Hi,
I have a question regarding thermal conductivity. My simulation is on strontium titanate in solid phase. I have calculated the thermal conductivity of strontium titanate (by Muller-Plathe method) at different simulation length ranging from 23 nm (3x3x60 unit cell) to 47 nm (3x3x120 unit cell). I found out the calculated thermal conductivity increases with length. To my knowledge, phonon scattering at boundaries of heat source (hottest slab) and heat sink (coldest slab) might give rise to finite size effect, if the simulation length is not significantly larger than the phonon mean free path. The phonon mean free path of strontium titanate is about 1.5 nm. Since the simulation length is significantly larger than the phonon mean free path (about an order larger), therefore I would expect to see no size effect. So I think there might be some other factors that give rise to this size effect. The only thing that comes in my mind is may be the simulation length have not reached a convergence. May I have your opinions?
Thank you.
Regards,
Christopher
All of your arguments are reasonable. I would try
running bigger systems and longer runs and see
if you get better convergence.
Steve
Hi Chris,
Where is the phonon MFP 1.5 nm from? Experiment on k measurement? The measured k may be a low value due to the defects in the sample. Another thing you may need to concern is the M-P method to calculate k of strontium titanate. I don’t know the lastest procedure of M-P method in lammps, but in old version, the velocity exchange can only apply on the same type of atom, which may not fit your case.
As Steve suggested, you’d better try running larger system and longer runs.
Xiaopeng
Dear Steve, Xiaopeng and Axel,
The phonon mean free paths (MFP) of STO were estimated from the experimental measurement of thermal conductivity (kappa) and thermal diffusivity. 3 experiments affirmed that it range from 1.5 - 2.0 nm at room temperature. Worth mentioning that the Debye temperature of STO is 693K. Yup, kappa may be a low value due to to defects in the sample, but I think in real the phonon MFP will not exceed this value by so much, furthermore I think the experimentalists will try to reduce the impurities as much as possible. The manual (latest version) says swaps conserved both momentum and kinetic energy, even the masses of the swapped atoms are not equal. Thus I think kinetic exchange for different types of atom should be fine. I am currently making the system as large as possible and see if I get better convergence. However, 3x3x220 unit cell is the largest size I can get without running out of memory.
In fact, there is one method from the literature to make correction for finite size effect, i.e. plot 1/kappa vs 1/Lz and extrapolate to infinite Lz (Lz is the simulation cell length where heat flows), i.e. 1/Lz=0, the ‘bulk’ kappa will be the interception of the extrapolated line with the y axis. I tried this method, but the calculated kappa was over predicted by about 30% at room temperature. Many papers also reported their over predictions by this method too. This method was derived from the kinetic theory of thermal conductivity and it decomposed the calculated phonon relaxation time into a ‘bulk’ one and a ‘box’ (contribution due to boundaries) one. I doubt how it relates the ‘box’ relaxation time to phonon velocity and Lz as mentioned in the paper (Jiang et al J. Phys. Chem. B, 112, 10207, 2008). If I follow the method as mentioned but replace 1/kappa with kappa, I get exact value.
Thank you.
Regards,
Christopher
Dear Steve, Xiaopeng, Axel and all,
The phonon mean free paths (MFP) of STO were estimated from the experimental measurement of thermal conductivity (kappa) and thermal diffusivity. 3 experiments affirmed that it range from 1.5 - 2.0 nm at room temperature. Worth mentioning that the Debye temperature of STO is 693K. Yup, kappa may be a low value due to to defects in the sample, but I think in real the phonon MFP will not exceed this value by so much, furthermore I think the experimentalists will try to reduce the impurities as much as possible. The manual (latest version) says swaps conserved both momentum and kinetic energy, even the masses of the swapped atoms are not equal. Thus I think kinetic exchange for different types of atom should be fine. I am currently making the system as large as possible and see if I get better convergence. However, 3x3x220 unit cell is the largest size I can get without running out of memory.
In fact, there is one method from the literature to make correction for finite size effect, i.e. plot 1/kappa vs 1/Lz and extrapolate to infinite Lz (Lz is the simulation cell length where heat flows), i.e. 1/Lz=0, the ‘bulk’ kappa will be the interception of the extrapolated line with the y axis. I tried this method, but the calculated kappa was over predicted by about 30% at room temperature. Many papers also reported their over predictions by this method too. This method was derived from the kinetic theory of thermal conductivity and it decomposed the calculated phonon relaxation time into a ‘bulk’ one and a ‘box’ (contribution due to boundaries) one. I doubt how it relates the ‘box’ relaxation time to phonon velocity and Lz as mentioned in the paper (Jiang et al J. Phys. Chem. B, 112, 10207, 2008). If I follow the method as mentioned but replace 1/kappa with kappa, I get exact value. I appreciate your opinions.
Thank you.
Regards,
Christopher
Hi Chris,
You made a good mention that the Debye temperature of STO is 693 K, which means you need to do quantum correction on the temperature calculated by energy equipartition theorem if you do simulation at room temperature. The corrected temperature gradient along the length will become a little larger and results in a smaller kappa. However, this won’t help to explain your problem about the finite size effect.
Actually, I just notice that the other two dimensions (3X3) of your system are both smaller than the MFP of STO. This will definitely cause finite size effect even you’re using periodic boundary conditions in these two directions. You can try to run some cases with larger scale than MFP in these two dimensions. I hope it will help.
Best,
Xiaopeng
Dear Xiaopeng,
Actually the calculated kappa of STO with correction made for finite size effect shows deviations (over-predicted) with the experiment below debye temperature, hence it gives me a motivation to diminish the deviations. Since the calculated kappa increases with lengths, so I employed an extrapolation (created myself), i.e. plot kappa vs 1/Lz and extrapolate to 1/Lz=0, and I found out this extrapolating method shows good agreement with the experiment for all temperatures (above 298K) even below deybe temperature. However, I have troubles convincing others about the reliability of this extrapolating method. I’m glad you mentioned that quantum corrected temperature will become larger and results in smaller kappa, but what makes you so sure that the corrected temperature will become larger? I’m interested in knowing how to make quantum corrections to the temperature.
About the other 2 dimensions, I had made a convergence test ranging from 2x2 to 6x6, and found out it converges at 3x3. Thus 3x3 is chosen.
Xiaopeng, I really appreciate your help. Thanks.
Regards,
Christopher