Preferential Nucleation during Polymorphic Transformations

Polymorphism is the ability of a solid material to exist in more than one phase or crystal structure. Polymorphism may occur in metals, alloys, ceramics, minerals, polymers, and pharmaceutical substances. Unresolved are the conditions for preferential nucleation during polymorphic transformations in which structural relationships or special crystallographic orientation relationships (OR’s) form between the nucleus and surrounding matrix grains. We measured in-situ and simultaneously the nucleation rates of grains that have zero, one, two, three and four special OR’s with the surrounding parent grains. These experiments show a trend in which the activation energy for nucleation becomes smaller – and therefore nucleation more probable - with increasing number of special OR’s. These insights contribute to steering the processing of polymorphic materials with tailored properties, since preferential nucleation affects which crystal structure forms, the average grain size and texture of the material, and thereby - to a large extent - the final properties of the material.

: Schematic geometry of the sample. The location of the X-ray beam is highlighted.
The dimensions are in mm.
3 Figure S3. Schematic of the heat-treatment carried out on the sample. During the interrupted heating from 830 °C to 852 °C in 2 °C steps, after a stabilization step for 10 min, a Friedel 3D measurement is carried out at each step. At the isothermal annealing temperature of 900 °C, Friedel 3D measurements were carried out a total of 10 times.

Intermediate results and discussion
Three intermediate results are used in fitting the classical nucleation theory to the measured nucleation rates: 1) the density of potential nucleation sites, 2) the austenite fraction, and 3) the driving force for nucleation.

Density of potential nucleation sites
The density of potential nucleation sites N n as a function of temperature is estimated from the measurements in the following way. Before the ferrite-to-austenite transformation, the density of potential nucleation sites is equal to or higher than the number of nuclei that we experimentally observe at the end of the transformation. By increasing the temperature with one step of 2°C, the number of potential nucleation sites decreases by the number of actual nuclei that appear in that temperature interval during the experiment. An example of the density of potential nucleation sites ' n N that we measured in this way as a function of temperature is shown in Fig. S4 for the case that the austenite nuclei have 1 special OR with the neighboring ferrite grains. Here, we assume that the actual density of potential nucleation sites N n changes with temperature in the same way as the measured density of potential

Austenite fraction
In order to calculate the fraction transformed at each point during the phase transformation, all the diffraction images recorded during a Friedel 3D measurement were summed in order to 11 get powder-like diffraction images. After a radial integration step on the summed image, the formed austenite fraction (see supplementary Fig. S5) is calculated according to procedure described in ref. S1. A polynomial curve is fit to the measured austenite fraction. This polynomial curve for the austenite fraction is used in Eqn.
(3) to calculate the austenite fraction at each temperature during the phase transformation.

Driving force for nucleation
The equilibrium temperature at which the ferrite to austenite transformation starts is calculated under ortho-and para-equilibrium conditions with the thermodynamic software package Thermo-Calc 28 . Ortho-equilibrium indicates that all alloying elements distribute according to equilibrium during the transformation. Para-equilibrium means that the carbon atoms have enough time to reach an equilibrium distribution, but the other atoms in the solidsolution do not have enough time to reach an equilibrium distribution. The transition temperature according to ortho-equilibrium conditions is 1118 K. The transition temperature according to para-equilibrium conditions is 1024 K. The experimentally observed transition temperature is 1105 K. From this we derive that the nucleation does not take place under ortho-equilibrium conditions, because the experimentally observed transition temperature is below the ortho-equilibrium transition temperature, which is physically not possible. The nucleation can take place under para-equilibrium conditions, because the experimentally observed transition temperature is above the para-equilibrium transition temperature.
The driving force for the transformation (under para-equilibrium conditions) as a function of temperature is calculated with the thermodynamic software package Thermo-Calc 28 . The result is shown in supplementary Fig. S6. Supplementary Fig. S6 shows that the driving force for the transformation is approximately 1.1⋅10 7 J/m 3 at 1105 K. A strain energy of 10 7 J/m 3 as 12 estimated in reference (19) could therefore explain the increase in the experimentally observed transition temperatures compared to the equilibrium para-equilibrium transition temperature.

Physical interpretation values fit-parameter A of austenite nucleation rate
The experimentally observed nucleation rate that is shown in Fig. 4 is calculated by dividing the measured number of austenite grains at a particular temperature by the volume of the 'cylinder' and by the isothermal holding time of 10 minutes. However, the nucleation process ceased within shorter times than ten minutes. Therefore, the absolute nucleation rates that are reported in Fig. 4 are in fact higher by an approximately constant factor C. The relative change in nucleation rate with temperature remains approximately the same as reported in Fig. 4. The fitted A-parameter is therefore equal to A = (S/C)Zexp(-τ/t), with C ≥ 1 and S ≥ 1.
The fitted values for the A-parameter are given in supplementary Table S1. In order to assess the physical meaning of these fitted values, we perform the following analysis. The value for the frequency factor is estimated to be β * =(kT/h)exp[-Q D /(kT)] ≈ 229 s -1 (13), for T = 1115 K and Q D = 3.93⋅10 -19 J (26). The incubation time is given by τ=κ/(2πβ * Z 2 ). Given that 0.5< κ <5 (4) and 0.01 < Z < 1 (14), the incubation time is (7 ± 15)⋅10 -3 s. Therefore, the factor exp(τ/t) = 1.00000±0.00003 for t = 600 s. From exp(-τ/t)=1 and the fitting parameter A, we derive the variation in the S/C-ratio as given in supplementary Table S1. Generally, the S/C-ratio is below 1, except for OR1, where the S/C-ratio varies between 0.05 and 5.3. An S/C-ratio >1 means that S > C and S >1, which means that the density of potential nucleation sites is higher than the density of nuclei with OR1 that actually appeared in the experiment. In the case of OR1, S can be larger than 1, because there are a large number of potential nucleation sites in 13 the microstructure where austenite nuclei with OR1 can form, since there are essentially no crystallographic restrictions. However, there is an upper bound to S, because C ≥ 1. The maximum for S is 5.3, which is obtained for the case C = 1. This corresponds to a maximum time t end of 10 minutes at which nucleation ended during the isothermal holding.
Supplementary Table S1 shows a trend in which the S/C-ratio decreases with increasing number of OR's. This could be related to the fact that with increasing number of OR's the number of potential nucleation sites becomes more restricted due to increasing crystallographic restrictions. This means that the value for S decreases with increasing the number of OR's. The fact that the S/C-ratio is below 1 for OR's 2-4, means that C > S. In case the S/C-ratio = 0.2, which is the upper bound for OR2 and OR3, and S = 1, then C = 5, which means the nucleation process ended latest after a time t end of 2 minutes. In case the S/C-ratio = 0.04, which is the upper bound for OR4, and S = 1, then C = 23, which means the nucleation process ended latest after t end = 26 seconds.
Supplementary Fig. S7 shows the maximum time t end at which the nucleation ended during isothermal holding as a function of the number of crystallographic OR's. Supplementary Fig.   S7 shows a trend in which t end decreases with increasing number of special OR's. This is logically related to Fig. 4, which shows that the activation energy for nucleation decreases with increasing number of special OR's. In other words, a lower activation energy for nucleation leads to a shorter time t end in which the nucleation process ends during isothermal holding.

Number of atoms in the critical nucleus
Supplementary Fig. S8 shows the number of atoms n * in the critical nucleus, which is given by 14 n * = 2Ψ/[ν γ (∆g v -g s ) 3 ], where ν γ =(1/4)(a γ ) 3 is the atomic volume of iron in the austenite phase and a γ = 0.366396 nm is the lattice parameter of the austenite. The values for Ψ are taken from Fig. 4. The driving force for nucleation is taken at the temperature at which the nucleation rate is maximum. Supplementary Fig. S8 shows that the number of iron atoms in the critical nucleus is about seven thousand in case Ψ = 10 -7 J 3 /m 6 and g S = 10 7 J/m 3 . Given the uncertainty in the absolute value of Ψ, which varies between 10 -4 to 10 -8 J 3 /m 6 as we change g S from 0 to 1.1⋅10 7 J/m 3 , respectively, the number of atoms in the critical nucleus lies between 2,500 and 10,000 atoms. 15

Details 3DXRD data-analysis Raw Data: peak searching and fitting
A low threshold of 150 counts was used in order to identify regions consisting of diffracted intensity. After peak searching and fitting, almost 99 % of the diffraction spots were consistently identified. The global parameters of the experiment were determined from the diffraction data itself and no calibration sample was required, as described in references (24,25).

Indexing: correlating diffraction spots to individual grains
In order to keep the processing time low, indexing and refinement routines were executed on each Friedel 3D measurement with the following parameters: Diffraction spots belonging to the {200}-planes were chosen as starting points for indexing. Since the {110}-ring of α and {111}-ring of γ overlap, the diffraction spots belonging to these two rings were omitted from analysis in case both phases are present in the microstructure. In the analysis, a 5 μm step in the position in the sample and 0.2° step in the rotation of the diffracting plane normal were used. The margins used for the position on the detector and in ω were 60 μm and 0.4°, respectively. Detection of both the mixed and the un-mixed Friedel pairs is taken into account. All the grains for which the mean difference in ω was less than 0.06° and the mean difference in the position of the observed and the simulated diffraction spots on the detector was less than 25 μm were identified and stored as grains.

Identifying the grains in the 'cylinder'
Due to the horizontal size of the X-ray beam being smaller than the diameter of the sample during rotation, the part of the sample illuminated by the X-ray beam changed continuously, even though a smaller cylinder of diameter equal to the width of the beam was always in the 16 beam. Due to this, firstly, the set of grains at the edges of the smaller cylinder was different each time the grains diffract in different orientations of the sample. Secondly, for the grains located outside this cylinder, not all diffraction spots are observed because the grains are not in the beam during the entire angular scanning. Thus, three filtering steps were used here: (i) the minimum ratio of observed number of diffraction spots divided by the expected number of diffraction spots was chosen to be 0.8; (ii) the difference between grain sizes as calculated from a starting diffraction spot and the diffraction spots being matched during indexing was chosen to be less than 10 μm in grain radius and; (iii) a grain was accepted only if the refined centre-of-mass position of the grain was within the smaller cylinder defined by the beam size.
In the case of the grains close to the edges of the cylinder, the uncertainty in the centre-ofmass position, the orientation and the volume of the grains is accordingly higher than for that for the grains located completely in the cylinder.

Detection limits
At each point during the phase transformation the grains were observed after the nucleation stage, because of the detection limit and because the Friedel 3D measurements were carried out only after the equilibrium phase fractions had been reached. The volume of the smallest grains that could be detected at any time during the experiment (determined by the threshold used) is ~ 120 μm 3 (~ 3 μm radius), so that all the grains above this size could successfully be identified. Thus, even though all the grains larger than 3 μm in radius could be identified, no nuclei of such small size were found because the nuclei had already grown before the Friedel 3D measurements were carried out. In the results shown here, the minimum size of the γ grains when first detected (calculated from volume assuming spherical shape) is between 4.8 to 12 μm, since some nuclei had already grown before the start of the Friedel 3D measurements. Compared to the much coarser initial α microstructure (average grain size of ~ 17 105 μm) and the final γ microstructure (average grain size of ~ 98 μm), this is not expected to have a significant effect on identifying which are the neighboring ferrite grains close the moment an austenite grain nucleated. Moreover, we can determine at which temperature interval a nucleus forms, since the heat-treatment consists of discrete temperature steps ∆T and we anneal at a particular temperature until the moment that the austenite grains stop growing.

Identification of neighboring grains
In the experimental setup, diffraction images were recorded using the medium resolution FReLoN-4M detector, due to which the geometry of the grain boundaries could not be determined. However, it is not needed for this study to reconstruct the morphology of the grain boundaries, which in this case reduces the measurement time from many hours (if not days) to 13 minutes. The shape of the grains were approximated by ellipsoids with small aspect ratios. The strategy to identify which are the neighboring ferrite grains close the moment an austenite grain nucleated is as follows: The distance between the centre-of-mass of an austenite grain (at the time when it was first detected and thus relatively small) and that of each ferrite grain in the illuminated volume is calculated first including a factor to account for the uncertainty in the position of the grains. This distance is normalized by the radius of the corresponding ferrite grain to calculate a so-called 'sphere of influence' of the ferrite grain. The ferrite grains for which this sphere of influence had a value smaller than 1.5 were identified as possible neighbours of the austenite grain under consideration, henceforth referred to as the candidate ferrite grains. This step makes sure that if the ferrite grain is identified as a neighbour in the end, the shape of the grain is still close to an ellipsoid with a maximum aspect ratio of 1.5. Repeated calculations showed that the choice of the maximum aspect ratio had no significant influence up to a value of 2. 18 For each of the candidate ferrite grains, the 3D space was divided into half. This was done by transforming the coordinates of the ferrite grains in such a way that the z-axis of position coordinates was aligned in the direction from the centre-of-mass of the austenite grain towards the candidate ferrite grain. In case the sphere of influence of the ferrite grain in question was less than the sphere of influence of the remaining ferrite grains (with positive value of transformed z-position) that are closer to the austenite grain (in absolute distance) as compared to the ferrite grain in consideration, the ferrite grain was identified as a neighbour of the austenite grain. In case there were ferrite grains with the sphere of influence smaller than the original ferrite grain, the absolute distance between the ferrite grains and the austenite grain was compared and the ferrite grain positioned at the smallest absolute distance from the austenite grain was saved as a neighbour of the austenite grain. 19