Parameter determination of critical nucleation frequency in solidification of undercooled metallic melts

Abstract

A simple method was proposed to calculate the essential parameters correlated with the critical nucleation frequency of undercooled metals and alloy melt. Numerical results show that the calculation accuracy from this method can be improved using the experimental data either with high undercooling or with low undercooling range (the difference of undercoolings between two solidification events). The calculations of the interfacial energy for high undercooling of silver and of the catalytic factor f(θ) for high undercooling of Al, Cu and Al–30 wt-Cu alloy indicate that the results are consistent with the experimental measurements and with the results of Jian's model [Metall. Trans. A, 2001, 32A, 391–395]. In addition, by analysing the differential scanning calorimetry data of pure Sn subjected to different cooling rates, similar values of catalytic factor f(θ) are obtained. This further indicates the validity of the current method.

Keywords

Solidification Nucleation Undercooling Interfacial energy

Introduction

Generally, the most important parameters involved in the classical theory of homogeneous and heterogeneous nucleations derived by Turnbull1 are the solid/liquid interfacial energy σ and the catalytic factor f(θ). Owing to the experimental difficulty of measuring the solid/liquid interfacial energy and the wetting angle θ,2 the available experimental data with respect to σ and θ are rather sparse.2,3 So far, the interfacial energy of metal or alloy has always been estimated by calculating the nucleation frequency in terms of Spaepen's model.3,4

From the classical nucleation theory, an important parameter affecting the microstructure development during solidification is actually the critical nucleation frequency (i.e. the nucleation frequency satisfying the formation of the first nucleus in the undercooled melt).1,5 However, in order to calculate the critical nucleation frequency, the nucleation time, which is affected by the real undercooling value before nucleation of the melt, and the cooling rate must be considered.1^–4 On this basis, Jian et al. propose a method to calculate the critical nucleation frequency6 using the experimentally measured parameters, i.e. the volume, the undercooling and the cooling rate. Although the critical nucleation frequency from Jian's model is in agreement with the experimental result, application of the method is limited due to the requirement of solving a non-linear formula, which cannot lead to a unique (physically realistic) solution.

In the present paper, we aim to propose another method to deduce the critical nucleation frequency and, in turn, to determine the solid/liquid interfacial energy σ and the effective catalyst factor f(θ) of the undercooled melt.

Mathematical formulations

Homogeneous nucleation

The classical theory gives the homogeneous nucleation frequency per unit volume6 (1) with (2) (3) (4) where A_V is estimated to be a constant (A_V = 10^41±1 m⁻³ s⁻¹) for the solidification process,6 ΔG_A is the activation energy for transporting an atom across the liquid/solid interface, k is Boltzmann's constant, T_m is the equilibrium melting point of metal, ΔT is the undercooling, α is the factor determined by the shape of the nucleus (for spherical nucleus, α = 16π/3), σ is the interfacial energy between solid and liquid and ΔH_V is the latent heat of fusion per unit volume of metal.6

Upon solidification, at least one nucleus forms (to induce solidification) as soon as the critical undercooling ΔT* is reached. Suppose the sample volume and the cooling rate of the metallic melt are V and R_c respectively. According to the continuous cooling kinetics, i.e. (where is the critical nucleation frequency for homogeneous nucleation, and t is the time to achieve nucleation), the following relation results6 (5) If the values of R_c1, V₁ and and R_c2, V₂ and can be measured using two solidification events, the following equations can be deduced from equation (5) (6a) and (6b) The integral terms in equations (6a) and (6b), related to the shaded area in Fig. 1, cannot be analytically calculated. However, the ratio of the above two integral terms can be reasonably approximated as a ratio of the two triangle's areas (see Fig. 1 and the section on ‘Numerical calculations’) (7) The combination of equations (1) and (7) gives (8) with and Subsequently, equation (8) can be generalised as (9) Combing equations (1), (4) and (9) gives the critical nucleation frequency and the interfacial energy σ.

Figure 1

Homogeneous nucleation frequency as function of undercooling for aluminium calculated by equation (1)

Heterogeneous nucleation

The classical theory gives the expression for heterogeneous nucleation as6 (10) where I_S is the nucleation frequency per unit area of catalyst surface, and A_S is the constant (A_S = 10^31±1 m⁻² s⁻¹) for the solidification process.6

Actually, the heterogeneous nucleation in the continuous cooling process is influenced not only by the value of but also by S_V (the effective surface area of the catalyst in a unit volume of the liquid). As soon as the first nucleus forms in the undercooled melt, equation (5) must be replaced by6 (11) Following the analogy with the homogeneous nucleation case, two equations for heterogeneous nucleation can be derived (12) and (13) For experimentally obtained R_c1, V₁ and and R_c2, V₂ and , the value of φf(θ) can be calculated by equation (13) and subsequently the value of (i.e. the heterogeneous critical nucleation frequency per unit area of catalysis surface) by equation (10). Now that φf(θ) has been calculated, if σ is known, the value of f(θ) can be determined by equations (4) and (13). In a definite melting condition, the kind and amount of catalyst often are constant, so S_V and φf(θ) are constant. From equation (11), the effective surface area of the catalyst per unit volume of the liquid S_V can be determined by (14) With the calculated values of S_V and φf(θ), the critical nucleation frequency and the undercooling of liquid metal or alloy at different R and V can be forecasted from equations (10) and (11).6 In addition, from Jian's work, the nucleation mode can be judged, i.e. if S_V is equal to A_V/A_S (10^10±1 m⁻¹), then the liquid has nucleated homogeneously, and if S_V is less than A_V/A_S, then it has nucleated heterogeneously, where A_V = 10^41±1 m⁻³ s⁻¹ m, and A_S = 10^31±1 m⁻² s⁻¹.6

Numerical calculations

From equations (9) and (13), one can see that the current calculation is very simple without dealing with a non-linear equation. In the following, the applicability of the current method (incorporating equation (7)) will be demonstrated.

From equations (6a) and (6b), two symbols S₁ and S₂ are defined as (15) As mentioned above, S₁ stands for the exact value of the ratio between the two integral terms (which can be calculated numerically using MATLAB software), whereas S₂ stands for the ratio between the two triangles’ areas (see Fig. 1). On this basis, the applicability of the current method is mainly decided by the difference between S₁ and S₂, i.e. the smaller the difference, the more effective the method.

Given two nucleation events with = 20 K and f(θ) = 1, the changes in S₂/S₁ with undercooling was calculated for silver, copper and aluminium (Fig. 2a). The needed thermophysical parameters are shown in Table 1.6^–10 It can be found that the value of S₂/S₁ tends to be 1·1 as the undercooling increases, i.e. a larger ΔT* favours a smaller difference between S₁ and S₂. In fact, S₂/S₁ will be closer to 1 when f(θ)<1. On the other hand, when the varieties of S₂/S₁ range from 1 to 1·5, it does not affect remarkably on the value of φf(θ) (see section on ‘Discussion’). Generally, in the real experiment for rapid solidification of highly undercooled metal melts, the critical undercooling for solidification is always sufficiently high to guarantee the validity of equation (7).

Figure 2

Assuming homogeneous nucleation, plots S₂/S₁ versus ΔT

Table 1

Thermophysical parameters of silver, copper, aluminium, tin and Al–30 wt-Cu alloy used for nucleation frequency calculation6^–10

Metal (alloy)	T_m/K	ΔG_A/meV atom⁻¹	ΔH_V/J m⁻³	σ/J m⁻²
Silver	12336	3486	9·61×10⁸ (Ref. 7)	0·1268
Copper	13566	4966	1·64×10⁹ (Ref. 7)	0·1778
Aluminium	9336	2596	9·26×10⁸ (Ref. 7)	0·1139
Tin	5057	1127	4·16×10⁸ (Ref. 7)	0·05458
Al–30 wt-Cu	8216	2596	1·31×10⁹ (Ref. 10)	0·0889

Applying the thermophysical parameters for aluminium, the changes in S₂/S₁ with undercooling subjected to different values of can be calculated (Fig. 2b). Obviously, the applicability of the current method is guaranteed by a sufficiently small undercooling range ( ). Therefore, it would be reasonable to estimate σ, f(θ) and or using the present proposed method, when the undercooling data from the experiment are sufficiently high and the undercooling range ( ) is sufficiently small.

Applications

Solid/liquid interfacial energy calculation

From the above discussions, the solid/liquid interfacial energy can be determined from the experimental data for the maximum undercooling (>0·20T_m), assuming that the obtained data are consistent with homogeneous nucleation.

Table 2 lists the values of maximum undercooling for silver droplets (with different sample volumes and cooling rates), obtained experimentally by Jian and Jie.6 Since the obtained values for the critical undercooling ΔT* are >0·20T_m, homogeneous nucleation would be reasonable. On this basis, the value of φ is, according to equation (9), determined to be 2·31, and then the solid/liquid interfacial energy σ of silver from equation (4) is determined as 0·129 J m⁻², which is consistent with the measurement (0·126 J m⁻²) from Turnbull and Cech.8 Meanwhile, the critical nucleation frequency can be accordingly estimated to be 10^11·25 and 10^10·53 m⁻³ s⁻¹ for the undercooling of 260 and 257 K using equation (1).

Table 2

Experimental data and calculated values of Ag, Cu, Al and Al–30 wt-Cu alloy

Method of metals and alloys for undercooling6	Experiments data6				Calculated results			Jian's method
Method of metals and alloys for undercooling6	V/m³	R_c/K s⁻¹	ΔT*/K	ξ*	φf(θ)	f(θ)	or	φf(θ)6
Silver melted under a 70 soda lime glass+30Na₂CO₃	10^−6·17	3	260	0·211	2·31	1	10^11·25 m⁻³ s⁻¹	2·73
	10^−5·45	3	257	0·208			10^10·53 m⁻³ s⁻¹
Copper melted under 70 soda lime glass+30Na₂CO₃	10^−5·90	3	239	0·176	0·57	0·31	10^19·85 m⁻² s⁻¹	0·60
	10^−4·25	0·3	209	0·154			10^17·26 m⁻² s⁻¹
Copper melted under soda lime glass	10^−5·47	3·25	208	0·167	0·50	0·27	10^20·18 m⁻² s⁻¹	0·55
	10^−4·36	1	225	0·153			10^18·60 m⁻² s⁻¹
Aluminium droplets emulsified by sulphate	10^−14·38	500	174	0·187	0·77	0·35	10^18·18 m⁻² s⁻¹	0·74*
	10^−10·23	0·33	133	0·143			10^10·96 m⁻² s⁻¹
Al–30 wt-Cu droplets emulsified by chloride	10^−14·38	500	108	0·132	0·019	0·03	10^29·38 m⁻² s⁻¹
							10^26·50 m⁻² s⁻¹	0·012
	10^−14·38	0·25	41	0·050

Value of 0.31 given in error in Ref. 6

Catalytic factor f(θ) calculation

Generally, the contact angle often is hard to measure since it needs special experiments. From discussion above, when σ is known, f(θ) can be estimated by the combination of equations (4) and (13). Table 2 shows the experimental data of Cu, Al and Al–30 wt-Cu alloy and the corresponding results calculated by equations (4) and (13). It can be seen that all the calculated φf(θ) are close to Jian's results.6 Applying the given σ of the metal, such as 0·113 J m⁻² for aluminium,9 the corresponding catalytic factor f(θ) is calculated as 0·35 by φf(θ) = 0·77; the critical nucleation frequencies are calculated as 10^18·18 and 10^10·96 m⁻² s⁻¹ for the undercooling of 174 and 133 K respectively. The value of S_V is estimated to be 10^1·5 m⁻¹ from equation (14), so the nucleation form is heterogeneous.6 Actually, Mueller et al. also had performed similar undercooling experiments of aluminium droplets emulsified by sulphate. They estimated f(θ) to be 0·39 by the maximum measured undercooling for Al of 175 K, which is close to the current calculation.11

Differential scanning calorimetry measurement

The critical nucleation frequency during differential scanning calorimetry (DSC) measurement is difficult to be calculated for the unknown f(θ). However, the undercooling difference ( ) for DSC measurement with different cooling rates is small for most cases. As discussed in the section on ‘Numerical calculations’, the method in the present study is valid when the undercooling difference is sufficiently small (Fig. 2b). Therefore, this method can be used for DSC data analysis.

Figure 3 gives four DSC curves for pure Sn (each weighing 8 mg) with different cooling rates. Table 3 gives the undercooling data from the curves. Since the undercooling difference ( ) is small, φf(θ) and/or I_S can be estimated from equation (13). It can be seen that the value of φf(θ) calculated from any two groups of data is close to 0·016. This indicates that the real value of φf(θ) for DSC measurement can be calculated from any of the two group data by the present method. Furthermore, the values of f(θ), I_S and S_V can also be estimated from equations (4), (10) and (14) (see Table 3). It can be seen that the value of S_V is <10^10±1 m⁻¹; according to Jian's criterion,6 the nucleation mode for tin is heterogeneous, i.e. the nucleation is from impurity particles.

Figure 3

Differential scanning calorimetry curves of 8 mg pure Sn measured at different cooling rates

Table 3

Experimental data of 8 mg pure tin and calculated results

Cooling rate/K min⁻¹	0·5	1	5	40
Undercooling/K	22·9	23·9	26·85	32·80
φf(θ)	0·0161	0·0159	0·0159
f(θ)	0·0072	0·0071	0·0071
I_S/m⁻² s⁻¹	10^26·29	10^26·58	10^27·23	10^28·05
S_V/m⁻¹	10⁻¹⁵

Discussion

The numerical calculation in the previous section shows that S₂/S₁ approximates to 1·1 but not 1. It seems to lead to an error for parameter determination. In fact, equation (7) can be rewritten as (16) Then, for heterogeneous nucleation, we can further deduce (17) where k is equal to S₂/S₁ in the section on ‘Numerical calculations’. It can be found that on condition of ( )<50 K, in most cases, 1<k<1·5 (Fig. 2). Taking the experimental data in rows 2–5 of Table 2 as examples, we have obtained the values of φf(θ) and f(θ) for k = 1 (columns 6 and 7 of Table 2). For k = 1·5, from equation (17), φf(θ) are calculated to be 0·61, 0·56, 0·78 and 0·020, and then f(θ) are determined to be 0·33, 0·30, 0·36 and 0·03, respectively. Comparing with the case of k = 1, f(θ) = 0·31, 0·27, 0·35 and 0·03, the error of f(θ) can be acceptable, so that the variation of k from 1 to 1·5 has little influence on the results of f(θ). Therefore, equations (7), (9) and (13) are reasonable methods to estimate the parameters of critical nucleation frequency for the solidification of the undercooled metallic melt.

Conclusions

A simple method has been proposed to calculate the parameters of critical nucleation frequency for the solidification of the undercooled metallic melt. Numerical calculations indicate that it is a useful method to estimate the interfacial energy σ and/or catalytic factor f(θ) based on the experimental data with either high undercooling or small undercooling difference ( ). This method has also been validated through the experimental data with respect to Ag, Cu, Al, Sn and Al–30 wt-Cu alloy. A good agreement between theory prediction and experimental data is obtained.

Footnotes

Acknowledgements

The authors are grateful for the financial support of the Natural Science Foundation of China (grant nos. 50771084, 50901059 and 51071127), 111 project (B08040), the Huo Yingdong Yong Teacher Fund (grant no. 111502) and the National Basic Research Program of China (973 Program) (grant no. 2011CB610403). J. F. Xu is grateful to Professor Z. Y. Jian for indispensable instruction and cooperation in Xi'an Technological University.

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