Evaluation method for simultaneously measurable parameters in transient heat reflection method based on sensitivity matrix

Abstract

The application discloses an evaluation method for determining simultaneously measurable parameters in a transient thermal reflectance method based on a sensitivity matrix, and the method comprises the following steps: determining known parameters and unknown parameters required in fitting of a transient thermal reflectance method experiment and experimental variables according to a multilayer stacked sample structure; giving an estimated value of the unknown parameters; calculating the sensitivity parameters of all unknown parameters at all value points of the experimental variables, and constructing a sensitivity matrix; performing singular value decomposition on the sensitivity matrix S, and the number m of non-negative singular values satisfying the non-negative singular values in the non-negative singular values is the number of simultaneously measurable unknown parameters. The application is applicable to any transient thermal reflectance method experiment, can quickly identify simultaneously measurable physical quantities in the transient thermal reflectance method, can avoid incorrect fitting results and guide optimization of material measurement conditions, and is helpful for simultaneous measurement of multiple unknown parameters of new materials.

Description

Technical Field

[0001] This invention relates to the field of material thermophysical property measurement, and more particularly to an evaluation method for determining parameters that can be measured simultaneously in transient thermal reflection based on a sensitivity matrix. Background Technology

[0002] The measurement of material thermal properties has significant scientific importance and important engineering application value in industrial fields such as aerospace, energy utilization, microelectronics, and new material development, serving as the foundation for technological innovation, development, and scientific research. Transient thermal reflection methods based on material surface thermal reflection, such as frequency domain thermal reflection and time domain thermal reflection, are currently the most advanced methods for measuring material thermal properties, capable of measuring bulk materials and micro / nano-scale materials. This method relies on surface thermal reflection to collect temperature response signals under different experimental variables, and uses a known thermal conductivity model to fit the experimental data to determine unknown material parameters. This process often involves more than one unknown parameter. As the properties of the sample and experimental conditions change, the measurable parameters in a single transient thermal reflection experiment will also change. Taking the measurement of bulk materials as an example, in low-frequency measurements, thermal conductivity k and volumetric specific heat C can be measured simultaneously, but in high-frequency measurements, only... The measurable parameters can be determined through fitting. Determining these measurable parameters in transient thermal reflection experiments can help eliminate incorrect measurement results and guide the optimization of experimental conditions. However, a general-purpose and reliable method for determining measurable parameters is currently lacking. Summary of the Invention

[0003] To address the lack of methods for determining measurable parameters in existing transient thermal reflection methods, this invention proposes an evaluation method based on a sensitivity matrix to determine simultaneously measurable parameters in transient thermal reflection experiments. This method is based on parameter sensitivity analysis, constructing a sensitivity matrix containing the sensitivity parameters of all unknown parameters under all experimental variables, and then performing singular value decomposition to derive the number and form of measurable parameters in the transient thermal reflection method. This method is simple to operate, reliable, and versatile, and can be used for the simultaneous measurement of multiple unknown thermophysical parameters of new materials.

[0004] The objective of this invention is achieved through the following technical solution:

[0005] An evaluation method for determining simultaneously measurable parameters in transient thermal reflectance methods based on a sensitivity matrix, the method comprising:

[0006] Step 1: Based on the multi-layered stacked sample structure, determine the known and unknown parameters required for fitting in the transient thermal reflection method experiment, as well as the experimental variables; where the known parameters are represented as ζ=(B1,B2,…,B n The unknown parameters are represented as θ=(A1,A2,…,A…). n );

[0007] Step 2: Provide the estimated values ​​of the unknown parameters; calculate the sensitivity parameters of all unknown parameters at all values ​​of the experimental variables, and construct the sensitivity matrix S; wherein, the calculation formula of the sensitivity parameters is as follows:

[0008]

[0009] Where y = F(x,θ,ζ); y is the response signal collected in the transient thermal reflection method experiment for fitting, and F is the heat transfer model; x j Let A represent all possible values ​​of the experimental variable, j = 1, 2, ..., p; i For unknown parameters, i = 1, 2, ..., n;

[0010] The sensitivity matrix S is represented as follows:

[0011]

[0012] Step 3: Perform singular value decomposition on the sensitivity matrix S, S = U∑V T Where U and V are p×p and n×n unitary matrices, respectively; ∑ is a p×n matrix, and the diagonal elements of its first n rows are non-negative singular values ​​σ1,σ2,…,σ in descending order. n ;

[0013] Step 4: From the non-negative singular values ​​σ1, σ2, ..., σ n Selecting from those that meet the requirements The number of non-negative singular values ​​m is the number of unknown parameters that can be measured simultaneously; where t is a threshold value between 0 and 1.

[0014] Furthermore, the value of t ranges from 0.02 to 0.1.

[0015] Furthermore, the null space V of the sensitivity matrix S is obtained from the last nm columns of matrix V. null :

[0016]

[0017] Solving the following system of differential equations yields the form of the simultaneously measurable parameters among the unknown thermophysical properties:

[0018]

[0019] Where, l = 1, 2, ..., m, k = m + 1, ..., n; g l (A1,…,A n The solution is in the form of measurable parameters.

[0020] Furthermore, the estimated values ​​of the unknown parameters are determined as follows:

[0021] (1) Through simulation;

[0022] (2) Preliminary fitting of experimental data;

[0023] (3) By consulting literature.

[0024] Furthermore, the unknown parameter remains unchanged within the range of experimental variable values.

[0025] The beneficial effects of this invention are as follows:

[0026] (1) This invention is applicable to the analysis of any material and any parameter that remains unchanged within the range of experimental variables in the transient thermal reflection method. It is applicable to any range of experimental variables and is versatile and reliable.

[0027] (2) The number of non-zero elements in the singular values ​​of the sensitivity matrix directly gives the number of parameters that can be measured simultaneously in a single frequency domain thermal reflection experiment;

[0028] (3) The present invention can identify erroneous fitting results in transient thermal reflection method and guide the experimental design for simultaneous measurement of multiple thermophysical parameters in new materials. Attached Figure Description

[0029] Figure 1 This is a flowchart illustrating the evaluation method for determining simultaneously measurable parameters in the transient thermal reflection method based on the sensitivity matrix, according to an embodiment of the present invention.

[0030] Figure 2 This is a schematic diagram of a sample with a two-layer structure.

[0031] Figure 3 For k z ,k r The sensitivity parameters of C at different heating frequency values.

[0032] Figure 4 For k z ,k r And the sensitivity parameter of C at the time delay value point.

[0033] Figure 5 For k z ,k r And C's displacement x between the probe light and the pump light c Sensitivity parameters at the given value point.

[0034] Figure 6 The two-layer sample structure k in Example 1 z ,k r Normalized results of Monte Carlo simulations fitted simultaneously with C Detailed Implementation

[0035] The present invention will be described in detail below with reference to the accompanying drawings and preferred embodiments. The purpose and effects of the present invention will become clearer. It should be understood that the specific embodiments described herein are merely for explaining the present invention and are not intended to limit the present invention.

[0036] like Figure 1 As shown in the embodiment of the present invention, the evaluation method for determining the parameters that can be measured simultaneously in the transient thermal reflection method based on the sensitivity matrix includes the following steps:

[0037] Step 1: Based on the multi-layered stacked sample structure, determine the known and unknown parameters required for fitting in the transient thermal reflection method experiment, as well as the experimental variables. The known parameters include the thermal conductivity, volumetric specific heat, and thickness of each layer, the interfacial thermal conductivity of all interfaces, and the known value of the effective laser heating radius, etc. The known parameters are expressed as ζ = (B1, B2, ..., B...). n Unknown parameters include thermal conductivity, specific heat of volume, interfacial thermal resistance, etc., and are expressed as θ=(A1,A2,…,A…). n The unknown parameters here need to remain constant within the range of experimental variable values.

[0038] Step Two: Provide estimated values ​​for the unknown parameters through simulation, preliminary fitting of experimental data, or literature review. Calculate the sensitivity parameters of all unknown parameters at all values ​​of the experimental variables, and construct the sensitivity matrix S. The formula for calculating the sensitivity parameters is as follows:

[0039]

[0040] Where y = F(x,θ,ζ); y is the response signal collected in the transient thermal reflection method experiment for fitting, and F is the heat transfer model; x j Let A represent all possible values ​​of the experimental variable, j = 1, 2, ..., p; i For unknown parameters, i = 1, 2, ..., n;

[0041] The sensitivity matrix S is represented as follows:

[0042]

[0043] Step 3: Perform singular value decomposition on the sensitivity matrix S, S = U∑V T Where U and V are p×p and n×n unitary matrices, respectively; ∑ is a p×n matrix, and the diagonal elements of its first n rows are non-negative singular values ​​σ1,σ2,…,σ in descending order. n ;

[0044] Step 4: From the non-negative singular values ​​σ1, σ2, ..., σ n Selecting from those that meet the requirements The number of non-negative singular values, *m*, represents the number of unknown parameters that can be measured simultaneously; where *t* is a threshold value between 0 and 1. To improve prediction accuracy, the preferred value of *t* is 0.02 to 0.1. A smaller threshold *t* tends to fit more parameters, while a larger threshold *t* tends to fit fewer parameters. The value of the threshold *t* is related to the noise level of the transient thermal reflection method experimental platform; the higher the noise level, the higher the threshold *t* should be. The recommended value given in this application is 0.02.

[0045] In addition to providing the number of unknown parameters that can be measured simultaneously, this invention can also provide the form of the parameters that can be measured simultaneously, as follows:

[0046] The null space V of the sensitivity matrix S is obtained from the last nm columns of matrix V. null :

[0047]

[0048] Solving the following system of differential equations yields the form of the simultaneously measurable parameters among the unknown thermophysical properties:

[0049]

[0050] Where, l = 1, 2, ..., m, k = m + 1, ..., n; g l (A1,…,A n The solution is in the form of measurable parameters.

[0051] The following examples use three common transient thermal reflection methods: frequency domain thermal reflection, time domain thermal reflection, and spatial domain thermal reflection. Figure 2 The thermal properties of the two-layer material shown are further illustrated in the example.

[0052] Example 1: Frequency Domain Thermal Reflection Method

[0053] In the frequency domain thermal reflectance method, the experimental variable x is the heating frequency ω, and the response y used for fitting is the phase difference φ between the sample surface temperature rise and the heat flow.

[0054] In the frequency domain thermal reflection method, the heat transfer model φ=F(ω,θ,ζ) of φ and ω, known parameter ζ, and unknown parameter θ is determined by the following equations (5)-(12).

[0055] Sample surface temperature rise

[0056] ΔT=Ae iφ (5)

[0057] ΔT is determined by the sample parameters and the effective spot radius r of the heating laser through equation (6).

[0058]

[0059] Where P0 is the heating laser power, x is the Hankel transform parameter, ω is the heating frequency, and θ is the heating laser power. t and f t These are the sample's upper surface temperature and heat flux density, respectively. The sample in the transient thermal reflection method experiment has a multi-layered stacked structure, where the temperature of the upper surface of the i-th layer is... and heat flux density and the temperature of this lower surface and heat flux density The relationship between them is

[0060]

[0061]

[0062]

[0063] Where d i For thickness, Normal thermal conductivity, ρ is the in-plane thermal conductivity. i For density, c i Specific heat. For a single interface,

[0064]

[0065] Where h i It is the interfacial thermal conductivity. Multiplying all layers of the sample sequentially yields...

[0066]

[0067] Where n represents the sample sublayer. Based on the adiabatic boundary conditions of the sample sublayer, f b =C0θ t +D0f t =0, we can get

[0068]

[0069] The sample structure was determined to be two layers: the first layer was gold, and the second layer was lithium tantalate.

[0070] The experimental variable points for the frequency domain thermal reflection method were determined to be 10. 4 Hz to 10 7 Thirty frequency points are distributed at equal intervals on a logarithmic coordinate system between Hz.

[0071] The known parameters are shown in Table 1.

[0072] Table 1. Frequency (spatial) domain thermal reflectance method test Figure 2Known parameters in the thermophysical property experiment of the two-layer material shown

[0073]

[0074] The unknown parameter is determined to be the thermal conductivity of lithium tantalate, including the normal thermal conductivity k. z In-plane thermal conductivity k r And the volumetric specific heat C, based on literature estimates, their values ​​are k z =8.9W m -1 K -1 ,k r =7.2W m -1 K -1 C = 2.98 MJ m -3 K;

[0075] Calculate the sensitivity parameters of all unknown parameters at all heating frequencies.

[0076]

[0077] Where (i = 1, 2, ..., n; j = 1, 2, ..., p).

[0078] k is calculated based on the heat transfer model φ=F(ω,θ,ζ) and formula (13). z ,k r The sensitivity parameters of C at heating frequency are as follows: Figure 3 As shown, a 30×3 sensitivity matrix is ​​established.

[0079]

[0080] Singular value decomposition of S yields singular values ​​(σ1, σ2, σ3) = (1.41, 0.64, 5.5 × 10^6)^2. -4 ),according to The number of non-zero singular values ​​is determined to be 2;

[0081] Zero space is

[0082]

[0083] According to the results of its singular value decomposition, the number of non-zero singular values ​​of matrix S is 2, and therefore the number of unknown parameters that can be identified in the experiment is 2.

[0084] Solve the equation

[0085]

[0086] A set of independent solutions is g1 = k z k r and g2 = k zC, that is, the physical quantity measured simultaneously in this embodiment is k. z k r k z C.

[0087] Example 2: Time-Domain Thermal Reflectance Method

[0088] In the time-domain thermal reflectometry method, the experimental variable x is the time delay t between the probe light and the pulsed pump light. d The response y used for fitting is the acquired in-phase signal V. in and out-of-phase signal V out The ratio -V in / V out In the time-domain thermal reflection method, -V in / V out With t d A heat transfer model with known parameters ζ and unknown parameters θ - V in / V out =F(t) d In (θ,ζ), V in and V out They are the real and imaginary parts in equation (17), respectively.

[0089]

[0090] Where ω0 is the pump light modulation frequency, ω s ΔT is the pump light pulse frequency, i is the imaginary unit, and ΔT is determined by equation (6) as described in the frequency domain thermal reflection method.

[0091] The sample structure was determined to be two layers: the first layer was aluminum and the second layer was lithium tantalate.

[0092] The experimental variable points for the time-domain thermal reflection method were determined to be 30 points distributed at equal intervals on logarithmic coordinates between 0.1 ns and 10 ns; the known parameters are shown in Table 2.

[0093] Table 2. Time-Domain Thermal Reflectance Method Test Figure 2 Known parameters in the thermophysical property experiment of the two-layer material shown

[0094]

[0095] The unknown parameter is determined to be the thermal conductivity of lithium tantalate, including the normal thermal conductivity k. z In-plane thermal conductivity k r And the volumetric specific heat C, based on literature estimates, their values ​​are k z =8.9W m -1 K -1 ,k r =7.2W m -1 K -1 C = 2.98 MJ m-3 K;

[0096] Calculate all unknown parameters t under all time delays d Sensitivity parameters

[0097]

[0098] Where i = 1, 2, ..., n; j = 1, 2, ..., p.

[0099] According to the heat transfer model -V in / V out =F(t) d k is calculated using formula (18) and θ,ζ). z ,k r The sensitivity parameters of C under time delay are as follows: Figure 4 As shown, a 30×3 sensitivity matrix is ​​established.

[0100]

[0101] Singular value decomposition of S yields singular values ​​(σ1, σ2, σ3) = (0.85, 0.09, 1.4 × 10⁻⁶). -6 ),according to The number of non-zero singular values ​​is determined to be 2;

[0102] Zero space is

[0103]

[0104] According to the results of its singular value decomposition, the number of non-zero singular values ​​of matrix S is 2, and therefore the number of unknown parameters that can be identified in the experiment is 2.

[0105] Solve the equation

[0106]

[0107] A set of independent solutions is g1 = k z k r and g2 = k z C, that is, the physical quantity measured simultaneously in this embodiment is also k. z k r k z C.

[0108] Example 3 Spatial Domain Thermal Reflection Method

[0109] In the space thermal reflection method experiment, the variable x is the displacement x between the probe light and the pump light. c The response y used for fitting is the phase difference φ between the sample surface temperature rise and the heat flow;

[0110] In the spatial domain thermal reflection method, φ and xc Given the parameter ζ and the unknown parameter θ, the heat transfer model φ = F(x) c The method is consistent with the frequency domain thermal reflection method, and only the displacement between the pump light and the probe light needs to be considered based on equation (6).

[0111] The sample structure was determined to be two layers: the first layer was gold, and the second layer was lithium tantalate.

[0112] The experimental variable points for the time-domain thermal reflectometry method were determined to be 30 points distributed at equal intervals on linear coordinates between -9 μm and 9 μm.

[0113] The known parameters are shown in Table 1.

[0114] The unknown parameter is determined to be the thermal conductivity of lithium tantalate, including the normal thermal conductivity k. z In-plane thermal conductivity k r And the volumetric specific heat C, based on literature estimates, their values ​​are k z =8.9W m -1 K -1 ,k r =7.2W m -1 K -1 C = 2.98 MJ m -3 K;

[0115] Calculate all unknown parameters in x c Sensitivity parameters

[0116]

[0117] Where (i = 1, 2, ..., n; j = 1, 2, ..., p).

[0118] According to the heat transfer model φ=F(x) c k is calculated using formula (16) and θ,ζ). z ,k r and C in x c The sensitivity parameters are as follows: Figure 5 As shown, a 30×3 sensitivity matrix is ​​established.

[0119]

[0120] Singular value decomposition of S yields singular values ​​(σ1, σ2, σ3) = (2.68, 0.22, 1.1 × 10^3). -5 ),according to The number of non-zero singular values ​​is determined to be 2;

[0121] Zero space is

[0122]

[0123] According to the results of its singular value decomposition, the number of non-zero singular values ​​of matrix S is 2, and therefore the number of unknown parameters that can be identified in the experiment is 2.

[0124] Solve the equation

[0125]

[0126] A set of independent solutions is g1 = k z k r and g2 = k z C, that is, the physical quantity measured simultaneously in this embodiment is also k. z k r k z C.

[0127] To verify the validity of the above results, we used Monte Carlo simulation to validate the results in Example 1. This validation process can be extrapolated to Examples 2 and 3. Using the known parameter values ​​in Table 1 and the unknown parameters determined from the literature, namely the thermal conductivity of lithium tantalate, including the normal thermal conductivity k... z In-plane thermal conductivity k r and specific heat of volume C, k z =8.9W m -1 K -1 ,k r =7.2W m -1 K -1 C = 2.98 MJ m -3 K. The theoretical phase difference data at the experimental frequency in Example 1 was calculated using the heat transfer model φ=F(ω,θ,ζ). Random errors (noise in the simulation experiment) sampled from a normal distribution N(0,0.2°) with a mean of 0 and a standard deviation of 0.2° were added to this theoretical phase difference to simulate the phase difference data for measuring lithium tantalate in Example 1. Let k z0 =10W m -1 K -1 ,k r0 =10W m -1 K -1 C0 = 2MJ m -3 K is the initial value, and this data is processed by (k) z k r The three parameters of C) are fitted simultaneously, repeated 5000 times, and the mean and standard deviation of the fitted results are obtained. After normalization by their preset values, they are presented as follows. Figure 6 In the middle. By Figure 6 As can be seen, k z k r Both C and g1 deviate from their preset values, meaning they cannot be accurately measured simultaneously using the frequency domain thermal reflectance method. However, g1 = k z kr and g2 = k z The fitting result of C is consistent with the preset value, that is, they are the parameters that can be fitted simultaneously in Example 1. This Monte Carlo simulation result is consistent with the result of the method analysis in this invention, verifying the correctness of this invention.

[0128] It will be understood by those skilled in the art that the above descriptions are merely preferred examples of the invention and are not intended to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, those skilled in the art can still modify the technical solutions described in the foregoing examples or make equivalent substitutions for some of the technical features. All modifications and equivalent substitutions made within the spirit and principles of the invention should be included within the scope of protection of the invention.

Claims

1. An evaluation method for determining simultaneously measurable parameters in transient thermal reflectance methods based on a sensitivity matrix, characterized in that, The method includes: Step 1: Based on the multi-layered stacked sample structure, determine the known and unknown parameters required for fitting in the transient thermal reflection method experiment, as well as the experimental variables; where the known parameters are expressed as... The unknown parameter is represented as The unknown parameter remains constant within the range of experimental variable values. Step 2: Provide the estimated values ​​of the unknown parameters; calculate the sensitivity parameters of all unknown parameters at all values ​​of the experimental variables, and construct the sensitivity matrix S; wherein, the calculation formula of the sensitivity parameters is as follows: ； in, ; y is the response signal collected for fitting in the transient thermal reflection method experiment, and F is the heat transfer model; For all possible values ​​of the experimental variable, ; For unknown parameters, ; The sensitivity matrix S is represented as follows: ； Step 3: Perform singular value decomposition on the sensitivity matrix S. ,in and They are respectively and unitary matrix; for A matrix whose diagonal elements of the first n rows are non-negative singular values ​​in descending order. ; Step 4: From nonnegative singular values Selecting from those that meet the requirements The number of non-negative singular values ​​m is the number of unknown parameters that can be measured simultaneously; where t is a threshold value between 0 and 1.

2. The evaluation method for determining simultaneously measurable parameters in the transient thermal reflection method based on the sensitivity matrix according to claim 1, characterized in that, The value of t ranges from 0.02 to 0.

1.

3. The evaluation method for determining simultaneously measurable parameters in the transient thermal reflection method based on the sensitivity matrix according to claim 2, characterized in that, From the matrix After The null space of the sensitivity matrix S is obtained by columnar calculation. : ； Solving the following system of differential equations yields the form of the simultaneously measurable parameters among the unknown thermophysical properties: ； in, ; The solution is in the form of measurable parameters.

4. The evaluation method for determining simultaneously measurable parameters in the transient thermal reflection method based on the sensitivity matrix according to claim 2, characterized in that, The estimated values ​​of the unknown parameters are determined as follows: (1) Through simulation; (2) Preliminary fitting of experimental data; (3) By consulting literature.

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