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    In a single-component condensed system if degree of freedom is zero maximum number of phases that can co-exist ________A. 0 B. 1 C. 2 D. 3

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    In a single-component condensed system if degree of freedom is zero maximum number of phases that can co-exist ________A. 0 B. 1 C. 2 D. 3

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    In a single-component condensed system if degree of freedom is zero maximum number of phases that can co-exist  2.

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    In a single component condensed system if degree of freedom is zero maximum number of phase that can co exist

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    What is meant by degrees of freedom of a system?

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    The number of solutions of cos^(-1)x=e^(x)(A)1(B)2(C)3(D)0

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    Answer to Solved In a single-component condensed system, if degree of

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    A possible four

    For different phases to coexist in equilibrium at constant temperature T and pressure P, the condition of equal chemical potential μ must be satisfied. This condition dictates that, for a single-component system, the maximum number of phases that can coexist is three. Historically this is known as the Gibbs phase rule, and is one of the oldest and venerable rules of thermodynamics. Here we make use of the fact that, by varying model parameters, the Gibbs phase rule can be generalized so that four phases can coexist even in single-component systems. To systematically search for the quadruple point, we use a monoatomic system interacting with a Stillinger–Weber potential with variable tetrahedrality. Our study indicates that the quadruple point provides flexibility in controlling multiple equilibrium phases and may be realized in systems with tunable interactions, which are nowadays feasible in several soft matter systems such as patchy colloids. Gibbs' phase rule states that the maximum number of coexisting phases in a one-component system, and in absence of external fields, is three. Here, the authors show that directly controlling the Hamiltonian allows the extension of this rule to four phases.

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    Published: 25 August 2016

    A possible four-phase coexistence in a single-component system

    Kenji Akahane, John Russo & Hajime Tanaka

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    7, Article number: 12599 (2016) Cite this article

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    Abstract

    For different phases to coexist in equilibrium at constant temperature and pressure , the condition of equal chemical potential must be satisfied. This condition dictates that, for a single-component system, the maximum number of phases that can coexist is three. Historically this is known as the Gibbs phase rule, and is one of the oldest and venerable rules of thermodynamics. Here we make use of the fact that, by varying model parameters, the Gibbs phase rule can be generalized so that four phases can coexist even in single-component systems. To systematically search for the quadruple point, we use a monoatomic system interacting with a Stillinger–Weber potential with variable tetrahedrality. Our study indicates that the quadruple point provides flexibility in controlling multiple equilibrium phases and may be realized in systems with tunable interactions, which are nowadays feasible in several soft matter systems such as patchy colloids.

    Introduction

    When different phases are in thermodynamic equilibrium with each other at constant temperature and pressure , the chemical potentials of the phases must be equal. The number of equality relationships determines the number of degrees of freedom . This leads to the famous Gibbs phase rule1: =−+2, where is the number of chemically independent constituents of the system, and is the number of phases. The rule should be valid, provided that the equilibrium between phases is not influenced by external fields and there is no spatial constraint on the phases. The latter condition is known to be violated for coherent solids2. This rule tells us that for a pure substance, it is only possible that three phases can exist together in equilibrium (=3). For a one-component system, there are no degrees of freedom (=0) when there are three phases (, and ), and the three-phase mixture can only exist at a single temperature and pressure, which is known as a triple point. The two equations (, )=(, )=C(, ) are sufficient to uniquely determine the two thermodynamic variables, and . Four-phase coexistence should then be absent, as three chemical potential equations admit no solutions when there are only two independent variables and . Mathematically, however, this does not necessarily rule out the possibility that the set of equations may be solved in a special case. Here we seek such a possibility in a systematic manner by tuning the interaction potential, or the Hamiltonian of the system. Extending the dimensionality of the system will allow us to investigate what are the conditions for the existence of a quadruple point.

    The presence of a point where different phases coexist provides an interesting possibility of switching materials properties, including electric, magnetic, optical and mechanical properties, by a weak thermodynamic perturbation such as stressing or heating/cooling. The technological importance of a triple point has recently been shown for a popular candidate material for ultrafast optical and electrical switching applications, vanadium oxide (VO2) (ref. 3): it has been revealed that the well-known metal-insulator transition in this material actually takes place exactly at the triple point. Large piezoelectricity near a morphotropic phase boundary is another important example of the importance of multi-phase coexistence4,5,6. In these systems, structural transformations in lattice order are coupled with other orders such as dipole, spin, charge and orbital, which can be used for applications such as electromechanical or magnetoelectronic devices. Although the role of multi-phase coexistence in the ease of the transition is not so clear, the minimization of the volume change associated with a phase transition may be realized by combined nucleation of two phases with different signs of the volume change on the transition, which has been reported for a transition near a ferroelectric–anitiferroelectric–paraelectric triple point5. So, the presence of a multiple point may provide a novel kinetic pathway of phase transition, for which the barrier for phase transformation is much lower than an ordinary phase transition between two phases. Thus, the fundamental understanding of multiple-phase coexistence is not only of scientific interest but also of technological importance.

    To study the basics of multi-phase coexistence, we need a model system that shows rich polymorphism. In this context, it is well known that water exhibits a rich variety of crystal polymorphs (at least, 16 types of crystals7). Motivated by this, here we study systems interacting with tetrahedral interactions (for example, covalent bonding and hydrogen bonding). Tetrahedral interactions are the most important category of directional interactions found in nature, both in terms of abundance, and in terms of unique physical properties. They are ubiquitous in terrestrial and biological environments, and fundamental for technological applications. The disordered (liquid) phases of tetrahedral materials show unique thermodynamic properties, the most important being water’s anomalies8,9, like the density maximum, the isothermal compressibility and specific heat anomaly and so on. Ordered phases of tetrahedral materials are of fundamental importance in industrial application, as they include open crystalline structures, like the diamond cubic (dc) crystal, or the quartz crystal, with unique mechanical, optical and electronic properties. For example, in Si and Ge, the diamond cubic (dc) crystal is a semiconductor, whereas the liquid and body-centred cubic (BCC) crystal are metals. Furthermore, dc crystals of mesoscopic particles (like colloids) are also a promising candidate for photonic crystals10. It is thus not surprising that tetrahedral interactions are one of the focus of nanotechnology, with the aim of producing new generation of materials with properties that can be finely controlled by design10,11,12.

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