[Advanced] Nano particle energy¶
本節は発展編の位置付けとします。次章へ急ぎたい方はスキップし、後から読んでいただいても構いません。
本節では、Nano particle (クラスター構造) の過剰エネルギー(excess energy)計算と、担体上の吸着構造作成を行ってみます。
[1]:
import pfp_api_client
from pfp_api_client.pfp.calculators.ase_calculator import ASECalculator
from pfp_api_client.pfp.estimator import Estimator, EstimatorCalcMode
from pfcc_extras.visualize.view import view_ngl
print(f"pfp_api_client: {pfp_api_client.__version__}")
estimator = Estimator(calc_mode=EstimatorCalcMode.CRYSTAL_U0, model_version="v3.0.0")
calculator = ASECalculator(estimator)
pfp_api_client: 1.18.0
[2]:
from ase.cluster import Decahedron, Icosahedron, Octahedron, wulff_construction
from ase import Atoms
from ase.build import bulk
from ase.constraints import ExpCellFilter, StrainFilter
from ase.optimize import LBFGS
from ase.io import Trajectory
import numpy as np
import pandas as pd
過剰エネルギー - Excess energy¶
純金属に1種類以上他の元素を混ぜた材料を合金 (alloy) と呼びますが、 金属は2種類以上の元素を組み合わせることで、その性質を大きく変化させるようなものがあります。
例えば、アルミニウムと銅やマグネシウムを混ぜた、ジュラルミンはその軽くて硬い性質から航空宇宙機器などに使われています。
2種類の元素が混ざったものを2元系合金、3種類以上の元素が混ざったものは多元系合金と呼ばれます。
多元系合金は、新たな性質を持つような材料の発見も期待されていますが、 元素の種類が増えるほどその組成比の組み合わせが大きくなり、まだまだ解析が進んでいない領域です。 このような系もMatlantisではその汎用性を活かし、扱うことができます。
こういった合金の原子構造を考える際には、現実ではどのように混ざるのか、そもそもきれいに混ざることができる組み合わせなのかを知る必要があります。
このような合金の安定度合いを測る指標として、過剰エネルギー(excess energy)があります。 元素Aと元素Bを混ぜた2元系合金 (alloy)の過剰エネルギー(excess energy)は以下のように計算され、その合金のエネルギーが、元素A, Bがそれぞれ混ざっていなかった場合と比べてどのくらい安定かを示します。
\[E_{\rm{excess}} = \frac{1}{N_{\rm{alloy}}} \left( E_{\rm{alloy}} - \frac{N_{\rm{alloyA}}}{N_{\rm{alloy}}} E_{\rm{A}} - \frac{N_{\rm{alloyB}}}{N_{\rm{alloy}}} E_{\rm{B}} \right)\]
\(E_{\rm{alloy}}\)が合金のエネルギー、\(E_\rm{A}, E_\rm{B}\) は単原子AまたはBで構成された時のエネルギーで、\(N_{\rm{alloy}}, N_{\rm{alloyA}}, N_{\rm{alloyB}}\)はそれぞれ合金のすべての原子数、元素Aの原子数、元素Bの原子数です。
多元系の場合も同様に定義できます。
ここでは、PtとPdを混ぜた構造に対して、excess energyを評価することでその安定性を評価してみましょう。
参考文献:
Electronic Structure and Phase Stability of PdPt Nanoparticles
Electronic structure and phase stability of Pt3M (M = Co, Ni, and Cu) bimetallic nanoparticles
Calculations of Real-System Nanoparticles Using Universal Neural Network Potential PFP
Pt711 Nano particleに対して、Pdを混ぜた構造に対して、 2元系のNano particleを考えた場合にもその混ざり方は様々な可能性が考えられます。
今回は以下のような構造を作成し、そのexcess energyを評価してみます。
PtとPdが均等に混ざる形で存在する構造
Ptが内殻に、Pdが外殻に存在する構造
Pdが内殻に、Ptが外殻に存在する構造
Core shell構造を作成する関数定義です。読み飛ばしていただいて構いません。
[3]:
from typing import List, Tuple
from ase import Atoms, neighborlist
from ase.cluster import Cluster
import numpy as np
from ase.data import atomic_numbers
def cluster2atoms(cluster: Cluster) -> Atoms:
"""Convert ASE Cluster to ASE Atoms
Args:
cluster (Cluster): input cluster instance
Returns:
atoms (Atoms): converted output, atoms instance
"""
return Atoms(cluster.symbols, cluster.positions, cell=cluster.cell)
def calc_coordination_numbers(atoms: Atoms) -> Tuple[List[int], List[np.ndarray]]:
"""Calculates coordination number
Args:
atoms: input atoms
Returns:
cns (list): coordination number for each atom
bonds (list): bond destination indices for each atom
"""
nl = neighborlist.NeighborList(
neighborlist.natural_cutoffs(atoms), self_interaction=False, bothways=True
)
nl.update(atoms)
bonds = []
cns = []
for i, _ in enumerate(atoms):
indices, offsets = nl.get_neighbors(i)
bonds.append(indices)
cns.append(len(indices))
return cns, bonds
def substitute(scaffold: Atoms, sites: np.ndarray, elements: List[str]) -> Atoms:
"""Substibute `elements` to `sites` indices of `scaffold` atoms
Args:
scaffold (Atoms): Original input atoms
sites (np.ndarray): site indices to substitute `elements`
elements (list): elements to be substituted
Returns:
substituted (Atoms): substituted ase atoms
"""
substituted = scaffold.copy()
for site, element in zip(sites, elements):
substituted.numbers[site] = atomic_numbers[element]
return substituted
def make_core_shell(scaffold: Atoms, element: str) -> Atoms:
"""Make core shell structure
Input `scaffold` element is kept inside core shell,
and outside surface is replaced by `element`.
Note that it is not fully tested.
It was not guaranteed to work on all the cases.
Args:
scaffold (Atoms): input cluster structure
element (str): outside surface element to be replaced
Returns:
coreshell (Atoms): core shell structure
"""
# CN : vertex < edge < surface < core
cn = np.array(calc_coordination_numbers(scaffold)[0])
cn_set = np.unique(cn)
vertexes = None
surfaces = None
cores = None
if len(cn_set) == 1:
pass
elif len(cn_set) == 2:
cores = np.where(cn == cn_set[1])[0]
surfaces = np.where(cn == cn_set[0])[0]
elif len(cn_set) == 3 or len(cn_set) == 4:
cores = np.where(cn == cn_set[-1])[0]
surfaces = np.where(cn != cn_set[-1])[0]
vertexes = np.where(cn == cn_set[0])[0]
else:
cores = np.where(cn == cn_set[-1])[0]
surfaces = np.where(cn != cn_set[-1])[0]
vertexes = np.where(np.isin(cn, cn_set[:-3]))[0]
if surfaces is None:
core_shell = scaffold.copy()
else:
core_shell = substitute(scaffold, surfaces, [element] * len(surfaces))
return core_shell
まず今回ベースとなるNano particle 骨格構造(scaffold)を作成します。
[4]:
Pt711 = cluster2atoms(Octahedron("Pt", 11, cutoff=4))
view_ngl(Pt711)
make_core_shell 関数を用いて、ベース骨格の内殻のみ “Pd” 元素に置き換えた構造 Pd306Pt405構造を作成します。
[5]:
Pd306Pt405 = make_core_shell(Pt711, "Pd")
view_ngl(Pd306Pt405)
同様にして、外殻がPd, 内殻がPtとなる構造Pd405Pt306を作成します。
[6]:
Pd711 = Octahedron("Pd", 11, cutoff=4)
Pd405Pt306 = make_core_shell(Pd711, "Pt")
view_ngl(Pd405Pt306)
[7]:
symbols, counts = np.unique(Pd405Pt306.symbols, return_counts=True)
print(f"Pd405Pt306 structure contains {symbols} with counts {counts}", )
symbols, counts = np.unique(Pd306Pt405.symbols, return_counts=True)
print(f"Pd306Pt405 structure contains {symbols} with counts {counts}", )
Pd405Pt306 structure contains ['Pd' 'Pt'] with counts [405 306]
Pd306Pt405 structure contains ['Pd' 'Pt'] with counts [306 405]
ここで得られたそれぞれの構造を構造緩和し、エネルギーを求めます。
[8]:
from ase.optimize import FIRE
def get_opt_energy(atoms, fmax=0.001):
opt = FIRE(atoms)
opt.run(fmax=fmax)
return atoms.get_total_energy()
[9]:
print("Optimizing Pt711")
Pt711.calc = calculator
E_pt711 = get_opt_energy(Pt711)
print("Optimizing Pd306Pt405")
Pd306Pt405.calc = calculator
E_pd306pt405 = get_opt_energy(Pd306Pt405)
print("Optimizing Pd405Pt306")
Pd405Pt306.calc = calculator
E_pd405pt306 = get_opt_energy(Pd405Pt306)
print("Optimizing Pd711")
Pd711.calc = calculator
E_pd711 = get_opt_energy(Pd711)
Optimizing Pt711
Step Time Energy fmax
FIRE: 0 14:25:30 -3603.041877 1.106293
FIRE: 1 14:25:30 -3604.082400 1.049765
FIRE: 2 14:25:30 -3605.848908 0.930611
FIRE: 3 14:25:31 -3607.346101 0.799823
FIRE: 4 14:25:31 -3608.623109 0.664005
FIRE: 5 14:25:31 -3609.736314 0.532733
FIRE: 6 14:25:31 -3610.729497 0.414364
FIRE: 7 14:25:31 -3611.628875 0.354905
FIRE: 8 14:25:31 -3612.448010 0.343606
FIRE: 9 14:25:31 -3613.188483 0.317494
FIRE: 10 14:25:31 -3613.846765 0.266917
FIRE: 11 14:25:31 -3614.426036 0.212701
FIRE: 12 14:25:31 -3614.932004 0.218558
FIRE: 13 14:25:31 -3615.370187 0.212221
FIRE: 14 14:25:31 -3615.739076 0.193992
FIRE: 15 14:25:31 -3616.044967 0.167131
FIRE: 16 14:25:32 -3616.294819 0.138940
FIRE: 17 14:25:32 -3616.482616 0.118548
FIRE: 18 14:25:32 -3616.616606 0.126510
FIRE: 19 14:25:32 -3616.718223 0.123149
FIRE: 20 14:25:32 -3616.799659 0.123320
FIRE: 21 14:25:32 -3616.856451 0.149213
FIRE: 22 14:25:32 -3616.886184 0.136264
FIRE: 23 14:25:32 -3616.960276 0.079333
FIRE: 24 14:25:32 -3617.037810 0.082584
FIRE: 25 14:25:32 -3617.108626 0.058447
FIRE: 26 14:25:32 -3617.152649 0.069872
FIRE: 27 14:25:32 -3617.175519 0.058664
FIRE: 28 14:25:32 -3617.185417 0.052176
FIRE: 29 14:25:33 -3617.189734 0.045037
FIRE: 30 14:25:33 -3617.197330 0.037567
FIRE: 31 14:25:33 -3617.206121 0.034813
FIRE: 32 14:25:33 -3617.214139 0.035938
FIRE: 33 14:25:33 -3617.221831 0.033390
FIRE: 34 14:25:33 -3617.228216 0.025663
FIRE: 35 14:25:33 -3617.233478 0.023158
FIRE: 36 14:25:33 -3617.238206 0.019954
FIRE: 37 14:25:33 -3617.240954 0.015322
FIRE: 38 14:25:33 -3617.242835 0.016681
FIRE: 39 14:25:33 -3617.242636 0.015962
FIRE: 40 14:25:33 -3617.243507 0.014847
FIRE: 41 14:25:34 -3617.244221 0.013744
FIRE: 42 14:25:34 -3617.245183 0.012737
FIRE: 43 14:25:34 -3617.246162 0.011964
FIRE: 44 14:25:34 -3617.247463 0.010721
FIRE: 45 14:25:34 -3617.248033 0.009881
FIRE: 46 14:25:34 -3617.248958 0.009811
FIRE: 47 14:25:34 -3617.250371 0.008724
FIRE: 48 14:25:34 -3617.251432 0.008885
FIRE: 49 14:25:34 -3617.252173 0.008483
FIRE: 50 14:25:34 -3617.253084 0.007743
FIRE: 51 14:25:34 -3617.253663 0.006933
FIRE: 52 14:25:34 -3617.254801 0.007572
FIRE: 53 14:25:35 -3617.255755 0.006551
FIRE: 54 14:25:35 -3617.257078 0.006670
FIRE: 55 14:25:35 -3617.258328 0.005623
FIRE: 56 14:25:35 -3617.258520 0.004660
FIRE: 57 14:25:35 -3617.258338 0.005419
FIRE: 58 14:25:35 -3617.258259 0.004716
FIRE: 59 14:25:35 -3617.258837 0.003457
FIRE: 60 14:25:35 -3617.258737 0.001949
FIRE: 61 14:25:35 -3617.258360 0.002321
FIRE: 62 14:25:35 -3617.258359 0.002217
FIRE: 63 14:25:36 -3617.258826 0.002021
FIRE: 64 14:25:36 -3617.258391 0.001418
FIRE: 65 14:25:36 -3617.258426 0.002427
FIRE: 66 14:25:36 -3617.258023 0.002294
FIRE: 67 14:25:36 -3617.258018 0.002035
FIRE: 68 14:25:36 -3617.258499 0.001678
FIRE: 69 14:25:36 -3617.258468 0.001255
FIRE: 70 14:25:36 -3617.258770 0.001222
FIRE: 71 14:25:36 -3617.258111 0.001241
FIRE: 72 14:25:36 -3617.258517 0.001201
FIRE: 73 14:25:36 -3617.258467 0.001056
FIRE: 74 14:25:36 -3617.258608 0.000792
Optimizing Pd306Pt405
Step Time Energy fmax
FIRE: 0 14:25:37 -3128.362422 0.554781
FIRE: 1 14:25:37 -3128.718133 0.519479
FIRE: 2 14:25:37 -3129.327287 0.448112
FIRE: 3 14:25:37 -3130.025151 0.342119
FIRE: 4 14:25:37 -3130.656600 0.213162
FIRE: 5 14:25:37 -3131.125038 0.239736
FIRE: 6 14:25:37 -3131.516222 0.300291
FIRE: 7 14:25:37 -3131.877282 0.281049
FIRE: 8 14:25:37 -3132.205099 0.190794
FIRE: 9 14:25:37 -3132.466972 0.137278
FIRE: 10 14:25:37 -3132.647158 0.160543
FIRE: 11 14:25:37 -3132.767802 0.139157
FIRE: 12 14:25:38 -3132.856732 0.138932
FIRE: 13 14:25:38 -3132.915044 0.116241
FIRE: 14 14:25:38 -3132.932396 0.107701
FIRE: 15 14:25:38 -3132.953113 0.096480
FIRE: 16 14:25:38 -3132.987904 0.080990
FIRE: 17 14:25:38 -3133.025092 0.068376
FIRE: 18 14:25:38 -3133.057638 0.056209
FIRE: 19 14:25:38 -3133.083733 0.060529
FIRE: 20 14:25:38 -3133.105666 0.063915
FIRE: 21 14:25:38 -3133.124318 0.066711
FIRE: 22 14:25:38 -3133.140353 0.067630
FIRE: 23 14:25:38 -3133.150510 0.053313
FIRE: 24 14:25:39 -3133.154367 0.034847
FIRE: 25 14:25:39 -3133.155419 0.033043
FIRE: 26 14:25:39 -3133.157375 0.029604
FIRE: 27 14:25:39 -3133.160142 0.024925
FIRE: 28 14:25:39 -3133.163324 0.023466
FIRE: 29 14:25:39 -3133.166159 0.022205
FIRE: 30 14:25:39 -3133.168810 0.020751
FIRE: 31 14:25:39 -3133.171326 0.021625
FIRE: 32 14:25:39 -3133.173944 0.022589
FIRE: 33 14:25:39 -3133.176931 0.022479
FIRE: 34 14:25:39 -3133.179845 0.021033
FIRE: 35 14:25:39 -3133.182463 0.018295
FIRE: 36 14:25:40 -3133.184520 0.014787
FIRE: 37 14:25:40 -3133.186816 0.013854
FIRE: 38 14:25:40 -3133.188659 0.014411
FIRE: 39 14:25:40 -3133.191965 0.013155
FIRE: 40 14:25:40 -3133.195140 0.013690
FIRE: 41 14:25:40 -3133.197633 0.014567
FIRE: 42 14:25:40 -3133.200278 0.016927
FIRE: 43 14:25:40 -3133.201178 0.012301
FIRE: 44 14:25:40 -3133.201493 0.010042
FIRE: 45 14:25:40 -3133.201860 0.009081
FIRE: 46 14:25:40 -3133.202039 0.007727
FIRE: 47 14:25:41 -3133.202015 0.006072
FIRE: 48 14:25:41 -3133.202341 0.006020
FIRE: 49 14:25:41 -3133.202135 0.005921
FIRE: 50 14:25:41 -3133.202026 0.005769
FIRE: 51 14:25:41 -3133.202030 0.005554
FIRE: 52 14:25:41 -3133.202192 0.005277
FIRE: 53 14:25:41 -3133.202097 0.004928
FIRE: 54 14:25:41 -3133.201711 0.004488
FIRE: 55 14:25:41 -3133.201993 0.003916
FIRE: 56 14:25:41 -3133.202221 0.003203
FIRE: 57 14:25:41 -3133.202351 0.002568
FIRE: 58 14:25:42 -3133.201891 0.002143
FIRE: 59 14:25:42 -3133.202030 0.001219
FIRE: 60 14:25:42 -3133.202281 0.000738
Optimizing Pd405Pt306
Step Time Energy fmax
FIRE: 0 14:25:42 -2921.182322 0.808428
FIRE: 1 14:25:42 -2922.173551 0.778165
FIRE: 2 14:25:42 -2923.798310 0.717436
FIRE: 3 14:25:42 -2925.164911 0.649386
FIRE: 4 14:25:42 -2926.294831 0.574657
FIRE: 5 14:25:42 -2927.257897 0.494829
FIRE: 6 14:25:42 -2928.103585 0.412738
FIRE: 7 14:25:42 -2928.859962 0.331024
FIRE: 8 14:25:43 -2929.541733 0.273011
FIRE: 9 14:25:43 -2930.148465 0.215194
FIRE: 10 14:25:43 -2930.679256 0.194757
FIRE: 11 14:25:43 -2931.135037 0.185401
FIRE: 12 14:25:43 -2931.522678 0.173530
FIRE: 13 14:25:43 -2931.850095 0.156303
FIRE: 14 14:25:43 -2932.122774 0.138360
FIRE: 15 14:25:43 -2932.342372 0.125447
FIRE: 16 14:25:43 -2932.513091 0.119323
FIRE: 17 14:25:43 -2932.643051 0.123406
FIRE: 18 14:25:43 -2932.741551 0.109909
FIRE: 19 14:25:43 -2932.812201 0.103340
FIRE: 20 14:25:43 -2932.859174 0.117370
FIRE: 21 14:25:44 -2932.887554 0.109580
FIRE: 22 14:25:44 -2932.943548 0.083614
FIRE: 23 14:25:44 -2933.016410 0.066704
FIRE: 24 14:25:44 -2933.083822 0.061939
FIRE: 25 14:25:44 -2933.140639 0.048594
FIRE: 26 14:25:44 -2933.176532 0.042659
FIRE: 27 14:25:44 -2933.193233 0.042365
FIRE: 28 14:25:44 -2933.196132 0.038131
FIRE: 29 14:25:44 -2933.200875 0.030508
FIRE: 30 14:25:44 -2933.206880 0.029938
FIRE: 31 14:25:44 -2933.212638 0.029914
FIRE: 32 14:25:44 -2933.218642 0.029287
FIRE: 33 14:25:45 -2933.223844 0.027860
FIRE: 34 14:25:45 -2933.228650 0.025587
FIRE: 35 14:25:45 -2933.233940 0.022231
FIRE: 36 14:25:45 -2933.238736 0.017898
FIRE: 37 14:25:45 -2933.242426 0.017854
FIRE: 38 14:25:45 -2933.247045 0.022313
FIRE: 39 14:25:45 -2933.251806 0.020310
FIRE: 40 14:25:45 -2933.257739 0.015348
FIRE: 41 14:25:45 -2933.262457 0.019001
FIRE: 42 14:25:45 -2933.266097 0.017250
FIRE: 43 14:25:45 -2933.267625 0.009698
FIRE: 44 14:25:45 -2933.267971 0.008436
FIRE: 45 14:25:46 -2933.268189 0.006457
FIRE: 46 14:25:46 -2933.268305 0.004922
FIRE: 47 14:25:46 -2933.268735 0.004740
FIRE: 48 14:25:46 -2933.268920 0.005340
FIRE: 49 14:25:46 -2933.268624 0.005050
FIRE: 50 14:25:46 -2933.268842 0.003217
FIRE: 51 14:25:46 -2933.268917 0.002606
FIRE: 52 14:25:46 -2933.268878 0.002534
FIRE: 53 14:25:46 -2933.268976 0.002397
FIRE: 54 14:25:46 -2933.268877 0.002198
FIRE: 55 14:25:46 -2933.269080 0.001950
FIRE: 56 14:25:46 -2933.269169 0.001671
FIRE: 57 14:25:46 -2933.269039 0.001550
FIRE: 58 14:25:47 -2933.269291 0.001497
FIRE: 59 14:25:47 -2933.268987 0.001398
FIRE: 60 14:25:47 -2933.269336 0.001188
FIRE: 61 14:25:47 -2933.268870 0.000877
Optimizing Pd711
Step Time Energy fmax
FIRE: 0 14:25:47 -2415.405054 0.344279
FIRE: 1 14:25:47 -2415.612591 0.321856
FIRE: 2 14:25:47 -2415.978145 0.280675
FIRE: 3 14:25:47 -2416.425082 0.226757
FIRE: 4 14:25:47 -2416.883060 0.175707
FIRE: 5 14:25:47 -2417.313508 0.185636
FIRE: 6 14:25:47 -2417.704039 0.198317
FIRE: 7 14:25:47 -2418.041955 0.182545
FIRE: 8 14:25:48 -2418.336226 0.144903
FIRE: 9 14:25:48 -2418.583558 0.119375
FIRE: 10 14:25:48 -2418.783487 0.127179
FIRE: 11 14:25:48 -2418.940574 0.127709
FIRE: 12 14:25:48 -2419.064533 0.106266
FIRE: 13 14:25:48 -2419.160496 0.095096
FIRE: 14 14:25:48 -2419.227396 0.089455
FIRE: 15 14:25:48 -2419.261333 0.061273
FIRE: 16 14:25:48 -2419.268015 0.080886
FIRE: 17 14:25:48 -2419.278864 0.074140
FIRE: 18 14:25:48 -2419.297003 0.061625
FIRE: 19 14:25:48 -2419.317263 0.052965
FIRE: 20 14:25:49 -2419.335995 0.046415
FIRE: 21 14:25:49 -2419.352427 0.039599
FIRE: 22 14:25:49 -2419.368013 0.039988
FIRE: 23 14:25:49 -2419.382131 0.032588
FIRE: 24 14:25:49 -2419.394424 0.022818
FIRE: 25 14:25:49 -2419.401560 0.027586
FIRE: 26 14:25:49 -2419.404850 0.032415
FIRE: 27 14:25:49 -2419.405462 0.030809
FIRE: 28 14:25:49 -2419.406825 0.027701
FIRE: 29 14:25:49 -2419.408605 0.023284
FIRE: 30 14:25:49 -2419.410039 0.017863
FIRE: 31 14:25:49 -2419.411662 0.011849
FIRE: 32 14:25:50 -2419.412526 0.012844
FIRE: 33 14:25:50 -2419.413746 0.013750
FIRE: 34 14:25:50 -2419.414922 0.013153
FIRE: 35 14:25:50 -2419.415954 0.010720
FIRE: 36 14:25:50 -2419.417527 0.009610
FIRE: 37 14:25:50 -2419.419019 0.008786
FIRE: 38 14:25:50 -2419.419910 0.011812
FIRE: 39 14:25:50 -2419.420802 0.014921
FIRE: 40 14:25:50 -2419.422039 0.014570
FIRE: 41 14:25:50 -2419.422992 0.010232
FIRE: 42 14:25:50 -2419.424597 0.010144
FIRE: 43 14:25:50 -2419.425413 0.013389
FIRE: 44 14:25:51 -2419.426064 0.008441
FIRE: 45 14:25:51 -2419.426573 0.006695
FIRE: 46 14:25:51 -2419.426851 0.005802
FIRE: 47 14:25:51 -2419.426737 0.004154
FIRE: 48 14:25:51 -2419.426926 0.002776
FIRE: 49 14:25:51 -2419.426802 0.003845
FIRE: 50 14:25:51 -2419.426893 0.004072
FIRE: 51 14:25:51 -2419.426808 0.003231
FIRE: 52 14:25:51 -2419.426638 0.002835
FIRE: 53 14:25:51 -2419.426816 0.002023
FIRE: 54 14:25:51 -2419.426782 0.001955
FIRE: 55 14:25:51 -2419.426839 0.001820
FIRE: 56 14:25:52 -2419.426599 0.001631
FIRE: 57 14:25:52 -2419.426650 0.001392
FIRE: 58 14:25:52 -2419.426818 0.001256
FIRE: 59 14:25:52 -2419.426641 0.001145
FIRE: 60 14:25:52 -2419.426832 0.000994
それぞれのエネルギーが得られたので、ここからexcess energyを計算することができます。
[10]:
E_exess_pd306pt405 = (E_pd306pt405 - 306 / 711 * E_pd711 - 405 / 711 * E_pt711) / 711
E_exess_pd405pt306 = (E_pd405pt306 - 405 / 711 * E_pd711 - 306 / 711 * E_pt711) / 711
print(f"Excess energy of Pd306Pt405 = {E_exess_pd306pt405 * 1000:.2f} meV")
print(f"Excess energy of Pd405Pt306 = {E_exess_pd405pt306 * 1000:.2f} meV")
Excess energy of Pd306Pt405 = -44.26 meV
Excess energy of Pd405Pt306 = 2.36 meV
Excess energyが負となっているPd306Pt405は単独でPd, Pt nanoparticleが存在するよりも混合したほうが安定であるということを示唆しています。 一方で正となっているPd405Pt306の構造はPd306Pt405と比べると不安定であることになります。
この結果から、PdとPtの合金微粒子では、Pdが外殻に来るような構造が自然界では安定して存在しそうだという考察ができます。
最後に、Pt306個、Pd405個がランダムに配置されたような構造のexcess energyを求めてみましょう。
[11]:
Pd405Pt306_random = Pt711.copy()
# randomly choose 405 atoms to be replaced to Pd
pd_indices = np.random.choice(np.arange(711), 405, replace=False)
Pd405Pt306_random.numbers[pd_indices] = atomic_numbers["Pd"]
print(f"Replaced to Pd with indices {pd_indices}")
view_ngl(Pd405Pt306_random)
Replaced to Pd with indices [ 88 383 604 60 213 558 439 15 208 550 690 533 647 538 57 391 62 470
247 86 595 150 611 152 445 157 107 46 360 397 667 553 101 356 461 161
565 374 613 313 196 257 486 441 61 184 666 589 210 457 548 528 293 650
245 402 163 120 552 638 526 17 587 448 480 133 539 624 634 534 67 376
336 646 531 44 58 64 418 427 446 372 408 285 310 499 541 212 610 118
224 600 349 572 467 665 532 632 231 148 555 111 317 700 319 14 41 267
509 709 272 517 29 454 401 657 617 394 506 37 567 78 692 159 508 98
234 264 294 282 518 269 72 116 77 671 601 42 703 237 414 415 117 471
635 19 549 661 5 87 32 59 529 428 154 633 171 626 686 619 645 503
223 384 484 288 707 122 579 299 382 674 256 469 684 597 195 320 45 689
103 468 186 584 498 425 492 289 102 16 225 476 92 697 90 404 325 515
370 491 475 433 708 463 227 193 114 450 119 483 295 411 590 644 165 276
424 511 527 578 367 343 543 413 592 281 365 191 581 173 91 232 417 652
357 400 561 177 451 393 189 649 149 93 302 551 421 573 344 147 131 609
250 698 341 10 409 138 390 710 301 192 31 1 440 95 76 437 333 27
473 69 520 110 348 694 460 13 570 53 39 221 335 362 524 323 389 190
434 183 239 438 535 386 479 599 672 197 435 673 96 621 347 172 207 121
259 571 487 304 238 478 472 585 436 366 180 662 622 412 554 569 691 337
494 94 388 658 387 396 643 512 220 203 125 327 373 380 80 18 194 676
254 504 249 405 277 615 166 655 12 75 678 660 513 530 326 462 465 605
230 132 606 0 216 28 577 410 51 84 168 185 537 135 458 354 240 582
322 182 648 395 640 429 704 236 311 685 206 219 432 309 426 268 140 594
115 431 464 85 144 359 40 695 544]
[12]:
print("Optimizing Pd405Pt306_random")
Pd405Pt306_random.calc = calculator
E_pd405pt306_random = get_opt_energy(Pd405Pt306_random)
Optimizing Pd405Pt306_random
Step Time Energy fmax
FIRE: 0 14:25:52 -2951.506716 0.765766
FIRE: 1 14:25:52 -2951.937831 0.733224
FIRE: 2 14:25:52 -2952.682443 0.667993
FIRE: 3 14:25:52 -2953.554715 0.569734
FIRE: 4 14:25:52 -2954.299787 0.458146
FIRE: 5 14:25:53 -2954.871782 0.369099
FIRE: 6 14:25:53 -2955.320882 0.339895
FIRE: 7 14:25:53 -2955.682170 0.324078
FIRE: 8 14:25:53 -2955.974359 0.271887
FIRE: 9 14:25:53 -2956.200539 0.226704
FIRE: 10 14:25:53 -2956.364708 0.242435
FIRE: 11 14:25:53 -2956.480267 0.212748
FIRE: 12 14:25:53 -2956.566103 0.179629
FIRE: 13 14:25:53 -2956.634076 0.154079
FIRE: 14 14:25:53 -2956.685258 0.179475
FIRE: 15 14:25:53 -2956.714649 0.171600
FIRE: 16 14:25:54 -2956.765927 0.156315
FIRE: 17 14:25:54 -2956.826483 0.143411
FIRE: 18 14:25:54 -2956.885088 0.131068
FIRE: 19 14:25:54 -2956.934194 0.116116
FIRE: 20 14:25:54 -2956.971764 0.098397
FIRE: 21 14:25:54 -2956.998637 0.077978
FIRE: 22 14:25:54 -2957.016904 0.065541
FIRE: 23 14:25:54 -2957.025704 0.072795
FIRE: 24 14:25:54 -2957.027288 0.068062
FIRE: 25 14:25:54 -2957.029893 0.059009
FIRE: 26 14:25:54 -2957.033506 0.057342
FIRE: 27 14:25:54 -2957.037790 0.055508
FIRE: 28 14:25:54 -2957.042373 0.053160
FIRE: 29 14:25:55 -2957.046964 0.050259
FIRE: 30 14:25:55 -2957.051590 0.046763
FIRE: 31 14:25:55 -2957.056268 0.042200
FIRE: 32 14:25:55 -2957.060866 0.036376
FIRE: 33 14:25:55 -2957.065366 0.029223
FIRE: 34 14:25:55 -2957.069577 0.023238
FIRE: 35 14:25:55 -2957.073891 0.022994
FIRE: 36 14:25:55 -2957.077929 0.029810
FIRE: 37 14:25:55 -2957.083078 0.028504
FIRE: 38 14:25:55 -2957.088568 0.025424
FIRE: 39 14:25:55 -2957.094616 0.023899
FIRE: 40 14:25:55 -2957.100662 0.023016
FIRE: 41 14:25:56 -2957.106078 0.014227
FIRE: 42 14:25:56 -2957.109012 0.022076
FIRE: 43 14:25:56 -2957.109576 0.016555
FIRE: 44 14:25:56 -2957.109881 0.015553
FIRE: 45 14:25:56 -2957.110283 0.013603
FIRE: 46 14:25:56 -2957.110980 0.010819
FIRE: 47 14:25:56 -2957.111612 0.007799
FIRE: 48 14:25:56 -2957.111975 0.006767
FIRE: 49 14:25:56 -2957.112346 0.006141
FIRE: 50 14:25:56 -2957.112639 0.007015
FIRE: 51 14:25:56 -2957.112887 0.009040
FIRE: 52 14:25:57 -2957.113054 0.009885
FIRE: 53 14:25:57 -2957.113349 0.010308
FIRE: 54 14:25:57 -2957.113711 0.009055
FIRE: 55 14:25:57 -2957.114145 0.006029
FIRE: 56 14:25:57 -2957.114571 0.004789
FIRE: 57 14:25:57 -2957.114860 0.003731
FIRE: 58 14:25:57 -2957.114938 0.005924
FIRE: 59 14:25:57 -2957.114904 0.005552
FIRE: 60 14:25:57 -2957.114833 0.004836
FIRE: 61 14:25:57 -2957.115067 0.003832
FIRE: 62 14:25:57 -2957.115302 0.003199
FIRE: 63 14:25:57 -2957.115327 0.003109
FIRE: 64 14:25:58 -2957.115367 0.003030
FIRE: 65 14:25:58 -2957.115489 0.002909
FIRE: 66 14:25:58 -2957.115461 0.002807
FIRE: 67 14:25:58 -2957.115612 0.002987
FIRE: 68 14:25:58 -2957.115709 0.003293
FIRE: 69 14:25:58 -2957.115815 0.003012
FIRE: 70 14:25:58 -2957.115845 0.002345
FIRE: 71 14:25:58 -2957.116003 0.001799
FIRE: 72 14:25:58 -2957.116086 0.002503
FIRE: 73 14:25:58 -2957.116224 0.002297
FIRE: 74 14:25:58 -2957.116367 0.001981
FIRE: 75 14:25:59 -2957.116417 0.001914
FIRE: 76 14:25:59 -2957.116637 0.001993
FIRE: 77 14:25:59 -2957.116764 0.002053
FIRE: 78 14:25:59 -2957.116787 0.003677
FIRE: 79 14:25:59 -2957.116837 0.001182
FIRE: 80 14:25:59 -2957.116846 0.002599
FIRE: 81 14:25:59 -2957.116755 0.002064
FIRE: 82 14:25:59 -2957.116853 0.001125
FIRE: 83 14:25:59 -2957.116860 0.000959
[13]:
view_ngl(Pd405Pt306_random)
[14]:
E_exess_pd405pt306_random = (E_pd405pt306_random - 405 / 711 * E_pd711 - 306 / 711 * E_pt711) / 711
print(f"Excess energy of Pd306Pt405 random = {E_exess_pd405pt306_random * 1000:.2f} meV")
Excess energy of Pd306Pt405 random = -31.18 meV
この場合も負のexcess energyが得られ、この構造が安定になりうるという結果が得られました。
今回は完全にランダムな配置を行って構造を作成しましたが、実際は物質が混ざる場合には局所的にパターンを取ることでより安定な構造となる可能性もあります。 こういった分析を進める場合はモンテカルロ法を用いて配置パターンを最適化するということがよく行われますがここでは省略します。 （モンテカルロ法の説明及びその実例については将来後述されるかもしれません。）
このように、様々な構造を作成し、そのexcess energyを計算することで合金の安定構造がどのようになるかを解析する事ができます。
担体上への微粒子吸着構造作成¶
本チュートリアルでは、担体上に微粒子をおいたモデリングの簡単な計算事例を紹介するにとどめ、この構造の現実での妥当性検証までは行いません。
ここでは、担持貴金属担体としてSnO2を、Nano particleとしてはPt を用いて、担体上へPt Nano particleを乗せた構造を作成してみます。
[15]:
# bulk, support material
from ase.io import read
SnO2 = read("../input/SnO2_mp-856_conventional_standard.cif")
view_ngl(SnO2, representations=["ball+stick"])
ここではSlab構造の作成(Bulkからの特定ミラー面の切り出し)にpymatgenの SlabGenerator を用いています。 これはSlabGenerator のinstance化をする際の初期化パラメータとして作りたいSlab構造のパラメータを指定し、 get_slabs 関数を呼ぶことで指定された初期化パラメータにマッチするSlab構造が列挙されて得られます。
詳しくは、以下のMaterials Project Workshopでも紹介されています。
[16]:
# Generate slab
from ase.build import make_supercell
import numpy as np
from pymatgen.core.surface import SlabGenerator
from pymatgen.io.ase import AseAtomsAdaptor
slab_gen = SlabGenerator(
initial_structure=AseAtomsAdaptor.get_structure(SnO2),
miller_index=[1,1,0],
min_slab_size=5.0, # ここで層の出方が変わる
min_vacuum_size=15.0, # nanoparticleを載せるので大きめに
lll_reduce=False,
center_slab=True,
primitive=True,
max_normal_search=1,
)
slabs = slab_gen.get_slabs(tol=0.3, bonds=None, max_broken_bonds=0, symmetrize=False)
/home/jovyan/.py39/lib/python3.9/site-packages/spglib/spglib.py:115: DeprecationWarning: dict interface (SpglibDataset['wyckoffs']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead
warnings.warn(
/home/jovyan/.py39/lib/python3.9/site-packages/spglib/spglib.py:115: DeprecationWarning: dict interface (SpglibDataset['equivalent_atoms']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead
warnings.warn(
得られた slabs はpymatgenのinstanceです。これは、AseAtomsAdaptor を用いてASEのAtoms instanceに変換することができます。
得られた構造をASE Atomsに変換して、可視化してみます。
[17]:
slab_atoms_list = [AseAtomsAdaptor.get_atoms(slab) for slab in slabs]
view_ngl(slab_atoms_list, representations=["ball+stick"])
２つの構造が得られました。
以降では2つ目の構造を使用して、Slabを作成していきます。
ASEでsupercellを作成した後、z軸方向に平行移動して一番下にある原子(Layer)の高さが0になるよう設定しています。
[18]:
slab = slab_atoms_list[1].copy()
# make supercell: expand to xy-plane
#slab = make_supercell(slab, [[4, 0, 0], [0, 2, 0], [0, 0, 1]]) # this is same with below.
slab = slab * (4, 2, 1)
# shift `slab` to bottom of cell
min_pos_z = np.min(slab.positions, axis=0)[2]
slab.set_positions(slab.positions - [0, 0, min_pos_z])
[19]:
view_ngl(slab, representations=["ball+stick"], replace_structure=True)
担体のSlab構造を作成することができました。
次にこの担体上に乗せるNano particle clusterを作成します。
[20]:
Pt55 = Octahedron("Pt", 5, cutoff=2)
# cut cluster to make half-sphere
cluster = Pt55.copy()
# Rotate cluster to make triangle surface comes to top.
cluster.rotate([0, 0, 1], [1, 1, 1], center=cluster.get_center_of_mass())
# Cut bottom 2 layers
for _ in range(2):
target = np.round(cluster.positions, decimals=0)
del cluster[np.where(target[:,2]==np.min(target[:,2]))[0]]
[21]:
view_ngl(cluster, representations=["ball+stick"])
Nano particle clusterを担体の上に乗せましょう。
ここでは、プログラムを書いて担体真ん中の上にNano particleを乗せていますが、 5章で紹介するSurfaceEditorを用いると、InteractiveにNano particleを移動させて初期構造を作成することも可能です。
[22]:
from ase.data import atomic_numbers, chemical_symbols, covalent_radii
# Put Pt37 on top of SnO2. Both perpendincular lines goes z-axis.
slab_xy_size = np.min(slab.cell.cellpar()[0:2])
cluster_xy_size = np.max(
(np.max(cluster.positions, axis=0) - np.min(cluster.positions, axis=0))[0:2]
)
# Vacuum size
min_slab_xy_size = cluster_xy_size + 15
for i in range(1, 5):
print(f"i={i}")
if slab_xy_size * i < min_slab_xy_size:
pass
else:
slab_sc = make_supercell(slab, [[i, 0, 0], [0, i, 0], [0, 0, 1]])
break
slab_surface_xy_center = np.append(
slab_sc.cell.cellpar()[0:2] / 2, np.max(slab_sc.positions, axis=0)[2]
)
cluster_surface_xy_center = np.append(
np.mean(cluster.positions, axis=0)[0:2], np.min(cluster.positions, axis=0)[2]
)
cluster = Atoms(cluster.get_chemical_symbols(), cluster.positions - cluster_surface_xy_center)
slab_surface_covalent_radii = covalent_radii[
slab_sc.get_atomic_numbers()[np.argmax(slab_sc.positions, axis=0)[2]]
]
cluster_surface_covalent_radii = covalent_radii[
cluster.get_atomic_numbers()[np.argmin(cluster.positions, axis=0)[2]]
]
cluster.translate(
slab_surface_xy_center + [0, 0, slab_surface_covalent_radii + cluster_surface_covalent_radii]
)
supported = slab_sc.copy()
supported += cluster
i=1
i=2
[23]:
view_ngl(supported, representations=["ball+stick"])
[24]:
import pfp_api_client
from pfp_api_client.pfp.calculators.ase_calculator import ASECalculator
from pfp_api_client.pfp.estimator import Estimator, EstimatorCalcMode
from ase.optimize import BFGS, LBFGS, FIRE
from ase.io import Trajectory
print(f"pfp_api_client: {pfp_api_client.__version__}")
estimator = Estimator(calc_mode=EstimatorCalcMode.CRYSTAL, model_version="v3.0.0")
calculator = ASECalculator(estimator)
supported.calc = calculator
traj = Trajectory("./SnO2_Pt119.traj", "w", supported)
opt = FIRE(supported)
opt.attach(traj.write, interval=1)
opt.run(fmax=0.005)
traj.close()
pfp_api_client: 1.18.0
Step Time Energy fmax
FIRE: 0 14:26:01 -1860.090908 2.772337
FIRE: 1 14:26:01 -1866.458219 2.345127
FIRE: 2 14:26:01 -1871.904491 2.087904
FIRE: 3 14:26:01 -1876.434687 2.025570
FIRE: 4 14:26:01 -1880.091711 1.950061
FIRE: 5 14:26:01 -1882.953197 1.859120
FIRE: 6 14:26:01 -1885.147574 1.750495
FIRE: 7 14:26:01 -1886.867187 1.622280
FIRE: 8 14:26:01 -1888.359161 1.473233
FIRE: 9 14:26:01 -1889.852359 1.404868
FIRE: 10 14:26:01 -1891.473633 1.297790
FIRE: 11 14:26:01 -1893.201802 1.075850
FIRE: 12 14:26:02 -1894.905019 0.942317
FIRE: 13 14:26:02 -1896.434131 0.936909
FIRE: 14 14:26:02 -1897.707659 0.910064
FIRE: 15 14:26:02 -1898.742466 0.862185
FIRE: 16 14:26:02 -1899.634861 0.861930
FIRE: 17 14:26:02 -1900.494820 0.963558
FIRE: 18 14:26:02 -1901.381262 1.061127
FIRE: 19 14:26:02 -1902.273342 1.143998
FIRE: 20 14:26:02 -1903.099028 1.204080
FIRE: 21 14:26:02 -1903.779864 1.239105
FIRE: 22 14:26:02 -1904.287930 1.252026
FIRE: 23 14:26:02 -1904.671582 1.193156
FIRE: 24 14:26:02 -1905.006241 0.933255
FIRE: 25 14:26:02 -1905.331652 0.708003
FIRE: 26 14:26:02 -1905.632003 0.396677
FIRE: 27 14:26:03 -1905.853795 0.348697
FIRE: 28 14:26:03 -1905.981652 0.574539
FIRE: 29 14:26:03 -1906.069442 0.972596
FIRE: 30 14:26:03 -1906.185097 1.158863
FIRE: 31 14:26:03 -1906.327633 1.032108
FIRE: 32 14:26:03 -1906.441743 0.595610
FIRE: 33 14:26:03 -1906.530182 0.583615
FIRE: 34 14:26:03 -1906.646216 0.651942
FIRE: 35 14:26:03 -1906.779208 0.521820
FIRE: 36 14:26:03 -1906.857278 0.516498
FIRE: 37 14:26:03 -1906.938751 0.604114
FIRE: 38 14:26:03 -1906.936108 0.499259
FIRE: 39 14:26:03 -1907.073790 0.219470
FIRE: 40 14:26:03 -1907.089149 0.265532
FIRE: 41 14:26:04 -1907.099471 0.222812
FIRE: 42 14:26:04 -1907.114717 0.148420
FIRE: 43 14:26:04 -1907.127346 0.094075
FIRE: 44 14:26:04 -1907.133720 0.126109
FIRE: 45 14:26:04 -1907.136501 0.181823
FIRE: 46 14:26:04 -1907.141530 0.190427
FIRE: 47 14:26:04 -1907.151441 0.153244
FIRE: 48 14:26:04 -1907.166613 0.089478
FIRE: 49 14:26:04 -1907.182458 0.082058
FIRE: 50 14:26:04 -1907.193680 0.125017
FIRE: 51 14:26:04 -1907.203037 0.123199
FIRE: 52 14:26:04 -1907.216517 0.098550
FIRE: 53 14:26:04 -1907.231361 0.102635
FIRE: 54 14:26:04 -1907.241632 0.098295
FIRE: 55 14:26:05 -1907.250636 0.081453
FIRE: 56 14:26:05 -1907.262811 0.117063
FIRE: 57 14:26:05 -1907.273459 0.085623
FIRE: 58 14:26:05 -1907.286581 0.108768
FIRE: 59 14:26:05 -1907.303755 0.094357
FIRE: 60 14:26:05 -1907.324385 0.084778
FIRE: 61 14:26:05 -1907.342073 0.129564
FIRE: 62 14:26:05 -1907.366775 0.210234
FIRE: 63 14:26:05 -1907.376727 0.090460
FIRE: 64 14:26:05 -1907.376925 0.144653
FIRE: 65 14:26:05 -1907.378877 0.113235
FIRE: 66 14:26:05 -1907.381615 0.090670
FIRE: 67 14:26:05 -1907.383712 0.090889
FIRE: 68 14:26:05 -1907.384903 0.091418
FIRE: 69 14:26:05 -1907.386135 0.103487
FIRE: 70 14:26:05 -1907.388422 0.104706
FIRE: 71 14:26:06 -1907.391927 0.096948
FIRE: 72 14:26:06 -1907.396307 0.099185
FIRE: 73 14:26:06 -1907.400288 0.101718
FIRE: 74 14:26:06 -1907.404823 0.104522
FIRE: 75 14:26:06 -1907.411394 0.107619
FIRE: 76 14:26:06 -1907.419970 0.112768
FIRE: 77 14:26:06 -1907.429382 0.120250
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[25]:
SnO2_Pt119 = Trajectory("./SnO2_Pt119.traj")
view_ngl(SnO2_Pt119, representations=["ball+stick"])
[26]:
from ase.io import write
from IPython.display import Image
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
write("output/sno2_pt119_opt.png", SnO2_Pt119[-1], rotation="-90x,0y,0z")
fig, ax = plt.subplots(1, 1, figsize=(5, 5))
ax.imshow(mpimg.imread("output/sno2_pt119_opt.png"))
ax.set_axis_off()
ax.set_title("SnO2 - Pt119 optimized structure")
fig.show()
このように、担体上のNano particle 構造を作成することができました。
ここで、論文で示されているようなchargeの可視化も可能です。 Nano particleの吸着面に近いところでのみ電荷が０からずれていて、核の外側がプラスに、内側がマイナスになっていることが確認できます。
[27]:
v = view_ngl(SnO2_Pt119[-1], show_charge=True)
v.show_charge_label()
v
このように、SnO2担体上にPt Nano particleが吸着している構造を作成することができました。
実際には、本当にこの構造で正しいのか（微粒子の状態で吸着するのか、それとも全体に均質に伸びて吸着するような形のほうが安定なのかなど)の検証が必要となりますが、このTutorialでは省略します。
こういった構造が作成できる事により、ここから更に後述する反応探索やMDなどを組み合わせることで、
様々な面が露出している触媒下での反応
触媒と担体の境界面での反応
なども扱えるようになる可能性があり、より現実に近い状態での触媒反応シミュレーションができる可能性があります。
参考文献
Calculations of Real-System Nanoparticles Using Universal Neural Network Potential PFP
Structural Stability of Ruthenium Nanoparticles: A Density Functional Theory Study
Electronic Structure and Phase Stability of PdPt Nanoparticles
Electronic structure and phase stability of Pt3M (M = Co, Ni, and Cu) bimetallic nanoparticles