Effect of Phonon Dispersion on Thermal Conduction Across Si/Ge Interfaces1
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HEAT TRANSFER and the SECOND LAW Thus Far We’Ve Used the First Law of Thermodynamics: Energy Is Conserved
HEAT TRANSFER AND THE SECOND LAW Thus far we’ve used the first law of thermodynamics: Energy is conserved. Where does the second law come in? One way is when heat flows. Heat flows in response to a temperature gradient. If two points are in thermal contact and at different temperatures, T1 and T2 then energy is transferred between the two in the form of heat, Q. Heat Transfer and The Second Law page 1 © Neville Hogan The rate of heat flow from point 1 to point 2 depends on the two temperatures. Q˙ = f(T1,T2) If heat flows from hot to cold, (the standard convention) this function must be such that Q˙ > 0 iff T1 > T2 Q˙ < 0 iff T1 < T2 Q˙ = 0 iff T1 = T2 Heat Transfer and The Second Law page 2 © Neville Hogan In other words, the relation must be restricted to 1st and 3rd quadrants of the Q˙ vs. T1 – T2 plane. Q T1-T2 NOTE IN PASSING: It is not necessary for heat flow to be a function of temperature difference alone. —see example later. Heat Transfer and The Second Law page 3 © Neville Hogan HEAT FLOW GENERATES ENTROPY. FROM THE DEFINITION OF ENTROPY: dQ = TdS Therefore Q˙ = TS˙ IDEALIZE THE HEAT TRANSFER PROCESS: Assume no heat energy is stored between points 1 and 2. Therefore Q˙ = T1S˙ 1 = T2S˙ 2 Heat Transfer and The Second Law page 4 © Neville Hogan NET RATE OF ENTROPY PRODUCTION: entropy flow rate out minus entropy flow rate in. S˙ 2 - S˙ 1 = Q˙ /T2 - Q˙ /T1 = Q(T˙ 1 - T2) /T1T2 Absolute temperatures are never negative (by definition). -
Group Velocity
Particle Waves and Group Velocity Particles with known energy Consider a particle with mass m, traveling in the +x direction and known velocity 1 vo and energy Eo = mvo . The wavefunction that represents this particle is: 2 !(x,t) = Ce jkxe" j#t [1] where C is a constant and 2! 1 ko = = 2mEo [2] "o ! Eo !o = 2"#o = [3] ! 2 The envelope !(x,t) of this wavefunction is 2 2 !(x,t) = C , which is a constant. This means that when a particle’s energy is known exactly, it’s position is completely unknown. This is consistent with the Heisenberg Uncertainty principle. Even though the magnitude of this wave function is a constant with respect to both position and time, its phase is not. As with any type of wavefunction, the phase velocity vp of this wavefunction is: ! E / ! v v = = o = E / 2m = v2 / 4 = o [4] p k 1 o o 2 2mEo ! At first glance, this result seems wrong, since we started with the assumption that the particle is moving at velocity vo. However, only the magnitude of a wavefunction contains measurable information, so there is no reason to believe that its phase velocity is the same as the particle’s velocity. Particles with uncertain energy A more realistic situation is when there is at least some uncertainly about the particle’s energy and momentum. For real situations, a particle’s energy will be known to lie only within some band of uncertainly. This can be handled by assuming that the particle’s wavefunction is the superposition of a range of constant-energy wavefunctions: jkn x # j$nt !(x,t) = "Cn (e e ) [5] n Here, each value of kn and ωn correspond to energy En, and Cn is the probability that the particle has energy En. -
A Homemade 2 Dimensional Thermal Conduction Apparatus Designed As
AC 2008-292: A HOMEMADE 2-DIMENSIONAL THERMAL CONDUCTION APPARATUS DESIGNED AS A STUDENT PROJECT Robert Edwards, Pennsylvania State University-Erie Robert Edwards is currently a Lecturer in Engineering at The Penn State Erie, The Behrend College where he teaches Statics, Dynamics, and Fluid and Thermal Science courses. He earned a BS degree in Mechanical Engineering from Rochester Institute of Technology and an MS degree in Mechanical Engineering from Gannon University. Page 13.49.1 Page © American Society for Engineering Education, 2008 A Homemade 2-Dimensional Thermal Conduction Apparatus Designed as a Student Project Abstract: It can be fairly expensive to equip a heat transfer lab with commercially available devices. It is always nice to be able to make a device that provides an effective lab experience for the students. It is an extra bonus if the device can be designed as a student project, giving the students working on the device both a real design experience and a better understanding of the principles involved with the device and the associated lab exercise. One example of such a device is a 2-dimensional heat conduction device which was designed and built as a student senior design project by mechanical engineering technology students at Penn State Erie, The Behrend College. The device described in this paper allows the students to determine the thermal conductivity of several different materials, and to graphically see the effect of contact resistance in the heat conduction path. The students use the conductivity information to try to determine what material the test samples are made of. This paper describes the design of the device and the basic theory including a description of how it applies to this particular device. -
Photons That Travel in Free Space Slower Than the Speed of Light Authors
Title: Photons that travel in free space slower than the speed of light Authors: Daniel Giovannini1†, Jacquiline Romero1†, Václav Potoček1, Gergely Ferenczi1, Fiona Speirits1, Stephen M. Barnett1, Daniele Faccio2, Miles J. Padgett1* Affiliations: 1 School of Physics and Astronomy, SUPA, University of Glasgow, Glasgow G12 8QQ, UK 2 School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK † These authors contributed equally to this work. * Correspondence to: [email protected] Abstract: That the speed of light in free space is constant is a cornerstone of modern physics. However, light beams have finite transverse size, which leads to a modification of their wavevectors resulting in a change to their phase and group velocities. We study the group velocity of single photons by measuring a change in their arrival time that results from changing the beam’s transverse spatial structure. Using time-correlated photon pairs we show a reduction of the group velocity of photons in both a Bessel beam and photons in a focused Gaussian beam. In both cases, the delay is several microns over a propagation distance of the order of 1 m. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves. Introducing spatial structure to an optical beam, even for a single photon, reduces the group velocity of the light by a readily measurable amount. One sentence summary: The group velocity of light in free space is reduced by controlling the transverse spatial structure of the light beam. Main text The speed of light is trivially given as �/�, where � is the speed of light in free space and � is the refractive index of the medium. -
Introduction to Thermal Transport
Introduction to Thermal Transport Simon Phillpot Department of Materials Science and Engineering University of Florida Gainesville FL 32611 [email protected] Objectives •Identify reasons that heat transport is important •Describe fundamental processes of heat transport, particularly conduction •Apply fundamental ideas and computer simulation methods to address significant issues in thermal conduction in solids Many Thanks to … •Patrick Schelling (Argonne National Lab Æ U. Central Florida) •Pawel Keblinski (Rensselaer Polytechnic Institute) •Robin Grimes (Imperial College London) •Taku Watanabe (University of Florida) •Priyank Shukla (U. Florida) •Marina Yao (Dalian University / U. Florida) Part 1: Fundamentals Thermal Transport: Why Do We Care? Heat Shield: Disposable space vehicles Cross-section of Mercury Heat Shield Apollo 10 Heat Shield Ablation of polymeric system Heat Shield: Space Shuttle Thermal Barrier Coatings for Turbines http://www.mse.eng.ohio-state.edu/fac_staff/faculty/padture/padturewebpage/padture/turbine_blade.jpg Heat Generation in NanoFETs SOURCE ELECTRON 20 nm FINS RELAXATION G A D ~ 20 nm 10 3 T Q”’ ~ 10 eV/nm /s E DRAIN HOTSPOT EDGE Y-K. Choi et al., EECS, U.C. Berkeley BALLISTIC DRAIN TRANSPORT • Scaling → Localized heating → Phonon hotspot • Impact on ESD, parasitic resistances ? Thermoelectrics cold junction n-type p-type hot junctions Heat Transfer Mechanisms Three fundamental mechanisms of heat transfer: • Convection • Conduction •Radiation ¾ Convection is a mass movement of fluids (liquid or gas) rather than -
Heat Transfer and Thermal Modelling
H0B EAT TRANSFER AND THERMAL RADIATION MODELLING HEAT TRANSFER AND THERMAL MODELLING ................................................................................ 2 Thermal modelling approaches ................................................................................................................. 2 Heat transfer modes and the heat equation ............................................................................................... 3 MODELLING THERMAL CONDUCTION ............................................................................................... 5 Thermal conductivities and other thermo-physical properties of materials .............................................. 5 Thermal inertia and energy storage ....................................................................................................... 7 Numerical discretization. Nodal elements ............................................................................................ 7 Thermal conduction averaging .................................................................................................................. 9 Multilayer plate ..................................................................................................................................... 9 Non-uniform thickness ........................................................................................................................ 11 Honeycomb panels ............................................................................................................................. -
Lecture 2. BASICS of HEAT TRANSFER ( )
Lecture 2. BASICS OF HEAT TRANSFER 2.1 SUMMARY OF LAST WEEK LECTURE There are three modes of heat transfer: conduction, convection and radiation. We can use the analogy between Electrical and Thermal Conduction processes to simplify the representation of heat flows and thermal resistances. T q R Fourier’s law relates heat flow to local temperature gradient. T qx Axk x Convection heat transfer arises when heat is lost/gained by a fluid in contact with a solid surface at a different temperature. TW Ts TW Ts q hAs TW Ts [Watts] or q 1/ hAs Rconv 1 Where: Rconv hAs Radiation heat transfer is dependent on absolute temperature of surfaces, surface properties and geometry. For case of small object in a large enclosure. 4 4 q s As Ts Tsurr Kosasih 2012 Lecture 2 Basics of Heat Transfer 1 2.2 CONTACT RESISTANCE In practice materials in thermal contact may not be perfectly bonded and voids at their interface occur. Even a flat surfaces that appear smooth turn out to be rough when examined under microscope with numerous peaks and valleys. Figure 1. Comparison of temperature distribution and heat flow along two plates pressed against each other for the case of perfect and imperfect contact. In imperfect contact, the “contact resistance”, Ri causes an additional temperature drop at the interface Ti Ri qx (1) Ri is very difficult to predict but one should be aware of its effect. Some order‐ of‐magnitude values for metal‐to‐metal contact are as follows. Material 2 Contact Resistance Ri [m W/K] ‐5 Aluminum 5 x 10 Copper 1 x 10‐5 Stainless steel 3 x 10‐4 Kosasih 2012 Lecture 2 Basics of Heat Transfer 2 We use grease or soft metal foil to improve contact resistance e.g. -
Theory and Application of SBS-Based Group Velocity Manipulation in Optical Fiber
Theory and Application of SBS-based Group Velocity Manipulation in Optical Fiber by Yunhui Zhu Department of Physics Duke University Date: Approved: Daniel J. Gauthier, Advisor Henry Greenside Jungsang Kim Haiyan Gao David Skatrud Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2013 ABSTRACT (Physics) Theory and Application of SBS-based Group Velocity Manipulation in Optical Fiber by Yunhui Zhu Department of Physics Duke University Date: Approved: Daniel J. Gauthier, Advisor Henry Greenside Jungsang Kim Haiyan Gao David Skatrud An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2013 Copyright c 2013 by Yunhui Zhu All rights reserved Abstract All-optical devices have attracted many research interests due to their ultimately low heat dissipation compared to conventional devices based on electric-optical conver- sion. With recent advances in nonlinear optics, it is now possible to design the optical properties of a medium via all-optical nonlinear effects in a table-top device or even on a chip. In this thesis, I realize all-optical control of the optical group velocity using the nonlinear process of stimulated Brillouin scattering (SBS) in optical fibers. The SBS- based techniques generally require very low pump power and offer a wide transparent window and a large tunable range. Moreover, my invention of the arbitrary SBS res- onance tailoring technique enables engineering of the optical properties to optimize desired function performance, which has made the SBS techniques particularly widely adapted for various applications. -
Phase and Group Velocities of Surface Waves in Left-Handed Material Waveguide Structures
Optica Applicata, Vol. XLVII, No. 2, 2017 DOI: 10.5277/oa170213 Phase and group velocities of surface waves in left-handed material waveguide structures SOFYAN A. TAYA*, KHITAM Y. ELWASIFE, IBRAHIM M. QADOURA Physics Department, Islamic University of Gaza, Gaza, Palestine *Corresponding author: [email protected] We assume a three-layer waveguide structure consisting of a dielectric core layer embedded between two left-handed material claddings. The phase and group velocities of surface waves supported by the waveguide structure are investigated. Many interesting features were observed such as normal dispersion behavior in which the effective index increases with the increase in the propagating wave frequency. The phase velocity shows a strong dependence on the wave frequency and decreases with increasing the frequency. It can be enhanced with the increase in the guiding layer thickness. The group velocity peaks at some value of the normalized frequency and then decays. Keywords: slab waveguide, phase velocity, group velocity, left-handed materials. 1. Introduction Materials with negative electric permittivity ε and magnetic permeability μ are artifi- cially fabricated metamaterials having certain peculiar properties which cannot be found in natural materials. The unusual properties include negative refractive index, reversed Goos–Hänchen shift, reversed Doppler shift, and reversed Cherenkov radiation. These materials are known as left-handed materials (LHMs) since the electric field, magnetic field, and the wavevector form a left-handed set. VESELAGO was the first who investi- gated the propagation of electromagnetic waves in such media and predicted a number of unusual properties [1]. Till now, LHMs have been realized successfully in micro- waves, terahertz waves and optical waves. -
Chapter 3 Wave Properties of Particles
Chapter 3 Wave Properties of Particles Overview of Chapter 3 Einstein introduced us to the particle properties of waves in 1905 (photoelectric effect). Compton scattering of x-rays by electrons (which we skipped in Chapter 2) confirmed Einstein's theories. You ought to ask "Is there a converse?" Do particles have wave properties? De Broglie postulated wave properties of particles in his thesis in 1924, based partly on the idea that if waves can behave like particles, then particles should be able to behave like waves. Werner Heisenberg and a little later Erwin Schrödinger developed theories based on the wave properties of particles. In 1927, Davisson and Germer confirmed the wave properties of particles by diffracting electrons from a nickel single crystal. 3.1 de Broglie Waves Recall that a photon has energy E=hf, momentum p=hf/c=h/, and a wavelength =h/p. De Broglie postulated that these equations also apply to particles. In particular, a particle of mass m moving with velocity v has a de Broglie wavelength of h λ = . mv where m is the relativistic mass m m = 0 . 1-v22/ c In other words, it may be necessary to use the relativistic momentum in =h/mv=h/p. In order for us to observe a particle's wave properties, the de Broglie wavelength must be comparable to something the particle interacts with; e.g. the spacing of a slit or a double slit, or the spacing between periodic arrays of atoms in crystals. The example on page 92 shows how it is "appropriate" to describe an electron in an atom by its wavelength, but not a golf ball in flight. -
Module 2 – Linear Fiber Propagation
Module 2 – Linear Fiber Propagation Dr. E. M. Wright Professor, College of Optics, University of Arizona Dr. E. M. Wright is a Professor of Optical Sciences and Physic sin the University of Arizona. Research activity within Dr. Wright's group is centered on the theory and simulation of propagation in nonlinear optical media including optical fibers, integrated optics, and bulk media. Active areas of current research include nonlinear propagation of light strings in the atmosphere, and the propagation of novel laser fields for applications in optical manipulation and trapping of particles, and quantum nonlinear optics. Email: [email protected] Introduction The goal of modules 2-5 is to build an understanding of nonlinear fiber optics and in particular temporal optical solitons. To accomplish this goal we shall refer to material from module 1 of the graduate super-course, Electromagnetic Wave Propagation, as well as pointing to the connections with several undergraduate (UG) modules as we go along. We start in module 2 with a treatment of linear pulse propagation in fibers that will facilitate the transition to nonlinear propagation and also establish notation, and in module 3 we shall discuss numerical pulse propagation both as a simulation tool and an aid to understanding. Nonlinear fiber properties will be discussed in module 4, and temporal solitons arising from the combination of linear and nonlinear properties in optical fibers shall be treated in module 5. The learning outcomes for this module include • The student will be conversant with the LP modes of optical fibers and how to calculate the properties of the fundamental mode. -
Phase and Group Velocity
PHASEN- UND GRUPPENGESCHWINDIGKEIT VON ULTRASCHALL IN FLÜSSIGKEITEN Phase and group velocity of ultrasonic waves in liquids 1 Goal In this experiment we want to measure phase and group velocity of sound waves in liquids. Consequently, we measure time and space propagation of ultrasonic waves (frequencies above 20 kHz) in water, saline and glycerin. 2 Theory 2.1 Phase Velocity Sound is a temporal and special periodic sequence of positive and negative pressures. Since the changes in the pressure occur parallel to the propagation direction, it is called a longitudinal wave. The pressure function P(x, t) can be described by a cosine function: 22ππ Pxt(,)= A⋅− cos( kxωω t )mit k = und = ,(1) λ T where A is the amplitude of the wave, and k is the wave number and λ is the wavelength. The propagation P(x,t) c velocity CPh is expressed with the help of a reference Ph point (e.g. the maximum).in the space. The wave propagates with the velocity of the points that have the A same phase position in space. Therefore, Cph is called phase velocity. Depending on the frequency ν it is P0 x,t defined as ω c =λν⋅= (2). Ph k The phase velocity (speed of sound) is 331 m/s in the air. The phase velocity reaches around 1500 m/s in the water because of the incompressibility of liquids. It increases even to 6000 m/s in solids. During this process, energy and momentum is transported in a medium without requiring a mass transfer. 1 PHASEN- UND GRUPPENGESCHWINDIGKEIT VON ULTRASCHALL IN FLÜSSIGKEITEN 2.2 Group Velocity P(x,t) A wave packet or wave group consists of a superposition of cG several waves with different frequencies, wavelengths and amplitudes (Not to be confused with interference that arises when superimposing waves with the same frequencies!).