Ind. Eng. Chem. Fundam. 1902, 21, 484-485
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COMMUNICATIONS Thermal Conductivity of Liquid Mercury Considering the liquid state to be made up of clusters of distorted microcrystallites and assuming a predominant role of structure scattering of phonons, a theory has been developed for the cluster contribution to the total thermal conductivity of a metallic liquid. The total thermal conductivity is then given by the addition of cluster thermal conductivity and electronic thermal conductivity. Calculations have been made for liquid mercury over a wide range of temperature. The calculated values are found to be in excellent agreement with experimental data.
mn'is the average mass of a cluster, and ((V - V,)/n)is the volume available to the cluster for translatory motion. Due to the large size of clusters and the hindrance caused by other clusters, it can move only in the free space available around it. In other words, the surrounding cluster create a cage around the given cluster which keeps it confined in a given region (Hirshfelder et al., 1954). Using the Stirling formula for n!,eq 2 reduces to
Introduction Bagchi (1970,1972),through the analysis of radial distribution functions of liquids, has suggested that the liquid state may consist of clusters of distorted microcrystallites. We here use this model to obtain an expression for the lattice contribution to the total thermal conductivity of a liquid metal. The total thermal conductivity is obtained by addition of cluster thermal conductivity and electronic thermal conductivity. Groose (1966a,b)has calculated the thermal conductivity of metallic liquids assuming the lattice contribution to be negligible on the basis of calculations of McLaughlin (1964) for liquid metals. A careful analysis of the calculated results and experimental data, however, reveals that the lattice contribution, although small, is not negligible. Deviations to the extent of about 10% exist between the two results. There is a definite trend, particularly at high temperature, indicating the significant lattice contributions. Here we attempt to correlate the thermal conductivity of mercury, a metallic liquid, for which experimental data exist over a wide range of temperature. Development of Theory The thermal conductivity of a metallic liquid can be treated as the sum of two components, phonon exchange controlled by structure scattering Prabhuram and Saksena (1981) and heat transport due to the free electrons. According to Bagchi (1970, 1972), liquid consists of clusters of distorted microcrystallites randomly oriented and distributed in the volume of the liquid. Inside the clusters the molecules have solid-like behavior, while the cluster as a whole may have gas-like random translatory motion. Thus the partition function of the liquid can be written as
ZL = (Zclg)n(Zc,tr)"/n! ZL = [ e ~ p ( - P E ~ n ' / N ) ( T / 6 ' ) ~ x "'1"
[(
vi
The number of clusters n at a given temperature and volume is now calculated from eq 3 using the variational method. the Helmholtz free energy of the system is given by A = -kT In ZL (4) Minimizing A for the equilibrium, the average number of clusters a t a temperature T and volume V is given by (Kozak and Rice, 1968)
Considering the scattering of phonon due to the mosaic blocks of microcrystallites-the mean free path can be related to the wave vector K as (Pomeranchuk, 1942)
where a is lattice constant and M is a constant dependent on the size of the cluster, which is inversely proportional to the equilibrium number of clusters at a given temperature and volume. Thus, one can write A' M= (7)
(1)
")In/
n! (2)
T3/7(
where A' is a constant. Further, the phonon exchange is limited by vacant sites between microcrystallites. The energy exchange as well as the number of interactions will be proportional to the concentration of the solid-like molecules present in a given volume. Thus the cluster contributions to the thermal conductivity of liquid will be given by
where n'is the average number of molecules in a cluster and n is the number of clusters in a liquid of volume V a t temperature T. n and n' are related to each other by the condition that the number of molecules N in a liquid is constant; i.e., nn' = N . Zc? is the partition function for the translational motion of the cluster and Zc
