By: Aaditya A. Candadai, Aalok Gaitonde, Shanmukhi Sripada, Justin A. Weibel, Amy M. Marconnet
Introduction
New high thermal conductivity materials have been developed to enable enhanced heat transport within electronics packages and other thermal management applications. Many of these engineered materials have anisotropic thermal properties inherent to their fabrication, especially the sheet-like materials that are commonly used as heat spreaders. [1] For example, large-area graphene/graphite sheets fabricated in roll-to-roll processes have high in-plane thermal conductivity (but lower through-thickness conductivity)[2] and reinforced polymer composites that include high-strength carbon fibers have high thermal conductivity along the axis of the fibers, but poorer thermal conductivity in the direction perpendicular to the fibers.[3] Generally, these materials offer significantly improved hot spot dissipation compared to conventional heat spreaders, but their anisotropic properties must be characterized to allow for accurate design of thermal solutions.
Most measurement techniques evaluate thermal conductivity along a single direction. Mature and commercially available techniques including reference bar methods, transient plane source techniques, or laser flash analyses are commonly used for through-thickness thermal conductivity characterization. However, there are comparatively fewer measurement techniques for in-plane thermal conductivity characterization, with examples being the Ångström method or modified versions of the laser flash method. Measuring the in-plane thermal conductivity in different directions (e.g., parallel fibers versus perpendicular to fibers) generally requires fabricating multiple samples and conducting multiple tests focusing on a single direction each. Hence, there is a pressing need for methods that can accurately characterize the anisotropic thermal conductivity of thin heat-spreading materials, ideally with a single measurement of a single sample in a non-destructive and potentially non-contact manner.
To facilitate the evaluation of thin film or sheet-like materials with in-plane thermal anisotropy, we have recently developed a new technique that leverages a laser to periodically heat a spot on one surface of a suspended sample, while the temporal evolution of the spatial temperature response is monitored on the opposite surface with infrared thermal imaging.[4] Our analysis and experimental results have demonstrated that the technique is accurate across a wide range of thermal conductivities spanning 0.1 to 2000 W/(m K) and anisotropy ratios up to 100, with the frequency of heating and the sample geometry being tuning parameters to optimize measurement sensitivity. One notable challenge for high thermal conductivity samples, such as graphite sheets of interest in electronics cooling applications, is that the limited power of the optical heat source leads to relatively small amplitudes in the temperature oscillations, which in turn leads to low signal-to-noise ratios. To overcome this challenge, we have further developed a robust fitting approach leveraging physics-informed neural networks (PINNs) to overcome the limitations of a traditional least squares fitting approach.[5] In this article, we briefly introduce our measurement approach including both conventional least squares fitting and novel PINNs data analysis approaches to illustrate the versatility of the measurement technique across a range of sample parameters.
Measurement System Overview
To briefly describe the laser .ngstrom method developed by our team[4], it involves periodically heating a sample at a fixed central point using a laser, while infrared (IR) imaging captures the resulting two-dimensional in-plane temperature distribution across the suspended portion of the thin film (see Figure 1a). The spatially-varying amplitude, T˜(x,y), and phase, ϕ(x,y), of the temperature oscillations in the suspended portion of the sample are analyzed to deduce the in-plane thermal properties. Importantly, this process does not rely on knowledge of laser power, heating frequency, or specific boundary conditions at the edge of the sample, which reduces the measurement uncertainty. The technique enables characterizing the in-plane thermal properties simultaneously for both in-plane directions.
In the physical test facility developed to demonstrate the technique (see Figure 1b), periodic heating is generated via a squarewave modulated laser beam incident on a small central spot on the bottom surface of the sample. A graphite-coated circular metal disk attached to the bottom surface of the sample acts as an absorber for the incident laser beam, ensuring uniform heating at the spot, and preventing any laser radiation from potentially penetrating through the sample and reaching the IR detector. A fiber-coupled laser attaches to the base of the heat sink, fully enclosing the beam when it is in free space.
The thin sample is suspended over a circular opening in a heat sink base. This maintains the outer portion of the sample at the temperature of the heat sink. An IR camera or IR microscope records the transient temperature response of the entire top surface of the sample. During the measurement, the sample is heated at a specified frequency and power while the temperature response is recorded, yielding data for the amplitude and phase of the first harmonic throughout the imaging region. In the suspended region of the sample that is not directly receiving incident radiation or above the absorber disk, the temperature data are analyzed to quantitatively extract the in-plane thermal properties.
Data Analysis Approach
The process of analyzing the thermal movies to extract thermal properties is briefly overviewed in Figure 2.
This method analyzes the transient two-dimensional temperature profile, T(x,y,t), of the top surface of the sample in the suspended region between the edge of the metal absorber disk and the edge of the heat sink. Neglecting convection losses and assuming the sample is thermally thin (uniform in temperature across the thickness, which can be achieved by tuning the heating frequency such that the thermal penetration depth far exceeds the sample thickness[4]), the temperature response must satisfy the governing 2D heat diffusion equation:
where kx and ky are the thermal conductivities in two orthogonal directions (x and y), ρ is the density, and cp is the specific heat of the sample. Assuming the density and heat capacity are known or separately measured, there are two unknown parameters: kx and ky. Note that an unknown convection coefficient can be included as a fitted value to account for heat losses to the ambient, as is described in more detail in our previous work.[4]
Because the temperature data analyzed are measured at steady periodic conditions, a time-periodic temperature solution is assumed for the suspended domain of the sample, which can be written in complex form in the frequency domain as:
where P(x,y) and Q(x,y) represent the real and imaginary parts of the complex amplitude, and the eiωt term accounts for the time-periodic behavior of the solution. Figure 3 shows the amplitude of the thermal response for isotropic and anisotropic materials and similar plots can be obtained for phase, as well as the in-phase and out-of-phase components of the signal.
By substituting this solution for temperature into the heat diffusion equation, two sets of partial differential equations (PDEs) are obtained by equating the real and imaginary terms of the equation. Further, these sets of equations are valid at all spatial points in the measurement domain. Assuming that the material is homogeneous (kx and ky do not depend on position) throughout the measured domain, these equations are solved as a system of algebraic equations across all the points inside the domain:
where n is the number of data points (pixels) in the region of analysis and the subscripts indicate each pixel’s index in the image. The unknown thermal conductivities in the two in-plane directions, kx and ky, that best satisfy this set of equations are then extracted using a numerical least-squares fitting[4] or an inverse parameter fitting using PINNs.[5] The techniques are detailed in the associated publications and the following section will highlight the PINNs approach, which can more effectively fit given noisy data.
Robust Extraction Of Thermal Conductivity From Noisy Data Using Pinns
Measurement data is inevitably influenced by sources of noise such as sensor inaccuracies, measurement errors, and inherent variability. The requirement to numerically evaluate derivatives from the pixelated measurement data in the least squares fitting approach makes the thermal property extraction potentially sensitive to noise in the measurement, particularly when dealing with low-magnitude temperature oscillations and short phase delays. To address this issue, an alternative method using physics- informed neural networks (PINNs) is introduced for inverse parameter fitting of the thermal conductivity, enhancing tolerance to noise in the measurement data. PINNs are deep learning frameworks that integrate machine learning algorithms with fundamental physics principles to efficiently solve partial differential equations (PDEs) or systems of PDEs governing a physical system. By incorporating physics-based equations, PINNs enhance training accuracy and speed by ensuring the neural network learns from experimental or computational data while adhering to universally valid physical laws. Additionally, PINNs can utilize automatic differentiation, a computational technique that efficiently evaluates derivatives using the chain rule, to compute PDE derivatives directly within the neural network, which eliminates the need for numerical differentiation and potential errors.[7]
To demonstrate the performance of the PINNs approach, noise is added explicitly to synthetic measurement data generated from numerical simulations that mimic the testing conditions. White Gaussian noise is introduced to the transient temperature data in the time domain (before taking the Fourier transform). This case of noise addition simulates the impact of inherent noise in the IR detector response over time.
The primary aim of employing a PINNs framework for inverse parameter fitting is to minimize the loss function (L), which considers both the neural network training errors and the residual errors of the governing physics equations. Figure 4 shows the architecture of the fully connected feed-forward neural network used in this framework. This network consists of an input layer with two neurons representing the spatial coordinates x and y, and an output layer with two neurons corresponding to the variables P and Q. These outputs are then utilized to calculate the residuals of the physics PDEs using automatic differentiation and thus, solving the inverse problem to extract the in-plane conductivities (kx and ky).
The implementation of this model is carried out using DeepXDE [7], a Python library, with Google Colaboratory services utilized for executing the code necessary for training the PINNs model and extracting the inverse parameters. The process of selecting model hyperparameters, including depth (number of hidden layers), width (number of neurons in each hidden layer), learning rate, and loss weights, was guided by an initial series of training trials. These trials involved using numerically generated data with known inverse parameters (kx, ky), exploring various combinations of hyperparameters to identify the optimal set that could accurately predict inverse parameters while minimizing training time.
The comparison between the least squares approach and approach for inverse thermal conductivity fitting is evaluated across various noise levels. Figure 5 illustrates this comparison for an anisotropic sample. The PINNs approach (black solid triangles) accurately predicts thermal conductivity values with just 5 time-periodic cycles of data, maintaining errors below 1% across all noise levels. In contrast, the least squares method (solid light blue and red circles) shows higher errors and sensitivity to the number of cycles.
When using 5 cycles, the least squares method performs poorly, with > 85% error for Signal to Noise Ratio (SNR) <10. The accuracy of the least squares approach is significantly impacted by time-domain noise when SNR < 30 due to discretization errors from numerical differentiation. In contrast, PINNs use automatic differentiation for robustness under high noise levels (up to SNR = 1), outperforming least squares even with increased cycle numbers. Increasing the number of cycles mitigates the impact of noise for the least squares approach, but still results in 50-70% error at SNR = 1 with 1000 data cycles recorded. Thus, PINNs clearly offer a dual benefit for parameter extractions: (1) increased accuracy in the extracted parameters when the data is noisy; and (2) reduced number of measurement cycles required for accurate parameter extraction, which, in turn, reduces the measurement time.
Summary
Inspired by the Ångström method for characterizing the thermal diffusivity of thin and long samples in one direction, our new laser-based Ångström method advances the field of thermal metrology by enabling the characterization of the thermal properties of thin sheet-like materials, including heat spreaders and composites, while also capturing effects of in-plane anisotropy within a single measurement of a single sample. From thermal maps of the surface of the sample, anisotropy is clearly visible in the isotherms, which are circular for the isotropic material, but elliptical for the anisotropic material (see Figure 3). This measurement technique was initially designed and validated numerically using simulated data.
Experimentally, this method has demonstrated accurate measurement spanning a wide spectrum of thermal conductivity, ranging approximately from 0.1 to 2000 W/(m K) as demonstrated by characterization of materials with known thermal properties such as Teflon (PTFE) and synthetic graphite (see Figure 6).[4]
The technique requires no special sample preparation if the samples are opaque at the laser wavelength and highly emissive and opaque in the infrared. For samples that that are not opaque and emissive, the only sample preparation required is a thin graphite layer to increase the emissivity of the sample for infrared imaging and/or the attachment of the thermally black absorber disk for laser absorption. The compact measurement system can be assembled at a low cost and operated on a benchtop without the complexities required for other methods [such as vacuum chambers (since convection is accounted for in the data analysis)]. Leveraging the tunable parameters in this method, such as the frequency of heating, laser power, size of the heat sink, etc., materials across a wide range of material properties can be accommodated and measured using this technique. Continued work on this method aims to standardize the measurement workflow and to further reduce the measurement uncertainties. Ultimately, improved measurement capabilities that enable the characterization of materials used in the electronics cooling domain will improve thermal design capabilities.
Acknowledgments
Financial support for this work provided in part by members of the Cooling Technologies Research Center, a graduated National Science Foundation Industry/University Cooperative Research Center at Purdue University, is gratefully acknowledged. S.S. appreciates the financial support from the Adelberg Fellowship (awarded by the School of Mechanical Engineering at Purdue University). The authors would like to thank Ritwik V. Kulkarni, Rohan M. Dekate, and Pranay P. Nagrani, graduate researchers at Purdue University for their discussions and assistance with implementation of the PINN approach.
Authors
Aaditya A. Candadai is an R&D engineer at Intel Corporation (Chandler, AZ), working as part of a team within the Technology Development organization focused on developing next-generation semiconductor electronics packaging technologies. He received a bachelor’s in Mechanical Engineering (2015) from BITS Pilani (Rajasthan, India), and an MS (2017) and PhD (2021) in Mechanical Engineering, both from Purdue University (West Lafayette, IN). His graduate research was primarily focused on heat transfer and thermal metrology techniques, working as part of the MTEC Lab and Cooling Technologies Research Center at Purdue University. His work on thermal metrology development was recognized with an Emerging Technologies Outstanding Paper Award at IEEE ITherm (2019), and related research on characterization of high-performance polymer fabrics was featured in scientific press articles.
Aalok Gaitonde is a PhD student at the School of Mechanical Engineering at Purdue University in West Lafayette, Indiana. He develops thermal metrology techniques for advanced semiconductor packaging technologies, including the characterization of anisotropic in-plane thermal properties of heat spreaders and composite materials, and the measurement of high conductance across deeply buried interfaces found in the next-generation packaging and heterogenous integration. His research is supported by the Cooling Technologies Research Center (CTRC) and the Semiconductor Research Corporation (SRC). Prior to enrolling in the PhD program, he spent 4 years working with the R&D wing of 3D Systems, developing thermal systems and architecture of Selective Laser Sintering (SLS) additive manufacturing platforms. Aalok completed his master’s degree in mechanical engineering from Purdue University in 2016.
Shanmukhi Sripada is a PhD student at the School of Mechanical Engineering at Purdue University in West Lafayette, Indiana. She received her Bachelor of Technology in Mechanical Engineering from Indian Institute of Technology, Delhi (IIT Delhi) in 2021. After finishing her undergraduate education, she worked for a brief period as a software engineering associate at Dassault Systemes Solutions Lab, Bengaluru, India. She joined Purdue University in Fall 2022 and is currently working on projects under the supervision of Dr Amy Marconnet, Dr Justin Weibel, and Dr Chelsea Davis. On the experimental side, she is currently working on developing thermally conducting and electrically insulating polymers for their application in motors, generators, power electronics, and power distribution. On the computational side, she is working on developing a physics informed neural networks model for accurate thermal characterization of materials.
Justin A. Weibel is a professor of Mechanical Engineering at Purdue University and Director of the Cooling Technologies Research Center (CTRC) a graduated NSF I/UCRC that addresses research and development needs of companies and organizations in the area of high-performance heat removal from compact spaces. He received his PhD in 2012 and BSME in 2007, both from Purdue University. Dr. Weibel’s research group explores methodologies for prediction and control of heat transport to enhance the performance and efficiency of thermal management technologies and energy transfer processes. He has been recognized as an Outstanding Engineering Teacher and Outstanding Faculty Mentor in the College of Engineering at Purdue University. He received the 2020 ASME Electronic & Photonic Packaging Division (EPPD) Young Engineer Award, 2021 ASME K-16 Outstanding Early Faculty Career in Thermal Management Award, and in 2023 was elected a Fellow of the ASME. Dr. Weibel is on the IEEE ITherm Executive Committee and is Associate Editor of the IEEE Transactions on Components Packaging and Manufacturing Technology.
Amy M. Marconnet is a professor of Mechanical Engineering and associate professor of Materials Engineering (by Courtesy), as well as a Perry Academic Excellence Scholar, at Purdue University. She received a B.S. in Mechanical Engineering from the University of Wisconsin – Madison in 2007, and an M.S. and a PhD in Mechanical Engineering at Stanford University in 2009 and 2012, respectively. She then worked briefly as a postdoctoral associate at the Massachusetts Institute of Technology, before joining the faculty at Purdue University in August 2013. Research in the MTEC Lab intersects heat transfer, energy conversion, and materials science to enable advances in technologies where energy conversion and thermal transport are key factors in performance. In 2017, she won the Woman in Engineering Award from the ASME Electronics & Photonics Packaging Division (EPPD). In 2020, she won the Bergles-Rohsenow Young Investigator Award in Heat Transfer and the Outstanding Graduate Student Mentor from the Official Mechanical Engineering Graduate Association (OMEGA) and the College of Engineering. She won a Humboldt Fellowship for Experienced Researchers and conducted research at Karlsruhe Institute of Technology in the 2021-22 academic year.
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