Leyes de la Termodinámica, Fases de la Materia y ... · Por sólo esas razones, es una parte muy importante en la educación de un físico. ... 2 Adrián Huerta Leyes de la Termodinámica,

Leyes de la TermodnamicaFases de la Materia

Propiedades Estructurales

Leyes de la Termodinmica, Fases de laMateria y Propiedades Estructurales.

Adrin [email protected]

Departamento de Fsica, FFIAUniversidad Veracruzana, Mxico.

Adrin Huerta Leyes de la Termodinmica, F. de la Materia y P. Estructurales.



Contenido

1 Leyes de la TermodnamicaConceptosLey Cero (Temperatura y Ecuacin de Estado Trmica)Primera Ley (Energa Interna y Ec. de Estado Calrica)Segunda Ley (Entropa y Procesos Cclicos)

2 Fases de la MateriaFases en EquilibrioFases Fuera de Equilibrio

3 Propiedades EstructuralesMecanismo de cajas para un sistema de discos durosDinmica




ConceptosLey Cero (Temperatura y Ecuacin de Estado Trmica)Primera Ley (Energa Interna y Ec. de Estado Calrica)Segunda Ley (Entropa y Procesos Cclicos)

A. Einstein

Una teora es mas impresionante cuantomayor sea la simplicidad de sus postulados,el nmero de cosas que relacione y la exten-sin de su campo de aplicacin. De aqu laimpresin tan profunda que me ha causado latermodinmica. Es la nica teora fsica decontenido universal de la cual estoy conven-cido que, por lo que respecta al campo de apli-cacin de sus conceptos bsicos, nunca serdestituida. Por slo esas razones, es una partemuy importante en la educacin de un fsico.





Conceptos

Sistema Termodinmico: Est constituido por algunaporcin del universo fsico que nosotros consideramospara su estudio.Fronteras: El mecanismo que lo separa del resto deluniverso.Alrededores: La interaccin entre el sistema y susalrededores estar caracterizada por los intercambiosmutuos de energa, en sus diversas formas. El grado deinteraccin con sus alrededores depender obviamente dela naturaleza de sus fronteras.





Intercambio de energa en sus diversas formasPresin (P), Volumen (V ),

Campo Magntico (~H), Magnetizacin (~M)

Alrededores

Sistema Termodinmico

Fronteras

AislantesAdiabticasDiatrmicas

Equilibrio Trmico

Aislantes: No permiteninteraccin con sus alrede-dores.Adiabticas: Impidenel intercambio de calorentre el sistema y susalrededores.Diatrmicas: Son aquel-las que no son adiabticas.





Equilibrio Trmico

f1(PA,PC ,VA,VC) = 0, f2(PB,PC ,VB,VC) = 0

A C B

A C B

Existe: f3(PA,PB,VA,VB) = 0Equilibrio Trmico

"Si tres o ms cuerposse encuentran en con-tacto trmico, uno a uno,por medio de paredes di-atrmicas y se encuen-tran en equilibrio todos jun-tos, entonces dos cua-lesquiera tomados sepa-radamente se encuentranen equilibrio uno con elotro."





Equilibrio Trmico

f1(PA,PC ,VA,VC) = 0f2(PB,PC ,VB,VC) = 0

PC = g1(PA,VA,VC)PC = g2(PB,VB,VC)

g1(PA,VA,VC) = g2(PB,VB,VC)

f3(PA,PB,VA,VB) = 0

Estas ltima son expresionesque describen la misma situacinfsica. Proponemos una funcintal que se elimine VC

g1 = 1(PA,VA)(VC) + (VC)g2 = 2(PB,VB)(VC) + (VC)

1(PA,VA) = 2(PB,VB)

Analogamente:2(PB,VB) = 3(PC ,VC)





Temperatura Emprica y Ecuacin de Estado Trmica

Existe una funcin que es comn para todos los sistemas en equilibrio.1(PA,VA) = 2(PC ,VC) = 3(PB,VB)

Temperatura emprica .(P,V ) =

Ecuacin de Estado Trmica!





Ecuacin de Estado Trmica

P = P(V ,T )dP =

(PV

)T dV +

(PT

)V dT

dPP =

1PVT

dV + PT dT

Compresibilidad IsotrmicaT = 1V

(VP

)T

1P

Coef. de Expansin Trmico = 1V

(VT

)P

1T

dPP =

1V dV +

1T dT

lnP = lnV + lnT + lnClnP = lnCT/VPV = CT ; Gas Ideal

Observaciones experimentales:- Ley Charles, 1780(o Gay-Lussac, 1802)P = cte (Isobricas): V T

- Ley de Boyle-Mariotte, 1660T = cte (Isotermas): P 1V





Ecuacin de Estado TrmicaExperimentos de Thomas Andrews, 1861-1869

The Bakerian Lecture for 1869: "The continuity of

the gaseous and liquid states of matter"

La Tesis de Johannes Diderik van der Waals 1873, "The

continuity of the gaseous and liquid states of matter"

posiblemente en respuesta a los experimentos de Andrews

(Nobel 1910).

P = NKT(VNb) a(

NV )

2





Diferenciales inexactas, variables de estado

A C

P

V

A B

CD

W =

PdVABC PdV 6=

ADC PdV

A diferencia de las vari-ables termodinmicascomo P, V , T , U,El trabajo W y el calor Qno son variables de estadopor lo que sus diferencialesson inexactas.

dU = Q WdU = Q pdV





Procesos Isotrmicos y adiabticos

Presin (P), Volumen (V ), Temperatura (T ),Energa (Ui Uf )

Alrededores


Fronteras

AislantesAdiabticasDiatrmicas

P

1

1.5

2

2.5

3

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

isotermaadiabatica

V





Procesos Isotrmicos y adiabticosPresin (P), Volumen (V ), Temperatura (T ),

Energa (Ui Uf )

Alrededores


WQ

dU = Q W

P

1

1.5

2

2.5

3

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

isotermaadiabatica

V





Procesos Isotrmicos y adiabticosPresin (P), Volumen (V ), Temperatura (T ),

Energa (Ui Uf )

Alrededores


W

dU = Wad

P

1

1.5

2

2.5

3

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

isotermaadiabatica

V





Experimento de Kelvin (Ecuacin Calrica)

dU = Q W = 0

Q = 0W = 0

Ui (Vi ,Ti ) = Uf (Vf ,Tf )Ti = Tf ;Vi 6= VfU(T )CV = QdT =

dUdT

U(T ) = U(To) + T

ToCV dT

U(T ) = U(To)+CV (TTo)U(T ) = CV T

Ecuacin de estadoCalrica.





Entropa

Ley Cero: TPV = NkT (trmica)Primera Ley: UU(T ) = Uo + CV T , (calrica)

dU = Q WQ = dU + WQ = Cv dT + pdV

QT = Cv

dTT + p

dVT

QT = Cv

dTT + Nk

dVV

= CV d(ln(T )) + Nkd(ln(V ))= d(ln(T CV V Nk )) = dSQT = dSQ = TdSPor lo que:dU = TdS pdVPara procesos reversibles!!





Ciclo de CarnotPresin (P), Volumen (V ), Temperatura (T ),

Energa (Ui Uf )

Alrededores


W

dU = 0

P

V

isotermaisoterma

adiabaticaadiabatica





Ciclo de Carnot

= WQ1= Q1Q2Q1

= 1 Q2Q1 = 1 T2ST1S

= 1 T2T1





Teorema y corolario de Carnot

"Ninguna mquina trmica, operando enciclos entre dos fuentes de temperaturas fijas,

tiene una eficiencia mayor que la de unamquina reversible operando entre las

mismas dos fuentes""Todas las mquinas reversibles (e.g.

mquinas de Carnot con cualquier sustancia)operando entre dos recipientes trmicos a

temperaturas dadas, tienen la mismaeficiencia" = 1

T2T1





Segunda ley: Procesos irreversibles

Experimento de Joule

Pelota rebotando





Segunda ley: Procesos irreversibles

Experimento de KelvinQ 6= TdS

Produccin de entropaQ1 = 0

S1 S2isotrmico pero irreversible




Fases en EquilibrioFases Fuera de Equilibrio

Transiciones de Fase en Equilibrio





Analoga con un Sist. Mecnico

X

E





The Pitch Drop Experiment

The experiment taking placeat University of Queenslandin Brisbane, Australia, is a

long-term experiment whichmeasures the flow of a piece

of pitch. The experimentbegan in 1927 when

Professor Thomas Parnellset it up. Demonstrating that

pitch has a viscosityapproximately 1011 times

that of water.




Mecanismo de cajas para un sistema de discos durosDinmica

History of Freezing Transition (FT)FT was first seen in the computer simulation studies of 870 hard disks by Alderand Wainwright approx. 50 years ago:

Equation of state and traces

B.J. Alder and T.E. Wainwright, Phys. Rev. 127, 359 (1962)





KTHNYAlthought the system of 2D HD is one of the simplest models of a fluid itexhibits a rich behaviour, e.g. still is not clear if this transformation is...

K. Binder, et al, J. Phys.: CM 14 (2002) 23232333

"KTHNY theory predicts that the crystal undergoes a con-

tinuous melting transition into a possibly metastable hex-

atic phase. Hexatic phase: have short range translational

order and long-range orientational order. Of course, the

KTHNY scenario does not rule out the possibility of a first-

order transition occurring by another mechanism."

Trusket et al, PRE, 58, 3083 (1998)





KTHNYDifferent Scenario

http:www.lps.ens.fr/~krauth/index.php/

Bernard_Krauth_2011

A different possibility: Using large-scale simulations

with event-chain Monte Carlo algorithm Bernad and

Krauth have report "a first-order scenario between a

liquid and a solid, and the KTHNY scenario with an

intermediate hexatic phase separated by continuous

transitions from liquid and the solid."


http:www.lps.ens.fr/~krauth/index.php/Bernard_Krauth_2011http:www.lps.ens.fr/~krauth/index.php/Bernard_Krauth_2011




Orientational order parameterf = 0.69, m = 0.716

6 = 1Nnn jk ei6jk





Structural Precursor of Order

"...As the dense fluid approaches the freez-

ing transition, the ordered arrangements form

large embryonic domanis, commensurate with

those seen in the crystal at the melting point.

Contrary to the notion that the split second

peak is a signature of the amorphous solid."

Trusket et al, PRE, 58, 3083 (1998)





Caging Mechanismf = 0.69, m = 0.716

a) b)

From the Voronoi hexagonal cells.Select the three alternating nearest neighbors withdistance < 2.a) Near freezing more single triangular cagedb) Near melting more double triangular caged

A. Huerta, A. Trokhymchuck, D. Henderson PRE 74, 061106 (2006)





Caging Mechanismf = 0.69, m = 0.716, 6 =

1Nnn jk ei6jk

Fraction of caged disk follows two criteria:i) Voronoi hexagonal neighbors < 2

ii) Voronoi alternating hexagonal neighbors < 2 Eyring J. Chem. Phys. 4, 283 (1936)






af2

=

3

4(d/)2

2af2

=

3

4(d/)2

2+ 3cos1(

d/2

) 34(d/)

4 (d/)2





Equation of State





Caging Mechanism

Eyring J. Chem. Phys. 4, 283 (1936)






The Pair Distribution Function

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1.6 1.8 2 2.2 2.4 2.6

g(r)

r

Second oscillation of the pair distribution function. For = 0.600, 0.620, 0.640 in red and = 0.660, 0.680, 0.690, 0.723 in blue.






g(r,) for HD f = 0.69, m = 0.716

= 0.500 = 0.686





The mean square displacement.

0.01

0.1

1

10

100

0.1 1 10

msd

t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.1 1 10

msd

t

Figure: The mean square displacement. for = 0.600,0.620,0.640in red and = 0.660,0.680,0.690,0.723. The behaviour at = 0.723near melting are included for comparison proposes also a line withslope 1 is included to compare with diffusive behaviour.





The intermediate self-scattering function.

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000 10000 100000

Fs(k

,t)

t

Figure: The intermediate self-scattering function. All for = 0.600,0.620,0.640 in red and = 0.660,0.680,0.690,0.723.





The intermediate self-scattering function.

100

1000

10000

100000

1e+06

1e+07

1e+08

0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

tao

eta

Figure: Vogel fulcher fit





Fraction Semi-Caged Particles

0

0.2

0.4

0.6

0.8

1

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0

1fra

ctio

n

eta

singledouble

totalX0+X1

X2+X3+X4+x5+x6Semicaged/Xi-total

'./aniso.dat'

Figure: Fraction of single and double caged particles, the sum(particles with at least one cage. Fraction of particles with one andzero constraints, the fraction of particles with 2 and more constraints.And finally the fraction of semi-caged particles.





Thesis Josefina Vicente Santiago UVFrustration of the caging mechanism using Polidispersity

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10

./d0000/mcgr0723_d000.dat./d0200/mcgr0723_d0200.dat





Thesis Virginia Carrasco Fadanelli UVFrustration of the caging mechanism using Bidispersity

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8

g(r)

r

0

1

2

1.5 2 2.5 0

1

2

1.5 2 2.5

Visit the Virginia Poster





Thanks!





Thesis Virginia Carrasco Fadanelli UV

A frustration study of the freezing mechanism

Virginia Carrasco Fadanelli, [email protected], Facultad de Fsica e I.A. Depto.Fsica, Universidad Veracruzana; Adrin Arturo Huerta Hernndez,

[email protected], Facultad de Fsica e I.A. Depto. Fsica, Universidad Veracruzana

ABSTRACTIn the present work we report a systematic study of the frustration mechanismoriginally proposed in Huerta et al, Phys. Rev. E, 061106 (2006). The later wascarried out by tuning the diameter ratio of a bidisperse system. We make aninterpretation of the thermodynamic, structural and dynamic properties in termsof its energy landscape. The later helps us to give an interpretation of the glassybehavior due to the frustration of the originally proposed freezing mechanism.





Frustracin del mecanismo de cajas

I. MODEL

In this work we use an equimolar mixture of hard disks of diameters .where helps us to tunned the ratio of the diameters mixture. The =0corresponds to the monodisperse case, the increase of the values increasesthe hard disks asymmetry of the mixture.

II. METHODOLOGY

We employed Monte Carlo Metropolis simulations to study the thermodynamicsand structural behavior of different diameter ratios of hard disks mixture. To studythe dynamical behavior we use even driven molecular dynamics simulations.To plot the Fself we use the wave vector k=2.14 for all the reported packing fractions.We employed 400 hard disks and a rectangular simulation cell which sides have asquare root of 3 ratio and we use periodic boundary conditions.






Figure 1: The phase diagram for: (green line) monodisperse hard diskssystem with radius 1.00; (light blue) bidisperse hard disks radius 0.95-1.05and (blue) bidisperse hard disks radius 0.8-1.20. The red asterisks indicatesthe first four msd ploted in figure 2.

III. THERMODYNAMICS

0.95 1.05

0.80 - 1.20

1.00






Figure 2: The mean square displacement (msd) as a function of time forthe densities indicated in figure 1 in red, for the system with radius 0.8-1.20,the last two lines in red corresponds to eta=0.782, 0.800, non shown in thephase diagram. We compare with the msd of the monodispese system at freezing andmelting points (lines in green).The blue lines with slope two and one respectively are shown to guide the eye atballistic and diffusive regimes, indicating a sub-diffusive regime between them. Inall cases the density increase from top left to bottom right.

IV. DYNAMICS

melting = 0.723

= 0.800






Figure 3: The self intermediate scattering function (F_self) for the same situation described in figure 2.

melting = 0.723






Figure 4: Radial distribution function of the bidisperse system at = 0.619 and 0.782, thin and thick red lines respectively; and monodisperse system = 0.686 and 0.723 thin and thick green lines, respectively.

V. STRUCTURE

= 0.782

= 0.619

= 0.723

= 0.686






a) b)

Figure 5: Snapshot of the final configurations of the bidisperse system at a) =0.619 and b) =0.782.






a) b)

Figure 6: Snapshot of the final configurations of the monodisperse system ata) =0.686 and b) =0.723.






References:-Freezing of two-dimensional hard disks a Huerta al et. Phys.Rev. E 2006.-One liquid two glasses, F. Sciortino Nature Materials, November 2002.

VI. CONCLUSIONS

In this work we have evaluated the thermodynamics, dynamics and structuralproperties of a hard disks mixture for several hard disks ratios. For a large enoughratios we observe the avoidance of the crystalline phase driven the systemto a high density liquidlike phase similar to the liquid pockets


Leyes de la TermodnamicaConceptosLey Cero (Temperatura y Ecuacin de Estado Trmica)Primera Ley (Energa Interna y Ec. de Estado Calrica)Segunda Ley (Entropa y Procesos Cclicos)

Documents

Leyes de la Termodinámica, Fases de la Materia y ... · Por sólo esas razones, es una parte muy importante en la educación de un físico. ... 2 Adrián Huerta Leyes de la Termodinámica,