Dependence of crystal nucleation on prior liquid overheating by differential fastscanning calorimeterBin 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

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40

THE JOURNAL OF CHEMICAL PHYSICS 140, 104513 (2014)

Dependence of crystal nucleation on prior liquid overheating by differentialfast scanning calorimeter

Bin Yang,1 John H. Perepezko,2 Jürn W. P. Schmelzer,1 Yulai Gao,3

and Christoph Schick1,a)

1Institute of Physics, University of Rostock, Wismarsche Str. 43-45, 18051 Rostock, Germany2Department of Materials Science and Engineering, University of Wisconsin-Madison,1509 University Avenue, Madison, Wisconsin 53706, USA3School of Materials Science and Engineering, Shanghai University,Shanghai 200072, People’s Republic of China

(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 tosubsequent undercooling, it determines the nucleation and growth behavior and the properties of thefinal crystalline products. However, the dependence of accessible undercooling of different bulk meltsamples on prior liquid overheating has been reported to exhibit a variety of specific features whichcould not be given a satisfactory explanation so far. In order to determine uniquely the dependence ofaccessible undercooling on prior overheating and the possible factors affecting it, the solidificationof a pure Sn single micro-sized droplet was studied by differential fast scanning calorimeter withcooling rates in the range from 500 to 10 000 K/s. It is observed experimentally that (i) the degreeof undercooling increases first gradually with increase of prior overheating; (ii) if the degree of priorsuperheating exceeds a certain limiting value, then the accessible undercooling increases alwayswith increasing cooling rate; in the alternative case of moderate initial overheating, the degree ofundercooling reaches an undercooling plateau; and (iii) in latter case, the accessible undercoolingincreases initially with increasing cooling rate. However, at a certain limiting value of the cooling ratethis kind of response is qualitatively changed and the accessible undercooling decreases strongly witha further increase of cooling rate. The observed rate dependent behavior is consistent with a kineticmodel involving cavity induced heterogeneous nucleation and cavity size dependent growth. Thismechanism 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 sub-sequent undercooling and resulting crystallization has beendemonstrated to play a significant role in determining the so-lidification behavior, in particular, for the nucleation processand the accessible undercooling.1–4 Yet, our understanding ofthe relationship between prior overheating and undercoolingis still rather incomplete. This statement refers especially torapid solidification processes and is due in part to limitationsof measurement capability. On the other hand, based on theassumption of the existence of cavities, Turnbull5 developeda theory of cavity induced heterogeneous nucleation in orderto describe typical features of such processes. His approachpredicts a linear relationship between accessible solidifica-tion undercooling (�T−) and prior liquid overheating (�T+).Here �T+ = Tmax − Tm is the difference between the max-imum temperature, Tmax, reached in prior overheating of theliquid and the melting temperature, Tm, �T− = Tm − Ts isthe difference between melting temperature and the particulartemperature Ts where rapid solidification is observed to initi-ate experimentally. The meaning of both these parameters is

a)Electronic mail: christoph.schick@uni-rostock.de

illustrated in more detail in Fig. 1(a) for one particular heatingand cooling run.

Turnbull’s predictions have been tested in a variety of ex-perimental analyses. Most of these studies, however, only ad-dress nucleation at low cooling rates measured by conven-tional thermal analysis devices2, 6, 7 or without cooling ratecontrol.5, 8 However, cooling rate is known to significantly af-fect the undercooling.9, 10 Moreover, most of the former stud-ies examined bulk samples of the melt where extraneous nu-cleants of different kind can control the response of the sys-tem to undercooling. Hence, the relationship between �T−

and �T+ has not been specified or explained in a sufficientlycomprehensive form.

With the development of nanotechnology and micro-electro mechanical systems (MEMS), it is possible to fabri-cate calorimeter sensors that are able to measure samples withnanogram masses and energies less than one nanojoule11–13

or even picojoule.14 These developments enable the possi-bility of considerably enlarging and controlling the scanningrates. In particular, Zhuravlev et al.15, 16 developed a powercompensated differential fast scanning calorimeter (DFSC)and analyzed the heat capacity change during scanning. Thisapproach provides a good chance to approach an in situundercooling measurement at high cooling rates spanningfour orders of magnitude for one single droplet.9, 10, 17, 18 In

0021-9606/2014/140(10)/104513/7/$30.00 © 2014 AIP Publishing LLC140, 104513-1

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40

104513-2 Yang et al. J. Chem. Phys. 140, 104513 (2014)

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 themeasuring area of the calorimeter sensor prior to the measurements. The sim-ple cartoon of the Sn droplet (with an oxide layer) is shown in the inset.

the present paper, extending our earlier brief report,19 wepresent the results of application of this method to measurethe accessible undercooling at the solidification of one sin-gle Sn droplet at a series of cooling rates between 500 and10 000 K/s. The drop has a diameter of 23.5 μm and is coatedwith an oxide layer. The experimental results on the depen-dence of undercooling on liquid overheating at different cool-ing rates are interpreted here based on a modification of thecavity induced heterogeneous nucleation mechanism as de-veloped by Turnbull.

The paper is structured as follows. In Sec. II, we present abrief description of the experimental procedure. The results ofthe measurements are given in Sec. III. A theoretical analysisand interpretation of the experimental data is given in Sec. IV.The main conclusions of the present analysis are summarizedin Sec. V.

II. EXPERIMENTAL PROCEDURE

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

ical shape, as shown in Fig. 1(b) They are always covered byan oxide layer. This fact can be understood easily realizingthat Sn oxide has a large heat of formation (�H for SnO2

= −577 kJ/mole). Since the loading of the sample into theDFSC involves exposure to air it is virtually impossible to ob-tain an atomically clean Sn surface. DFSC10, 15, 16, 20 based onthin film sensors was employed to investigate the relationshipbetween �T− and �T+. The nanocalorimeter sensors, XEN-39395 (Xensor Integrations, Netherlands), consist of an amor-phous silicon-nitride membrane with a film-thermopile and aresistive film-heater placed at the center of the membrane.

These sensors can effectively promote high heating ratesthrough a preferable ratio between addenda and sample heatcapacity and applicable heating power. Under non-adiabaticconditions the same holds for high cooling rates, even thoughthe cooling capability of the whole system is restricted dueto a finite heat transfer rate from the sample. For a mem-brane sample system, the heat transfer is limited by the ther-mal properties of the surrounding cooling agent.12, 21 In a μmscale system, the most efficient cooling agents are gases be-cause of their small heat capacity.22 The power compensa-tion scheme of the DFSC15, 20 takes care of the heat lossesand both sensors follow the predefined temperature programvery 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 μmand placed it into the center of the heating zone of the sensorusing an optical microscope (Olympus SZX16), as shown inFig. 1(b). To improve the thermal contact between the sam-ple and the membrane silicone oil was used. The micro-sizeddroplet was heated up in air to the desired maximum temper-ature (Tmax) above the melting temperature at a fixed heatingrate of 2300 K/s. After holding the droplet at that maximumtemperature 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 cool-ing rates in the range from 500 to 10 000 K/s. The heating-cooling cycles were repeated 10 times for each temperatureprofile, i.e., each Tmax, cooling rate and holding time, respec-tively, in order to appropriately account for the stochastic na-ture of the nucleation process. It should be pointed out thatthe outer layer of the Sn droplet did not break when the in-ner metal Sn expands as temperature increases. The reasonis that for SnO2 the Pilling-Bedworth ratio (volume of ox-ide to volume of metal oxidized) is equal to 1.3. Values ofthe Pilling-Bedworth ratio in the range between 1 and 2 areconsidered as protective and adherent. On the other hand, themelting temperature did not change in the course of the exper-iments. This result implies that during the whole experimentsthe Sn droplet keeps stable.

III. EXPERIMENTAL RESULTS

A typical DFSC curve showing the differential temper-ature between sample and reference sensors versus time isshown in Fig. 1(a). The onset temperatures of the meltingpeak (plotted downwards) and solidification peak (plottedupwards) were taken as the melting (Tm) and solidificationtemperatures (Ts), respectively. The thermal lag between the

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40

104513-3 Yang et al. J. Chem. Phys. 140, 104513 (2014)

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

regular shaped spherical sample and the membrane is directlyproportional to the scanning rate. Therefore the melting andsolidification temperatures were corrected according to theprocedure detailed in Ref. 20. The thermal lag occurring dur-ing melting is also responsible for the broadening of the melt-ing peak compared to the solidification peak, which is notheat transfer limited because of the large undercooling. As al-ready mentioned briefly, the undercooling �T− is calculatedby �T− = Tm − Ts, and the overheating �T+ is determinedas �T+ = Tmax − Tm.

In Fig. 2, the dependence of the mean value of the acces-sible undercooling of the single Sn droplet on process con-ditions is shown. In Fig. 2(a), the results are presented forthe case of various cooling rates for different overheating at aconstant holding time of 5 ms. Fig. 2(b) shows the respectiveresults for fixed heating and cooling rates, but different dura-tions of overheating at which the melt was held at Tmax. It isfound that, in both cases above a critical overheating level, a

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

plateau develops for the undercooling at 99 ± 2 K. As shownin Fig. 2(a), a strong decrease of accessible undercooling isfound when the overheating is lower than some critical valuewhich 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 thatthe limiting overheating for reaching the undercooling plateauin Fig. 2(b) decreases with increasing holding time.

Fig. 3 shows the relationship between the measured un-dercooling and the cooling rate. Note that for overheatingbelow 14 K it was not possible to obtain the undercoolingat higher cooling rates because of the melting kinetics, i.e.,in these cases the droplet solidified immediately during thesubsequent cooling because it was not enough time to fullymelt the droplet matrix. As a consequence, in such cases theundercooling is equal to zero. It should be pointed out thatthe time the sample stays above the melting temperature de-creases with increasing cooling rate. It is found that the un-dercooling for the overheating of 31 K and above increasesmoderately with increasing cooling rate which is consideredas a plateau within the range of ∼4 K. Such kind of depen-dence is the usually expected behavior,9, 10 as will be alsodemonstrated in Sec. IV. For overheating of 21 K and be-low, which is below the onset of the undercooling plateau at99 ± 2 K, however, above a certain cooling rate the under-cooling decreases strongly with increasing cooling rate. Fur-thermore the cooling rate for this observed transition (qt) de-creases with decreasing overheating.

IV. THEORETICAL INTERPRETATION

The experimentally observed dependences (mainlyshown in Fig. 3), i.e., that above a certain value of the cool-ing rate the accessible undercooling decreases strongly witha further increasing of the cooling rate, cannot be completelydescribed by the model derived in Refs. 9 and 10. This modelyields a shelf-like dependence of crystal nucleation on un-dercooling. On the other hand, Tong et al.6, 7 investigated theeffect of thermal history on this dependence. They invoked a

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40

104513-4 Yang et al. J. Chem. Phys. 140, 104513 (2014)

time dependent structural change in the liquid as the expla-nation. However, the nature of heterogeneous nucleation be-havior and the role of regrowth were ignored in this approach.In the present study, in order to understand such kind of be-havior, we assume, advancing our model described in detailin Refs. 9 and 10, the existence of the cavity induced hetero-geneous nucleation mechanism as suggested in its principalfeatures first by Turnbull.5 In this approach, followed also inour earlier brief communication,19 the thermal history effecton the value of the accessible undercooling can be explainedby considering the details involved in the evolution of a cavitygenerated nucleus in differently sized cavities during cooling.In the development of this model, the following assumptionsare made about the nucleation mechanism which is expectedto dominate the heating-cooling process.

Let us consider a cylindrical or conical cavity on thedroplet surface in its oxide layer. This, of course, is an ideal-ized geometry since the liquid-oxide layer interface is likelyto exhibit an irregular roughness. Unfortunately, we are notaware of any technique capable to characterize the surfacetopology between the liquid tin and the oxide layer. It shouldbe pointed out that if the Sn oxide layer were thick andrigid there may be an effect on the undercooling. In otherwords, a thick, rigid, and adherent oxide would undergo dif-ferent expansion and contraction as the temperature changesand this could exert a pressure on the Sn. However, this ef-fect would also influence the melting temperature which didnot change over the course of our experiments. Thus, theeffect of the oxide aside from the role in catalyzing nucle-ation seems to be negligible. Therefore the idealized geome-try will be shown to account for all of the main features of theobserved overheating-undercooling relationship. During theheating process some unmelted solid is retained in these smallcavities whose size are typically of the order of nano-size, asschematically shown in Fig. 4.5, 19 The melting temperature ofnano-sized particles in an oxide matrix, contrary to free parti-cles, in some cases increases when the particle size decreasesor by applying a very high heating rate.23 This is due to thefact that particles embedded in an oxide matrix will be subjectto hydrostatic pressure since oxides have a lower coefficientof thermal expansion (CTE) as compared to metals. This ef-fect results in an increase of the melting temperature Tm ac-cording the Clausius-Clapyeron relation. However, in the caseof cavities due to the curvature of the liquid-solid interface asillustrated in Fig. 4,5, 8 the melting temperature of the resid-ual Sn crystal in small cavities is higher than the bulk meltingtemperature.

It should be pointed out that for each overheating with afixed holding time there is a stable position of the solid-liquidinterface determined by the maximum value of the radius, γ ,of the conical or cylindrical cavities. During the subsequentcooling, the residual Sn crystal can regrow and become a nu-cleus if it reaches the mouth of the cavity. However, the resid-ual Sn crystal will not act as a nucleus upon cooling unlessthe radius of the mouth of the cavities equals or exceeds thecritical radius for heterogeneous nucleation (see Fig. 4).

As the overheating increases, the amount of solid mate-rial inside the cavity decreases or even disappears for over-heating larger than 26 K. Consequently, for the residual Sn

FIG. 4. Schematic illustration of remaining crystals in conical and cylindri-cal cavities.5, 19

crystal material in the cavity, the distance to reach the cavitytop increases so that an increasing time for regrowth is re-quired. Similarly, as the cooling rate increases, there is lesstime available for the solid to grow and to fill the cavity com-pletely. As a result, the usual nucleation behavior dominates.For moderate overheating (21 K or less), a residual Sn crys-tal remains in the cavity. During the subsequent cooling at amodest rate, the residual Sn crystal in the cavities will regrowto the mouth of the cavities. The height, h (see Fig. 4), of theremaining solid in the cavity increases with decreasing over-heating, so that the time for regrowth the solid to reach the topand to act as a nucleus becomes shorter.

In order to explain the different changing trend of under-cooling below and above the transition cooling rate (shown inFig. 3) we must also consider the actual size of the cavities.For the nanometer size range nanowire growth rate measure-ments show that the growth rate decreases with decreasingnanowire diameter. This is related to the reduction in driv-ing force at constant undercooling due to the Gibbs-Thomsoneffect.24 Thus, let us consider this case with higher overheat-ing. As a result, the radius of the remaining partly filled cavi-ties is smaller which yields a slower growth rate. Meanwhile,the amount of residual Sn crystal in the cavity will be reducedtoo. Consequently the residual Sn crystal needs more timefor the regrowth to the mouth of the cavity and to act as anucleus. Both of these mutual effects would yield the largerundercooling with increasing overheating, and vice versa. Inother words, the nucleation due to the regrowth of remainingsolid in the cavities is more sensitive to cooling rate than theusual nucleation behavior.

In order to determine the critical cavity size for differ-ent overheating (below 26 K), we make two assumptions.(i) When the cooling rate equals qt (Fig. 3), the nucleationprocess is controlled by the regrowth of the remaining solidSn crystal just at the bottom of the cavity. This means thatduring cooling the remaining solid Sn crystal regrows fromthe bottom to the top of the cavity and then acts as a nucleus.This would yield the undercooling at qt ≈ 99 ± 2 K (under-cooling plateau) which is shown as the solid curve in Fig. 3.(ii) According to the results of previous investigation (detailedin Refs. 9 and 10), the nucleation of the single Sn droplet iscaused by both bulk and surface heterogeneous nucleation.For the cavity induced nucleation of the single Sn droplet, thesurface nucleation mechanism is dominating due to the cav-ities on the droplet surface, and the total number of nuclei,

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40

104513-5 Yang et al. J. Chem. Phys. 140, 104513 (2014)

N, formed in the system in cooling it from Tm down to Tm −�T− is given by

N = 1

qc

∫ �T −

0JSSd�T −, (1)

where JS is the surface nucleation rate; qc is the cooling rate;and S is the surface area of the droplet. The crystallization ofthe droplet proceeds fast, when the first supercritical crystalnucleus is formed. By this reason, the accessible undercoolingis determined by Eq. (1) if in this relation N is equal to one(N = 1). The expression for the nucleation rate of the singledroplet during a rapid cooling process on the surface of thedroplet was given in Ref. 9 as

JS = nS� (1 − cos θ0)

(�T −)2exp

(− �f (θ0)

(Tm − �T −) (�T −)2

),

(2)with

f (θ0) = 1

4(2 − 3 cos θ0 + cos3 θ0), (3)

Rc = 2σlsTm

�HV �T − , (4)

where nS = CiS/a20 is the number of heterogeneous nucle-

ation sites of the catalytic surface per unit surface. This num-ber can be also denoted as the density of nucleation sites(CiS is the surface impurity concentration); � and � arethe constants related with the physical properties (detailed inRef. 9); Rc is the critical radius of nucleus; θ is the contactangle between the undercooled melt and the heterogeneousnuclei; f(θ0) is the catalytic activity which describes the cat-alytic potency of the heterogeneous nuclei and θ0 is the con-stant contact angle on the surface when the critical radius ofthe crystal nucleus is less than the radius of the heterogeneousnucleation site with fixed size, 2RS for surface nucleation, asshown in Fig. 5. For a small radius of the nucleus, the con-tact angle, θ0 is constant (Fig. 5(a)). The contact angle doesnot change with increasing radius of the nucleus (Fig. 5(b)),until the border of the nucleus reaches the edge of the nucle-ation site. There are two contact angles in this case for thesame radius of nucleus: θ0 and π − θ0 (Figs. 5(c) and 5(d)).Undoubtedly, this case is realized only if θ0 < π /2. Whenthe critical radius of nucleus Rc is larger than the crossoverradius Rθ = RS/sinθ0, the contact angle increases sharply(Fig. 5(e)). In such case, the catalytic activity of such nucle-ation 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), thesedata are employed in the computations shown in Fig. 3 (solidcurve) for the undercooling plateau. Then the fitting resultscan be obtained as: nS = 3.73 × 1014 m−2, RS = 1.72 nm, andθ0 = 58.8◦. In Fig. 3, we can obtain the point of intersection qt

between the undercooling plateau and the linear dependencyof undercooling and log(qc) listed in Table I. As shown inFig. 5, when the radius of the cavity rc is smaller than RS,rc can be calculated by rc = Rc · sin θ0 (Figs. 5(a) and 5(b)).Until the border of the cavity reaches the edge of the nucle-ation site, rc is equal to RS = Rθ · sin θ0 (Figs. 5(c)–5(e)). And

FIG. 5. Schematic illustration of the change of the contact angle.

the critical cavity size for different overheating can be calcu-lated according to Eq. (4) and the fitting results (RS = 1.72 nmand θ0 = 58.8◦) and listed in Table I which is in agreementwith the analysis outlined above.

On the other hand, due to incomplete knowledge of ei-ther the cavity size distribution or the growth velocity (relatedto the cavity size distribution and cooling rate), the time as ameasure of the regrowth process was used to demonstrate therelation between the time (tg) for regrowth, i.e., the ratio of thesum of the undercooling plus the overheating (�T++�T−)to cooling rate (qc), and undercooling for different overheat-ing levels below the undercooling plateau onset is shown inFig. 6. It is found that at a fixed overheating with increas-ing undercooling the time for regrowth increases. It shouldbe pointed out that in Fig. 6 with higher overheating, there isless solid material in the cavity and it shows a slower growthrate, but the time for regrowth of the residual Sn crystal up tothe top of the cavity is shorter. The reason is that the growthrate is also related to cooling rate and a higher cooling rateshows a higher growth rate which is not shown in Fig. 6. Thissupports the competition between the amount of the residualSn crystal in cavity and its regrowth rate related to the cavitysize and cooling rate. Fig. 6 also shows a linear relation be-tween log(tg) and undercooling which gives further support toregrowth process.

TABLE I. Critical cavity size obtained by fitting the model for differentoverheating.

�T+ (K) qt (K/s) �T− (K) Rc (nm) rc (nm)

21 1600 98.2 1.965 1.68116 1200 97.8 1.972 1.68715 860 97.5 1.978 1.69214 830 97.4 1.979 1.693

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40

104513-6 Yang et al. J. Chem. Phys. 140, 104513 (2014)

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

To illustrate further the model employed here we com-pare different shapes as shown in Fig. 7. In this case we con-sider that �T−(A) > �T−(B), because site B has a largerdiameter and so the growth rate, VB > VA. With increasingcooling rate, the heterogeneous sites like site B will becomeactivated, i.e., the cavity solid will reach the top of the cavityB before the heterogeneous sites like site A are activated anda low undercooling will be observed. For a given overheating,above the transition cooling rate hB < hA in the cylinder casebut since VB > VA site B can still control nucleation whichcauses the undercooling to decrease with increasing coolingrate. However, below the transition cooling rate the solid insite B has melted completely and the undercooling increaseswith increasing cooling rate as expected. In addition, in or-der to explain the different changing trend of undercoolingbelow and above the transition cooling rate in more detail, wemust also consider the time the droplet is hold above the melt-ing temperature. Fig. 8 shows the relation between the timeabove bulk melting temperature and undercooling. It shouldbe pointed out that the bulk melting temperature was used forcalculation of the time above the bulk melting temperaturedue to a lack of knowledge of the real melting temperature ofthe residual Sn crystal in small cavities.

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

FIG. 8. Relation between the time above bulk melting temperature and un-dercooling with enlarged inset.

In Fig. 8, the time above the bulk melting temperature(ttotal) is plotted for all the holding time, overheating and cool-ing rates, i.e., �T+/qh + th + �T+/qc, where qh is the heatingrate. It is found that the undercooling increases gradually withincreasing ttotal when ttotal is shorter than ∼20 ms and reachesan undercooling plateau. A more detailed quantitative analy-sis requires a comprehensive information on the cavity sizedistribution and the real melting temperature of the residualSn crystal in small cavities. Such analysis will be performedin future.

V. CONCLUSIONS

The dependence of accessible undercooling on priorliquid overheating of a pure Sn single micro-sized dropletwas investigated by differential fast scanning calorimetrywith cooling rates in the range from 500 to 10 000 K/s. Theundercooling increases gradually with increasing overheatingand eventually reaches an undercooling plateau. Moreover,it is the normal behavior that the undercooling increases withincreasing cooling rate, but for a given overheating levelwhich is lower than the level for the onset of the undercoolingplateau, the undercooling decreases with increasing coolingrate. The observed rate dependent behavior was successfullyexplained by the cavity induced heterogeneous nucleationmechanism with the incorporation of size dependent growthkinetics. This first theoretical attempt, developed based onthe advanced surface heterogeneous nucleation mechanismand supplemented here by the cavity induced heterogeneousnucleation model and cavity size dependent growth, is be-lieved to be able to direct further research to shed more lighton the nucleation mechanisms for metal droplet solidificationor related rapid solidification processes also beyond theparticular system analyzed in the present study.

ACKNOWLEDGMENTS

J.H.P. is grateful to the Alexander von Humboldt Foun-dation for an award to support research collaboration. YulaiGao acknowledges the National Natural Science Foundation

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40

104513-7 Yang et al. J. Chem. Phys. 140, 104513 (2014)

of China (Grant Nos. 50971086 and 51171105) for researchsupport.

1P. Rudolph, H. J. Koh, N. Schäfer, and T. Fukuda, J. Cryst. Growth 166,578 (1996).

2H. J. Koh, P. Rudolph, N. Schäfer, K. Umetsu, and T. Fukuda, Mater. Sci.Eng. B 34, 199 (1995).

3Z. Zhou, W. Wang, and L. Sun, Appl. Phys. A 71, 261 (2000).4M. Mühlberg, P. Rudolph, M. Laasch, and E. Treser, J. Cryst. Growth 128,571 (1993).

5D. Turnbull, J. Chem. Phys. 18, 198 (1950).6H. Y. Tong and F. G. Shi, Appl. Phys. Lett. 70, 841 (1997).7H. Y. Tong and F. G. Shi, J. Chem. Phys. 107, 7964 (1997).8W. Luzny, J. Phys.: Condens. Matter. 2, 10183 (1990).9B. Yang, A. S. Abyzov, E. Zhuravlev, Y. Gao, J. W. P. Schmelzer, andC. Schick, J. Chem. Phys. 138, 054501 (2013).

10B. Yang, Y. Gao, C. Zou, Q. Zhai, A. Abyzov, E. Zhuravlev, J. Schmelzer,and C. Schick, Appl. Phys. A 104, 189 (2011).

11S. L. Lai, J. Y. Guo, V. Petrova, G. Ramanath, and L. H. Allen, Phys. Rev.Lett. 77, 99 (1996).

12A. A. Minakov and C. Schick, Rev. Sci. Instrum. 78, 073902 (2007).13M. Zhang, M. Y. Efremov, F. Schiettekatte, E. A. Olson, A. T. Kwan,

S. L. Lai, T. Wisleder, J. E. Greene, and L. H. Allen, Phys. Rev. B 62,10548 (2000).

14M. Y. Efremov, E. A. Olson, M. Zhang, F. Schiettekatte, Z. Zhang, andL. H. Allen, Rev. Sci. Instrum. 75, 179 (2004).

15E. Zhuravlev and C. Schick, Thermochim. Acta 505, 1 (2010).16E. Zhuravlev and C. Schick, Thermochim. Acta 505, 14 (2010).17B. Yang, Y. Gao, C. Zou, Q. Zhai, E. Zhuravlev, and C. Schick, Mater. Lett.

63, 2476 (2009).18B. Zhao, L. Li, Q. Zhai, and Y. Gao, Appl. Phys. Lett. 103, 131913 (2013).19B. Yang, J. Perepezko, E. Zhuravlev, Y. Gao, and C. Schick, in TMS2013

Supplemental Proceedings (John Wiley & Sons, Inc., 2013), p. 485.20Y. Gao, E. Zhuravlev, C. Zou, B. Yang, Q. Zhai, and C. Schick, Ther-

mochim. Acta 482, 1 (2009).21S. A. Adamovsky, A. A. Minakov, and C. Schick, Thermochim. Acta 403,

55 (2003).22A. A. Minakov, S. A. Adamovsky, and C. Schick, Thermochim. Acta 432,

177 (2005).23Q. S. Mei and K. Lu, Prog. Mater. Sci. 52, 1175 (2007).24V. Schmidt, J. Wittemann, and U. Gosele, Chem. Rev. 110, 361 (2010).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

31.18.249.112 On: Thu, 13 Mar 2014 15:23:40