JP2008008793A - Thermophysical property measuring method and measuring device of high-temperature melt conductive material - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To measure directly and accurately a thermal conductivity κ and an emissivity ε of a high-temperature melt conductive material. <P>SOLUTION: In thermophysical property measurement using a high frequency coil having a space for fusing the high-temperature melt conductive material as a liquid drop at a center part and performing electromagnetic floating thereof, a laser heating means for applying heat energy from an upper part in a modulation mode, a means for applying a static magnetic field for suppressing rocking of the fused material and suppressing convection inside the material, and a means for measuring the temperature of the fused material from a lower part, a relation between an angular frequency of laser periodic heating acquired by measurement and a phase difference is fitted based on a nonlinear least-squares method by a numerical analysis result of a mathematical expression derived from an unsteady state heat conduction equation, to thereby acquire the thermal conductivity κ and the emissivity ε which are parameters. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention relates to the measurement of thermal properties (hemispherical total emissivity and thermal conductivity) of high-temperature melt conductive materials such as metal melts and semiconductor melts.

In fields such as crystal growth of silicon single crystals, casting of heat-resistant alloys used for turbine blades, welding of automobiles and structural materials, etc., speed and precision are required, and international competition is based on conventional intuition and experience alone. It has become impossible to deal with. In particular, in the IT field and aerospace field where high quality is required, product development by computer simulation analysis has become essential.
There is a strong demand from the industry for highly accurate thermophysical values of the melt (liquid state) and crystals (solid state) of materials used in such simulation analysis. However, for example, a semiconductor material such as silicon or an ultra-high temperature heat resistant material such as a nickel-based alloy is in a liquid state (high temperature melt) is chemically extremely active, and its thermophysical value is difficult to measure. There is no reliable value for thermophysical values used so far.

The conventional measurement of thermophysical properties is schematically described below. The thermophysical value when the substance is in a solid state is relatively easy to measure. For example, the specific heat can be measured by a method called an alternating current calorimetry method. The AC calorimetry method is a method in which a sample is heated with AC vibration and measured by the amplitude of the temperature response (the amount of change in temperature from a certain temperature). The specific heat of a solid can be measured by this method. However, the AC calorimetry method cannot be applied to a sample in a molten state because heat conduction to a container holding the melt occurs.
Therefore, a method for obtaining a specific heat from a melt by dropping a melted sample into an indirect medium such as oil and determining the temperature of the indirect medium is used. In this method, there is a problem of low accuracy due to temperature drop during the fall, reaction with the indirect medium, contamination of the indirect medium resulting therefrom, and indirect measurement.

In order to accurately measure the thermophysical value of the object being measured at a high temperature as described above, various floating methods (electromagnetic floating, electrostatic floating, gas floating, sonic floating, etc.) It is considered to use a technique for keeping the contactless. By measuring the thermophysical value by suspending the sample, it is expected to prevent the sample from being contaminated from the wall surface of the container and to remove measurement noise from the container. However, in most thermophysical properties measurements that are carried out in combination with the floating method, there are no accurate measured values.
For example, in the measurement of the thermal conductivity of silicon, in the above method, in addition to Marangoni convection due to surface tension distribution and natural convection due to non-uniform density in the silicon melt, Fast convection occurs and true thermal conductivity is not obtained.

  Wunderlich and Fecht (Non-Patent Document 1) established a method to measure constant-pressure molar heat capacity and hemispherical total emissivity by performing AC calorimetry on electromagnetically suspended high-temperature melts. Measuring. According to their method, in principle, the thermal conductivity of the sample to be measured can also be measured, but usually the droplet sample has natural convection due to density difference and Marangoni convection due to surface tension difference, so the measured heat Conductivity is an apparent value affected by them, and true thermal conductivity cannot be measured. In Non-Patent Document 2 and Non-Patent Document 3, the thermal conductivity of the melt sample is measured by electromagnetic levitation in a static magnetic field, but the influence of convection due to electromagnetic stirring of the temperature distribution is not considered. The method of calculating thermal conductivity is unknown.

In addition, as a conventional technique for measuring thermal conductivity, a method of heating one end of a rod-shaped sample and measuring a temperature rise due to thermal conduction at a point away from the heating point is common. The measurement of the thermal conductivity of the melt is difficult because the thermal diffusion to the container or the like holding the melt occurs, similar to the specific heat measurement.
As described above, a method of measuring the thermal conductivity in a state where the melt is suspended is also conceivable. However, as with the specific heat measurement, various means for suppressing convection must be taken. For example, in order to suppress the natural convection due to the density difference, it is conceivable to use a microgravity environment such as a rocket or a fall tower. However, there are currently few opportunities to conduct experiments in a microgravity environment that can ensure sufficient experiment time.

There is a method of deriving thermal conductivity by measuring thermal diffusivity from temperature change corresponding to instantaneous sample heating by laser (laser flash method). The relational expression between the thermal conductivity and the thermal diffusivity is expressed as follows, and the thermal conductivity is derived from this formula.
κ = c _p a ρ
Where κ: thermal conductivity (W m ⁻¹ K ⁻¹ ), a: thermal diffusivity (m ² s ⁻¹ ), c _p : mass heat capacity (J kg ⁻¹ K ⁻¹ ), ρ: density ( kg m ^-3 ). Since this method has a short measurement time, it is said that the measurement can be completed before convection occurs. However, when converting into heat conductivity as mentioned above, the data of the heat capacity and density measured by others are required (nonpatent literature 4).
The unsteady hot wire method can directly determine the thermal conductivity. The measurement principle is that a constant current is passed through a heating wire installed in the sample, and the thermal conductivity is determined from the change over time of the temperature rise of the heating wire. However, when the unsteady hot wire method is applied to a metal melt, an insulating film must be applied to the thin wire, and the influence of the insulating film must be evaluated (Non-Patent Document 5).

Another method is to estimate the thermal conductivity from the electrical resistivity using the Wiedemann-Franz rule. The Wiedemann-Franz rule is expressed by the following equation.
κ = LT / ρE
Here, L: Lorentz number (2.45 × 10 ⁻⁸ WΩK ⁻² ), T: temperature (K), and ρE: electrical resistivity (mΩ). The Wiedemann-Franz law is based on the assumption that the relaxation time in the electrical process is equal to the relaxation time in the thermal process, and whether or not this law holds must be confirmed by measurement. In particular, caution is required for semiconductor melts such as alloys and silicon. This calculation method is based on estimation from other physical property values, and does not directly measure thermal conductivity.

  Next, with respect to emissivity, a technique of measuring emissivity by measuring infrared light or the like is generally used. However, when measuring the emissivity of a high-temperature substance, the surrounding materials other than the object to be measured are also in a high-temperature state, so that the heat radiation from them becomes noise, and accurate physical property values cannot be measured. In particular, when the object to be measured is in a molten state at a high temperature, it is always in contact with a container or the like that holds the object, so that it is difficult to measure the thermal emissivity of the melt. It is the same as the measurement.

  The electromagnetic levitation method is a type of levitation technique, in which an induction current is generated in a conductive sample by a high-frequency coil placed around the sample, and the sample is floated using Lorentz force due to interaction with an alternating magnetic field. . Since the high-temperature melt can be held in a non-contact manner, it is used for various thermophysical properties measurements and supercooling experiments. For example, in the metal melt suspended by the electromagnetic levitation method, in addition to Marangoni convection and natural convection, convection due to electromagnetic force is generated, and therefore, thermal conductivity cannot be measured as it is. When a strong static magnetic field (approx. Several Tesla) is applied to the metal melt in this state from the outside, Lorentz force is caused by the interaction between the metal melt and the static magnetic field to move the center of gravity of the melt or in the melt. It works in the direction of suppressing convection. That is, in a strong static magnetic field, the flow of liquid is strongly suppressed and behaves in the same way as a solid in terms of heat. Yasuda et al. (Non-Patent Document 6) report that when a static magnetic field is superimposed on an electromagnetic floating droplet, the droplet's vibration and surface convection are suppressed by the Lorentz force, and the droplet behaves like a hard sphere rotates. did.

Next, the inventors have published a method that can directly measure the thermal conductivity and emissivity of a molten material that could only be measured indirectly.
That is, a high-frequency coil having a space in which a high-temperature melt conductive material is melted at the center and electromagnetically suspended, a laser device that applies thermal energy to the material in a modulation mode from the top of the material, and oscillation of the molten material In the thermophysical property measurement method of a high-temperature molten conductive material using a magnet device that provides a static magnetic field to suppress and suppress convection inside the molten material, and a radiation thermometer that measures the temperature of the material melted from the bottom,
The unsteady heat conduction equation in a spherical coordinate system with the center of gravity of the droplet as the origin, for measuring the temperature of the lower part of the material by laser heating from the upper part of the material,
1) The system is axisymmetric.
2) The thermophysical property value is constant within the range of the temperature amplitude of periodic heating.
3) The incident laser light is absorbed on the surface of the droplet according to the absorption rate of the substance and does not pass through.
4) The intensity distribution of incident laser light follows a Gaussian distribution.
5) Heat radiation from the droplet surface is only radiation.
6) The average temperature rise and temperature amplitude are small compared to the initial temperature.
Obtained under the conditions of 1) to 6)
Temperature response amplitude ΔT _{AC in} AC steady state, temperature response delay from laser heating (phase difference φ _s ), external thermal relaxation time τ ₁ due to radiation, internal thermal relaxation time τ ₂ due to heat conduction Step 1 for preparing 8) to (11),

(Where ε _s : spectral emissivity, ω: angular frequency of laser periodic heating, P ₀ : laser intensity amplitude, C _p : constant pressure heat capacity, ε: hemispherical total emissivity, R: molten droplet radius, T ₀ : (Initial temperature, κ: thermal conductivity, σ _SB : Boltzmann constant)
The melt is heated by laser periodic heating of the molten material, and after measuring the temperature response amplitude ΔT _AC in the steady state of _AC and the temperature response delay (phase difference φ _s ) from laser heating, the laser periodic heating is stopped. Step 2 for measuring the radiation cooling curve of the melt,
Step 3 for determining the numerical value of ω that gives the maximum value of the product ΔT _AC * ω of the temperature response amplitude ΔT _AC and angular frequency of the measured ω, and calculating the constant pressure heat capacity C _{p of the} melt from the equation (8),
For the radiative cooling curve of the melt measured in step 3, the following relational expression T _{(t = t)} = T ₀ between the temperature T _{(t = t)} during radiative cooling and the external thermal relaxation time τ ₁ due to radiation + ΔT _DC exp (−t / τ ₁ ) (12)
(Here, T _{0 is} the initial temperature before cyclic heating, ΔT _{DC is} the average temperature increased during cyclic heating, and t is time.)
To obtain τ ₁ by fitting and calculating the hemispherical total emissivity ε from equation (10),
By fitting the value of the phase difference φ _s measured in step 1 and the equation (9) in which the value of τ ₁ obtained in step 4 is substituted, τ ₂ is obtained, and the constant pressure heat capacity C _p obtained in step 3 is obtained. Step 5 for calculating the thermal conductivity κ from the equation (11) using
A method for measuring the thermal properties of a high-temperature melt conductive material containing benzene has already been proposed (Non-Patent Document 7).
RK Wunderlich and H.-J. Fecht, Meas. Sci. Technol., 16 (2005), 402-416. Satoshi Onishi, Space Utilization Res. 21 (2005), p39 Onishi Shimonori, Proceedings of the 26th Japan Thermophysical Symposium, 475 pages, lecture number C314 T. Nishi, A. Ikari, H. Shibata and H. Ohta, Materials. Trans., 44 (2003), 2369-2374. E. Yamasue, M. Susa, H. Fukuyama, and K. Nagata, J. Crys. Growth, 234 (2002), 121-131. H. Yasuda, I. Ohnaka, R. Ishii, S. Fujita and Y. Tamura: ISIJ Int., 45, No. 7 (2005), 991-996. H. Kobatake, H. Fukuyama, I. Minato, T. Nakamura and S. Awaji: The 26th Japan Symposium on Thermophysical Properties, 2005, Tsukuba C309 460-462.

  The above-mentioned thermophysical property measurement method for high-temperature melt conductive materials is evaluated as a method that can directly measure the thermal conductivity and emissivity of molten materials that could only be measured indirectly. However, the above-described method for measuring the thermal properties of a high-temperature melt conductive material is based on the premise of a thermal conduction relaxation process from the center of the sphere to the outside as a process of thermal conductance due to thermal conduction inside the sample. Therefore, the result was different from the actual measurement method. In view of this, an object of the present invention is to derive a basic equation of heat conduction that faithfully expresses a thermophysical property measurement method by laser heating, and thereby directly measures the true thermophysical property of a high-temperature melt conductive material.

Means for solving the above-described problems are as follows.
1. A high-frequency coil having a space in which high-temperature melt conductive material is melted as droplets at the center and electromagnetically suspended, laser heating means for applying thermal energy in a modulation mode from above the high-temperature melt conductive material, and molten high-temperature melt A means for applying a static magnetic field for suppressing fluctuation of the body conductive material and suppressing vertical convection of the high temperature melt conductive material and a means for measuring the temperature of the high temperature melt conductive material melted from the lower part were used. In the method for measuring thermophysical properties of a high-temperature melt conductive material, the thermal conductivity κ and the hemispherical total emissivity ε of the high-temperature melt conductive material are determined by the following first to third steps. Method for measuring thermophysical properties of materials.
Temperature response amplitude (AC component) ΔT _AC , thermal conductivity κ, hemispherical total emissivity ε, phase difference φ _s which is a delay in temperature response from laser heating, partial differential equation (1) (2) First boundary condition equations (3) and (4) for the laser irradiation part, boundary condition expressions (5) and (6) for the laser non-irradiation part, and a phase difference defining expression (7) are prepared. Step,

here,
c _p : constant pressure mass heat capacity (specific heat) [J / kg.K]
e _laser : Unit vector κ: thermal conductivity [W / mK] indicating the incident direction of laser
n: Distance in the normal direction of the droplet surface [m]
n: Normal unit vector P _{0 of the} droplet surface: Laser power [W]
R _s : radial distance of the droplet surface [m]
r: Radial distance in spherical coordinates [m]
r _laser : e- ² radius of laser beam [m]
T: Temperature [K]
T ₀ : Initial temperature [K]
ρ: Density [kg / m ³ ]
σ _SB : Stefan-Boltzmann constant [W / m ² K ⁴ ]
θ: Polar angle in spherical coordinates [rad]
α: Absorption rate ε: Hemisphere total emissivity φ _s : Phase difference [rad]
ω: Angular frequency of laser periodic heating [rad / s]
[Delta] T _AC ^in: temperature of the amplitude [Delta] T _AC of the laser and the same phase (in-phase) component [Delta] T _AC ^out: amplitude [Delta] T from the _AC laser [pi / 2 deviation has phase (out-of-phase) temperature component
averageΔT _AC ⁱⁿ : average value of ΔT _AC ⁱⁿ the measurement area of the radiation thermometer
averageΔT _AC ^out: the average value of [Delta] T _AC ^out in the measurement area of the radiation thermometer.

The melt is heated to the molten high-temperature melt conductive material by laser periodic heating, the temperature response amplitude (AC component) ΔT _AC in the steady state, the temperature response delay (phase difference φ _s ) from the laser heating, and the laser A second step for measuring the angular frequency ω of the periodic heating;
Further, by fitting the ω dependence of the phase difference φ _s measured in the second step based on the nonlinear analysis method of the numerical expressions (1) to (7) above, the heat conduction that is a parameter Third step to obtain the rate κ and the hemispherical total emissivity ε.

2. A high-temperature melt conductive material including a step of obtaining a constant pressure heat capacity C _{p of the} melt from a maximum value of a product of ΔT _AC and ω with respect to the laser modulation frequency ω between the second step and the third step. Thermophysical property measurement method.
3. The high temperature melt conductive material is a method for measuring a thermal property of a high temperature melt conductive material which is silicon.
4). The said static magnetic field for suppressing the convection inside a high temperature melt conductive material is a thermophysical property measuring method of a high temperature melt conductive material which is 2T or more.

5. A thermophysical property measuring apparatus used for the thermophysical property measuring method of any one of the high temperature melt conductive materials according to any one of 1 to 4 above, wherein the high frequency coil has a space in which the high temperature melt conductive material is melted at the center and electromagnetically suspended. A laser device that applies thermal energy to the high-temperature melt conductive material from the top in a modulation mode; and a static device for suppressing fluctuation of the molten high-temperature melt conductive material and suppressing vertical convection in the high-temperature melt conductive material. A thermophysical property measuring device for a high-temperature melt conductive material, comprising: a magnet device that applies a magnetic field; and a radiation thermometer that measures the temperature of the high-temperature melt conductive material melted from below.
6). The laser device is a semiconductor laser device, and the radiation thermometer is a two-color radiation thermometer.

  According to the present invention, an epoch-making method capable of directly and accurately measuring the thermophysical property measurement of a high-temperature melt, which has been extremely difficult until now, has been developed. According to the present invention, for example, it is possible to accurately measure thermophysical values necessary for numerical simulation of silicon crystal growth and solidification process of silicon for solar cells. Helps improve, save energy and reduce costs. Moreover, since this measuring method can be applied to other metals, it can also provide thermophysical values necessary for numerical simulation such as precision casting of super heat-resistant alloys and the like and precision welding of automobiles and structural materials. Therefore, it is possible to speed up casting, welding, high accuracy, and high reliability.

(Measurement principle)
FIG. 1 shows a measurement principle diagram of static magnetic field application electromagnetic floating AC calorimetry according to the present invention.
A high-frequency coil having a space in which a high-temperature melt conductive material is melted as droplets at the center and electromagnetically suspended; laser heating means for applying thermal energy from the upper portion of the high-temperature melt conductive material in a modulation mode P0 (1 + cosωt); Means such as a superconducting magnet that provides a static magnetic field to suppress oscillation of the molten high-temperature melt conductive material and suppress convection inside the high-temperature melt conductive material, and the temperature of the high-temperature melt conductive material melted from below The thermal conductivity κ and the hemispherical total emissivity ε, which are thermophysical properties of the high-temperature melt conductive material, are directly measured by a laser irradiation alternating current calorimetry method using a means for measuring the temperature.
Hereinafter, the present invention will be described in detail by exemplifying silicon as a high-temperature melt conductive material.

(measuring device)
FIG. 2 shows a schematic diagram of the measuring apparatus. In the present invention, an electromagnetic levitation furnace is used for floating and heating the sample. The electromagnetic levitation furnace consists of a coil, a high frequency power supply, a vacuum chamber, a control panel, and a cooling water circulation device. The coil is composed of a floating coil that floats the sample with an upward force against gravity and a stabilization coil that stabilizes the sample with a downward force from above. The high-frequency power source that supplies power to the coil is a transistor inverter type, and the output power is 15 kW and 200 kHz. The vacuum chamber is connected to a turbo molecular pump via a rotary pump, and can be depressurized to the order of 10 ⁻⁴ Pa by using two vacuum pumps in combination. Since the sample temperature depends on the floating position of the sample, the temperature can be adjusted by adjusting the output power of the high-frequency power source. The sample is periodically heated using a semiconductor CW laser (140 W, 808 nm).

As the sample, high-purity single crystal silicon cut into a cubic shape so as to be about 0.8 g was used. The sample was held at the center of the high-frequency coil with a quartz rod, the inside of the chamber was replaced with high-purity Ar gas (99.9999%), and then the pressure was reduced to the order of 10 ⁻³ to 10 ⁻² Pa with a turbo molecular pump. After preheating the silicon using a semiconductor laser, the sample was floated and melted in a high-frequency coil installed in a superconducting magnet. The position and temperature of the molten silicon were controlled by applying a static magnetic field (0.5 to 4 T) and adjusting the high frequency current. The radius of molten silicon is 4 mm.
Periodic heating was performed by irradiating a semiconductor CW laser from above the sample. Periodic modulation of laser output was performed using a function generator. The laser irradiation diameter on the sample surface is 4 mm in diameter. The temperature response of the sample was measured from the bottom of the sample using a two-color radiation thermometer (λ = 0.9 μm, 1.35 μm). In order to selectively pass only the radiated light from the sample, a diaphragm having a height of 60 mm and a hole having a diameter of 6 mm was installed at a position of about 610 mm below the sample.

(Derivation of relational expression from unsteady heat conduction equation)
By simplifying the unsteady heat conduction equation in the spherical coordinate system with the origin at the center of gravity of the droplet for heating from the top and measuring from the bottom with a laser under the assumptions of 1) to 6) described later , Temperature response amplitude ΔT _AC in steady state, thermal conductivity κ, hemispherical total emissivity ε, temperature response delay from laser heating: partial differential equations (1), (2) below for phase difference φ _s , Boundary condition equations (3) and (4) for the laser irradiated portion, boundary condition equations (5) and (6) for the laser non-irradiated portion, and a definition equation (7) for the phase difference were obtained.

here,
c _p : constant pressure mass heat capacity (specific heat) [J / kg.K]
e _laser : Unit vector κ: thermal conductivity [W / mK] indicating the incident direction of laser
n: Distance in the normal direction of the droplet surface [m]
n: Normal unit vector P _{0 of the} droplet surface: Laser power [W]
R _s : radial distance of the droplet surface [m]
r: Radial distance in spherical coordinates [m]
r _laser : e- ² radius of laser beam [m]
T: Temperature [K]
T ₀ : Initial temperature [K]
ρ: Density [kg / m ³ ]
σ _SB : Stefan-Boltzmann constant [W / m ² K ⁴ ]
θ: Polar angle in spherical coordinates [rad]
α: Absorption rate ε: Hemisphere total emissivity φ _s : Phase difference [rad]
ω: Angular frequency of laser periodic heating [rad / s]
[Delta] T _AC ^in: temperature of the amplitude [Delta] T _AC of the laser and the same phase (in-phase) component [Delta] T _AC ^out: amplitude [Delta] T from the _AC laser [pi / 2 deviation has phase (out-of-phase) temperature component
averageΔT _AC ⁱⁿ : average value of ΔT _AC ⁱⁿ the measurement area of the radiation thermometer
averageΔT _AC ^out: the average value of [Delta] T _AC ^out in the measurement area of the radiation thermometer.

The above equations are obtained in the following (a) to (c), respectively.
(A) Assumption in numerical simulation 1) The system is axisymmetric.
2) The thermophysical property value is constant within the range of the temperature amplitude of periodic heating.
3) The incident laser light is absorbed on the surface of the droplet according to the emissivity of the substance and does not pass through.
4) The intensity distribution of incident laser light follows a Gaussian distribution.
5) Heat radiation from the droplet surface is only radiation.
6) Average temperature rise (DC component) ΔT _DC and temperature amplitude (AC component) ΔT _AC are smaller than the initial temperature T ₀ (ΔT _DC << T ₀ , ΔT _AC << T ₀ ).

(B) Basic Expression and Boundary Conditions Under the above assumption, the unsteady heat conduction equation in a spherical coordinate system (see FIG. 3) with the origin as the center of gravity of the droplet is given as follows.

Where ρ [kg / m ³ ] is density, c _p [J / kg.K] is constant pressure mass heat capacity (specific heat), T [K] is temperature, t [s] is time, κ [W / mK] Is the thermal conductivity, r [m] and θ [rad] are the radial distance and polar angle in spherical coordinates.
The boundary condition is given by
Droplet surface (part heated by laser):

Droplet surface (not heated by laser):

Axis of symmetry:

Where T _∞ [K] is the ambient temperature, σ _SB [W / m ² .K ⁴ ] is the Stefan-Boltzmann constant, ε [-] is the hemispherical total emissivity, α [-] is the absorptance, and I ₀ [ W / m ² ] is the laser intensity on the central axis, R _s [m] is the radial distance of the droplet surface, r _laser [m] is the laser beam radius, n is the normal unit vector of the droplet surface, e _laser is a unit vector indicating the direction of incidence of the laser is given in the case of the analyzed e _laser = cosθe _r -sinθe _θ. Further, I ₀ has the following relationship with the so-called laser intensity P [W].

Here, ω [rad / s] is an angular frequency of laser periodic heating.
The initial condition is
T = T ₀ (18)
It becomes.

(C) Model simplification The temperature T is divided into a direct current component ΔT _DC and an alternating current component ΔT _AC as follows.
T (r, θ, t) = T ₀ + ΔT _DC (r, θ, t) + ΔT _AC (r, θ, t) (19)
Now, the AC component is rewritten as follows.
T (r, θ, t) = T ₀ + ΔT _DC (r, θ, t) + ΔT _AC ⁱⁿ (r, θ) cosωt + ΔT _AC ^out (r, θ) sinωt
(20)
Here, ΔT _AC ⁱⁿ and ΔT _AC ^out are the same phase, in-phase and phase shifted by π / 2 from the laser of ΔT _AC , and an out-of-phase component, respectively.
Substituting equation (19) into equation (13),

And then

Thus, it can be divided into partial differential equations relating to the DC component and the AC component. Here, when equation (20) is applied to equation (23),

Therefore, when cosωt and sinωt are arranged, the following partial differential equations regarding ΔT _AC ⁱⁿ and ΔT _AC ^out can be obtained.

Next, consider boundary conditions. Substituting equation (19) into equation (14),

Is obtained. here,

And ignoring higher order terms based on Assumption 6) of (a), Equation (27) becomes

It becomes. Substituting equation (20) into equation (29),

It becomes. Here, when cosωt and sinωt of ΔT _AC are arranged,

Is obtained. If the boundary conditions of equations (15) and (16) are handled in the same way,

Solving equations (22), (25), and (26) under boundary condition equations (31) to (36),
ΔT _AC ⁱⁿ and ΔT _AC ^out are obtained.
Using the above result, the phase difference φs corresponding to the measurement result is obtained by the following equation.

here,

S _pyrometer is the dimensionless area of the spot of the radiation thermometer.

In order to obtain ΔT _AC ⁱⁿ and ΔT _AC ^out by solving equations (22), (25), and (26) under boundary condition equations (31) to (36), the following (d) Derivation can be realized by using means of dimensioning and (e) discretization.
(D) Non-dimensionalization As representative values of length, temperature and thermal conductivity, the sphere equivalent radius R [m], initial temperature T ₀ [K] and κ _ref [W / mK] of the droplet are used, respectively. When (25) and (26) are made dimensionless, the following equation is obtained.

Here, the superscript * indicates a dimensionless value.
The boundary condition is as follows.
Droplet surface (part heated by laser):

Droplet surface (not heated by laser):

Axis of symmetry:

Here, Ra ^* (= 4σ _SB εRT ₀ ³ / κ _ref ) is the number of radiations. Further, dimensionless laser intensity P ₀ ^* is _{2αP 0 / (πT 0 κ ref} R), non-dimensional frequency omega ^* is defined by ωR ² ρc _p / κ _ref.

(E) Discretization Here, the basic equation and the boundary condition equations (40) to (47) are discretized by the Galerkin finite element method. However, the superscript * is omitted. The Galerkin finite element equation of Equation (40) is given by the following equation, where φ ⁱ is the weight function.

Applying partial integration,

Here, n _r and n _θ are each component of the normal direction unit vector n, and the final term of the above formula represents the surface integral term at the boundary. For example, on the droplet surface (laser irradiation part), from the equation (42),

Therefore, the equation (50) is as follows.

here,

It is. Similarly, when equation (41) is discretized,

It becomes.
In solving the equations (52) and (54), the target region (0 ≦ r ≦ r (θ), 0 ≦ θ ≦ π) is divided into a finite number of nine-node quadrangular elements, and ΔT _AC ⁱⁿ and Approximate ΔT _AC ^out as follows.

Here, φ ⁱ is a biquadratic interpolation function. Substituting the equations (55-1, 2) into the equations (52) and (54), the following algebraic equations are finally obtained.

By solving the equations (56) and (57) by the direct method, ΔT _{AC, j} ⁱⁿ and ΔT _{AC, j} ^out at each node can be obtained.

(Acquisition of measurement data)
In the measurement apparatus shown in FIG. 2, when the periodic heating by the laser starts, the sample temperature gradually increases and reaches a steady state. Measurement was performed continuously by changing the laser frequency in a steady state. In the measurement, necessary data was acquired by changing the initial temperature To, the static magnetic field, and the angular frequency ω of the periodic heating of the laser within the following ranges.
Initial temperature T ₀ : 1700 <T ₀ <2050K
Static magnetic field: 0-4T
Angular frequency ω: 0 <ω <2 rad / s

FIG. 4 shows a typical temperature response in an AC steady state when the angular frequency ω is 0.1256 rad / s (0.02 Hz). By measuring by changing the angular frequency ω of the periodic heating of the laser, the angular frequency ω dependence of the phase difference φs and the temperature amplitude [Delta] T _AC is measured.

(Calculation of thermal conductivity and hemispherical total emissivity)
Since the spatial distribution of the phase difference φs is obtained from Equation (7) using ΔT _{AC, j} ⁱⁿ and ΔT _{AC, j} ^out , the relationship between the angular frequency ω obtained by measurement and the phase difference φs (see FIG. 5). ) With the numerical analysis results of the above formulas (1) to (7) based on the nonlinear least square method, the parameters of thermal conductivity κ and emissivity ε are obtained. Fitting was performed using a self-made program. In addition, a commercially available computer can be used without limitation for the calculation.

Note numerical constant pressure heat capacity C _p of the melt required for the calculation of the thermal conductivity κ and emissivity ε is good to adopt the existing numeric but, a known method by Non-Patent Document 7 was also introduced paragraph 0013 Thus, the heat conductivity κ and emissivity ε can be calculated from the measured values prior to calculation. That is, the numerical value of ω that gives the maximum value of the product of the temperature response amplitude ΔT _AC and the angular frequency ω with respect to the measured ω is determined, and the constant pressure heat capacity C _{p of the} melt can be calculated from the equation (8). In this case, there is an advantage that all the thermophysical values such as the constant pressure heat capacity C _p , the thermal conductivity κ, and the emissivity ε of silicon in the melt state can be directly measured.

(Calculation result of hemispherical total emissivity)
The numerical value of the hemispherical total emissivity of the molten silicon obtained by this measurement is shown in FIG. The numerical values measured in the range of 1T to 4T for the strength of the static magnetic field and 1750 to 1930K for the temperature are shown. There is no effect on the total emissivity of the hemisphere by the static magnetic field. By taking the average of the measured values, 0.25 was obtained as the total emissivity of the hemisphere of silicon. There are few literature values for the total emissivity of the hemisphere, and there is a certain value near the melting point as shown in the figure.

(Calculation result of thermal conductivity)
Next, the numerical value of the thermal conductivity of the molten silicon obtained by this measurement is shown in FIG. The numerical values measured in the range of 0.5 T to 4 T for the strength of the static magnetic field and 1750 to 2050 K for the temperature are shown. As a result of measuring by changing the strength of the static magnetic field, it can be seen that when the magnetic field is increased, the value of the thermal conductivity gradually decreases. And at 2T or more, no significant difference is recognized. This is considered to be because convection in the molten silicon was suppressed by the effect of suppressing the flow of the static magnetic field at 2T or more.
For reference, the past measured values (Cusack to Yamasue) are shown in FIG. Past measured values are limited to measured values in a temperature range lower than 1800K.

Apart from this measurement, FIG. 8 shows the difference in the surface behavior of silicon droplets depending on the presence or absence of a static magnetic field. FIG. 8 shows the result (Side view) of the observation of the difference in the flow of the silicon droplet surface depending on the presence / absence of a static magnetic field (0T and 3T) by frame advance. In the same figure, both of the images immediately after the start of recurrence after floating and dissolving using a silicon sphere having a diameter of 10 mm are solidified silicon. Such island-shaped silicon crystals play a role of a tracer on the surface of the silicon melt, and the horizontal and vertical longitudinal velocities were calculated from the movement distance of the silicon crystals indicated by arrows.
From FIG. 8, it can be seen that when the static magnetic field strength is 3T, the movement in the vertical and vertical direction is considerably suppressed, and the rotation is performed with the vertical axis of the sample as the rotation axis. It has been confirmed that this tendency is observed in a static magnetic field of 2T or more.
For this reason, 62 Wm ⁻¹ K ⁻¹ was obtained as the thermal conductivity of silicon by averaging the thermal conductivity values of magnetic field strengths of 2 to 4T.

In this measurement, silicon was used as the high-temperature melt conductive material, but metals such as nickel and platinum may be used. In other words, the present invention can be applied to any material that can be a high-temperature melt having conductivity that allows Lorentz force to melt and float in a high-frequency coil.
Moreover, the thermophysical property measuring method of the present invention can be applied to droplets having an arbitrary shape. That is, the high-temperature melt conductive material to be measured can be applied to a shape that is not ideally spherical due to gravity or other influences, but is distorted in the vertical direction, for example.

It is a measurement principle figure of a static magnetic field application electromagnetic floating alternating current calorimetry. It is the schematic of a static magnetic field application electromagnetic floating alternating current calorimetry apparatus. It is a figure showing a spherical coordinate system. It is a temperature response figure by laser irradiation (0.02Hz) of molten silicon. It is a figure which shows the relationship between angular frequency (omega) and phase difference (phi) s. It is a figure which shows the measured value of the hemispherical total emissivity of molten silicon. It is a figure which shows the measured value of the heat conductivity of molten silicon. It is a figure which shows the difference in the surface behavior of the silicon droplet by the presence or absence of a static magnetic field.

Claims (6)

A high-frequency coil having a space in which high-temperature melt conductive material is melted as droplets at the center and electromagnetically suspended, laser heating means for applying thermal energy in a modulation mode from above the high-temperature melt conductive material, and molten high-temperature melt High temperature fusion using means for applying a static magnetic field to suppress oscillation of the body conductive material and to suppress convection inside the high temperature melt conductive material, and means for measuring the temperature of the high temperature melt conductive material melted from below. In the method for measuring thermophysical properties of a body conductive material, the thermal conductivity κ and the hemispherical total emissivity ε of the high temperature melt conductive material are determined by the following first to third steps. Thermophysical property measurement method.
Temperature response amplitude (AC component) ΔT _AC , thermal conductivity κ, hemispherical total emissivity ε, phase difference φ _s which is a delay in temperature response from laser heating, partial differential equation (1) (2) First boundary condition equations (3) and (4) for the laser irradiation part, boundary condition expressions (5) and (6) for the laser non-irradiation part, and a phase difference defining expression (7) are prepared. Step,
here,
c _p : constant pressure mass heat capacity (specific heat) [J / kg.K]
e _laser : Unit vector κ: thermal conductivity [W / mK] indicating the incident direction of laser
n: Distance in the normal direction of the droplet surface [m]
n: Normal unit vector P _{0 of the} droplet surface: Laser power [W]
R _s : radial distance of the droplet surface [m]
r: Radial distance in spherical coordinates [m]
r _laser : e- ² radius of laser beam [m]
T: Temperature [K]
T ₀ : Initial temperature [K]
ρ: Density [kg / m ³ ]
σ _SB : Stefan-Boltzmann constant [W / m ² K ⁴ ]
θ: Polar angle in spherical coordinates [rad]
α: Absorption rate ε: Hemisphere total emissivity φ _s : Phase difference [rad]
ω: Angular frequency of laser periodic heating [rad / s]
[Delta] T _AC ^in: temperature of the amplitude [Delta] T _AC of the laser and the same phase (in-phase) component [Delta] T _AC ^out: amplitude [Delta] T from the _AC laser [pi / 2 deviation has phase (out-of-phase) temperature component
averageΔT _AC ⁱⁿ : average value of ΔT _AC ⁱⁿ the measurement area of the radiation thermometer
averageΔT _AC ^out: the average value of [Delta] T _AC ^out in the measurement area of the radiation thermometer.
The melt is heated to the molten high-temperature melt conductive material by laser periodic heating, the temperature response amplitude (AC component) ΔT _AC in the steady state, the temperature response delay (phase difference φ _s ) from the laser heating, and the laser A second step for measuring the angular frequency ω of the periodic heating;
Further, by fitting the ω dependence of the phase difference φ _s measured in the second step based on the nonlinear analysis method of the numerical expressions (1) to (7) above, the heat conduction that is a parameter Third step to obtain the rate κ and the hemispherical total emissivity ε. The high-temperature melt according to claim 1, further comprising a step of obtaining a constant pressure heat capacity C _{p of the} melt from a maximum value of a product of ΔT _AC and ω with respect to ω between the second step and the third step. A method for measuring the thermal properties of conductive materials.   The method for measuring thermal properties of a high-temperature melt conductive material according to claim 1 or 2, wherein the high-temperature melt conductive material is silicon.   The method for measuring a thermal property of a high-temperature melt conductive material according to any one of claims 1 to 3, wherein the static magnetic field for suppressing convection inside the high-temperature melt conductive material is 2T or more.   It is a thermophysical property measuring apparatus used for the thermophysical property measuring method of the high-temperature melt conductive material according to any one of claims 1 to 4, wherein the high-temperature melt conductive material is melted in the center and electromagnetically suspended. A high-frequency coil having a space, a laser device that applies thermal energy to the high-temperature melt conductive material from the top in a modulation mode, and suppressing fluctuations in the molten high-temperature melt conductive material and suppressing convection inside the high-temperature melt conductive material A thermophysical property measuring device for a high-temperature melt conductive material, comprising: a magnet device for applying a static magnetic field for the measurement; and a radiation thermometer for measuring the temperature of the high-temperature melt conductive material melted from below.   6. The thermophysical property measuring apparatus for a high-temperature melt conductive material according to claim 5, wherein the laser device is a semiconductor laser device, and the radiation thermometer is a two-color radiation thermometer.