Dependence of crystal nucleation on prior liquid overheating by ...

Dependence of crystal nucleation on prior liquid overheating by differential fast scanning calorimeter Bin Yang, John H. Perepezko, Jürn W. P. Schmelzer, Yulai Gao, and Christoph Schick Citation: The Journal of Chemical Physics 140, 104513 (2014); doi: 10.1063/1.4868002 View online: http://dx.doi.org/10.1063/1.4868002 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/10?ver=pdfcov Published by the AIP Publishing

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THE JOURNAL OF CHEMICAL PHYSICS 140, 104513 (2014)

Dependence of crystal nucleation on prior liquid overheating by differential fast scanning calorimeter Bin Yang,1 John H. Perepezko,2 Jürn W. P. Schmelzer,1 Yulai Gao,3 and Christoph Schick1,a) 1

Institute of Physics, University of Rostock, Wismarsche Str. 43-45, 18051 Rostock, Germany Department of Materials Science and Engineering, University of Wisconsin-Madison, 1509 University Avenue, Madison, Wisconsin 53706, USA 3 School of Materials Science and Engineering, Shanghai University, Shanghai 200072, People’s Republic of China 2

(Received 8 January 2014; accepted 26 February 2014; published online 13 March 2014) The degree of overheating of a melt often plays an important role in the response of the melt to subsequent undercooling, it determines the nucleation and growth behavior and the properties of the final crystalline products. However, the dependence of accessible undercooling of different bulk melt samples on prior liquid overheating has been reported to exhibit a variety of specific features which could not be given a satisfactory explanation so far. In order to determine uniquely the dependence of accessible undercooling on prior overheating and the possible factors affecting it, the solidification of a pure Sn single micro-sized droplet was studied by differential fast scanning calorimeter with cooling rates in the range from 500 to 10 000 K/s. It is observed experimentally that (i) the degree of undercooling increases first gradually with increase of prior overheating; (ii) if the degree of prior superheating exceeds a certain limiting value, then the accessible undercooling increases always with increasing cooling rate; in the alternative case of moderate initial overheating, the degree of undercooling reaches an undercooling plateau; and (iii) in latter case, the accessible undercooling increases initially with increasing cooling rate. However, at a certain limiting value of the cooling rate this kind of response is qualitatively changed and the accessible undercooling decreases strongly with a further increase of cooling rate. The observed rate dependent behavior is consistent with a kinetic model involving cavity induced heterogeneous nucleation and cavity size dependent growth. This mechanism is believed to be relevant also for other similar rapid solidification nucleation processes. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4868002] I. INTRODUCTION

The degree of overheating of a melt prior to its subsequent undercooling and resulting crystallization has been demonstrated to play a significant role in determining the solidification behavior, in particular, for the nucleation process and the accessible undercooling.1–4 Yet, our understanding of the relationship between prior overheating and undercooling is still rather incomplete. This statement refers especially to rapid solidification processes and is due in part to limitations of measurement capability. On the other hand, based on the assumption of the existence of cavities, Turnbull5 developed a theory of cavity induced heterogeneous nucleation in order to describe typical features of such processes. His approach predicts a linear relationship between accessible solidification undercooling (T − ) and prior liquid overheating (T + ). Here T + = Tmax − Tm is the difference between the maximum temperature, Tmax , reached in prior overheating of the liquid and the melting temperature, Tm , T − = Tm − Ts is the difference between melting temperature and the particular temperature Ts where rapid solidification is observed to initiate experimentally. The meaning of both these parameters is

a) Electronic mail: [email protected]

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illustrated in more detail in Fig. 1(a) for one particular heating and cooling run. Turnbull’s predictions have been tested in a variety of experimental analyses. Most of these studies, however, only address nucleation at low cooling rates measured by conventional thermal analysis devices2, 6, 7 or without cooling rate control.5, 8 However, cooling rate is known to significantly affect the undercooling.9, 10 Moreover, most of the former studies examined bulk samples of the melt where extraneous nucleants of different kind can control the response of the system to undercooling. Hence, the relationship between T − and T + has not been specified or explained in a sufficiently comprehensive form. With the development of nanotechnology and microelectro mechanical systems (MEMS), it is possible to fabricate calorimeter sensors that are able to measure samples with nanogram masses and energies less than one nanojoule11–13 or even picojoule.14 These developments enable the possibility of considerably enlarging and controlling the scanning rates. In particular, Zhuravlev et al.15, 16 developed a power compensated differential fast scanning calorimeter (DFSC) and analyzed the heat capacity change during scanning. This approach provides a good chance to approach an in situ undercooling measurement at high cooling rates spanning four orders of magnitude for one single droplet.9, 10, 17, 18 In

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FIG. 1. (a) Typical temperature profile (blue line) and measurement curve (black line) and (b) photograph of the single Sn droplet in the center of the measuring area of the calorimeter sensor prior to the measurements. The simple cartoon of the Sn droplet (with an oxide layer) is shown in the inset.

the present paper, extending our earlier brief report,19 we present the results of application of this method to measure the accessible undercooling at the solidification of one single Sn droplet at a series of cooling rates between 500 and 10 000 K/s. The drop has a diameter of 23.5 μm and is coated with an oxide layer. The experimental results on the dependence of undercooling on liquid overheating at different cooling rates are interpreted here based on a modification of the cavity induced heterogeneous nucleation mechanism as developed by Turnbull. The paper is structured as follows. In Sec. II, we present a brief description of the experimental procedure. The results of the measurements are given in Sec. III. A theoretical analysis and interpretation of the experimental data is given in Sec. IV. The main conclusions of the present analysis are summarized in Sec. V.

II. EXPERIMENTAL PROCEDURE

Micro-sized Sn (5N) droplets were prepared by the consumable-electrode direct current arc (CDCA) technique in various sizes.9, 20 The droplets showed a nearly perfect spher-

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ical shape, as shown in Fig. 1(b) They are always covered by an oxide layer. This fact can be understood easily realizing that Sn oxide has a large heat of formation (H for SnO2 = −577 kJ/mole). Since the loading of the sample into the DFSC involves exposure to air it is virtually impossible to obtain an atomically clean Sn surface. DFSC10, 15, 16, 20 based on thin film sensors was employed to investigate the relationship between T − and T + . The nanocalorimeter sensors, XEN39395 (Xensor Integrations, Netherlands), consist of an amorphous silicon-nitride membrane with a film-thermopile and a resistive film-heater placed at the center of the membrane. These sensors can effectively promote high heating rates through a preferable ratio between addenda and sample heat capacity and applicable heating power. Under non-adiabatic conditions the same holds for high cooling rates, even though the cooling capability of the whole system is restricted due to a finite heat transfer rate from the sample. For a membrane sample system, the heat transfer is limited by the thermal properties of the surrounding cooling agent.12, 21 In a μm scale system, the most efficient cooling agents are gases because of their small heat capacity.22 The power compensation scheme of the DFSC15, 20 takes care of the heat losses and both sensors follow the predefined temperature program very closely (the temperature difference between reference (TR ) and sample sensor (TS ) obeys the inequality, TR −TS < 0.1 K) even at 104 K/s. Considering these circumstances, we have chosen a single droplet with a diameter of 23.5 μm and placed it into the center of the heating zone of the sensor using an optical microscope (Olympus SZX16), as shown in Fig. 1(b). To improve the thermal contact between the sample and the membrane silicone oil was used. The micro-sized droplet was heated up in air to the desired maximum temperature (Tmax ) above the melting temperature at a fixed heating rate of 2300 K/s. After holding the droplet at that maximum temperature for a period of time, th , i.e., 10, 5, 0.7, and 0 ms, respectively, the droplet was cooled down to 310 K at cooling rates in the range from 500 to 10 000 K/s. The heatingcooling cycles were repeated 10 times for each temperature profile, i.e., each Tmax , cooling rate and holding time, respectively, in order to appropriately account for the stochastic nature of the nucleation process. It should be pointed out that the outer layer of the Sn droplet did not break when the inner metal Sn expands as temperature increases. The reason is that for SnO2 the Pilling-Bedworth ratio (volume of oxide to volume of metal oxidized) is equal to 1.3. Values of the Pilling-Bedworth ratio in the range between 1 and 2 are considered as protective and adherent. On the other hand, the melting temperature did not change in the course of the experiments. This result implies that during the whole experiments the Sn droplet keeps stable. III. EXPERIMENTAL RESULTS

A typical DFSC curve showing the differential temperature between sample and reference sensors versus time is shown in Fig. 1(a). The onset temperatures of the melting peak (plotted downwards) and solidification peak (plotted upwards) were taken as the melting (Tm ) and solidification temperatures (Ts ), respectively. The thermal lag between the


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FIG. 3. Undercooling dependence on cooling rate and overheating: solid curve – surface nucleation9 and dashed curve – linear fit (for details see text). The inset shows a close-up of the undercooling plateau (99 ± 2 K). Each point is the average of 10 identical measurements and the error bars are the standard error of 10 identical measurements for each temperature course.

FIG. 2. Dependences of undercooling on overheating for different cooling rates and constant holding time at Tmax (a) and for fixed heating and cooling rates but different durations of melt overheating at Tmax (b). Each point is the average of 10 identical measurements and the error bars are the standard error of 10 identical measurements for each temperature course.

regular shaped spherical sample and the membrane is directly proportional to the scanning rate. Therefore the melting and solidification temperatures were corrected according to the procedure detailed in Ref. 20. The thermal lag occurring during melting is also responsible for the broadening of the melting peak compared to the solidification peak, which is not heat transfer limited because of the large undercooling. As already mentioned briefly, the undercooling T − is calculated by T − = Tm − Ts , and the overheating T + is determined as T + = Tmax − Tm . In Fig. 2, the dependence of the mean value of the accessible undercooling of the single Sn droplet on process conditions is shown. In Fig. 2(a), the results are presented for the case of various cooling rates for different overheating at a constant holding time of 5 ms. Fig. 2(b) shows the respective results for fixed heating and cooling rates, but different durations of overheating at which the melt was held at Tmax . It is found that, in both cases above a critical overheating level, a

plateau develops for the undercooling at 99 ± 2 K. As shown in Fig. 2(a), a strong decrease of accessible undercooling is found when the overheating is lower than some critical value which is slightly dependent on cooling rate, e.g., equal to ∼26 K for a cooling rate of 10 000 K/s. It is also found that the limiting overheating for reaching the undercooling plateau in Fig. 2(b) decreases with increasing holding time. Fig. 3 shows the relationship between the measured undercooling and the cooling rate. Note that for overheating below 14 K it was not possible to obtain the undercooling at higher cooling rates because of the melting kinetics, i.e., in these cases the droplet solidified immediately during the subsequent cooling because it was not enough time to fully melt the droplet matrix. As a consequence, in such cases the undercooling is equal to zero. It should be pointed out that the time the sample stays above the melting temperature decreases with increasing cooling rate. It is found that the undercooling for the overheating of 31 K and above increases moderately with increasing cooling rate which is considered as a plateau within the range of ∼4 K. Such kind of dependence is the usually expected behavior,9, 10 as will be also demonstrated in Sec. IV. For overheating of 21 K and below, which is below the onset of the undercooling plateau at 99 ± 2 K, however, above a certain cooling rate the undercooling decreases strongly with increasing cooling rate. Furthermore the cooling rate for this observed transition (qt ) decreases with decreasing overheating. IV. THEORETICAL INTERPRETATION

The experimentally observed dependences (mainly shown in Fig. 3), i.e., that above a certain value of the cooling rate the accessible undercooling decreases strongly with a further increasing of the cooling rate, cannot be completely described by the model derived in Refs. 9 and 10. This model yields a shelf-like dependence of crystal nucleation on undercooling. On the other hand, Tong et al.6, 7 investigated the effect of thermal history on this dependence. They invoked a


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time dependent structural change in the liquid as the explanation. However, the nature of heterogeneous nucleation behavior and the role of regrowth were ignored in this approach. In the present study, in order to understand such kind of behavior, we assume, advancing our model described in detail in Refs. 9 and 10, the existence of the cavity induced heterogeneous nucleation mechanism as suggested in its principal features first by Turnbull.5 In this approach, followed also in our earlier brief communication,19 the thermal history effect on the value of the accessible undercooling can be explained by considering the details involved in the evolution of a cavity generated nucleus in differently sized cavities during cooling. In the development of this model, the following assumptions are made about the nucleation mechanism which is expected to dominate the heating-cooling process. Let us consider a cylindrical or conical cavity on the droplet surface in its oxide layer. This, of course, is an idealized geometry since the liquid-oxide layer interface is likely to exhibit an irregular roughness. Unfortunately, we are not aware of any technique capable to characterize the surface topology between the liquid tin and the oxide layer. It should be pointed out that if the Sn oxide layer were thick and rigid there may be an effect on the undercooling. In other words, a thick, rigid, and adherent oxide would undergo different expansion and contraction as the temperature changes and this could exert a pressure on the Sn. However, this effect would also influence the melting temperature which did not change over the course of our experiments. Thus, the effect of the oxide aside from the role in catalyzing nucleation seems to be negligible. Therefore the idealized geometry will be shown to account for all of the main features of the observed overheating-undercooling relationship. During the heating process some unmelted solid is retained in these small cavities whose size are typically of the order of nano-size, as schematically shown in Fig. 4.5, 19 The melting temperature of nano-sized particles in an oxide matrix, contrary to free particles, in some cases increases when the particle size decreases or by applying a very high heating rate.23 This is due to the fact that particles embedded in an oxide matrix will be subject to hydrostatic pressure since oxides have a lower coefficient of thermal expansion (CTE) as compared to metals. This effect results in an increase of the melting temperature Tm according the Clausius-Clapyeron relation. However, in the case of cavities due to the curvature of the liquid-solid interface as illustrated in Fig. 4,5, 8 the melting temperature of the residual Sn crystal in small cavities is higher than the bulk melting temperature. It should be pointed out that for each overheating with a fixed holding time there is a stable position of the solid-liquid interface determined by the maximum value of the radius, γ , of the conical or cylindrical cavities. During the subsequent cooling, the residual Sn crystal can regrow and become a nucleus if it reaches the mouth of the cavity. However, the residual Sn crystal will not act as a nucleus upon cooling unless the radius of the mouth of the cavities equals or exceeds the critical radius for heterogeneous nucleation (see Fig. 4). As the overheating increases, the amount of solid material inside the cavity decreases or even disappears for overheating larger than 26 K. Consequently, for the residual Sn

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FIG. 4. Schematic illustration of remaining crystals in conical and cylindrical cavities.5, 19

crystal material in the cavity, the distance to reach the cavity top increases so that an increasing time for regrowth is required. Similarly, as the cooling rate increases, there is less time available for the solid to grow and to fill the cavity completely. As a result, the usual nucleation behavior dominates. For moderate overheating (21 K or less), a residual Sn crystal remains in the cavity. During the subsequent cooling at a modest rate, the residual Sn crystal in the cavities will regrow to the mouth of the cavities. The height, h (see Fig. 4), of the remaining solid in the cavity increases with decreasing overheating, so that the time for regrowth the solid to reach the top and to act as a nucleus becomes shorter. In order to explain the different changing trend of undercooling below and above the transition cooling rate (shown in Fig. 3) we must also consider the actual size of the cavities. For the nanometer size range nanowire growth rate measurements show that the growth rate decreases with decreasing nanowire diameter. This is related to the reduction in driving force at constant undercooling due to the Gibbs-Thomson effect.24 Thus, let us consider this case with higher overheating. As a result, the radius of the remaining partly filled cavities is smaller which yields a slower growth rate. Meanwhile, the amount of residual Sn crystal in the cavity will be reduced too. Consequently the residual Sn crystal needs more time for the regrowth to the mouth of the cavity and to act as a nucleus. Both of these mutual effects would yield the larger undercooling with increasing overheating, and vice versa. In other words, the nucleation due to the regrowth of remaining solid in the cavities is more sensitive to cooling rate than the usual nucleation behavior. In order to determine the critical cavity size for different overheating (below 26 K), we make two assumptions. (i) When the cooling rate equals qt (Fig. 3), the nucleation process is controlled by the regrowth of the remaining solid Sn crystal just at the bottom of the cavity. This means that during cooling the remaining solid Sn crystal regrows from the bottom to the top of the cavity and then acts as a nucleus. This would yield the undercooling at qt ≈ 99 ± 2 K (undercooling plateau) which is shown as the solid curve in Fig. 3. (ii) According to the results of previous investigation (detailed in Refs. 9 and 10), the nucleation of the single Sn droplet is caused by both bulk and surface heterogeneous nucleation. For the cavity induced nucleation of the single Sn droplet, the surface nucleation mechanism is dominating due to the cavities on the droplet surface, and the total number of nuclei,


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N, formed in the system in cooling it from Tm down to Tm − T − is given by − 1 T N= JS SdT − , (1) qc 0 where JS is the surface nucleation rate; qc is the cooling rate; and S is the surface area of the droplet. The crystallization of the droplet proceeds fast, when the first supercritical crystal nucleus is formed. By this reason, the accessible undercooling is determined by Eq. (1) if in this relation N is equal to one (N = 1). The expression for the nucleation rate of the single droplet during a rapid cooling process on the surface of the droplet was given in Ref. 9 as nS (1 − cos θ0 ) f (θ0 ) , exp − JS = (Tm − T − ) (T − )2 (T − )2 (2) with 1 (3) f (θ0 ) = (2 − 3 cos θ0 + cos3 θ0 ), 4 FIG. 5. Schematic illustration of the change of the contact angle.

Rc =

2σls Tm , HV T −

(4)

where nS = CiS /a02 is the number of heterogeneous nucleation sites of the catalytic surface per unit surface. This number can be also denoted as the density of nucleation sites (CiS is the surface impurity concentration); and are the constants related with the physical properties (detailed in Ref. 9); Rc is the critical radius of nucleus; θ is the contact angle between the undercooled melt and the heterogeneous nuclei; f(θ 0 ) is the catalytic activity which describes the catalytic potency of the heterogeneous nuclei and θ 0 is the constant contact angle on the surface when the critical radius of the crystal nucleus is less than the radius of the heterogeneous nucleation site with fixed size, 2RS for surface nucleation, as shown in Fig. 5. For a small radius of the nucleus, the contact angle, θ 0 is constant (Fig. 5(a)). The contact angle does not change with increasing radius of the nucleus (Fig. 5(b)), until the border of the nucleus reaches the edge of the nucleation site. There are two contact angles in this case for the same radius of nucleus: θ 0 and π − θ 0 (Figs. 5(c) and 5(d)). Undoubtedly, this case is realized only if θ 0 < π /2. When the critical radius of nucleus Rc is larger than the crossover radius Rθ = RS /sinθ 0 , the contact angle increases sharply (Fig. 5(e)). In such case, the catalytic activity of such nucleation site diminishes. For pure tin, Tm = 505 K, σ ls = 84 × 10−3 J/m2 , HV = 4.4 × 108 J/m3 , Dl = 2.7 × 10−9 m2 /s, and a0 = 2.81 × 10−10 m.9 According to Eqs. (1)–(4), these data are employed in the computations shown in Fig. 3 (solid curve) for the undercooling plateau. Then the fitting results can be obtained as: nS = 3.73 × 1014 m−2 , RS = 1.72 nm, and θ 0 = 58.8◦ . In Fig. 3, we can obtain the point of intersection qt between the undercooling plateau and the linear dependency of undercooling and log(qc ) listed in Table I. As shown in Fig. 5, when the radius of the cavity rc is smaller than RS , rc can be calculated by rc = Rc · sin θ 0 (Figs. 5(a) and 5(b)). Until the border of the cavity reaches the edge of the nucleation site, rc is equal to RS = Rθ · sin θ 0 (Figs. 5(c)–5(e)). And

the critical cavity size for different overheating can be calculated according to Eq. (4) and the fitting results (RS = 1.72 nm and θ 0 = 58.8◦ ) and listed in Table I which is in agreement with the analysis outlined above. On the other hand, due to incomplete knowledge of either the cavity size distribution or the growth velocity (related to the cavity size distribution and cooling rate), the time as a measure of the regrowth process was used to demonstrate the relation between the time (tg ) for regrowth, i.e., the ratio of the sum of the undercooling plus the overheating (T + +T − ) to cooling rate (qc ), and undercooling for different overheating levels below the undercooling plateau onset is shown in Fig. 6. It is found that at a fixed overheating with increasing undercooling the time for regrowth increases. It should be pointed out that in Fig. 6 with higher overheating, there is less solid material in the cavity and it shows a slower growth rate, but the time for regrowth of the residual Sn crystal up to the top of the cavity is shorter. The reason is that the growth rate is also related to cooling rate and a higher cooling rate shows a higher growth rate which is not shown in Fig. 6. This supports the competition between the amount of the residual Sn crystal in cavity and its regrowth rate related to the cavity size and cooling rate. Fig. 6 also shows a linear relation between log(tg ) and undercooling which gives further support to regrowth process.

TABLE I. Critical cavity size obtained by fitting the model for different overheating. T + (K) 21 16 15 14

qt (K/s)

T − (K)

Rc (nm)

rc (nm)

1600 1200 860 830

98.2 97.8 97.5 97.4

1.965 1.972 1.978 1.979

1.681 1.687 1.692 1.693


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FIG. 6. Relation between the time for regrowth and undercooling. The droplet was heated at a fixed heating rate (2300 K/s) to Tmax which corresponds to different overheating and held there for 5 ms, then cooled with different cooling rates in the range from 10 000 to 850 K/s as shown in Fig. 6. Each dotted line corresponds to the same cooling rate. The error bars are the standard error.

To illustrate further the model employed here we compare different shapes as shown in Fig. 7. In this case we consider that T − (A) > T − (B), because site B has a larger diameter and so the growth rate, VB > VA . With increasing cooling rate, the heterogeneous sites like site B will become activated, i.e., the cavity solid will reach the top of the cavity B before the heterogeneous sites like site A are activated and a low undercooling will be observed. For a given overheating, above the transition cooling rate hB < hA in the cylinder case but since VB > VA site B can still control nucleation which causes the undercooling to decrease with increasing cooling rate. However, below the transition cooling rate the solid in site B has melted completely and the undercooling increases with increasing cooling rate as expected. In addition, in order to explain the different changing trend of undercooling below and above the transition cooling rate in more detail, we must also consider the time the droplet is hold above the melting temperature. Fig. 8 shows the relation between the time above bulk melting temperature and undercooling. It should be pointed out that the bulk melting temperature was used for calculation of the time above the bulk melting temperature due to a lack of knowledge of the real melting temperature of the residual Sn crystal in small cavities.

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FIG. 8. Relation between the time above bulk melting temperature and undercooling with enlarged inset.

In Fig. 8, the time above the bulk melting temperature (ttotal ) is plotted for all the holding time, overheating and cooling rates, i.e., T + /qh + th + T + /qc , where qh is the heating rate. It is found that the undercooling increases gradually with increasing ttotal when ttotal is shorter than ∼20 ms and reaches an undercooling plateau. A more detailed quantitative analysis requires a comprehensive information on the cavity size distribution and the real melting temperature of the residual Sn crystal in small cavities. Such analysis will be performed in future.

V. CONCLUSIONS

The dependence of accessible undercooling on prior liquid overheating of a pure Sn single micro-sized droplet was investigated by differential fast scanning calorimetry with cooling rates in the range from 500 to 10 000 K/s. The undercooling increases gradually with increasing overheating and eventually reaches an undercooling plateau. Moreover, it is the normal behavior that the undercooling increases with increasing cooling rate, but for a given overheating level which is lower than the level for the onset of the undercooling plateau, the undercooling decreases with increasing cooling rate. The observed rate dependent behavior was successfully explained by the cavity induced heterogeneous nucleation mechanism with the incorporation of size dependent growth kinetics. This first theoretical attempt, developed based on the advanced surface heterogeneous nucleation mechanism and supplemented here by the cavity induced heterogeneous nucleation model and cavity size dependent growth, is believed to be able to direct further research to shed more light on the nucleation mechanisms for metal droplet solidification or related rapid solidification processes also beyond the particular system analyzed in the present study.

ACKNOWLEDGMENTS

FIG. 7. Schematic illustration of the regrowth of the residual Sn crystal for differently sized cylindrical cavities.19

J.H.P. is grateful to the Alexander von Humboldt Foundation for an award to support research collaboration. Yulai Gao acknowledges the National Natural Science Foundation


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of China (Grant Nos. 50971086 and 51171105) for research support. 1 P.

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