http://molecularmodelingbasics.blogspot.com

1. Heat Capacity and Stability
How do you define and measure the stability of nanosystems (that you can’t see)
The simula:on shows that you can find the mel:ng temperature by measuring
the heat capacity as a func:on of T
h=p://propka.ki.ku.dk/~jhjensen/calorimetry.html
1

2. The simula:on computes T as a func:on of U
0.4
Here we switch the axes
0.2
0
0 500 1000 1500 2000 2500 3000
‐0.2
U (eV)
‐0.4
‐0.6
‐0.8
‐1
T (K)
2

3. The data is noisy primarily because of the fast hea:ng rate
We smooth it by fiPng it to a polynomial (blue curve)
0.4
0.2
0
0 500 1000 1500 2000 2500 3000
‐0.2
U (eV)
‐0.4
‐0.6
‐0.8
‐1
T (K)
3

4. From the smoothed data (blue curve) we can compute the
0.4 heat capacity (red curve, right y‐axis) 0.0016
0.2 0.0014
0.0012
0
0 500 1000 1500 2000 2500 3000
0.001
⎛ ∂U ⎞
‐0.2
Cv (eV/K)
U (eV)
CV = ⎜
⎝ ∂T ⎟ V
0.0008
‐0.4 ⎠
0.0006
U(T2 ) − U(T1 )
‐0.6 ≈
T2 − T1 0.0004
‐0.8 0.0002
‐1 0
T (K)
4

5. Real life applica:ons
Figure from: h=p://dx.doi.org/10.1016/S1631‐0705(02)01326‐9
This ar:cle is also intes:ng: h=p://dx.doi.org/10.1021/ja802389d
5

6. Why is the heat capacity highest at a phase transi:on?
Heat capacity and energy fluctua:ons
⎛ ∂U ⎞ (E − E ) 2
CV = ⎜ ⎟ =
⎝ ∂T ⎠ V kT 2
(Page 230‐231 in Molecular Driving Forces
U= E
remember that )
The system changes most at a phase transi:on
so the energy fluctua:ons are largest here
6

7. 0.4 0.0016
0.2 0.0014
0.0012
0
0 500 1000 1500 2000 2500 3000
0.001
Cv (eV/K)
‐0.2
U (eV)
0.0008
‐0.4
0.0006
‐0.6
0.0004
‐0.8 0.0002
‐1 0
T (K)
h=p://propka.ki.ku.dk/~jhjensen/cvandt.html 7

8. 0.4 0.0016
0.2 0.0014
0.0012
0
0 500 1000 1500 2000 2500 3000
0.001
Cv (eV/K)
‐0.2
U (eV)
0.0008
‐0.4
0.0006
‐0.6
0.0004
‐0.8 0.0002
‐1 0
T (K)
h=p://propka.ki.ku.dk/~jhjensen/cvandt.html 8

9. 0.4 0.0016
0.2 0.0014
0.0012
0
0 500 1000 1500 2000 2500 3000
0.001
Cv (eV/K)
‐0.2
U (eV)
0.0008
‐0.4
0.0006
‐0.6
0.0004
‐0.8 0.0002
‐1 0
T (K)
h=p://propka.ki.ku.dk/~jhjensen/cvandt.html 9

10. 0.4 0.0016
0.2 0.0014
0.0012
0
0 500 1000 1500 2000 2500 3000
0.001
Cv (eV/K)
‐0.2
U (eV)
0.0008
‐0.4
0.0006
‐0.6
0.0004
‐0.8 0.0002
‐1 0
T (K)
10