The Study Of Statistical The Study Of Statistical Thermodynamics In Thermodynamics In Melting of Atomic Melting of Atomic ClusterCluster

The Study Of Statistical The Study Of Statistical Thermodynamics In Thermodynamics In Melting of Atomic Melting of Atomic ClusterCluster

Pooja ShresthaPooja ShresthaPooja ShresthaPooja Shrestha

Over ViewOver View

What is Phase Transition?

Phase Transition in Finite System Caloric curve

Microcanonical Ensemble

Canonical Ensemble

Phase Transition in Bulk System

Model and Calculation

Result and Conclusion

What is Phase Transition?What is Phase Transition?

In thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another.

Analytic discontinuities or singularities in the thermodynamic functions corresponds to occurrence of various kinds of phase transition.

Phase Transition In Finite System (Cluster)Phase Transition In Finite System (Cluster)

Within finite range of temperature small cluster exhibit a coexistence of solid and liquid state.

Before cluster evaporates, three ranges of temperature exist:

Low-temperature solid region

The coexistence range

High-temperature liquid region

Continued…Continued…

Dynamical coexistence is observed which is indicated by potential energy fluctuation between relatively high and low values.

The lower potential energy corresponds solid state.

Higher potential energy corresponds liquid state.

Coexistence of liquid-like and solid-like states in cluster implies two phases coexisting at different times rather than coexisting in contact.

55 atoms T*=0.30

Fig. (1): Potential energy; the horizontallines corresponds to maxima and minima [5]

E

S

ET

1

Caloric CurveCaloric Curve

In Microcanonical Ensemble

Temperature,

Entropy,

Fig. (2): Microcanonical caloric curve exhibiting S-bend

EkES B ln

Continued…Continued…

In Microcanonical Ensemble

Potential energy versus temperature curve, the derivative of which corresponds specific heat, Cv(T).

Van der Waal’s loop or “S-bend”

Temperature decreases as energy increases at certain region which correspond negative specific heat capacity.

In canonical EnsembleIn canonical Ensemble

EdEkESZ B

exp

Partition function,

Internal energy,

Fig. (3): Canonical caloric curve

EZ

TU

ln

)(

Continued…Continued…

In Canonical Ensemble

Plot of mean energy against temperature of heat bath.

Curve is monotonically increasing and there is sharp increase in slope at transition region as a result of peaking of Cv.

Heat capacity is always positive.

Phase Transition in Bulk SystemPhase Transition in Bulk System

Phase transition is ordinarily defined for infinite homogeneous systems, making use singular behavior of e.g. specific heat at phase transition.

Melting is first order transition which is controlled by nonanalyticity in the free energy.

Both phases coexist at the same time (coexistence).

Central part of the S-bend becomes straight line joining two branches giving positive value of specific heat.

ModelModel

We simulated and computed 55-atoms cluster in canonical ensemble, using the Lennard-Jones potential of the form

and

612

4jiji rr

V

ji

jirVE

Continued… Continued…

The heat capacity was computed using

Where,

2*2*

2**

* 1)( EE

TT

ETCv

Tk

T B*

E

E *

Continued…Continued…

The Flow chart

Configuration E = V (ri,j)Choose a particle at random and

Move small amount

New configuration

Metropolis condition: Accept or Reject

Iterate

Output Data

ResultsResults

Exhibit three regions:solid region-E increases steadily.

transition region-single sharp increase in slope.

liquid region-the slope is again low.

Curve agrees with Refs.[6]

0.26 0.27 0.28 0.29 0.3 0.31 0.32T

-232

-230

-228

-226

E

Fig. (4): Energy Vs Temperature curve as a canonical result.

N = 55

Continued…Continued…

Cv peak around transition region at T = 0.3.

intermediate range of temperatures exists where clusters show both solid and liquid behavior as a consequence of which Cv shows a smooth peak.

Fig. (5): Cv Vs T curve as a canonical result at transition region.

0.28 0.29 0.3 0.31 0.32T

1.7

1.8

1.9

2

2.1

2.2

Cv

N = 55

Continued…Continued…

Dynamical coexistence occurs at T = 0.3, which is also known as transition period.

Lowest energy = -232

Highest energy = -2235000 10000 15000 20000 25000 30000

MC Step

-240

-235

-230

-225

-220

-215

E

Fig. (6): Energy of cluster as a function ofMC Steps in coexistence range

N = 55, T = 0.3

Continued…Continued…

Energy gap between a single lower-energy structure and higher-energy structure is small at T = 0.25

5000 10000 15000 20000 25000 30000

-240

-235

-230

-225

-220

-215

Energy

Fig. (7): Comparison of energy of cluster as a function of MC Steps at different temperature.

N = 55

T =0.3

T =0.25

ConclusionsConclusions

Phase transition in finite system has complicated thermodynamics.

Specific heat capacity has negative value in microcanonical ensemble.

The canonical caloric curve is monotonically increasing.

The existence of two types of structures, low-energy solid and high-energy liquid structures leads to the dynamical coexistence which is the effect of bulk first order transition.

Coexistence occurs at T = 0.3 at which solid expand more rapidly to form liquid.

ReferencesReferences

1. R. K. Pathria, Statistical Mechanics, Butterworth Heinemann, 2nd Edition, (1996).

2. Thomas L. Beck and R. Stephen Berry, J. Chem. Phys. 88 (6), 3910, 15 March (1988).

3. P. K. Jonathon Doye and David J. Wales, J. Chem. Phys. 102 (24), 9674, 22 June (1995)

4. David J. Wales, Phys Rev Letters, 73 (21), 2875, 21 November, (1994)5. R. M. Lynder-Bell and D. J. Wales, J. Chem. Phys. 88(6), 1460, 15 July

(1994)6. Pierre Labastie and Robert L. Whetten, Phys. Rev. Letts.,65(13), 1567, 24

Sept 1990.