the study of statistical thermodynamics in melting of atomic cluster pooja shrestha
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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
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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
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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.
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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
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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]
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E
S
ET
1
Caloric CurveCaloric Curve
In Microcanonical Ensemble
Temperature,
Entropy,
Fig. (2): Microcanonical caloric curve exhibiting S-bend
EkES B ln
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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.
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In canonical EnsembleIn canonical Ensemble
EdEkESZ B
exp
Partition function,
Internal energy,
Fig. (3): Canonical caloric curve
EZ
TU
ln
)(
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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.
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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.
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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
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Continued… Continued…
The heat capacity was computed using
Where,
2*2*
2**
* 1)( EE
TT
ETCv
Tk
T B*
E
E *
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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
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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
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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
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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
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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
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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.
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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.
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