Enthalpy measurements of undercooled melts by levitation calorimetry: the pure metals nickel, iron, vanadium and niobium

MATERIALS SCIENCE & ENGINEERING ELSEVIER

Materials Science and Engineering A197 (1995) 83 90

A

Enthalpy measurements of undercooled melts by levitation calorimetry: the pure metals nickel, iron, vanadium and niobium K. Schaefers, M. R6sner-Kuhn, M.G. Frohberg Institute of General Metallurgy, Technical University Berlin, Joachimstalerstr. 31/32, D-10719 Berlin, Germany Received 26 October 1994; in revised form 22 December 1994

Abstract

A circulating gas cooling (CGC) system is added to a combination of an electromagnetic levitation apparatus and a drop calorimeter to measure the enthalpies of the pure metals nickel, iron, vanadium and niobium in the undercooled temperature range. The CGC system extends the experimental temperature range to lower temperatures. The effect of the CGC system on the heat losses during the drop of the sample is discussed and the evaluation procedure presented. The measured enthalpies of the metals in the undercooled range confirm the temperature dependences of the enthalpies of the liquid phase above their melting points. Keywords: Levitation calorimetry; Nickel; Iron; Vanadium; Niobium

1. Introduction Different kinds of special heat treatment are frequently used to achieve required structures in materials production. For many technical applications, the metastable state is the aim of the treatment, which is mostly realized by rapid solidification. Such conditions are able to transform metallic melts into the glassy state, which enlarges the technological properties of the materials considerably. Within this important field, data of experimental calorimetry are of interest for heat balances of the production processes and the estimation of materials properties. Measurements of the enthalpy and the heat capacity of undercooled melts [1 15] have been preferentially performed with low melting metals and alloys [1-14]. Considerable difficulties arise with conventional calorimeters during measurements in the liquid state of high melting systems, as a result of the reactivity of the materials and the consequent contamination of the samples. Therefore, contactless electromagnetic levitation is used to guarantee the highest experimental temperatures without side-reactions. Moreover, the absence of nucleation-promoting conditions makes it possible to investigate the thermodynamic properties in the liquid undercooled region. In addition to high cooling rates, 0921-5093/95/$9.50 © 1995 SSDI 0921-5093(95)09764-3

Elsevier Science S.A. All rights reserved

undercooling is a prerequisite condition for the realization of metastable phases. The enthalpies presented for the pure metals nickel, iron, vanadium and niobium are measured in the u~dercooled state by use of the electromagnetic levitation technique and a drop calorimeter. The apparatus is also equipped with a circulating gas cooling (CGC) system to attain the temperature range below the melting points. Without the CGC system, the same apparatus was successfully employed for enthalpy measurements of refractory metals in the solid and liquid states [16 21]. The heat capacities can be calculated from the experimentally determined values of the enthalpies.

2. Experimental procedure and evaluation 2.1. Leviation apparatus The apparatus consists of a high frequency (HF) generator (65 kW) with selective frequencies of 450 and 900kHz, an H F transformer (1:16), a stainless steel vessel connected to the vacuum system (rotary vane and turbomolecular pump), the drop calorimeter and the CGC system. The experiments presented were performed using a frequency of 450 kHz. Fig. 1 gives a schematic view of the equipment (without the generator

K. Schaejers et al. / Materials Science and Engineering A 197 (1995) 83 90

84

.

partialradiation

~-G-~ -s-Y~t-e-mi ~""

~

Table 1 Hemispherical spectral emissivities of the pure metals nickel, iron, vanadium and niobium

manipulater

I I'i /~____ _____

lent

Wavelength (nm)

Nickel

Iron

Vanadium

Niobium

547 650

0.492 0.423

0.4535 0.391

0.401 0.385

0.404 0.355

computer kNt

.... ~

[2222222222-,,

i

iV2"~s-]K~] [~-----.=/!]

Cma~tr; vacuumsystem

Fig. 1. Experimental apparatus.

and transformer). It is possible to fill the vessel directly with helium, hydrogen and/or argon, through the valves V1-V3 or via the CGC system. The vacuum is electronically measured and the pressure reading of the gas-filled vessel is carried out using a precision barometer.

ment, and (2) the possibility to identify the melting point Tm as an arrest point in the time temperature diagram. Thus, we are able to measure the blackbody temperature Tb with the partial radiation pyrometer at the melting point and we obtain the hemispherical spectral emissivity e~(T,,) from Wien's approximation to Planck's laws as

Here, C2 is the second radiation constant (C2= 14.388 x 106nm K). The hemispherical spectral emissivities determined are shown in Table 1.

2.2. Stainless steel vessel 2.4. Drop calorimeter The cylindrical double-walled stainless steel vessel (inner diameter, 23 cm; depth, 25 cm) is water cooled. Its back wall is equipped with feedthroughs for the levitation coil, which is placed in the center. The front side of the vessel is constructed in the form of an observation window which can be dismantled. A manipulator handled from outside enables us to position the sample carrier inside the coil, and to move a small crucible or the nozzle of the CGC system under the coil. Two lateral quartz windows make contactless temperature measurements possible. A short tube (diameter, 15 cm) is fastened onto the bottom of the vessel; from here, flanges lead to the vacuum system and to the CGC system. Gate valves connect the vessel with the vacuum system (GVg) and the drop calorimeter (GVs).

2.3. Temperature measurements and recording Two kinds of pyrometer are used to measure the sample temperature. A partial radiation pyrometer working at the wavelengths of 547 and 650 nm gauges the true temperature; it is also used to determine the spectral emissivities at the melting points and to calibrate a quotient pyrometer. The quotient pyrometer, simultaneously working at 450 and 650 nm is connected to a data recorder (see Fig. 1) and offers two main advantages: (1) a very fast continuous temperature reading and recording during the undercooling experi-

The calorimeter consists of a nickel-plated copper block. The block itself rests on Teflon supports in a stainless steel container, i.e. the jacket. The whole device is located in a thermostat, which is filled with 301 of distilled water. The water equivalent (Cc) of the calorimeter was determined by electrical heating to be Cc = 2467.8 _+ 0.76 J K -I. The operation mode of the drop calorimeter is an isoperibolic mode. This expression was first introduced by Kubaschewski and Hultgren [22] in 1962. It is related to a calorimeter with a constant jacket temperature (also called the "convergence temperature" Tj) and varying temperature of the system (i.e. the copper block) during the main period of the measurement. The jacket temperature corresponds to the thermostat temperature, which is controlled to be Tj = 298 K before and after the experiment. During the experiment, the block temperature is measured by a quartz thermometer with an accuracy of about _+0.0001 K. The quartz cell is located 2 0 m m from the bottom and 10mm from the wall of the copper block. A conical hole is at the top of the block, within the receiver cup. The cup has a tungsten lining made out of foils and a sliding gate to prevent radiant heat losses of the dropped sample. Higher up, a movable convection plate prevents convective heat losses of the copper block. Furthermore, the gate valve (GVs)

K. Schae/ers et al. / Materials Science and Engineering A 197 (1995) 8 3 - 9 0

Db

T ~ jacket temperature ij

"I'c,mB Tc,e

$

Tc b . . . . . . . . .

L to,b

i mare- i

-I

i tb tm

k Cc

(T(',mb_

Tj)

(3)

De - dT("m~ k (Tc.me __ Tj) (4) dt Cc Here, Tc,mb and Tc.m, are the calorimeter temperatures at the times tm. b and tm,e. One obtains

~:t~n~=D~ pre-

.go

dTc'mbdt

85

Cc

postperiod

I~

i

t,

tin.

Zeit (t)

Fig. 2. Schematic time temperature curve of the calorimeter.

Tj = Db -k- + Tc,mb

(5)

Cc Tj = D e T +

(6)

Tc ....

Therefore, we have Db --

De

shields the block from radiation and the deposition of evaporation products during the period of levitation. The generator is switched off after the experimental temperature is adjusted, and the sample drops into the calorimeter, where the sliding gate closes automatically. The drop time is 0.32 s. The convection plate and the gate valve are immediately manually closed. The following heat exchange should not exceed a time of 20 min [23]. The average time of our measurements was 13 min.

k = Cc

2.5. Time-temperature curve of the calorimeter

A r c o r r = D e ( t e -- t b ) + \ - T c , ~ e ~ ~ C , m b /

Fig. 2 shows schematically the time-temperature curve of the calorimeter. One can divide this into a pre-period, main-period and post-period. During the experiment, the calorimeter block temperature Tc is below the jacket temperature Tj. The temperature rise in the pre- and post-periods follows Newton's law, i.e. an asymptotic approach of the block temperature to the jacket temperature occurs as

dT c

k

dt

Cc

( T o - Tj)

(2)

where Cc is the water equivalent of the calorimeter, k is the heating constant and t is the time. A smaller difference of 6 K between the block and jacket temperatures gives a guarantee of pure heat conduction, i.e. no heat transport by radiation or convection. A correction of the temperature difference ( T c , e - - T ( , b ) is necessary to calculate the time-lag free temperature rise of the calorimeter block. The corresponding evaluation is based on a proposal of Bonnel [24] and Stretz [25], and implies the equality of the areas A1 = A2 in Fig. 2. According to this, a linear increase in the block temperatures is assumed in the pre- and the post-periods ( O b = tan ~ and De = tan fl). These linear increases are determined before the beginning (tm.b) and after the end (tmx) of the main periods, i.e.

(7)

To,me-- Tc,mb

On substituting Eqs. (6) and (7) in Eq. (2), we obtain

dTc ( D b - D e ) (To - Tc,me ) dt - Dc - \To,me Tc,mb/

(8)

-

This formulation does not contain the water equivalent and the heating constant. The correction term of the temperature difference (ATcorr) is found by integration within t b and t~ as

X [Tc,me(te--tb)-- ft 7 Tcdt ]

(9)

The integral S~ Tc dt is determined with the aid of the trapezium rule

f

te

At

Tcdt=T(Tc,b+Tc,e)+At

b

i

l

~ Tc (tj)

(10)

.]= 1

with At = ( t e - - tb)/i and tj = ( t b + j A t ) . Therefore, the time-lag free corrected temperature rise of the calorimeter block is AT C = (Tc. e -- TC,b) -- ATcorr

(11)

The maximum relative error of this evaluation is no more than 10 4% [26].

2.6. Determination of the heat content The heat content of the samples can be determined from a knowledge of the corrected temperature difference and the water equivalent of the calorimeter. The water equivalent of the calorimeter must be corrected as a result of the tungsten foils inside the receiver cup, in the form C* = Cc + Cp,w nw

(12)

where Cp,w is the heat capacity and nw is the mole number of the tungsten foils.

86

K. Sehaefers et al. / Materials Science and Engineering A 197 (1995) 83 90

The reference temperature (298.15 K) for the enthalpy is higher than the calorimeter temperature at the end of the main period (Tc.e). This fact should be accounted for by an enthalpy correction term, i.e. Ah . . . .

=

Cp,p

np(Tc, e -- 298.15)

(13)

with Cp.p as the heat capacity and np the mole number of the sample. The heat capacity is taken from the literature or, in the case of alloys, estimated with the aid of the rule of N e u m a n n and Kopp. The mass of the sample is determined after the experiment. The final result of the enthalpy is Ahc = C* ATc @ Ah . . . .

(14)

3. GCG system The C G C system is necessary to control the temperature of the sample independently from the generator output. In m a n y cases of metals and alloys, the minim u m output required to levitate the sample leads to temperatures considerably higher than the melting point.

Gr

gfl A TDp 3 v2

(21)

Here the following are properties of the surrounding gas: 2, thermal conductivity; Cp.c, heat capacity; r/, dynamic viscosity; v, kinematic viscosity; fl, volumetric expansion coefficient ( f l = 1/273.15K l for an ideal gas); w, gas velocity; AT, temperature difference between sample and gas; Dp, characteristic length (i.e. sample diameter). On the condition that the heat transfer is purely convective, the flow conditions can be expressed using two equations by Ranz and Marshall [29,30] (see also ref. [31]): Nu = 2 + 0.6Grl/4pr 1/3

(22)

Nu = 2 + 0.6Rel/2prl/3

(23)

Eq. (22) is valid for free (natural) convection and Eq. (23) for forced convection. On introducing Eq. (18) into Eq. (22), the heat transfer coefficient becomes = ~

(2 + 0.6Grl/4prl/3)

(24)

3. I. Concept o f the C G C system

For the stationary state, the inductive input Pi is equal to the heat current of radiation (QR) and convection (Qc)- The radiation losses are calculated according to the law of Stefan and Boltzmann, and the convection losses according to Newton's equation. It follows that

Pi = OR + Qc

(15)

Pi = et aA (Tp4 -- Ts4) + o ~ A ( T p - Ts)

(16)

where Tp is the sample temperature, Ts is the surrounding temperature, A is the sample area, a is the S t e f a n Boltzmann constant (a = 5.6697 x 10 -8 W m : K - a ) , is the total hemispherical emissivity of the sample material (for vanadium, for example, e~ = 0.26) and e is the heat transfer coefficient (in watts per square metre per kelvin). To calculate the heat transfer coefficient, one needs the Nusselt number (Nu), which is a function of the Reynolds (Re), Prandtl (Pr) and G r a s h o f (Gr) numbers, i.e. Nu = f ( R e , Pr, Gr)

(17)

Analysis of the energy balance of the heat transfer gives the numbers as follows [27,28]:

For the calculation of the heat transfer coefficient, the dimensionless numbers refer to the so-called film temperature Tf, i.e.

T,-

Tp + Ts

(25)

2

The concept of the C G C system is underpinned on the basis of a vanadium sample with D p - - 8 mm. The resulting temperature for levitating the sample is Tp--2200 K in a helium atmosphere and an estimated surrounding temperature of 298 K inside the vessel. The aim was to arrive at a temperature reduction of 500 K. With Eq. (25), one obtains Tf~2~o0)= 1249K and TroT00) = 999 K. The necessary physical data of helium, together with the corresponding dimensionless numbers are listed in Table 2. Then, from Eq. (24), one obtains the heat transfer coefficient as Table 2 Properties of helium and dimensionless numbers T(K) 298

Nu

= 0~Op

2 Pr

=

qCp'G

(19)

2

Re =

wDp V

999

1249

Re~

(18)

(20)

Cp,G(kJ kg -t K -1) 5.1931 2(Wm -t K -1) a(kg m- s) 0.083975 q(Nsm 2) 3.01x10 -5 Pr [1] Gr [1]

5.1931 5.1931 417 x 10-3 354 x 10-3 0.048255 0.039355 4.357x 10.5 5.047 x 10 5 0.63 0.639 21.27 31.64

[32] [33] [33] [34]

87

K. SchaeJers et al. / Materials Science and Engineering A 197 (1995) 83-90

354 x 10 - 3 (2 + 0.6 x 21.271/4 x 0.631/3) 8x10 3

c~

=137.38

Wm -2K

4. Heat losses of the sample during the drop

1

Also, the input inductive energy is calculated from the radiant heat losses and convection losses as P~ = 69.288 + 52.54 = 121.83

W

A nozzle of a diameter Dn = 5 mm provides a cooling gas stream directly to the sample. The heat transfer coefficient of the now forced convection is determined by Eq. (23). The dimensionless numbers of this equation refer to the film temperature T f = 999 K. The quotient of the pumping speed (12~) and the cross-section of the nozzle (Dn2~r/4) is introduced instead of the gas velocity in the Reynolds number. With this, the heat transfer coefficient given by D e I/~4

1,.2

1..3

During the drop, the heat losses (QD) of the sample can be divided into radiative (QR) and convective (Qc) quantitites, i.e. QD = QR + Qc

For the radiant heat losses, the law of Stefan and Boltzmann is valid, i.e. QR = et aA tD ( Tp4 -- Ts 4)

Qc = A(Tp - Ts)

(2gh) 1/2

Wp . . . .

A ( T e - - Ts)

-

g

g

//2h~ 1/2 - ~g)

(31)

(27) v

From this equation, the pumping speed is calculated. For our example, one obtains I/G=2.78 x 10 4m3s

(30)

Introducing the maximum drop velocity of the sample (wp.... =(2sh) I/2 into the Reynolds number (Eq. (20)) yields

Dr 12G4 1..2 1,'3 2 [2 + 0.6 ( ~ - n 2 ~ - V ) 0 . 6 3 9 - ] Pi -- ~t o-A ( Tp 4 -- Ts 4)

c~ dt

The drop time tD is obtained with the law of a freefalling particle, i.e. tD

=

(29)

The convective heat losses are described by the equation of Newton, i.e.

7=~pp When introduced into Eq. (16), it follows that

(28)

\g/

\

v /

or, using Eq. (31), we have

1=999.261h-1

Therefore, the flow conditions in the nozzle are laminar with a value of Re = 197.24.

,33, Introducing the Eqs. (33) and (23) into Eq. (29) leads to expressions for the convective heat losses as

3.2. Construction and operation o f the CGC system

/t

An oil-free pump ensures that no oil mist is added to the gas stream. Two cold traps (LNt) filled with liquid nitrogen are placed before and behind the dry pump to protect the pump and the vessel against evaporation products (Fig. 1). The circulating gas stream flows through valves V4 and V5, the first liquid nitrogen trap, suction valves V6 and V9, then through a small tank (Tk) to equalize the discontinuously pumped stream, through the second liquid nitrogen trap and, finally, via valve V8 into the vessel. Dependent on the tuning of valve V7, a certain share of the gas stream passes valve V8. In this way, it is possible to control the gas stream to the sample and to adjust its temperature. During the evacuation process of the vessel, the oil-free pump is isolated by closing valves V6, V7 and V9. The inert gas atmosphere inside the pump is prepared by closing valve V7 while the pump is working and by opening and closing of valve V6 several times during the fresh gas flow into the vessel.

Qc = A(r~ - rs) ~2

tF

/0[ [

O c = A ( T p - - Ts ) ~--~o

12prX/3 dt

l

2 + 0.6tD '/2

(34)

2prl/3

2tD + 0.4tD 3!2

l

(35)

Therefore, the total heat losses during the drop of the sample are QD = ~ t c r A t D ( T p 4 x ~

Ts 4) + A ( T p -

2t D + 0.4tD 3/2

Ts) P r I/3

(36)

The CGC system affects the heat losses, because of the flow velocity of the gas inside the vessel. Because of this, the relative velocity of the falling sample increases. This fact is taken into account by the Reynolds number (Eq. (20)), i.e. Re =--DP [(2gh)l/2 + w] = tv v

+ -- w v

(37)

K. Schaefers et al./ Materials" Science and Engineering A 197 (1995) 83 90

88

Forced convection is only present inside the vessel. Between the vessel and the calorimeter, the atmosphere is motionless. The drop time in the vessel is 0.153 s. For a vanadium sample of diameter 8 m m at a temperature of T = 1700 K, the convective heat losses without the C G C system are calculated to be Qc = 15.05 J and, with the C G C system, they are calculated to be Qcw = 15.18 J. The difference of about 0.13 J (about 1%) is negligibly small, i.e. the sample is hardly affected by the C G C system inside the vessel.

5. Enthalpies of the pure metals nickel, iron, vanadium and niobium in the undercooled state The heat content of the sample (Ah) results from the heat transferred to the calorimeter (Eq. (15)) added to the heat losses during the drop, i.e.

Enthalpies of the pure metals nickel, iron, vanadium and niobium, and comparison of the measured (AH(T)) and calculated (AHi,caI(T)) values using Eqs. (41) (44) Metal

Temperature (K)

Nickel

1463 1500 1565 1681 1889

52610.8 54386.7 57749.4 62388.1 71509.1

53971 55265.5 58066.7 63065.7 72029.4

- 1.97 - 1.59 -0.55 - 1.07 -0.61

1613 1649 1654 1711 1737 1773 1819 1904 2035

63316.6 64806.6 65207.7 68681.6 69520.1 70870.0 73829.8 76969.2 81876.5

63414.3 65071.2 65301.3 67924.6 69121.3 70778.1 72895.2 76807.3 82836.4

-0.15 -0.41 -0.14 -0.01 0.58 -0.11 1.28 0.21 - 1.16

2016 2031 2035 2072 2103 2120 2136 2153 2178 2228 2315 2403

75528.4 76002.6 77710.5 77471.2 80445.2 81805.5 81201.4 83109.7 84284.8 85989.6 90411.8 93254.3

76033.1 76733.9 76920.8 78649.4 80083.7 80873.3 81658.2 82447.7 83625.1 85937.7 90002.4 94113.7

-0.66 -0.95 1.03 1.5 0.45 1.15 -0.56 0.8 0.79 0.6 0.45 -0.91

2468 2525 2582 2630 2667 2680 2739 2818

97456.2 99619.1 101497.3 105065.2 105277.9 104267.5 106593.4 112907.9

96602.6 98967.4 101386.5 103362.7 104908.6 105443.4 107941.9 111225.9

0.88 0.66 0.11 1.65 0.35 - 1.11 1.25 1.51

Vanadium

Niobium

=

Ahc + QD

(38)

The enthalpies presented are extensive quantities which must be converted into molar quantities by division by the mole number of the sample. Therefore, the molar enthalpy (AH) of the sample is Ah

Ah = H ( T ) -- H(298) = - -

(39)

Hp

Here, H ( T ) and H(298) are the enthalpies at T = T and T = 298 K respectively. For the experiments, the nickel, iron and vanadium were of 99.9% purity, and the niobium 99.6% purity. For the stable liquid state, the thermodynamic data of Barin [35] were taken for nickel and iron, those of Lin and Frohberg [20] for vanadium, and those of Betz and Frohberg [17] for niobium. The temperature-dependent enthalpy for the liquid state (AH(T, 1)) is formally described by A H i ( T , 1) = AHi(rm, 1) + C p , , ( T - - Tm)(J mol

Table 3

Iron

Ah

AH,(T) AHix,l(T ) ( J m o l ') ( J m o l - I )

DH (%)

"6 E

(40)

90000 Barin

L--I 8 0 0 0 0 o~ o4

l)

[ 3 5 3 ~

70000

I

I-

60000

II I--

Q.

r ¢-

50000 40000

-['m,Ni

30000 1300

1500

1700

1900

temperature

2100

2300

[K]

Fig. 3. Enthalpy of pure nickel: O, this work; ©, after Barth et al. [15]. n 0

100000

E

--) i._J

90000

o~ (-4

80000

Barin

I

I-

70000

II I--

60000

<3 _%

50000

Tm,F e

c" 4OOO0 o

1400

1600

1800

2000

temperature

2200

2400

[K]

Fig. 4. Enthlapy of pure iron: 0 , this work; 0 , after Barth et al. [15].

K. Sehaelers et al.

0

110000

0

E -3

I

130000

E 100000

0"~ Cxl I

90000

,=,

110000

/

80000

v I I

100000

I'-

70000

ll

t--

<3

L

120000

u

t--

89

Materials Seienee and Engineering A 197 (1995) 83-90

90000

I-'-

60000

<3

80000

>.

50000

0..

AZ

40000

-

i

1700

i

i

i

i

1900

i

2100

i

!

I

i

,

,

2300

temperature

, I , 2500

, 2700

(Jmol

AHNi(T, 1) = 65 005 + 4 3 . 0 9 5 ( T - 1726)

(J mol

AHvc(T, 1) = 72 435 + 46.024(T-- 1809)

')

1) (42)

(J m o l - )

AHyb(T, 1) = 10 7967 + 4 1 . 7 8 1 ( T - 2740)

(43)

(J mol -~) (44)

With average sample weights between 2.0 and 3.5 g, the enthalpies were determined down to the following undercoolings (AT = Tm - T): nickel, AT = 263 K; iron, A T = 196 K; vanadium, A T = 186 K; niobium, A T = 2 7 2 K . The measured enthalpies (AH~(T)) are presented in Table 3, together with the calculated enthalpies of Eqs. (41)-(44) and the corresponding deviation (DH), where

AHLcd ( T)

2300

2500

2700

2900

3100

I-K]

Fig. 6. Enthalpy of pure niobium: Q, this work.

(41)

AH~ (T) - AHi.caI(T)

i 1 [ 1 1 1

60000

temperature

with Tm the melting temperature, AHi(T m, 1) the enthalpy of the liquid metal i at the melting point and Cp,~ the heat capacity of the liquid metal. The equations for the four metals are

DH--

o @

[K]

Fig. 5. Enthalpy of pure vanadium: 0 , this work.

AHv(T, 1) = 83 835 + 4 6 . 7 2 ( T - 2183)

Tm , N b

70000

c-

6. Conclusions A combination of electromagnetic levitation apparatus and a drop calorimeter is presented; the system is also equipped with a CGC system to reduce the experimental temperatures. With this technique, the enthalpies of the pure metals nickel, iron, vanadium and niobium have been measured in the liquid undercooled range. The presentation of the results in the form of temperature enthalpy diagrams enables us to compare the stable liquid state above the melting point with the undercooled state. It is shown that the enthalpies in the undercooled range follow the temperature functions of the liquid stable state.

Acknowledgements We wish to thank the Deutsche Forschungsgemeinschaft for their financial support.

References 100

(%)

(45)

The enthalpies are shown in Fig. 3 6. The first two figures for nickel (Fig. 3) and iron (Fig. 4) also include the data of Barth et al. [15], who also used electromagnetic leviation with an adiabatic drop calorimeter. Furthermore the figures contain the temperature functions of the enthalpies for the liquid and solid states, taken from Refs. [17], [20] and [35]. The accuracy of our values is + 5 % [36]. It can be seen that the enthalpies of the undercooled metals can be described by the same temperature dependences that are valid for the stable liquid states. This behaviour corresponds to other physical properties extrapolated from the stable liquid state into the undercooled state.

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