MODELING AND ANALYSIS OF TEMPERATURE MODULATED DIFFERENTIAL SCANNING CALORIMETRY (TMDSC) XU SHENXI (B. Eng., M. Eng., HUAZHONG UNIV. OF SCI. AND TECHNO., CHINA) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATERIALS SCIENCE NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements First of all, I sincerely thank my research project supervisors. Prof. Li Yi’s knowledge and patience were very important throughout my research in the Department of Materials Science, National University of Singapore (NUS). Many thanks are given to Prof. Feng Yuanping for his reviews and careful inspection of my work. I am grateful to Lu Zhaoping, Hu Xiang, Tan Hao, Xu Wei, Irene Lee, Annie Tan, Mitchell Ong, and all other team members, as well as to those who helped me during the long course of the Ph.D. program. I have enjoyed myself at NUS during the last few years and believe that it has been an important part of my life, not only because of the generous offering of a full scholarship by NUS that helped me to complete the research project, but also because of the hospitality and beauty of Singapore, factors that make this country so lovely and energetic. I am grateful to Prof. Li Z. Y., who was my mentor at Huazhong University of Science and Technology (Wuhan, China) and encouraged me to take the Ph.D. program back in early 1997, when I was still working in southern China. I would also like to extend my gratitude to the teachers and other staff members of my hometown-schools who helped me throughout the 20-year-long learning career. Although I cannot possibly list every one of them here, I thank them all for the help they gave in various forms. In addition, I would like to thank my parents for their never dwindling love and care. Last, but by no means least, I wish to thank my wife and daughter. They have consistently provided me with warmth and sweet distractions, and were able to forgive me for my long absences from time to time. Xu Shenxi, Oct. 2006 at the National University of Singapore i Modeling and Analysis of Temperature Modulated Differential Scanning Calorimetry Table of Contents Page Acknowledgements i Table of Contents ii Summary v List of Tables vii List of Figures viii List of Symbols xii List of Publications xv Chapter Literature Review 1.1 Review of dynamic thermal calorimetry 1.2 The 3-ω method: A milestone in dynamic thermal calorimetry 1.3 Comparison between the conventional DSC and TMDSC 1.3.1 Principles and advantages of TMDSC 1.3.1.1 Better temperature resolution and ability to measure specific heat in a single run 1.3.1.2 Ability to separate the reversing and non-reversing heat flows 1.3.2 Current status and limitations of TMDSC 11 1.3.2.1 Accurate calibration for heat capacity measurement 19 1.3.2.2 Influence of low sample thermal conductivity 23 1.3.2.3 The applicability of TMDSC 28 1.3.2.4 Heat capacity and complex heat capacity 33 1.3.2.4.1 Heat capacity, Debye and Einstein theories 1.3.2.4.2 14 18 33 Complex heat capacity and phase angle: definition and calculation 1.3.2.5 Calibration and system linearity of TMDSC 44 1.3.2.6 System linearity inspection 49 1.3.3 Progress in light modulation technique 39 50 ii 1.4 Summary 53 1.5 Objectives of the research 55 References Chapter Sample Mass, Modulation Parameters vs. Observed Specific Heat, and Numerical Simulation of TMDSC with an R-C Network 2.1 Introduction 2.2 Modeling and experiments of TMDSC 57 65 65 67 2.2.1 Indium melting experiment in the conventional DSC 67 2.2.2 A resistance-capacitance network model that takes into account the thermal contact resistance 2.2.3 TMDSC experimental conditions 70 2.3 Results and discussion 72 73 2.3.1 The effect of sample mass 74 2.3.2 TMDSC system output characteristics 79 2.3.3 The effect of modulation period 83 2.3.4 The effect of modulation amplitude 83 2.3.5 Calibration factor of sapphire and mass dependence 84 2.3.6 Possible effect of temperature profile in metallic samples 2.4 Comparison of heat capacity measurements in the conventional DSC and TMDSC 2.5 Conclusions 86 References Chapter Study of Temperature Profile and Specific Heat in TMDSC with a Low Sample Heat Diffusivity 3.1 Introduction 3.2 A TMDSC model with thermal diffusivity 3.2.1 An analytical solution to the heat conduction equation 3.2.2 A numerical approach 87 91 92 93 93 95 96 102 3.3 Experimental procedures for temperature profile study 103 3.4 Results and discussion 105 3.5 Conclusions 115 iii References Chapter Numerical Modeling and Analysis of TMDSC: On the Separability of Reversing Heat Flow from Non-reversing Heat Flow 4.1 Introduction 116 118 118 4.2 Model of TMDSC for numerical calculations 120 4.3 Simulation procedure and data treatment 122 4.4 Simulation results and discussions 125 4.4.1 Temperature dependent NHF 125 4.4.2 Time dependent NHF 127 4.4.3 Effect of the underlying heating rate 130 4.5 Conclusions References 132 132 Chapter System Linearity and the Effect of Kinetic Events on the Observed Specific Heat 5.1 Introduction 134 5.2 Analysis of complex heat capacity and the effect of kinetic events 5.3 Case studies on several kinetic models 135 5.4 Experimental analysis on several melt-spun amorphous alloys 163 5.5 Considerations in the selection of experiment parameters 172 5.6 Conclusions 173 References Chapter Overall Conclusions and possible future work 134 141 174 176 Appendix Fourier Transform and Phase Angle Calculation APP-1 Appendix Steady State Analytical Solution of the Model under Linear Heating Conditions in conventional DSC Appendix Finite Difference Method for One-dimensional Steady State Heat Transfer Problems Appendix Program Listings of Chapter to (on the floppy disk) APP-4 APP-7 iv Summary In this thesis, different aspects of TMDSC are studied and the main results are given below. (1) Effects of the contact thermal resistance on the observed specific heat • The relationship among the measured heat capacity, the actual heat capacity and temperature modulation frequency of heat flux type TMDSC is similar to that of a low-pass filter. • Careful sample preparation is important because too large or too small a sample mass (relative to the mass of the calibration reference) will lead to increased errors in the measured specific heat. • When TMDSC device works in the conventional differential scanning calorimetry (DSC) mode, the measured specific heat of the sample is not affected by the contact resistance. (2) Effects of the internal thermal resistance of the sample with a low heat diffusivity • A model that takes into account the thermal diffusivity of the sample is used and an analytical solution is derived. • To improve the accuracy of measured specific heat, we may use a longer temperature modulation period, or reduce the sample thickness and mass. (3) Effects of the non-reversing heat flow on the separability of the reversing heat flow and non-reversing heat flow The separability of non-reversing heat flow (NHF) and reversing heat flow (RHF) by TMDSC depends on the NHF and temperature modulation conditions. Two v different types of NHF are considered: time dependent NHF and temperature dependent NHF. • Time dependent NHF: The measurement of specific heat (cp), is applicable for the steady state where there is no NHF. While inside the NHF temperature range, if the modulation frequency is high enough, it still allows deconvolution of cp, HF, RHF, and NHF by Fourier transform. • Temperature dependent NHF: The NHF will be modulated by the temperature modulation and the NHF will contribute to the modulated part of the total heat flow (HF). This in turn can affect the linearity of the entire TMDSC system. (4) Study of the general situation and comparison with experimental results • A general case that takes into account a kinetic reaction that is both time and temperature dependent is studied. • Several kinetic models are used to demonstrate the importance of the selection of the experimental parameters as well as their effects on the system linearity. • TMDSC experiments with several melt spun Al-based amorphous alloys are carried out to demonstrate the unique capabilities of TMDSC. These include the ability to measure the differences between the specific heats of a sample in a fully amorphous, partially crystallized, or fully crystallized state. • The imaginary part of the complex heat capacity can be defined as C" ≅ -fT'/ω, where fT' is the temperature derivative of the kinetic heat flow and ω is the angular frequency of temperature modulation. It should be pointed out that this definition only holds true when the TMDSC system linearity satisfies (fT'/ Csω)[...]... see Eq (1.27) ρ Density xiv List of Publications Xu SX, Li Y, Feng YP Numerical modeling and analysis of temperature modulated differential scanning calorimetry: On the separability of reversing heat flow from non-reversing heat flow THERMOCHIMICA ACTA 343: (1-2) 81-88 JAN 14 2000 Xu SX, Li Y, Feng YP Study of temperature profile and specific heat capacity in temperature modulated DSC with a low sample... temperature (in K) AHF Amplitude of modulated heat flow A∆T Amplitude of the temperature difference between the sample and reference AT b Amplitude of heating block temperature ATs Amplitude of sample temperature ATr Amplitude of reference temperature B Reaction constant (in s-1) Ci Heat capacity (upper case, in J/K) of the heat transfer path or observed heat capacities at different temperature sensing positions... Nagel [14, 15], and Dixon [16] using the 3-ω approach has been further developed [47―51] The advances in temperature modulated calorimetry in the 1970s and 1980s finally saw the integration of the modulation technique with the widely used conventional DSC instrument, which is now known as temperature modulated differential scanning calorimetry (TMDSC) [3] Some references on dynamic calorimetry are... 131-140 SEP 28 2000 Xu SX, Li Y, Feng YP Temperature modulated differential scanning calorimetry: on system linearity and the effect of kinetic events on the observed sample specific heat THERMOCHIMICA ACTA 359: (1) 43-54 AUG 21 2000 Xu SX, Li Y, Feng YP Some elements in specific heat capacity measurement and numerical simulation of temperature modulated DSC (TMDSC) with R/C network THERMOCHIMICA ACTA... function of temperature modulation period in TMDSC Simulated heat flow amplitude as a function of temperature modulation period and amplitude in TMDSC Experimentally obtained heat flow amplitude in TMDSC PET sample mass: 19.6 mg Experimentally obtained amplitude of the sample temperature Ts as a function of temperature modulation period and amplitude The "measured" specific heat as a function of the temperature. .. modulation The modulated part of the sample temperature is Ts (t ) cyclic = ATs sin (ωt ) , (1.12) and the modulated part of the heat flow is HFcyclic = ms c p ATs ω cos(ωt ) = AHF cos(ωt ) (1.13) Comparing Eq (1.12) with Eq (1.13), we notice that if the amplitude of the sample temperature, ATs, and the amplitude of modulated heat flow, AHF, are obtained simultaneously, we can find the specific heat of the... A∆T is the amplitude of the temperature difference between Ts0 and Tr0, Cs is the heat capacity of the sample, and ATs is the amplitude of the sample temperature The phase angle between the modulated heat flow and the time derivative of sample temperature satisfies sin (ϕ ) = 1 + ω 2 τ sτ 0 (1 + ω τ )(1 + ω τ ) 2 2 s 2 2 , (1.27) 0 where τs=Cs/K', τ0=C0/K, C0 is the heat capacity of the support plate... ratio between the modulated heat flow and the modulated temperature The total heat flow is an average of the modulated heat flow over a sliding transform window [3] The reversing heat flow is the product of the heat capacity and the heating rate, and the non-reversing heat flow is the difference between the total heat flow and the reversing heat flow [166] The difference between Figs 1.7 and 1.8 is that... calculation T0 Initial temperature (in K) Tb Heating block temperature (in K) xiii Tg Glass transition temperature Ti Temperature of a certain sample unit, i=1,2,3… Tr Sample temperature (in K) Ts Reference temperature (in K) ∆T The difference between the sample and reference temperature ∆T= Tr -Ts ∆Tcyclic The cyclic part of ∆T V(t) Voltage V3ω(t) The third harmonic component of V(t) W Weight Y Heat... Adapted from [165] When the furnace temperature is modulated to follow Aeiωt (the complex form of temperature is used here), the cyclic part of the temperature difference between the sample and reference can be derived [165] (Tr ⎡ e iωt - Ts )cyclic = AωCs Re ⎢ ⎣ (α1 + α 2 )k + iωC r ⎤ ⎥, ⎦ (1.18) where α1 and α2 are two constants determined by the structure of the calorimetry device that can either . MODELING AND ANALYSIS OF TEMPERATURE MODULATED DIFFERENTIAL SCANNING CALORIMETRY (TMDSC) XU SHENXI (B. Eng., M. Eng., HUAZHONG UNIV. OF SCI. AND TECHNO.,. National University of Singapore ii Modeling and Analysis of Temperature Modulated Differential Scanning Calorimetry Table of Contents Page Acknowledgements i Table of Contents ii . Publications Xu SX, Li Y, Feng YP Numerical modeling and analysis of temperature modulated differential scanning calorimetry: On the separability of reversing heat flow from non-reversing heat