Seetharaman S. Fundamentals of metallurgy
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detaches. The maximum pressure is determined and used to calculate the
interfacial tension. This method has not been widely used for measurements
at high temperatures.
Results
Slag±metal reactions are ionic in nature (e.g. equation 4.10):
S
metal
O
2ÿ
slag
O
metal
S
2ÿ
slag
(4.10)
Rapid mass transfer between the metal and slag has been found to be associated
with low interfacial tension (Fig. 4.13) and when the rate of mass transfer
slowed down the interfacial tension was found to increase to a high value once
more.
75,76
It has been suggested that that the decrease in interfacial tension
occurs when the rate of oxygen transfer from slag to metal exceeds a critical
value. Similar ph enomena have been observed in organic systems
77
and it was
found that the decrease in interfacial tension did not occur when the volumes of
the two liquids were similar. Consequently, there is a possibility that the
observed phenomenon is due to the fact that the volume of the metal is much
lower than that of the slag in the X-ray sessile drop method.
4.5 Properties related to heat transfer
4.5.1 Heat capacity (Cp), enthalpy (H
T
ÿ H
298
)
The temperature dependence of the heat capacity is usually expressed in the form:
Cp a bT ÿ c/T
2
(4.11)
The enthalpy at a specific temperature, T, relative to 298K, can be derived by
integrating Cp between 298 and T.
H
T
ÿ H
298
Z
T
298
CpdT aT ÿ 298
b=2T
2
ÿ 298
2
c=T ÿ c=298 4:12
which can be expressed as a power series:
H
T
ÿ H
298
aT b=2T
2
ÿ c=T ÿ d 4:13
where d contains all 298K terms.
Methods
The following methods have been used to measure heat capacities and
enthalpies:
· Calorimetry is a well-established tec hnique but is difficult and time-
consuming at higher temperatures. Detailed descriptions of the methods
136 Fundamentals of metallurgy
adopted have been reported elsewhere.
78
Enthalpies have been studied
extensively by drop calorimetry. In this technique a crucible containing the
sample (of known mass and temperature) is dropped into a copper or silver
block and the temperature rise of the block (typically 1±5K) is measured very
accurately. In levitated drop calorimetry the sample is levitat ed by EML,
ESL, ADL or USL and in the case of high melting metals (e.g. W) a laser
heating can be used to augment the EM he ating. The experimental
uncertainty in enthalpy measurements is usually around 1%.
·
In recent years the development of commercial, differential scanning
calorimeters (DSC) has resulted in the wide use of these instruments to
measure to Cp and enthalpies. There are two types of DSC: (i) in which the
temperature difference between the sample and reference pans are monitored
(differential temperature scanning calorimeter (DTSC)) and (ii) in which
the power required to keep the two pans at the same temperature
(differential power scanning calorimeter (DPSC)). DPSC can be used to
temperatures to <1000K and Cp values have an experimental uncertainty of
1±2%. DTSC is used for temper atures in the range 1000K to 1800K and
the experimental uncertainties are probably around 5%. Enthalpies of
fusion or transition are manifested as a peak in the apparent Cp±T Curve (see
Section 4.2.4 and Fig. 4.9) and are determin ed by integrating under the curve
(experimental uncertainty < 10%). Fraction solid values are usually
derived in the cooling cycle by integrating under the curve to a specific
temperature and dividing this value by the total area under the curve for the
solidification process. One problem with dynamic measurements (DPSC,
DTSC) is that in the fusion (or solidification) process the calorimeter must
suddenly provide (or remove) a large amount of energy (latent heat). Since
the temperature is not static, the apparent Cp is `smeared' over a few
degrees. The single pan calorimeter avoids this problem. It consist s of two
con centric crucib les
79
with thermocouples p laced on the walls. The
temperature difference between inner and outer crucibles is monitored
continuously. The sample is placed in the inner crucible. Measurements are
made with the inner crucible (i) empty (ii) filled with a calibrant of known
Cp (e.g. Al
2
O
3
) and (iii) filled with the sample. Accurate results were
obtained with Al alloys.
79
· In the exploding wire method the specimen is heated very rapidly
24,25
and
the enthalpy is derived from the known current and potential values and the
measured temperature. As mentioned above, solid state transitions tend to be
slow and consequently enthalpies tend to get `smeared out' in regions where
solid state transitions occur. Experimental uncertainties in the enthalpy are
probably around 5%.
Problems
The principal problems encountered are:
Measurement and estimation of physical properties of metals 137
·
Measurement of temperatures since (i) there can be a considerable gradient
between the measurement site and sample temperatures in conventional drop
calorimetry and (ii) in levitat ed drop calorime try the measurements are made
by pyrometry and consequently either the emissivity must be known
accurately or the temperature scale should be calibrated.
·
When measuring the enthalpy of metallic samples using conventional drop
calorimetry, it is neces sary to use oxide (e.g. Al
2
O
3
) crucibles and these have
a high heat capacity (cf. metallic crucibles used to hold slags and glasses) and
this reduces the accuracy of the results obtained for the metal.
·
The rapid quenching of slags and glasses can result in the sample adopting a
metastable state (i.e. a non-equilibrium state).
·
It is necessary to differentiate enthalpy data to obtain Cp values from
(H
T
ÿ H
298
) valu es; more accurate values of Cp can be derived if Cp values
are available at low temperatures but the presence of high-temperature, phase
transitions (e.g. in steels, Ni-superalloys and Ti-alloys) makes differentiation
very difficult.
·
Solid state transitions tend to be sluggish and when using dynamic tec hniques
(such as DSC) the sample does not have enough time to disorder when
passing through the transition range. This is an especial problem where the
transition is followed by a second transition (e.g. in Ni-superalloys (Fig. 4.9)
where the coarsening of the
0
phase transition is followed by the dissolution
of the
0
phase
23
).
· In some slags and glasses a glassy phase converts into a supercooled liquid at
the glass transition temperature and then subsequently crystallises at a higher
temperature; measured properties in this region where glass and crystal co-
exist are in an undefined (non-equilibrium) state and refer solely to that
sample for that specific temperature.
4.5.2 Thermal conductivity (k) thermal diffusivity (a) thermal
effusivity (e)
The thermal conductivity (k) is related to thermal diffusivity through the
equation
k Cp.a. (4.14)
Thermal conductivity can be defined as the heat flow (q) across a unit area per
second per unit temperature gradient (dT/dx). Thus the units are Wm
ÿ1
K
ÿ1
k (ÿq/A)(dT/dx) (4.15)
The principal conduction mechanism in metals at high temperatures is through
the transport of electrons, although, phonon (or lattice) conduction can make a
significant contribution at ambient temperatures. Consequently, there is a
relation between the electrical and thermal conductivity of metals at high
138 Fundamentals of metallurgy
temperatures since both involve electron transport. This relation is known as the
Wiedemann±Franz±Lorenz (WFL) rule
k L
o
.T./R (4.16)
where L
o
is a constant with a value of 2.445 10
ÿ8
WK
ÿ2
, R is the electrical
resistivity and T is in K. The WFL rule has been shown to be valid for a large
number of pure metals in both the solid and liquid phases near the melting
point.
80
Consequently, thermal conductivity values can be derived from
measurements of the electrical resistivity. This method is particularly useful
for measurements in the (solid + liquid) or `mushy' region (see Section 4.2.5). It
has been proposed
81
that the temperature dependence takes the form
k (A. L
o
T/R
el
) B (4.17)
where A and B are constants. However, it has been pointed out
94
that measured
thermal conductivities of molten Ge, Si, Sn and Pb decrease with increasing
temperature while the values calculated from electrical resistivity values and
Equation 4.16 show the opposite trend. This was attributed to an inelastic scattering
of electrons (possibly due to electron±electron scattering) at higher temperatures.
94
There is a large increase in electrical resistivity when a solid transforms to a
liquid; the ratio (R
m
l
/R
m
s
) has a value of about 2 for most metals but transition
metals have much smaller values (Fe, 1.06, Co, 1.15 and Ni, 1.40)
7
and thermal
conductivities of most metals show a marked decrease at the melting point.
There are two major problems affecting thermal conductivity and diffusivity
measurements, namely, convection and radiation conduction in semi-transparent
media such as glasses and some slags in both liquid and solid phases.
Convection
As described in Section 4.2.3, convection is difficult to eliminate. Methods used to
measure the thermal conductivity or diffusivity fall into two classes, namely,
steady state and transient methods. Convective contributions to the thermal
conductivities (or diffusivities) tend to be large in steady state measurements.
However, in transient methods (where the temperature±time curves are determined
following a short heat pulse) the convective contributions are much smaller,
especially if the measurements can be carried out in a time period shorter than the
time required (< 1 sec) for convection to be initiated. Consequently, steady state
experiments are not very suitable for measurements on the liquid phase and
transient methods are preferred. Convection in transient experiments can be
minimised by taking the various measures outlined in Section 4.2.3 into account.
Radiation conductivity
In the 1960s it was noted that measurements of the thermal conductivities of
glasses were dependent upon the thickness of the specimen used (Fig. 4.14). In
Measurement and estimation of physical properties of metals 139
order to explain this observation it was proposed
82
that in addition to the phonon
(or lattice) conductivity (k
c
) there were also contributions from a heat transfer
mechanism known as radiation conductivity (k
R
). This radiation conduction
arises by a mechanism of absorption and re-emission. Consider a pulse of energy
focused onto the surface of a glass specimen. The surface layer will increase in
temperature and this will be at a higher temperature than the next layer in the
glass and thus heat will be radiated from the surface layer to this second layer.
This second layer will absorb the heat and will be, consequently, at a higher
temperature than the third layer and thus will radiate heat to the third layer and so
on. It was also found that k
R
increased with increasing thickness until it came to a
certain point where it remained constant. The sample is said to be `optically thick'
at this point. Optically thick is usually understood to occur when the product of
absorption coefficient (*) thickness is > 3. For optically thick conditions, k
R
can be calculated using equation 4.18 where n is the refractive index and is the
Stefan Boltzmann constant with a value of 5.67 10
ÿ8
Wm
ÿ2
K
ÿ4
.
k
R
16.n
2
.T
3
/3* (4.18)
However, it is difficult to calculate k
R
if it lies outside the optically thin region.
The effective conductivity k
eff
is given by equation 4.19.
k
eff
k
c
k
R
(4.19)
where k
c
is the phonon (or lattice) conductivity.
Radiation conductivity is diminished by the presence of crystallites in the
sample which scatter the radiation (i.e. photons). Radiation conductivity is also
decreased by the presence of transition metallic oxides such as FeO, NiO or
Cr
2
O
3
which increase the absorption coefficient significantly. For solids the
4.14 Effective thermal conductivit y of glass as a funct ion of temp era tur e
showing the effect of sample thickness.
140 Fundamentals of metallurgy
absorption coefficient in equation 4.18 should be replaced by the extinction
coefficient E * s* where s* is the scattering coefficient.
Although, steady state methods can be used to calculate the effective
conductivity of semi-transparent media such as glasses in the solid state it is
essential that the sample be optically thick. Furthermore, it is necessary to measure
the absorption coefficient and refractive index of the sample if the contributions
from k
R
and k
c
need to be determined. However, transient techniques are preferred
for measuring k
eff
since contributions from k
R
tend to be much smaller.
The thermal conductivities of slags and glasses are dependent upon the
polymeric structure of the melt. The thermal resistance (1/k
m
, i.e. reciprocal
thermal conductivity at the melting point) has been shown to be a linear function
of struct ural parameters (e.g. (NBO/T) or the corrected optical basicity).
11,83
The
thermal resistance was found to decrease as the polym erisation increased (or the
conductivity increases as the chain length of the slag increas es (i.e. SiO
2
)
increases). Susa et al.
83
showed that the thermal resistance terms were smaller
for the Si-O bonds in the chain than those for ionic (NBO) bonds at the end of
the chain (denoted as O-O bonds). There is also some evidence that a change in
slope in the k-temperature plot for some solid slags may be due to the change of
a glass into a supercooled liquid which occurs at the glass transiti on temperature.
Thermal effusivity (e) is a measure of the ability of a body to exchange
thermal energy with its surroundings (thermal impedance) and is given by the
following equation and has units of Js
0.5
m
ÿ2
K
ÿ1
.
84±86
e (k.Cp.)
0.5
(4.20)
It has been reported that thermal effusivity measurements carried out on a drop
of liquid at ambient temperatures were free from convection contributions
84
and
this may provide a way in the future of obtaining thermal conductivity/
diffusivity values which are free from convective contributions.
Methods for thermal conductivity and diffusivity measurement
1. Steady state techniques
The conventional steady state techniques such as the concentric cylinder,
parallel plate and axial and radial flow methods
78
are all suitable for
measuring thermal conductivities of metals in the solid state. However, they are
not particularly suitable for measurements (i) in the liquid state because of the
difficulties of minimising convection at high temperatures, and (ii) on semi-
transparent solid unless the samples are optically thick (see above).
2. Non-steady and transient techniques
The laser pulse apparatus is manufactured commercially by several companies and
is a very popular technique.
78
The sample is a disc-shaped specimen 10±12 mm in
Measurement and estimation of physical properties of metals 141
diameter and usually 2±3 mm thick with parallel faces. For measurements on solids
the sample is placed horizontally and a pulse of energy is directed onto the front
face of the specimen. The temperature of the back face is monitored continuously
(usually with an infra-red detector). The temperature±time curve goes through a
maximum (T
max
) because of radiation losses and it is customary to determine the
time (t
0.5
) needed for 0.5 (T
max
). The thermal diffusivity is derived from the
relation a 0.1388L
2
/t
0.5
where L is the thickness of the specimen.
For measurements on liquids and semi-transparent media a metallic (e.g. Pt
or C) disc is usually placed on the upper surface of the sample and the
temperature transient can be measured by monitoring the temperature of the disc
or by the temperature of the back face.
87
Liquid samples are usually held in
sapphire or silica crucibles
88,89
and it is desirable to carry out the measurements
very rapidly (< 1 sec) to avoid the initiation of convection.
In the transient hot wire (THW or line source) method a current is applied to
a fine wire (circa 0.1 mm diameter) of known length which acts as both a
heating element and a resistance thermocouple.
90,22,33,80
The wire is immersed
in the melt and a current is applied then the temperature rise of the wire (T) (or
strip) is measured continuously as a function of time . The thermal conductivity
is derived from the reciprocal of the slope of the linear portion of the plot of T
versus ln (time). Convective contributions to k
c
can be detected as departures
from the linear relation. The current should be applied for < 1 sec to avoid the
establishment of buoyancy driven convection. Experiments have been carried
out in 0 g using a drop tower to minimise convection.
22
Thermo-capillary
convection can be minimised by floating a lid on top of the test liquid.
22
There is
some evidence indicating that radiation conduction contributions in semi-
transparent liquids (and solids) are smaller in the THW method than in the laser
pulse method. This is probably due to the much smaller surf ace area of the wire
compared with that of the metal disc used in the laser pulse. When the method is
applied to metals it is necessary to insulate the metallic probe from the melt and
coatings of Al
2
O
3
or other oxides are applied to the wire or strip. Recent work
has shown that thermal conductivity values for liquid metals must be corrected
for the thickness of the insulating coating on the probe.
91
In the radial temperature wave (RTW) method,
92,80
a modulated heat flux
is applied along the centre of a cylindrical sample. The variations in temperature
are monitored on the o utside of the specimen. There is a phase lag between the
input and output and this is related to the thermal diffusivity. Thermal
diffusivities can be calculated from (i) the amplitude of temperature oscillations
and (ii) from phase differences. Values calculated by the two methods are
usually in good agreement.
92
In the plane temperature wave (PTW) method
93,80
the plane temperature
waves are generated by bombarding the specimen with a harmonically-
modulated electron beam. PTWs are directed onto one face of a disc-shaped
sample and the temperature transient is recorded on the other face. The method
142 Fundamentals of metallurgy
can be used either at constant temperature or in a dynamic mode (and heating
rates of up to 1000K per second have been used) to derive thermal diffusivity
values. For measurements on liquids only the central portion of the disc was
allowed to melt and high heating rates ensure that measurements are carried out
very quickly thereby avoiding the problem of buoyancy-driven convection.
93
With temperature gradients across the free surface some contributions to the
thermal diffusivity from thermo-capilla ry convection might be expected.
Problems
Solids
·
In metallic alloys phonons can be scattered by grain boundaries, precipitated
particles, etc. and consequently, the thermal conductivity or diffusivity value
depends on microstructure of the sample and this is dependent upon both the
heat and mechanical treatment. Differences in values of k or a of 10±20% can
occur when repeating the measurements on the same sample.
·
Corrections should be applied to account for the effect of the insulating oxide
layer on the wire (or strip) used in the THW method
91
when making
measurements on metals.
·
The determination of the magnitude of the contribution from radiation
conduction (k
R
)
94
is the principal problem with measurements on slags and
glasses in the solid state (and there may be considerable difference in the
contributions to k
R
for the experimental and the industrial conditions). Some
indication of the magnitude of k
R
in the experiments could possibly be
obtained by: (i) doping the slag or glass with FeO which increases the
absorption coefficient and decreases k
R
for optically thick specimens (but for
optically thin specimens it could also have the effect of making the specimen
more optically thick), and (ii) by using materials with different emissivities
for the wire in the THW method on the disc or the top of melt in the laser
pulse method (e.g. Pt, C, W).
Liquids
The principal problems with thermal conductivity and diffusivity measurements
lie in eliminating convection and minimising radiation conduction (in the case of
molten slags and glasses). These problems and ways of overcoming them have
been described above.
4.5.3 Electrical resistivity (R)
Electrical resistivity increases with increasing temperat ure in both the solid and
liquid phases. The resistivity of the liquid metal at the melting point is
approximately twice that of the solid for many metals but (R
m
l
/R
m
s
) ratios for Fe,
Co and Ni are between 1 and 1.4.
Measurement and estimation of physical properties of metals 143
Methods
Electrical resistivity (R) measurements on melts can be classified as either direct
or indirect measurements.
Direct measurements
The electrical resistivity is determined from measurements of current and
potential and the use of Ohm's Law.
Capillary methods have been used for measurements on molten metals. The
4-probe method is usually adopted in which the potential drop across the liquid
column is determined with two electrodes and the other two electrodes are used
to measure the current.
7,78
The cell constant is usually determined using a liquid
of known resistivity (e.g. Hg). The two principal problems lie in the selection of
(i) the capillary material (e.g. glass, quartz) and (ii) the electrode since C, W or
Pt may dissolve in (or react with) the metal (one solution is to use a solid
electrode of the same material as the sample by cooling the ends of the cell).
The exploding wire method has been used for measurements on the solid
state and also for measurements on the liquid phase.
24,25
Values obtained for the
liquid are frequently in good agreement with values obtained with 4-probe
studies.
25
Capillary methods have been little used for measurements on slags. This is
due to the fact that resistance values are much smaller and are also more prone to
polarisation and other effects. Two-electrode probes have been used with
different geometries:
· central electrode where one electrode is immersed in a crucible and the
crucible serves as the other electrode;
18,33
· two electrodes are immersed in a cell;
18,33
· ring electrode cells where the electrodes are in the form of two concentric
cells.
18,33
However, 4-electrode cells are preferred since they largely elim inate the effects
of polarisation on the current-carrying electrodes (by measuring the potential
drop and the current at different electrodes).
Resistances have been measured on molten slags using (i) AC bridges (ii)
potentiometric methods and (iii) phase sensitive techniques.
18,33
The effect of frequency on the measured resistance/conductivity has been
discussed by several workers.
33
Indirect method
The rotating magnetic field method is based on the principle that a torque is
developed when a liquid metal is subjected to a rotating magnetic field; the
torque is inversely proportional to the electrical resistivity.
7,96
Values of the
liquid density are needed to cal culate the resistivity from the torque.
144 Fundamentals of metallurgy
4.5.4 Emissivity (), absorption coefficient (*)
Emissivity () is defined as the ratio of radiated energy emitted by a surface to
that emitted by a black body. However, the radiated energy is dependent upon
the nature of the surface, temper ature, wavelength () and the direction.
Spectral emissivity (
) denotes the emissivity of a body at a specific
wavelength. Total emissivity (
T
) is the ratio of (intensity of radiation emitted at
all wavelengths by a body/intensity of radiation emitted at all wavelengths by a
black body).
With respect to direction emissivities these can be measured either (i) normal
to the plane of the surface (denoted by subscript, N) or (ii) for the hemisphere
containing all the radiated energy (denoted by subscript H). Thus (
N
) and (
TH
)
refer to the normal spectral emissivity and the total hemispherical emissivity,
respectively.
The nature of the surface also affects the measured value, and measured
values depend upon whether a surface is rough or smooth or whether the surface
has been oxidised (or nitrided) by reaction with the atmosphere. Most cited
values of emissivity refer to polished surfaces.
Methods
Emissivities are usually measured (using a spectroscope) by determining the
ratio of the energies emitted by the surface divided by that emitted by a
neighbouring black body. Values have been obtained using both electro-
magnetic
97
and cold crucible levitation.
98
Emissivities for the solid and liquid
phases have also been obtained using the exploding wire method using a
submicrosecond-resolution laser polarimeter.
99
However, they can also be determined by measur ements of other optical
constants (refract ive indices, refl ectivities, etc.) using rotating analyser
ellipsometry to determine the polarisation state of monochromatic light reflected
from the surface at various angles.
100
Values of (Cp/
TN
) can be derived from the cooling curves of levitated
spheres in vacuum.
54,101
Values of emissivity given in Table 4.2 were obtained
from various sourc es.
Absorption coefficient (*) and extinction coefficients (E)
Absorption and extinction coefficients of semi-transparent media are needed for
calculations of the magnitude of the radiation conductivity (k
R
) in glasses and
slags.
Methods
There are two methods of measuring absorption coefficients.
Measurement and estimation of physical properties of metals 145