the perfect dislocations and the transformation hcp ! fcc by association of the SPs (see Figure 1). Studies of the microstructural evolution upon thermal (transformation) cycling showed that such ordered dislocation arrangements evolve indeed [6], establishing, by the ''back and forth'' movement of the same partial dislocations, the reversible hcp $ fcc transformation with preservation of the orientation, in the specimen frame of reference, of the hcp and fcc crystals, as validated for Co [1].
Each SP can be associated with one of six Burgers vectors of type 1=6 11 " 2 on a closest packed plane leading, upon glide of the SP, to a microscopic shear of the lattice (see Figure 2a). This shear can be nullified (no add up of microscopic to macroscopic shear) by the summation of a set of three successive dislocations in the ordered array of SPs, with Burgers vectors such thatb 1 þb 2 þb 3 ¼ 0 (see Figure 2b). Ifb 1 þb 2 þb 3 6 ¼ 0, macroscopic shear evolves (see Figure 2c, where the extreme case, all SPs have same b ! , is shown). (Note that for polymorphic Laves-phase transformations such macroscopic shear is impossible because of the glide of synchro-Shockley partial dislocation dipoles [22].) However, it is usually assumed [13,14] that microscopic shear cancels out over short distances and therefore the contribution of macroscopic shear to the phase transformation (kinetics) is considered to be negligible in this paper.
Irrespective of the micro/macroscopic shear discussed above, the allotropic transformation in cobalt is associated with a macroscopic distortion due to the change of the atomic distances. This macroscopic distortion for the fcc ! hcp transformation is þ0.021% parallel to and À0.242% perpendicular to the closest packed plane [13].
The hcp crystal lattice has only one closest packed set of {0001} planes; the fcc crystal lattice has four equivalent sets of closest packed {111} planes. Upon thermal cycling the dislocation structure is (re)arranged, from any initial state, such, that only one single set of {111} fcc planes is active and parallel to {0001} hcp (further see Section 6.1).

Nucleation
The thermodynamic model for nucleation must be compatible with the mechanism for the hcp $ fcc transformation by glide of an ordered array of Shockley partial dislocations (cf. Section 2.1). The model presented here is derived from an earlier description of nucleation of the martensitic fcc ! hcp transformation in metals by dissociation of perfect dislocations [19].
Consider the periodically arranged array of perfect dislocations in an fcc crystal as shown in Figures 1 and 3. The fcc $ hcp phase transformation is performed by glide of the periodically arranged Shockley partial dislocations through the crystal lattice (see Section 2.1). (a) Array of three perfect dislocations in the fcc Co phase (before dissociation); (b) two arrays each consisting of three Shockley partial dislocations; the two arrays build up two particles consisting of the hcp phase, each particle having a volume determined by the grain size D, the height of the defect structure (depending on the number of Shockley partial dislocations within the array) and half of the separation distance, 2r, realised by glide of the Shockley partial dislocations.  Initiation of an fcc ! hcp transformation requires dissociation of the array of perfect dislocations into two arrays of Shockley partial dislocations. The region in between these two arrays can be described as a stacking faulted region (with reference to the parent structure) or as transformed region (exhibiting the product, hcp crystal structure). As long as fcc is the stable phase, the dissociation is limited by the relatively high energy (with reference to the parent phase) of the faulted structure lying between the two arrays of Shockley partial dislocations. This is no longer the case if the hcp phase becomes the stable phase, i.e. by passing the hcp-fcc phase equilibrium temperature T 0 upon (isochronal) cooling from the fcc phase field.
The region between the two separated arrays of Shockley partial dislocations can be considered as two second-phase (hcp) particles each having the volume V and the interfacial area S. The total Gibbs energy change DG associated with the formation of this second phase particle can be given as where DG ch V (¼ DG ch m =V m ; DG ch m is the chemical Gibbs energy difference per mole of atoms between product phase and parent phase and V m is the molar volume) is the chemical Gibbs energy difference between product phase and parent phase per unit volume, E str V (¼ E str m =V m ; E str m is the elastic strain energy per mole of atoms) is the elastic strain energy per unit volume and is the particle/matrix interface energy per unit area. For small overheating/undercooling, i.e. in the vicinity of the equilibrium phase-transformation temperature T 0 , the chemical Gibbs-energy change per mole of atoms DG ch m can be approximately given as with DH hcp$fcc as the molar enthalpy of transformation (fcc ! hcp: DH50, hcp ! fcc: DH40) and DT as the undercooling/overheating. The molar chemical Gibbs energy change DG ch m corresponds to the chemical driving force for the transformation and, as evident from Equation (2), it changes its sign at T 0 . The elastic strain energy per unit volume, E str V , cannot be neglected since the product and the parent phases have unequal molar volumes (see Section 5).
The volume of a product-phase particle is given by with nc/2 as the height/size of the nucleus; n/2 is the number of dislocations within the array of dislocations oriented perpendicular to the stacking direction (cf. Figures 1 and 3; n is the number of closed packed layers in the stack considered and c is the distance between adjacent dislocations in the array parallel to the c-axis [34]), 2r is the separation distance of the partial dislocations (r is the distance passed by one array of Shockley partial dislocations) and D is the grain size of the parent crystal (cf. Figure 1).
The newly created interfacial area S of one particle is given by the top and the bottom side and the front and the rear side of the product-phase particle (as shown in Figure 1). Because D ! nc=2 the interfacial area S is approximately given by the top and bottom sides and thus: It follows from Equations (1), (3) and (4) that The term DG/rD corresponds to the energy difference between the product phase/particle and parent phase per unit area top/bottom interface, DG A . Note that both the volume chemical energy term as well as the interface energy term scale with r. Evidently, a critical size (a critical value of r) does not occur (cf. Equation (5)): the transformation can take place 'spontaneously', i.e. without overcoming an energy barrier by thermal activation, provided that the energy difference between the product phase (lying in-between the dissociated dislocation arrays) and parent phase per unit area, DG A , becomes negative, e.g. by a change of temperature (cf. Equations (2) and (5)). Hence, the product-phase particles develop by athermal nucleation.
The interfacial term 2 in Equation (5) is independent of the height nc/2 of the possibly operating dislocation array, whereas the chemical Gibbs energy change for operation of the same dislocation array ¼ ncDG ch m =2V m À Á increases with n. For a fixed value of n the energy difference per unit area top/bottom interface, DG A , equals zero at a finite value of undercooling, DT ¼ T 0 À T, for the fcc ! hcp transformation (or rather overheating, DT ¼ T 0 À T, for the hcp ! fcc transformation). A distribution of heights of the arrays of dislocations (corresponding with a varying number of closed packed layers, n, in the stack considered) is supposed to exist in the fcc crystal. It follows from Equation (5) that the larger the height of the dislocation array, i.e. the larger n, the lower the required undercooling in order that this dislocation array starts to produce by glide a product-phase (hcp) particle. In other words, the nucleation event can be described as a kind of site saturation at each temperature, where dislocation arrays of specific height start to operate.
At a given undercooling, DT(t) ¼ T 0 À T(t), the critical (minimal) value of n, i.e. n*, indicating the minimal height of the dislocation array for realising by glide the fcc ! hcp transformation, satisfies (see Equations (2) and (5)): Depending on the values of the parameters at the right-hand side of Equation (6), as , DH and E str m , a minimum number of stacked dislocations can be designated from n* in order to obtain a stable nucleus that can grow. For example, considering the hcp ! fcc transformation above T 0 ¼ 690 K, if DH hcp!fcc ¼ 501 J mol À1 , ¼ 10 mJ m À2 [8], E str m ¼ 0.838 J m 2 , it follows n* ¼ 32 at T ¼ 720 K and n* ¼ 24 at Upon decreasing temperature, more and more dislocation arrays of decreasing height can become active. From experimental data for the martensitic transformation in Fe 30.2 wt % Ni, it was proposed that the cumulative number of operating dislocation arrays, N (n*(DT(t))), obeys the empirical function [35]: with N tot as the total number of pre-existing Shockley partial dislocation arrays of variable height per unit volume. The total number of pre-existing SP arrays of variable height can be (over) estimated by approximating the grain volume by D 3 and recognising that D/c represents the number of dislocations covering a height D. Hence, N tot ¼ 2 D=c ð Þ=D 3 ¼ 2=cD 2 taking into account that every perfect dislocation by dissociation contributes to the development of two nuclei (see Figure 1). This estimation is rough as at the start of the transformation the number of SPs in a nucleus is larger than one. However, the values of the fit parameters of the transformation model do not depend strongly on the value of N tot (see Section 5). The above treatment focussed on the fcc ! hcp transformation occurring upon cooling. A parallel treatment holds for the hcp ! fcc transformation occurring upon heating.

Interface-controlled growth
When an ordered array of Shockley partial dislocations glides through the crystal (as shown in Figures 1 and 3) the product-phase particle grows. The dimensionality of the growth is one, i.e. the product-phase particle grows in one of the three possible 11 " 2 directions oriented perpendicular to the c-axis (cf. Figure 2). There is no composition change from parent to product phase in an allotropic phase transformation and thus the growth is controlled by atomic (jump) processes in the direct vicinity of the interface: interface-controlled growth.
The height of a product phase particle which starts to grow at time is given by n Ã ðDTðÞÞ Á c=2 (cf. Figure 1). Hence, at time t the volume Y(, t) of a product-phase particle, which starts to grow at time , is given by (see Figure 1 and Equations (3) and (6)): with as the interface/Shockley partial dislocation-glide velocity. For small undercooling or overheating the growth velocity is given by where M is the temperature dependent interface mobility, M 0 is the pre-exponential factor for growth and Q denotes the activation energy for growth. The net driving force DG m (T(t)) which is given as a molar quantity [21] amounts to (cf. Equations (1) and (5)): 2.4. Extended fraction, transformed fraction and impingement Adopting an appropriate impingement mode the real transformed fraction f model (T(t)) can be calculated from the extended transformed fraction x e (cf. beginning of Section 2). In the case of anisotropic growth, which pertains to the allotropic phase transformation considered here (one-dimensional growth; cf. Section 2.3), the (hard) impingement process can be phenomenologically described by the following equation [21]: where is a measure for the degree of anisotropic impingement. Integration of Equation (11) for the case 41 yields 3. Experimental

Alloy production
The cobalt rod with a diameter of 5 mm used in this study was obtained in the hammered, not annealed state from Alfa Aesar (Karlsruhe, Germany) and has a purity of 99.995 at. %. Disc-shaped specimens were produced by cutting pieces with a thickness of 750 mm. Both sides of the specimen discs were prepared by grinding with SiC paper and subsequently polishing using diamond paste down to 0.25 mm such that all specimens have approximately the same mass of about 100 mg. All specimens were cleaned ultrasonically in isopropyl. After the calorimetrical heat treatment the specimen discs were ground and polished again as mentioned above in order to reduce inhomogeneities of the surface as necessary for XRD and LM analysis.

Differential scanning calorimetry (DSC)
The isochronal annealing was carried out with a power-compensated differential scanning calorimeter Pyris Diamond by Perkin Elmer. The temperature was calibrated using the melting temperature of zinc (T m ¼ 692. 15 K [36]) measured for each heating rate used. Aluminium was used as pan material for both the specimen and reference container. Specimens of approximately the same mass were used in order to provide similar heat capacities. Pure argon gas with a constant flow was used as protective gas atmosphere. A measurement with empty pans served for determination of the baseline.

Philosophical Magazine
For each measurement in the temperature range from 523 K to 893 K at a heating rate varying from 10 K min À1 to 40 K min À1 a new specimen was used. In order to establish a microstructural reference state (see Section 4), each specimen used for the kinetic analysis, was initially exposed to 60 isochronal transformation cycles with a cooling/heating rate of AE50 K min À1 in a temperature range from 523 K to 893 K. The microstructure and phase composition of the specimens in the initial state and after the 1st, 2nd, 3rd, 10th, 20th, 40th and 60th isochronal transformation cycles were analysed by LM and XRD.
For the calculation of the cumulative enthalpy DH(T(t)), as function of temperature (time), the heat signal dDH(T(t))/dt was integrated for cumulative times. Previous to this integration, it is necessary to perform a baseline correction. This was done by subtracting the above mentioned DSC signal recorded with empty pans from the DSC signal recorded with the Co specimen for each heating rate [37]. This implies that the heat capacities of both phases are the same in the temperature region of the transformation, as holds for cobalt. The transformed fraction f exp (T(t)) as function of temperature then is given by where the total transformation enthalpy DH hcp$fcc was obtained by integration of the baseline corrected DSC signal over the entire temperature range of the transformation.

X-Ray diffraction
X-ray diffraction (XRD) was employed for phase analysis and to characterise the crystalline imperfection upon thermal cycling. The XRD measurements were performed with Mo K radiation employing a Bruker D8 Discover diffractometer operating in parallel-beam geometry equipped with an X-ray lens in the incident beam, a parallel-plate collimator in the diffracted beam and an energy-dispersive detector. The 2 range of 15-45 was measured with a step size of 0.015 and counting time per step of 10 s.

Light microscopy
The surface of the specimen discs before and after the 1st, 2nd, 3rd, 10th, 20th, 40th and 60th transformation cycles and of specimens used for the kinetic analysis were analysed using a Zeiss Axiophot light optical microscope. For that purpose the polished specimen discs were etched for 2-7 s using a fresh etching solution (

Results and evaluation
A baseline-corrected isochronal DSC scan of Co at a heating rate of 20 K min À1 showing a heat signal associated with the hcp ! fcc transformation is presented in Figure 4. T onset denotes the peak onset temperature, T peak is the peak maximum temperature and DH hcp!fcc is the enthalpy of transformation (40 for the endothermic hcp ! fcc transformation) given by the (hatched) area enclosed by the DSC signal and the baseline (dashed). In order to characterise the hcp ! fcc transformation behaviour and the microstructural evolution upon thermal cycling of initially heavily deformed Co, the change of the DSC heat signal (characterised by the parameters indicated above), XRD diffractograms and the change of the grain size were recorded as a function of the number of transformation cycles experienced (one cycle is hcp ! fcc followed by fcc ! hcp). Results are presented in Figure 5.
The variation of the parameters T onset , T peak and DH hcp!fcc for the hcp ! fcc transformation (see Figure 4) is shown in Figure 5a as a function of the number of transformation cycles at a heating rate of 50 K min À1 between 523 K and 893 K. Starting from the heavily deformed initial state, the parameters T onset , T peak and DH hcp!fcc decrease from the first to the second transformation cycle from 736.3 K, 746.9 K and 406 J mol À1 to 717.1 K, 735.6 K and 385 J mol À1 , followed by an increase between the second and third transformation cycle to 721.9 K, 738.7 K and 452 J mol À1 , respectively. After the third transformation cycle T onset and T peak pass through modest local maxima of about 722.3 K and 739.0 K, respectively. Eventually   suggest that the specimen only experiences a fractional hcp ! fcc transformation during the first cycles.
XRD measurements of Co specimens at room temperature in the initial state and after the first, second and third transformation cycles (inset in Figure 5b) reveal 111 and 002 reflexes of the fcc Co phase after (only) the first and second transformation cycle (Cards 89 4308 for hcp-Co and 15 0806 for fcc-Co of the powder diffraction file [38], were used for phase identification). In the initial state, after the third transformation cycle and during further cycling no fcc Bragg peaks could be detected. These XRD results support the above interpretation of the DH hcp!fcc changes during cycling.
Hence, the amount of hcp and fcc Co at room temperature after a specific number of transformation cycles can be deduced from the enthalpy data adopting direct proportionality of DH hcp!fcc with the amount of hcp present before the hcp ! fcc transformation takes place. The thus determined fractional amounts of hcp and fcc phase at room temperature are shown in Figure 5b as function of the number of transformation cycles. The amount of fcc Co at room temperature reaches a maximum of about 25% after two transformation cycles followed by a decrease towards nil reached at about the fifth transformation cycle.
The full width at half maximum (FWHM) was calculated for the 100, 002, 101, 102 110 and 103 reflexes of hcp Co. The FWHM results as obtained for the initial state and after the 1st, 2nd, 3rd, 10th, 20th, 40th and 60th transformation cycles are presented in Figure 5c as function of the corresponding transformation cycle number. For all reflexes a steep decrease can be observed from the values corresponding to the initial state to the values obtained after the second/third transformation cycle. This decrease can be ascribed to a reduction of lattice defects in the initially heavily deformed state upon annealing (recovery), grain growth and transformation cycling. For all reflexes, the FWHM and thus the defect structure remains about constant after the third transformation cycle.
The change of the mean grain size, D, as determined by LM, is shown in Figure 5d as function of the number of transformation cycles. It was not possible to measure the grain size of Co in the initial stage because the etched microstructure did not allow a clear identification of grain boundaries: the highly deformed initial state leads to an uncontrollable etching process. The grain size increases upon thermal cycling from initially less than 10 mm to about 87 AE 5 mm after the 40th cycle and remains constant thereafter.
It has been concluded from the above results that, to assure the same initial state for each experiment used for kinetic analysis, each such specimen will be subjected to 60 transformation cycles before a kinetic analysis is performed.
Important morphological characteristics are revealed by LM from the specimen surfaces, as shown for the 10th, 40th and 60th transformation cycles in Figures 6a-6c. After the 10th cycle ( Figure 6a) the etching suggests an underlying transformation structure exhibiting different, specific orientations of martensite plates within a Co grain. After the 40th cycle ( Figure 6b) the etching suggests that only a single specific orientation is associated with the martensitic transformation experienced by a grain. After the 60th transformation cycle no such etch effect is observed (cf. Figure 10 and its discussion in Section 6.1). Figure 6. Optical micrographs of the etched microstructure of the surface of Co specimens after the (a) 10th, (b) 40th and (c) 60th transformation cycle performed in the DSC at a rate of AE50 K min À1 in the temperature range from 523 K to 893 K. The etched microstructure suggests (see the arrows) that upon prolonged annealing the number of types of glide planes operating during the martensitic transformation (the fcc ! hcp experienced in the cooling part of the transformation cycle) in a single grain is reduced to one (see text and Section 6.1).

Analysis of the transformation kinetics
Baseline corrected isochronal DSC scans for the allotropic hcp ! fcc (upon heating) and fcc ! hcp (upon cooling) transformations in Co measured at heating/cooling rates in the range from 10 K min À1 to 40 K min À1 are presented in Figure 7. The enthalpy of transformation for the hcp ! fcc transformation (upon heating; independent of heating rate) is about þ 501 AE 7 J mol À1 and for the fcc ! hcp transformation (upon cooling; independent of cooling rate) about À512 AE 13 J mol À1 , i. e. within the experimental accuracy the hcp ! fcc and fcc ! hcp transformations exhibit the same absolute value for the enthalpy of transformation.
The cooling curves in Figure 7, i.e. for the fcc ! hcp transformation, have been included for the sake of completeness. Whereas the start temperature for the fcc ! hcp transformation (upon cooling) clearly depends on the cooling rate applied, this is much less the case (if at all) for the dependence of the start temperature of the hcp ! fcc transformation (upon heating) on heating rate (see Figure 7). In both cases, i.e. for the transformation upon cooling and the transformation upon heating, a cooling/heating rate independent start temperature is expected in view of the athermal value of the nucleation process (see Section 2.2). It must be noted, that no temperature calibration of the DSC for the cooling part of the cycle is possible and that the DSC signal during cooling is smeared distinctly, as is well known for powercompensated DSCs [37]. Therefore, the cooling-rate dependence of the start temperature of the transformation upon cooling has to be considered as an artefact from the measurement, and the cooling curves cannot be used for kinetic analysis.
The kinetic analysis of the allotropic hcp ! fcc transformation of Co was performed by applying the modular phase transformation model in the form as described in Section 2. The experimental transformation rates df exp =dT (df exp =dT ¼ 1=Èdf exp =dt) have been plotted in Figure 8 as function of the transformed fraction f exp (T(t)) for the different heating rates. The corresponding curves for the experimentally determined transformed fractions f exp (T(t)) (see Section 3.2, Equation (13)) are shown in Figure 9 as a function of temperature T(t). Evidently, the maximum transformation rate occurs at f exp 51 À 1/e. This is a strong indication for anisotropic growth [39] as expected for the hcp $ fcc transformation in Co (cf. Sections 2.2 and 2.3). The impingement mode for anisotropic growth (introduced in Section 2.4) has thus been used in the kinetic model of the phase transformation.
The kinetic model parameters , Q and were determined by simultaneously numerically fitting the model to all isochronal heating runs obtained for various heating rates in the range from 10 K min À1 to 40 K min À1 . The mean square error (MSE ) between the calculated (calc) and experimental (exp) transformed fraction curves was minimised by varying the fit parameters using a multidimensional unconstrained nonlinear minimisation fitting routine [40] as implemented in MATLAB for each ith of the applied heating rates: The results of the fitting as described above are shown in Figure 9 (experimental data: symbols; fit: lines) and Table 1 (i.e. the values thus determined for , Q and ).
Values used for the constants in the model have (also) been given in Table 1. The total number of pre-existing dislocations N tot within a grain of mean size D ¼ 87 AE 5 mm was calculated as described in Section 2.2 as 6.5 Â 10 17 m À3 . The total enthalpy of transformation was taken as DH hcp!fcc ¼ 501 AE 7 J mol À1 , as determined experimentally (cf. Section 4). The strain energy contribution expressed as E str m , corresponding with the macroscopic distortion discussed in Sections 2.1 and 2.2, was estimated according to [19] at about 0.838 J mol À1 under the assumption of linear elasticity and coherency (full elastic accommodation of volume misfit). It is supposed that dislocation glide is rate controlled by thermal activation as it generally holds for materials with metallic bonding type. In pure metals, it is assumed that the atomic structure within the closest packed glide plane represents a weak lattice resistance. The mobility of these dislocations is thus determined by thermal fluctuations characterised by an activation energy Q [41].
The temperature-independent dislocation/interface velocity 0 was estimated by using the well-known (empirical) expression for the pre-exponential factor for latticeresistance controlled glide of dislocations in metals (i.e. rate limited by weak discrete obstacles) (as validated for a wide range of metals [42]), which reads _ 0 ¼ 0 m b ¼ 10 6 s À1 with m as the dislocation density and b as the Burgers vector taken equal to 1/2 ffiffi ffi 2 p a fcc with a fcc ¼ 0.35447 nm [34]. The density of mobile dislocations can be estimated according to [42] as 10 13 m À2 lying in the range of values expected for fcc and hcp metals. The thus obtained value for v 0 is about 400 m s À1 . Thus the mobility M 0 ¼ 0 =RT [21] was assessed at 4.3 m mol J À1 min À1 , adopting a mean value for the temperature T ¼ 700 K. Note that the temperature dependence of M 0 is small in comparison to the temperature dependence of the exponential term in M (see Equation (9)) and thus the temperature dependence of M 0 can be neglected.
The influence of a variation of the constant parameters N tot , T 0 and M 0 on the fit parameters, as indicated by N tot ¼ 6.5 Â 10 17AE1 m À3 , T 0 ¼ 690 AE 1 K and M 0 ¼ 4.3 Â 10 0AE1 , yielded variations of and Q as indicated in Table 1; the impingement parameter is not influenced. (d, e) Eventually, upon prolonged thermal cycling, a dislocation structure emerges that realises the hcp $ fcc transformation on the basis of (ideally) one single stack of ordered perfect dislocations (in fcc) ¼ 2 parallel stacks of ordered Shockley partial dislocations (in hcp) implying that only one (instead of, maximally, four; see (a) þ (b)) glide plane system operates within a single grain, e.g. ð111Þ foc kð0001Þ hcp , ½ " 110 fcc k½ " 10 " 10 hcp .

Preceding transformation cycles
Each Co specimen used for kinetic analysis was subjected to a number of 60 preceding transformation cycles to assure similar starting conditions (same microstructure) and full transformation (see Section 4). The phenomenon can be interpreted as that the preceding transformation cycles are needed to stabilise the dislocation configuration (cf. Section 2.1) in the specimen that carries the forward (hcp ! fcc) and backward (fcc ! hcp) transformations. The initially incomplete transformation can partly be ascribed to the relatively strong interaction of the (partial) dislocations with the grain boundaries: upon increasing grain size during cycling (see Figure 5d) relatively more dislocations become available for establishing the transformation [1,2,10].
During the first transformation cycles the transformations occur in conjunction with recovery, possible (local) recrystallisation and grain growth in order to reduce the stored plastic deformation and grain-boundary surface. This leads to changes of the (initially disarranged) dislocation structure. A further complication is that after a first hcp ! fcc transformation upon heating, upon subsequent cooling, a completely reverse formation of hcp Co requires glide of SPs on the closest packed {111} fcc planes lying parallel to the previous {0001} hcp planes. However, at least initially SPs may be available as well on {111} fcc planes not parallel to the former {0001} hcp plane and thus the original hcp grain microstructure is not re-established. (Note the variously orientated hcp Co martensite 'plates' in a single grain in Figure 6a). As a consequence, this reasoning provides a further reason (see above) why the transformation cannot run to completion in such a grain; a small amount of parent phase is retained (see Figure 5b and its inset).
The above described transformation behaviour for the first transformation cycles is compatible with the DSC results. The decrease of T onset and T peak found for the second transformation cycle can be understood such that during the first complete hcp $ fcc cycle the disordered dislocation structure of the initial state evolves into a more ordered (in the sense of the discussion in Section 2.1) dislocation structure, thereby facilitating the transformation: less overheating is required (see Section 2.2).
Prolonged thermal cycling leads to an increase of the thickness of the martensite 'plates' (Figure 6b), i.e. the height of the operating dislocation array increases, and a single glide variant appears to become dominant (cf. Figures 6b and 6c) [6,10].
The above discussion leads to a summarising schematic presentation of the evolving dislocation structure in a grain during hcp $ fcc (thermal) cycling, as presented in Figure 10.

Kinetics
The evaluation of the kinetics of the allotropic hcp ! fcc phase transformation in Co performed in Section 5 demonstrates that this transformation can be well described as governed by the activation of pre-existing nuclei (stacked dislocation sequences) and thermally activated interface-controlled growth subjected to anisotropic impingement. The resulting values for the fit parameters , Q and agree well with data provided by theory and experiment (see what follows).
The investigation of dislocation nodes (the extension of a dislocation node depends amongst others on the stacking fault energy produced by the node) in pure Co [8] and Fe-Cr-Ni alloys [43] yielded stacking-fault energy values from 5 to 10 mJ m À2 . This agrees very well with the here determined value for the interface (¼stacking fault) energy resulting from model fitting of the transformation kinetics yielding ¼ 6.7 AE 0.4 mJ m À2 .
The ratio of the activation energy for diffusion and the activation energy for dislocation glide controlled by lattice resistance is for metals about 7 [42]. The activation energy for self diffusion of Co is Q Co ¼ 270 kJ mol À1 [2]. Indeed, 1/7Q Co % 38 kJ mol À1 , which is highly compatible with the here determined value of the activation energy for growth, Q ¼ 33 AE 15 kJ mol À1 , thereby validating the applied concept of growth controlled by dislocation glide.

Conclusions
. Extensive thermal cycling of Co in a fixed temperature range is necessary to establish full reversibility of the allotropic hcp $ fcc phase transformation (i.e. reaching constant values of T onset , T peak and DH hcp!fcc ). During thermal cycling, stabilisation of the dislocation structure is established such that in a single grain the hcp $ fcc transformation is established by (ideally) only one single stack of ordered perfect dislocations (in fcc) ¼ 2 parallel stacks of ordered Shockley partial dislocations (in hcp), implying that only one (of the maximally four) glide-plane types operates within a single grain. . The kinetics of the transformation can be well described on the basis of a modular transformation model adopting an athermal nucleation mode and an anisotropic interface-controlled growth mode. . Results obtained for the fit parameters (¼6.7 AE 0.4 mJ m À2 ) and Q (¼33 AE 15 kJ mol À1 ) are well compatible with the interpretation of the product/parent interface as a stacking fault and of the growth process as realised by thermally activated glide of Shockley partial dislocations.