Continuous Phase Transition

9 Key excerpts on "Continuous Phase Transition"

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  A Dynamical Systems Theory of Thermodynamics

  • Wassim M. Haddad(Author)
  • 2019(Publication Date)
  • Princeton University Press

    (Publisher)

  Chapter Eight Critical Phenomena and Continuous Phase Transitions 8.1 Introduction In the previous chapters, an implicit assumption of our large-scale dy-namical system model is that the thermodynamic state variables define a continuously differentiable flow on the nonnegative orthant of the state space. For systems that possess phase transitions and critical states, this assumption is clearly limiting. A phase transition is a phenomenon wherein an abrupt change between phases (i.e., a change between one state of matter into another) occurs as the system parameters (e.g., temperature, pressure, chemical composition, electric field, magnetic field) are varied. Phase transitions are not limited to thermodynamic systems, where temperature or pressure are the phase transition driving parameters. They include, for example, quantum, dynamic (i.e., bifurcation), and topological (i.e., structural) phase transitions. Phase transitions are ubiquitous in nature, with the different phases of water (i.e., vapor, liquid, and ice) perhaps being the most familiar, and with transitions involving change from one phase to the other. Other phase transi-tions include eutectic transformations, peritectic transformations, spinodal decompositions, mesophase transitions, ferromagnetic-paramagnetic phase transitions of magnetic materials, superconductivity, molecular structure (e.g., polymorph, allotrope, polyamorph) transitions, and quantum conden-sation. In the early universe (i.e., ∼ 10 -35 seconds after the burst and through inflationary cosmology and the initial cosmic horizon), symmetry-breaking transitions in the laws of physics due to ravaging pressures and temperatures prevented the permanent formation of elementary particles (e.g., bosons, quarks, leptons, antiquarks, and antileptons). Elementary and composite particles were unable to form stable constituents until the universe cooled beyond the supergravity phase, that is, gravity as predicted

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  Phases of Matter and their Transitions

  Concepts and Principles for Chemists, Physicists, Engineers, and Materials Scientists

  • Gijsbertus de With(Author)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  458 15 Continuous Phase Transitions: Liquids ↔ Gases In Chapters 13 and 14, we discussed disContinuous Phase Transitions. In this chapter, we discuss aspects of Continuous Phase Transitions, mainly focusing on the liquid–gas tran- sition. Continuous Phase Transitions provide a particularly rich part of chemical physics and considerable effort has been paid to describe and understand them. This has led to an image that captures the essentials well. In the sequel, we first discuss some experimental facts, thereafter mean-field theory, scaling, and renormalization. 15.1 Limiting Behavior We recall that thermodynamic equilibrium is determined by minimization of the appropri- ate potential with respect to the (relevant) internal variable(s). For the liquid–gas transition, the density difference Δ =  −  cri is single valued above T cri , has two values below T cri , and Δ is conventionally used as the internal variable. In connection with transitions, the inter- nal variable used is often called the order parameter. For liquids, both single-compound liquids and liquid mixtures exhibit continuous transitions. As an example, in Figure 15.1 the reduced density ( L −  V )/2 cri of CO 2 versus the reduced temperature t ≡ (T − T cri )/T cri is given, while Figure 15.2 shows the difference in volume fraction  (1) −  (2) for CCl 4 in C 7 F 14 versus t. In both cases, the behavior close to T cri is described by a power law [1, 2]. This type of experiment has led to the definition of various critical exponents. A critical exponent of a function, say a for a function f (x) of x, is defined by a ≡ lim x→0 [d ln f (x)∕d ln x] and we write f (x) ∼ x a . This does not imply that f (x) = Ax a but rather that f (x) = Ax a (1 + Bx b + …) with b > 0, and where Ax a is called the singular part of f (x) and 1 + Bx b + … the regular part or background function. The exponent a defines the rate of approach of f (x) to zero for a > 0 or to infinity for a < 0.

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  Phase Transitions in Foods

  • Yrjö H. Roos, Yrjo H Roos, Steve Taylor(Authors)
  • 1995(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER 1

  Introduction to Phase Transitions

  I. Introduction

  Phase transitions are changes in the physical state of materials, which have significant effects on their physical properties. Chemically pure compounds such as water or many organic and inorganic compounds in foods have exact phase transition temperatures. There are three basic physical states, which are the solid, the liquid, and the gaseous states. The term transition refers to the change in the physical state that is caused by a change in temperature or pressure. Heating of solid foods is often used to observe temperatures at which changes in thermal or physical properties, e.g., in heat capacity, viscosity, or textural characteristics, occur.

  Water is one of the most important compounds in nature and also in foods. It may exist in all of the three basic states during food processing, storage, and consumption. The effect of water on the phase behavior of food solids is of utmost importance in determining processability, stability, and quality. Well-known examples include transformation of liquid water into ice (freezing) or water into vapor (evaporation). These transitions in phase are the main physical phenomena that govern food preservation by freezing and dehydration. Engineering and sensory characteristics of food materials are often defined by the complicated combination of the physical state of component compounds. The main constituents of food solids are carbohydrates, proteins, water, and fat. These materials may exist in the liquid state and the solid crystalline or amorphous noncrystalline state. Many of the component compounds, e.g., sugars, fats, and water, when they are chemically pure, crystallize below their equilibrium melting temperature.

  Stability is an important criterion in food preservation. Materials in thermodynamic equilibrium are stable, i.e., they exist in the physical state that is determined by the pressure and temperature of the surroundings. However, most biological materials are composed of a number of compounds and they often exist in a thermodynamically nonequilibrium, amorphous state. Such materials exhibit many time-dependent changes that are not typical of pure compounds and they may significantly affect the shelf life of foods. The physical state of food solids is often extremely sensitive to water content, temperature, and time. This chapter introduces the basic terminology of thermodynamics and phase transitions, and describes the common thermodynamic principles that govern the physical state of foods.

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  Statistical and Thermal Physics

  An Introduction

  • Michael J.R. Hoch(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  173 C H A P T E R 9 Phase Transitions and Critical Phenomena 9.1 INTRODUCTION Phasetransitionsoccurwidelyinnatureandarepartofeverydayexperi-ence.Forexample,waterfreezesintoiceat0°C,orvaporizesintosteamat 100°Catsealevel.Overthepastseveraldecades,considerableinsightinto thenatureofphasetransitionshasbeengainedusingthemethodsofther-malandstatisticalphysics.AsdiscussedinChapter6,thethirdlawstates thattheentropy S tendstozeroasthetemperaturetendstozero.Asthe temperature is lowered, interactions between particles become increas-ingly important and may lead to the onset of some type of long-range orderaccompaniedbysymmetrybreakinginthesystem.Forexample,a magneticmaterial,suchasironornickel,undergoesachangeatitsCurie temperature T C fromdisorderandhighsymmetryintheparamagnetic phasetoorderandlowersymmetryintheferromagneticphase.Thesym-metrychangeisassociatedwiththeorientationofthemagneticdipoles associatedwiththeionsinthematerial;inzeromagneticfield,thereis nopreferreddirectionforthedipolemomentsintheparamagneticphase while,withinmagneticdomains,thedipolesarealignedwitheachother intheferromagneticphase.Weshallseethatitisconvenienttointroduce anorderparameter η todescribeorder–disordertransitions.Asanexam-ple,foraferromagnet,theorderparameterisdefinedas η = M / M 0 ,with M thebulkmagnetizationatthetemperatureofinterestand M 0 thelow-temperaturesaturatedmagnetization. 174 ◾ Statistical and Thermal Physics: An Introduction Phasetransitionsmaybeclassifiedintoseveraldifferenttypesonthe basisofthebehaviorwithtemperatureofthethermodynamicpotentials and the order parameter. The order parameter may change discontin-uously or continuously at the transition point.

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  Fundamentals of Condensed Matter and Crystalline Physics

  An Introduction for Students of Physics and Materials Science

  • David L. Sidebottom(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  We will fi nd that, in general, a phase transition is accompanied by some change in the amount of order as when, for example, liquid water freezes into crystalline ice. Moreover, we can describe this amount of ordering quantitatively by introducing an appropriate order parameter , whose value changes signi fi cantly only during the transition. Based upon the manner in which the order parameter changes, we can distinguish two different types of phase transitions: those of fi rst order for which the order parameter changes discontinuously and those of second order for which it changes continuously. Second-order transitions are possible for both the vapor-to-liquid transition and the paramagnetic-to-ferromagnetic transition and are of interest due to the way in which many properties diverge near the transition in a similar, power law manner. 15.1 Free energy considerations Phase transitions are governed by thermodynamics and thermodynamics is governed by two laws. The fi rst of these, d U ¼ T d S P d V , is a statement of conservation of energy during a reversible process. The second law is notori-ously imprecise, but for our current purposes can be interpreted to mean that a system will strive to minimize its Gibbs free energy, G ¼ U þ TS PV . In a conceptual way we can use this minimization principle to understand why and where a phase transition occurs. Recall that changes in the free energy are given by, d G ¼ V d P S d T ; ð 15 : 1 Þ 267 from which it follows that V ¼ @ G @ P T ; ð 15 : 2 Þ and, S ¼ @ G @ T P : ð 15 : 3 Þ Each of these last two equations is a statement regarding how the slope of either G(P) or G(T) depends on the system ’ s volume or entropy, respectively. Consider then how pressure affects a system of particles to produce either gas, liquid or solid phases. A schematic representation of the G(P) diagram would appear something like that shown in Fig. 15.1a , where the slope of each line segment is consistent with Eq.

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  Basic Statistical Physics

  • Nandita Rudra, P Rudra;;;(Authors)
  • 2009(Publication Date)
  • WSPC

    (Publisher)

  Phase Transitions 159 T H c H T c 0 Superconducting Phase Normal Phase Fig. 12.1.3 Critical field versus temperature curve for superconducting systems. 12.2 Ehrenfest’s Classification of Phase Transitions Early in the twentieth century Paul Ehrenfest gave a classification scheme for different classes of phase transitions. In spite of later and modern schemes of classifications, physicists still use Ehrenfest’s terminology. This scheme has the advantage that it refers to experimentally measured quan-tities like Latent Heat and Specific Heat. This classification scheme depends on the continuity properties of the Gibbs’ potential G and its derivatives of different order with respect to temperature T . If G ( r ) = ∂ r G ∂T r so that G (0) is the Gibbs’ potential itself, then according to this classification scheme for an n-th order transition from phase I to phase II G ( k ) I = G ( k ) II , for all 0 ≤ k ≤ n − 1 , (12.2.1) G ( n ) I = G ( n ) II . (12.2.2) This means that the n-th order derivative of the Gibbs’ potential is discon-tinuous while all the lower order derivatives including the function itself are continuous. 160 BASIC STATISTICAL PHYSICS T T c Gas Liquid − ρ ρ gas 0 Fig. 12.1.4 Density versus temperature curve of water. For a first order transition , n=1 and entropy is discontinuous , thus showing the presence of latent heat of transition ; and for a second order transition , n=2 and specific heat is discontinuous. As a matter of fact latent heat of transition is a function of temperature and the nature of this dependence is shown in Figure 12.2.1. In Ehrenfest’s classification derivatives of Gibbs’ potential with respect to other thermodynamic variables like pressure have no role. 12.3 Order Parameter, Continuous and Discontinuous Transitions Landau shifted the attention from changes in thermodynamic variables like entropy and specific heat to changes in properties like density and magne-tization as the system undergoes a phase transition.

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  Statistical Mechanics

  An Intermediate Course

  • G Morandi, E Ercolessi;F Napoli;;(Authors)
  • 2001(Publication Date)
  • WSPC

    (Publisher)

  Chapter 7 Phase Transitions and Critical Phenomena 7.1 Introduction to Phase Transitions 7.1.1 Phase Coexistence and Phase Diagrams: Classifica- tion of Phase Transitions. Although a large variety of physical systems exhibit critical phenomena, we will concentrate here, and for the time being, on the phase transitions of classical fluids (liquid-solid-vapor) and/or of magnetic systems. Consider an ordinary (classical) fluid composed of identical constituents (atoms or molecules) interacting via, say, a two-body potential. It is part of everyday's experience that such a system may exist in different phases, namely the solid, liquid and vapor (or gas) phase. There need not be a single phase of every kind: solid ice is known to have at least six different phases, carbon exists in at least three distinct solid phases (diamond, graphite and buckyballs), 4 He has at least two different liquid phases (normal liquid and superfluid) and so on. Anyhow, each phase is stable in well defined ranges of temperature and pressure. When can two distinct phases coexist and be in equilibrium? The appro- priate setting for discussing this problem is provided by the grand-canonical ensemble that we discussed in § 2.3, the two phases being considered as subsystems that can exchange both heat, mechanical work and particles. Then, denoting by the suffixes "1" and "2" the thermodynamic functions of the two phases, the conditions for thermal, mechanical and chemical equilibrium are: T x = T 2 = , Pl =p 2 = and fi x = /i 2 (7.1.1) 385 386 Phase Transitions and Critical Phenomena (remember that p, = /x(T,p)). Denoting by T and p the common values of temperature and pressure, we get the equation: Ml (T,p) = /x 2 (T,p) (7.1.2) that will define a line in the (T,p) plane. Three-phase equilibrium leads to two conditions, namely: Mi(T,p) = fi 2 {T,p) = /i 3 (T,p) (7.1.3) and hence to (isolated) triple points in the (T,p) plane.

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  Introduction to the Theory of Critical Phenomena

  Mean Field, Fluctuations and Renormalization

  • Dimo I Uzunov(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  It is simple, since the first order transitions are defined as previously suggested by Ehrenfest and all others are called continuous transitions in the sense that the first derivatives of the thermodynamic functions change continuously at the transition point. All possible singularities as finite jumps and divergences of various types are likely to be included in the Fisher classification and this fits well the pure thermodynamic approach, within which no particular predictions about the singularities can be made. This classification offers a stable basis for fur-ther analysis of continuous transitions, founded on the modern concept of universality (see Sections 5.9.2 and 7.11.3). Brout (1965) has proposed that the phase transitions can be classified by their symmetry characteristics. This idea binds together the problem of classification and the original Landau ideas about the close relation between the phase transition properties and the type of the symmetry change at the phase transition point (Section 4.9.2). Note that within the Brout classification the order parameter is the “response coordinate” of the system and this “coordinate” exhibits the loss of the symmetry below the transition point T c ; for details, see Brout (1965). 3.6 Compressible Systems So far our reasoning has been restricted in the limits of the simple case of phase transition of two phases in fluid variables. Using this example we have presented the main aspects of the thermodynamic theory of phase transi-tions. Now it can be applied to any substance where two phases change 86 Introduction to the Theory of Critical Phenomena one into another or to be directly generalized to complex systems where three or more phases may coexist. To do this one needs certain knowledge of the relationship between thermodynamics and other fields of theoretical physics, because some special forms of the fundamental equations have to be studied.

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  Statistical Mechanics of Phases and Phase Transitions

  • Steven A. Kivelson, Jack Mingde Jiang, Jeffrey Chang(Authors)
  • 2024(Publication Date)
  • Princeton University Press

    (Publisher)

  Continuous Phase Transitions share many properties with the critical point of water. There is a divergence in a susceptibility, indicating extreme sensitivity to external perturbations. Additionally, fluctuations (such as those responsible for critical opalescence) become relevant on all lengthscales. DisContinuous Phase Transitions, such as a liquid-gas transition, have a host of different properties. The defining characteristic is that there is a discontinuity in a first derivative of the free energy, e.g., as water vaporizes its density plummets discontinuously (Figure 1.2A). Another characteristic of a discontinuous transi- tion is a nonzero latent heat: upon crossing the phase transition, a particular amount of heat per unit volume is absorbed or released. 3 Furthermore, discontin- uous transitions can exhibit metastability: water can be chilled below its freezing point without the formation of any ice crystals, which is known as supercooling (Section 4.4.3). 2. Formally, the continuity of the free energy follows from the fact that it is a bounded, convex function. Physically, since its derivatives are state functions (such as the entropy or the magnetization), these must everywhere be well-defined (although possibly themselves discontinuous) functions. 3. The latent heat of water vapor is the reason why the steam over a pot of soup feels so hot: the water vapor releases heat as it condenses. It is also why it feels so cold to get out of a swimming pool on a windy day: the water absorbs heat as it evaporates off your skin, a process accelerated by the wind! 1.2. symmetries 5 For historical reasons, discontinuous and continuous transitions are also known as 1st-order and 2nd-order transitions, respectively. This nomenclature comes from thermodynamics: the density ρ is a first derivative of the free energy, whereas the compressibility κ is a second derivative.

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