# Intermediate phases

34 Pages · 2013 · 2.8 MB · English

Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY. LECTURE 10 .. GeSe. 2. Standard deviations for tetrahedral network glasses.

[email protected]Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY LECTURE 10 : RIGIDTY TRANSITIONS AND INTERMEDIATE PHASES Rigidity transitions and compositional trends Intermediate phases Cluster expansions MD based rigidity theory j Stretching constraintsa ij r/2 Bending constraintsb ijk 2r3 1 Remember (lecture 9) the enumeration of mechanical constraints Consider a rcoordinated atom  If r=2, there is only one angle Each time, one adds a bond, one needs to define 2 new a ngles  We consider a system with N species of concentration n r The number of constraints per atom is :  = ∑  ( 2 + (2 − 3)  ∑    [email protected] Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY A) CONSTRAINTS AND RIGIDITY TRANSITIONS [email protected] modeling of glass – LECTURE 9 TOPOLOGY Stretching Bending 2 Rigidity transition: Amorphous silicon and harmonic potential  Bond depletion ( decreases)  Calculation of the eigenmodes (vibrational frequencies) of the system  Example of simple eigenmode calculation: the linear chain A) CONSTRAINTS AND RIGIDITY TRANSITIONS [email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY Force acting on spring j:  Newton’s law gives: with frequencies  Normal mode solution : which actually reduces the problem to an eigenvalue ( W WW W ) problem :  More general : normal modes=eigenvalues Wof the dynamical matrix [email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY 2 Rigidity transition:  Thorpe (1983) found that bond depleted aSi with mean co ordination number < 2385 contain zero frequency normal (floppy) modes W  Their number f(rank of the W=0 block of the dynamical matrix) scales as > < = = r N f c 2 5 6 3  Flexible to rigid transition  Control parameter  Order parameter f  Powerlaw C ii=(24) p (p=1,5) in the stressed rigid phase Elastic phase transition  Isostatic glass n c=3 is at the R transition He and Thorpe, PRL 1985 A) CONSTRAINTS AND RIGIDITY TRANSITIONS 19792000: Anomalies at =24 Chalcogenide network glasses r(Se)=2 , r(Ge)=4 Ge xSe 1x GeSbSe, GeAsSe, etc… Varshneya et al, JNCS 1991 Boehmer and Angell, PRB 1994 [email protected] Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY A) CONSTRAINTS AND RIGIDITY TRANSITIONS [email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY A) CONSTRAINTS AND RIGIDITY TRANSITIONS Binary oxides : (1x)SiO 2xNa 2O Depolymerization with addition of Na r(Q 4)=4 , r(Q 3)=3,  Silica and germania: rigidity under pressure under pressure Micoulaut, Am Mineral 2008 Trachenko et al, PRL 2005 [email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY B INTERMEDIATE PHASE Experimentally Picture of rigidity transitions has been changed in d epth during the last decade Calorimetric measurements (modulated DSC) Use of modulated differential scanning calorimetry (MDSC) across the glass transition  Allows for the definition of 2 heat flows Reversible ( DC p) Nonreversible Nonreversing enthalpy ( DH nr) Georgiev et al PRB (2003 ) nonrev rev T H H H & & & + = Modulated DSC and reversibility windows [email protected] Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY Reversibility window B INTERMEDIATE PHASE Since the RW is found between the flexible and the stressed rigid phase, it is also often called the « Intermediate phase » Nonmean field phase otherwise one would have the single =24 transitions  Central idea: stress avoidance under increasing crosslinking density Network selforganization Nearly vanishing of DH nr at Tg  Stress free character (P c minimum, Raman)  Vibrational thresholds (Raman, IR)  Generic (chalcogenides, oxides) Wang et al, PRB 2005 Remarkable properties for selected glass compositions [email protected] Atomic modeling of glass – LECTURE 10 RIGIDITY THEORY [email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY “Marginality of nonreversible component of complex heat flow in MDSC experiments does not necessary reflect selforganized intermediate phase ” PSS 2011 « However, that (mDSC) measurement might also be subjected to a large experimental uncertainty ” PRB 2009 “ The observation of the reversibility window might be an experimental artifact ” PRB 2009 A subject which has not been without controversy… Micoulaut, Bauchy, PSS 2013 B INTERMEDIATE PHASE [email protected] modeling of glass – LECTURE 10 RIGIDITY THEORY B INTERMEDIATE PHASE Senapati, JNCS 1991 Bhosle et al, IJAGS 2012 Why has the RW been missed before ? And why are challengers missing it…and getting so annoye d ? Three

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8 Pages · 2003 · 589 KB · English

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on Pre-university Intermediate Level ESL Learners' Reading Speeds and students with the knowledge that computer-based reading rate

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INTRODUCTION. For most trades and especially commodities, certain categorizations are required to see the quality of the goods. For oil trade, this is done . the benchmark WTI crude oil has traded at discount to Brent benchmark. oil indices. This spread is detected on oil futures positions when.

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7 Pages · 2012 · 79 KB · English

English Learners. 5.90%. Foster Youth. 0.40%. Total Enrollment. 729. Student Achievement. Physical Fitness. In the spring of each year, Norco Intermediate School is required by the state to administer a physical fitness test to all students in grade seven. The physical fitness test measures each st