Continuous Transformations of the Nucleation Mechanism in the

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Continuous transformations of the nucleation mechanism in the undercooled state Yikun Zhang, Christian Simon, Thomas Volkmann, Matthias Kolbe, Yong Lei, Lingwei Li, and Gerhard Wilde Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01759 • Publication Date (Web): 30 Mar 2018 Downloaded from http://pubs.acs.org on March 31, 2018

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Crystal Growth & Design

Continuous transformations of the nucleation mechanism in the undercooled state Yikun Zhang

a, b, c, d, *

, Christian Simon c, Thomas Volkmann d, Matthias Kolbe d, Yong Lei e, Lingwei Li b, *, and Gerhard Wilde c, e, f, *

a

State Key Laboratory of Advanced Special Steels & Shanghai Key Laboratory of Advanced Ferrometallurgy &School of Materials Science and Engineering, Shanghai University, 200072, China b Key Laboratory of Electromagnetic Processing of Materials (Ministry of Education), Northeastern University, Shenyang 110819, China c Institute of Materials Physics, University of Münster, Wilhelm-Klemm-Straße 10, D-48149 Münster, Germany d Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Linder Höhe, 51147 Köln, Germany e Institute of Nanochemistry and Nanobiology, School of Environmental and Chemical Engineering, Shanghai University, Shanghai 200444, China f Herbert Gleiter Institute, Nanjing University of Science and Technology, Nanjing 210094, China

* corresponding authors: [email protected] (Y. Z.), [email protected] (L. L.), [email protected] (G. W.)

Abstract Studies on nucleation have relied for many decades on the classical nucleation theory. Within that picture, thermal fluctuations govern the formation of critically-sized, homogeneous nuclei of the newly developing phase. At the same time, structural inhomogeneities or impurities or extrinsic substrates such as surfaces or container walls can favor the formation of a critical-sized nucleus, leading to so-called heterogeneous nucleation. Specifically, according to this theoretical framework, a kinetic nucleation transition between heterogeneous and homogeneous is predicted to happen at a critical cooling rate. This underlying picture of nucleation has been applied since the development of classical nucleation theory, but this transition has rarely been observed experimentally for simple metallic systems. Now, with the quick development of fast scanning chip calorimetry and careful selection of a model alloy, we have been able to experimentally map the kinetic transition between these two fundamental modes of nucleation.

Keywords: Nucleation transitions; Fast DSC; Cooling rate; Undercooling; Immiscible alloy.

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Introduction Nucleation is the initial and often decisive fundamental process during initial phase formation [1-3] and is controlled by thermal fluctuations. Nucleation during the solidification of metallic melts presents a model situation, since metallic melts are a real-world approximation towards ideal monoatomic liquids. Determination of the nucleation kinetics experimentally, such as the energy barrier or the kinetic pre-factor of the corresponding rate equation, are very important but only scarcely been performed up to the present, due to the absence of experimental data [4-8]. Studies on nucleation have relied for many decades on the classical nucleation theory (CNT) initially developed by Farkas [10] together with Volmer and Weber [9] and as well as extended by Becker, Döring and Turnbull, Fisher [11, 12]. Within that picture, thermal fluctuations govern the formation of critically-sized, homogeneous nuclei of the newly developing phase. At the same time, structural inhomogeneities or impurities or extrinsic substrates such as surfaces or container walls can favor the formation of a critical-sized nucleus, leading to so-called heterogeneous nucleation. It is important to note that still thermal fluctuations control the nucleation process, whether is proceeding via homogeneous – or heterogeneous nucleation. Thus, within this framework, modifications of the synthesis - or processing pathways, as e.g. given by cooling rate variations, are described as sampling within the configurational space of subcritical clusters that can either be kinetically selected or deselected to develop into a “nucleus” of critical size that then grows rapidly into the new phase. Therefore, a kinetic nucleation transition from heterogeneous to homogeneous is expected to appear at a certain range of “critical” cooling rates. This kinetic transition is expected to occur rather continuously, i.e., heterogeneous nucleation would play a key role for cooling rates below the critical rate range while for larger cooling rates homogeneous nucleation appears. It is

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important to note that homogeneous nucleation may become dominant above a certain cooling rate, but heterogeneous nucleation still exists, as shown in [13]. This underlying picture of nucleation has been applied since the development of classical nucleation theory, but until now this transition has rarely been observed experimentally for simple metallic systems – particularly due to the limited range of controlled cooling rates that had been possible to be applied so far. On the other hand for non-metallic systems such as recently for e.g. water in the confinement of nanopores [14], this transition has been experimentally observed. Recently, nucleation kinetics analyses were performed based on the stochastic nature of nucleation and the nucleation rate was obtained from the statistical distribution of undercooling values [4-8, 15-18]. With the recent development of non-adiabatic fast scanning chip calorimetry (FSC), very high controlled heating/cooling rates exceeding 1×104 K/s can be adjusted for calorimetric measurements. The FSC technology has been used to investigate phase transformations, the size-effect on the melting point and enthalpy, as well as the crystallization behavior of materials [19-23]. While most of these works were based on indirect measurements of calorimetric properties, Schick et al. develop a chip-based fast scanning calorimeter that applies the differential scanning calorimetry (DSC) principle to micron-sized samples at high rates [24]. Such FSC offers unprecedented giant of heating and cooling rates ranges that are accessible for controlled experiments of nucleation kinetics based on a statistical approach, since such analyses require data sets with a sufficiently high statistical significance. Thus, applying high rates and performing many measurements per given time allows achieving two goals: high accuracy for the determined nucleation kinetics parameters and a wide rate range (and undercooling range) accessible to investigated the nucleation transition 3



for undercooled melts. In short, with this work we introduce a general strategy to achieve a large cooling rate range covering five orders of magnitude for the immiscible Bi40Ga60 alloy, which provides a wide undercooling window using FSC together with conventional DSC. In addition, the miscibility gap that is present for the chosen model alloy provides specific conditions that favour obtaining homogeneous nucleation conditions [7] and the phase separation transformation that is also initiated by nucleation presents a unique kinetic marker for the competition between the time scale for diffusional rearrangement and the experimental time scale dictated by the applied cooling rate. Based on a “survivorship function” statistical analysis and applying classical nucleation theory, two important nucleation kinetics parameters, the kinetic prefactor (Γ ) and the activation energy (∆G*) were obtained, and a kinetic transformation of the nucleation mechanism from heterogeneous to homogeneous has been experimentally observed.

Experimental The Bi40Ga60 alloy was synthesised by induction melting high purity (better than 99.999 wt.%) Bi and Ga elements under Ar atmosphere. Small-sized droplets with diameters (D) less than 1000 µm were obtained by a drop tube (13 meter) method. The FSC uses a high sensitivity and low addenda heat capacity thin film sensor (XEN-39392) [24, 25]. The cooling rate is effectively enhanced and well controlled by a heat sink at liquid nitrogen temperature. A droplet with D of 76 µm (0.00172 mg, 54872 µm3) was selected and heated at a heating rate of 1000 K/s in argon atmosphere from room temperature to 633 K and cooled down to 300 K at various cooling rates from 100 to 3000 K/s by using the FSC. Identical heating-cooling cycles were performed more than 4


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550 times at each cooling rate to obtain statistically significant data sets [4, 7]. The temperature was calibrated by using an indium particle with D of 20 µm which was placed close to the sample. The detailed calibration method of the FSC can be found in Refs. [26] and [27]. For comparison, a Bi40Ga60 droplet with D of 973 µm (3.6 mg) was selected for conventional DSC measurements. The droplet was sealed in an Al-crucible and heated up to 633 K with the heating rate of 0.83 K/s and cooled down to 370 K with the cooling rates of 0.33, 0.83, 1.67 and 3.33 K/s in Ar atmosphere. Identical heating-cooling cycles were performed more than 120 times at each cooling rate.

Results The undercooling ∆T of melts is usually defined as, ∆T = Tl - Tn, where Tl is the liquidus temperature and Tn is the nucleation temperature, i.e. for a given composition of the material. However, for the present Bi-Ga immiscible alloys where two different liquid phases occur during melting, consequently two undercooling values, corresponding to the liquidus temperatures of each liquid phase, have to be considered respectively. A similar behaviour has also been analysed for a Cu70Co30 immiscible alloy [7]. Thus the effective undercooling value of the Bi-rich phase (∆TBi-rich) in Bi40Ga60 with a liquid phase separation was determined by using the same method as described in Ref. [7] for the Co-rich phase in the Cu70Co30 immiscible alloy [7] based on the phase diagram of Bi-Ga alloys [28], i. e., ∆TBi-rich = Tl-(Bi-rich) - Tn (Bi-rich), where Tl-(Bi-rich) is the liquidus temperature of the actual composition of the Bi-rich liquid phase and Tn (Bi-rich) is the nucleation temperature, as illustrated in the left part of Fig. 1 (red dash-dotted rectangle). Additionally, a direct solidification without a liquid phase separation in the Bi40Ga60 melt occurred at cooling rates higher 1000 K/s (detailed explanation will be given in the discussion section). In this case the effective undercooling 5



∆TBi-rich was obtained by the temperature difference between the monotectic temperature Tmon = 222oC [28] and Tn (Bi-rich), [∆TBi-rich = Tmon - Tn (Bi-rich)], as shown in the right part of Fig.1 (blue dash-dotted rectangle). The values of ∆TBi-rich for Bi40Ga60 alloy which was undercooled with a cooling rate of 1000 K/s by FSC for 550 nucleation events and 1.67 K/s by DSC for 120 nucleation events, are shown in Fig. 2 (a) in chronological order, respectively. The absolute error of the temperature measurement in DSC measurements is less than ± 0.1 K. For FSC measurements, the accuracy of the temperature measurements, as obtained from calibration measurements with pure metal standards and also by measuring the rate-independent melting temperature of the alloy, amounts to ± 3 K. More information about the calibration procedure for the FSC measurements and on the determination of the experimental accuracy is given in the supplemental information. The obtained undercooling values at the cooling rate of 1000 K/s are obviously higher than those obtained at the cooling rate of 1.67 K/s. The cooling rate dependence of the average undercooling (∆Tav) of the Bi-rich phase is shown in Fig. 2 (b). The value of ∆Tav increases continuously and linearly with increasing cooling rate until reaching a cooling rate of approximately 1500 K/s. At that rate, the linear regressions of the data at the highest cooling rates and at cooling rates between 100 K/s and 1000 K/s do not match, since the two linear fits are separated by an increment in undercooling of about 30 K. This behavior is indicated in Fig. 2 (b) and will be discussed below. Nucleation events are stochastic events that are independent of each other and they appear randomly in time. Therefore, the nucleation process can be adequately described by inhomogeneous Poisson statistics and a corresponding “survivorship function” [4-7, 29-31], see supplemental material. The survivorship function that can be described as the fraction of experiments where crystallization did not yet happen at a given undercooling, ∆T, can be obtained 6


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as Fsur (∆T). The Fsur as a function of effective undercooling ∆TBi-rich for different cooling rates between 0.33 and 3000 K/s are displayed in Fig. 3 (a). Figure 3 (b) shows the comparison between the probability density distributions obtained from the experimental undercooling data measured at a cooling rate of 0.33 K/s by DSC and at 1000 K/s measured by FSC together with the theoretical fits based on describing nucleation as an inhomogeneous Poisson process. The excellent fit between experimental data and fit indicates the statistical significance of the obtained data sets as well as the appropriateness of describing nucleation in the framework of an inhomogeneous Poisson statistics. The survivorship functions shift to higher ∆T smoothly with increasing cooling rate. By employing the fact that nucleation events are independent in time and space and follow inhomogeneous Poisson statistics [4-6], the product of the nucleation rate J and the sample volume V can be directly determined from the following equation:

(

)

Fsur (∆T ) = 1 − exp − ∫ V ⋅ J (∆T ) dT .

(1)

The nucleation rates determined from Fsur (∆T) can provide high statistical accuracy [4-7]. The accuracy of the individual measurements was obtained by calibration measurements on pure In, Sn and Pb. Melting temperatures were determined with a relative accuracy of 0.15 K and an absolute accuracy of 1 K for conventional DSC experiments. The computed results of J as a function of ∆TBi-rich for Bi40Ga60 samples that were undercooled at different cooling rates are shown in Fig. 4 (a). The value of J increases continuously with increasing ∆TBi-rich, indicating that only one type of nucleation site in the Bi40Ga60 melts is active. The J(∆TBi-rich) curves at high cooling rates are almost parallel to each other. The value of J presents a fast increase with increasing undercooling for the cooling rates lower than 200 K/s. However, it increases only slightly for the cooling rates higher than 1000 K/s, although ∆TBi-rich increases 7



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continuously. This kinetic transition is likely due to a nucleation transition from heterogeneous at lower rates to homogeneous at higher rates, as discussed below. The present results also indicate that cooling rate and undercooling have very weak effects on the nucleation rate when homogeneous nucleation occurred in the undercooled Bi40Ga60 melts due to the abundant availability of possible nucleation sites that populate the entire configuration space and depend on thermal fluctuations only. According to classical nucleation theory (CNT), the steady state nucleation rate J for most undercooled metallic melts can be described as [4-17]:  ∆G *   BT 2  , J = Γ exp  − = Γ exp  − 2   k BT   k B ∆T 

(2)

where Γ is the kinetic prefactor, ∆G* is the activation barrier for the formation of a nucleus of critical size, kB is Boltzmann’s constant, B is proportionality factor. For a spherical cap that nucleates on a substrate via heterogeneous nucleation, ∆G* can be calculated as [4-7, 32]: ∆ G* =

16π σ 3 ⋅ ⋅ f (θ ) . 3 ∆GV2

(3)

σ is the interfacial energy between liquid and solid which is generally approximated as a linear function of temperature, f(θ) is the catalytic potency factor, θ is the contact angle, ∆Gv is the driving force for crystallization which can be approximated as ∆GV = ∆H f ⋅ ∆T / Tm , and ∆Hf is the enthalpy of fusion. Here the nucleation rate J per unit volume was analysed for the present Bi40Ga60 melts. A linear relation of ln J vs. T3 / ∆T2kBT plot can be observed, and then the intercept lnΓ and the slope B can be evaluated. The values of ∆G* are computed from the J(∆T) curves using Eq. (1) for all cooling rates. The evaluated values of Γ (left hand scale) and ∆G* (right hand scale) as a function of cooling rate are given in the inset of Fig. 4 (b). We can see that the values of ∆G* and 8


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Γ increase with the increase of the cooling rate, and Γ has a saturation tendency at high cooling rates. At cooling rates around 1000 K/s, the value of Γ and ∆G* deviate from the smooth cooling rate dependence, which is probably due to the coexistence and competition of the liquid phase separation and direct solidification without previous phase separation to occur (see below). The average value of the relative undercooling with respect to Tl (liquidus temperature), ∆Tav/Tl, increases continuously from 0.21 to 0.31 with the cooling rate increasing from 1000 to 3000 K/s, with all values larger than the critical ∆Tav/Tl value of 0.2) which is usually associated with the precondition for homogeneous nucleation [29]. Additionally, the values of Γ and ∆G* for cooling rates higher than 2000 K/s are also close to the values proposed for homogeneous nucleation (1039 m-3 s-1 and 60 kBT), respectively [33, 34]. These results clearly indicate that a nucleation transition from heterogeneous to homogeneous appears in a rather continuous way at cooling rates of about 2000 K/s for the present undercooled Bi40Ga60 melts. I. e., heterogeneous nucleation plays a key role for the cooling rates lower than 1000 K/s, while for cooling rates larger than 2000 K/s, homogeneous nucleation appears. It should be noted that this apparent transition appears within the regime of cooling rates that are accessible only by fast chip calorimetry and it occurs for the same sample. Thus, the observed transition is not related to any change of the measurement setup, protocol or sample.

Discussion The continuous nucleation transition from heterogeneous to homogeneous nucleation for Bi40Ga60 undercooled melts might appear unexpected at first glance. In fact, for a kinetic transition, a rather continuous behaviour is expected to occur upon cooling rate variation, as for example 9



well-known for the glass transition that is commonly described as a kinetic transition from a metastable liquid to a kinetically frozen non-equilibrium state. For the nucleation transition to occur, the cooling rate range translates into a range of critical times (or diffusion distances), meaning that for nucleants that have a smaller catalytic potency, i.e. which require more atoms to form the critical nucleus, at almost constant atomic mobility, the time required for the critical nucleus to form on such a heterogeneous substrate of lower catalytic potency is longer, resulting in larger undercooling, since both, homogeneous or heterogeneous nucleation are based on thermal fluctuations within the cluster configuration space. This scenario is in complete agreement with the observed experimental results of a continuous increase of undercooling with the increases of cooling rate. Thus, depending on the statistical availability of heterogeneous nucleation sites and their size/potency near the transition to homogeneous nucleation, either mechanism obtains similar probability, which results in an increased width of the transition region. In the regime of intermediate cooling rates where the kinetic transition from heterogeneous to homogeneous nucleation happens, apparently a kinetic nucleant de-selection occurs over a range of cooling rates by applying increasing cooling rates. At first sight, this observation might indicate the presence of a spectrum of different nucleation sites with an associated spectrum of catalytic potencies that are similar in magnitude. Thus, applying a higher cooling rate would serve to deselect slightly less potent nucleants and to “activate” nucleation by a nucleant that becomes active at larger undercooling, i.e. that presents higher values for Γ and ∆G*. Yet, the observation of a smooth cooling rate dependence of the nucleation rate does not support that interpretation. However, at such high cooling rates as applied here in the transition region between heterogeneous and homogeneous nucleation, the residence time of the undercooled melt in a given temperature 10


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interval at high undercooling becomes sufficiently small to effectively limit the diffusion flux towards heterogeneous nucleation sites. In order to test this assumption, the so-called “stationary interface growth model” proposed by Maxwell and Hellawell [35] was applied to yield the maximum size of a growing spherical particle under the given conditions, thus yielding an estimate of the largest radius, r, a nucleation site could reach for a given cooling rate. The maximum “possible” radius r of a spherical particle growing under diffusion-controlled growth can be given by:

r = λ ( D ⋅t )

1/2

(4)

where t is the growth time, t = ∆T/Ṫ (Ṫ is the cooling rate), D is the liquid solute diffusion coefficient, λ is the interfacial parameter which was estimated by: 1/2

 S   S2  λ =  − 1/2  +  −S  2π   4π 

.

(5)

and S is given by

S = 2(cl − c0 ) / (cs − cl ) ,

(6)

in which cl, cs, and c0 are the solute content at solid-liquid interface, the solid at the solid-liquid interface, and the bulk liquid, respectively. With the diffusion coefficient of Bi-rich liquid in a Ga-rich matrix D = 10-9 m2/s, cl and cs can be evaluated from the phase diagram [28]. The reciprocal radius, 1/r as a function of the cooling rate is given in Fig. 5 (left and top scales), the dashed line is a guide to the eyes. The value of 1/r increases abruptly with the increase of the cooling rate and shows a maximum around the cooling rate of 1000 K/s, and then decreases with further increase of the cooling rate. In general, the growth time decreases with the increase of the cooling rate, which results in a decrease of the radius, i.e., an increase of 1/r. Thus, together with the changes in ∆Tav that are specific for immiscible systems, the cooling rate dependence of 1/r indicates that the 11



nucleation mechanism of Bi40Ga60 changed at a cooling rate around 1000-1500 K/s. Under the assumption, that solidification at the highest undercooling level was initiated by homogeneous nucleation, classical nucleation theory can be applied to derive the interface excess energy density, σ(T), (often termed as “interface energy”) as a function of temperature (i.e. undercooling). In cases where heterogeneous nucleation initiated solidification, only the product of

σ(T) f(θ)1/3 can be determined due to the unknown catalytic substrate. In this particular case, where a transition between two nucleation regimes occurs, it might even be expected that the values for f(θ) change as a function of ∆T, which renders any analysis of σ(T) or f(θ) ambiguous. For the set of data that corresponds, according to our interpretation, to homogeneous nucleation, σ(T) can in principle directly be determined from the data if classical nucleation theory is applied. Due to the relatively narrow temperature regime where these conditions are supposed to hold and due to the involved experimental uncertainties, we have only determined an average value of σ(T) = 6.7 x 10-2 Jm-2 ± 4.9 x 10-2 Jm-2. This value was calculated for the conditions during cooling at 2000 K/s and lies within the expected range. However, one should be careful since a large uncertainty is introduced by deriving σ from the nucleation barrier, specifically since the melting entropy and the molar volume of the liquid at the respective undercooling temperature also include large uncertainties. For the heat of fusion, a value of about 50 ± 10 J/g was also determined that presents a value in the expected range for this alloy, if the respective values of the pure constituents are regarded. Yet, for such a monotectic system with large differences between the melting temperatures of the pure constituents, firstly it is expected that deviations from simple solution approaches occur and secondly, the accuracy of any experimental determination suffers from the large temperature range 12


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over which the signal is integrated, yielding a comparably large uncertainty. Thus, this value should be treated with some caution. A possible explanation is based on the assumption that the Bi40Ga60 melt solidified directly without a liquid phase separation at cooling rates higher than 1000 K/s. ∆Tav and 1/r were recalculated and the results are also shown in Fig. 5. The lines are again guides to the eye. It should be noted that the data obtained under the assumption of a phase separation occurring before nucleation overlaps with the data calculated for partitionless nucleation conditions up to a cooling rate of about 1000 K/s. The value of 1/r for the case that phase separation still occurs prior to nucleation also at higher cooling rates increases abruptly with increasing cooling rate, and has a maximum around the cooling rate of 1000 K/s. It then decreases with further increasing cooling rate (dash-dotted line). Generally, with increasing cooling rate, the growth time decreases, which results in a decrease of the radius, i.e, an increase of 1/r, indicating further that at about 1000 K/s a kinetic transition occurs due to the limited time for diffusion. As mentioned above, in this specific situation with a liquid-liquid miscibility gap present, not only nucleation of the crystalline phase is affected, but also the nucleation of the liquid-liquid phase separation that is also diffusion-controlled. If the phase separation is kinetically avoided, then the undercooling needs to be determined differently (see Fig. 1 and Fig. 5). As a result of this analysis, the ∆Tav values determined under the assumption of a kinetic avoidance of the phase separation increase linearly with increasing cooling rate, and 1/r increases abruptly at low cooling rates and has a saturation trend at cooling rates higher than 1000 K/s which is consistent with results obtained for several alloy systems, such as: Al-Ti-B, Mg-3Al-Zn or as-cast Al alloys [36-38]. This result and particularly the physically reasonable cooling rate dependence of ∆Tav and 1/r obtained under the 13



assumption that nucleation proceeded without prior phase separation indicate that the calculation method is reliable and further strengthen the hypothesis that the Bi40Ga60 melt solidified without liquid phase separation at high cooling rates. This result indicates that for this range of cooling rates and the associated undercooling levels the intrinsic time scales for long-range diffusional rearrangements and the experimental time scale dictated by the cooling rate become comparable, so that nucleation at existing heterogeneities and nucleation based only on thermal fluctuations of atomic configurations in the melt, i.e. heterogeneous - and homogeneous nucleation, are of similar probability.

Conclusions In summary, the repeated nucleation and crystallization behavior of undercooled Bi40Ga60 immiscible alloys have been investigated using ultra-fast scanning calorimetry together with conventional scanning calorimetry. Statistical analyses of the nucleation rate for the crystallization of the primary Bi-rich phase were performed independently for different cooling rates during 550 and 120 identical, subsequent cycles for FSC and conventional DSC, respectively. The results indicate that the Bi40Ga60 melt solidified directly without liquid-liquid phase separation for cooling rates higher than 1000 K/s. The quantitative analyses of the kinetic prefactor Γ and the activation energy ∆G* obtained from nucleation rate measurements and by applying classical nucleation theory indicate a continuous kinetic transition from heterogeneous to homogeneous nucleation for cooling rates around 1000 - 2000 K/s. Such a continuous transition that can be described as a cooling rate-dependent selection of the nucleation mechanism was experimentally observed. It is important to note that this cooling rate regime is also accessible in real-world processing, e.g. 14


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during added manufacturing and 3D printing, and thus presents a usable “tuning fork” for heterogeneous nucleant selection/de-selection and the corresponding microstructure formation, even in industrial processing environments.

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TOC-Synopsis

Continuous transformations of the nucleation mechanism in the undercooled state Yikun Zhang, Christian Simon, Thomas Volkmann, Matthias Kolbe, Yong Lei, Lingwei Li, and Gerhard Wilde

The repeated nucleation and crystallization behaviour of undercooled Bi40Ga60 immiscible alloys was investigated by using ultra-fast scanning calorimetry and conventional scanning calorimetry. A continuous kinetic transition from heterogeneous to homogeneous nucleation experimentally observed, which can be described as a cooling rate-dependent selection of the nucleation mechanism.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Nos. 51501036 and 51690162), Alexander von Humboldt (AvH) Foundation of Germany (Research stipend to Y. Zhang), Deutsche Forschungsgemeinschaft (G. Wilde and C. Simon), and Independent Research and Development Project of State Key Laboratory of Advanced Special Steel, Shanghai University. GW would like to thank Prof. C. Schick for his help with setting up the FSC.

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Supporting information I. Nucleation rate determination based on Poisson statistics, and II. Calibration procedures and data evaluation.

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Figure captions Figure 1. Bi-Ga phase diagram and designation of undercooling of the Bi-rich phase for a Bi40Ga60 immiscible alloy with and without liquid phase separation occurring prior to solidification. Figure 2. (a) The nucleation undercooling measured during 550 (top scale) at a cooling rate of 1000 K/s and 120 identical solidification experiments at a cooling rate of 1.67 K/s performed by FSC and DSC, respectively. (b) The cooling rate dependence of average undercooling, ∆Tav, for the Bi40Ga60 immiscible alloy. The dash-dotted lines indicate linear fits to the data obtained in different cooling rate ranges. Figure 3. (a) The survivorship function Fsur (∆T) as a function of undercooling ∆TBi-rich for various cooling rates between 0.33 and 3000 K/s obtained on the Bi40Ga60 immiscible alloy. (b) Probability density of undercooling obtained at cooling rates of 0.33 K/s by DSC and 1000 K/s measured by FSC. The filled circles represent the probability density obtained based on an analysis of the cumulative distribution function. The solid line represents the best fit to the data. Figure 4. (a) The nucleation rate J as a function of undercooling ∆TBi-rich for cooling rates ranging from 0.33 to 3000 K/s obtained on the Bi40Ga60 immiscible alloy. (b) The undercooling ∆TBi-rich dependence of the kinetic prefactor Γ (left hand scale) and the nucleation barrier ∆G* (right hand scale) for various cooling rates between 0.33 and 3000 K/s for the Bi40Ga60 immiscible alloy. Figure 5. Cooling rate dependence of the average undercooling, ∆Tav (right and bottom scales) and the reciprocal of the growth radius, 1/r (left and top scales) for Bi40Ga60 which was calculated with (solid lines) and without (dash lines) taking a prior liquid-liquid phase separation (P. S.) into account.

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(a) The nucleation undercooling measured during 550 (top scale) at a cooling rate of 1000 K/s and 120 identical solidification experiments at a cooling rate of 1.67 K/s performed by FSC and DSC, respectively. (b) The cooling rate dependence of average undercooling, ∆Tav, for the Bi40Ga60 immiscible alloy. The dashdotted lines indicate linear fits to the data obtained in different cooling rate ranges. 177x260mm (300 x 300 DPI)



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Continuous Transformations of the Nucleation Mechanism in the

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