Thermal Behavior

1
Thermal Behavior IChemical Processes
Transition State Theorybased on Chapter 6,
Sholl Steckel

• From zero K to warmer situations!
• Kinetics of processes
• Example Atomic diffusion on surfaces
• Transition state theory
• Determining transition states (or saddle-points)
  numerically
• The nudged elastic band method
• Connecting atomic level processes to overall
  dynamics
• The kinetic Monte Carlo (kMC) method
• Case studies
• Catalytic NO decomposition
• Catalytic CO oxidation
• Other catalytic reactions

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Key Dates/Lectures

• Oct 12 Lecture
• Oct 19 No class
• Oct 26 Midterm Exam
• Nov 2 Lecture
• Nov 9 Lecture
• Nov 16 Guest Lectures
• Nov 30 MRS week no class
• Dec 7?? In-class term paper presentations

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Term Paper/Presentation

• Choose topic close to your research.
• Cast the term paper/presentation like a
  proposal ? identify problem, provide
  background, discuss past DFT work, and identify
  open issues future work
• DFT has to be a necessary component of term
  paper.
• Literature search Phys. Rev. B, Phys. Rev.
  Lett., Appl. Phys. Lett., J. Phys. Chem., Nano
  Letters, etc., within the last 10 years.

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From zero K to warmer situations!

• All calculations considered so far deal with the
  ground state, meaning at absolute zero
  temperature
• Does this mean that all such results are
  meaningless?
• Not really, as these results correspond to the
  internal energy or enthalpy
• Many methods are available to incorporate
  temperature
• Through direct inclusion of entropic
  contributions
• Through Molecular Dynamics
• Through transition state theory kinetic Monte
  Carlo

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Example Surface diffusion

• Lets first consider surface adsorption, a
  necessary first step underlying many processes
  (e.g., crystal growth, catalysis)
• Specific example Ag atom on Cu(100) surface
• Three distinct adsorption sites what is the
  nature of each of these sites? (i.e., are they
  minima, maxima, etc.)

6
Potential energy surface

• Hollow site global minimum
• Bridge site 1st order saddle point
• On top site 2nd order saddle point

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Potential energy surface (PES)

• PES computed using DFT at zero K
• System is dynamic on this PES, and the kinetic
  energy determines the temperature
• Thus, zero K computations are very useful and
  relevant!

8
1-dimensional PES

• Only the lowest energy transition state (or
  barrier) will matter

9
1-dimensional transition state theory
Rate of transition from A to B
Probability of system being at transition state
Vibrational frequency at A 1012-1013 Hz
In this 1-d example, the minimum is characterized
by a real frequency, while the transition state
displays a imaginary frequency (why?) this is a
signature of a transition state What about in a
3-d system containing N atoms?
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3-d transition state theory

• One has to perform normal mode analysis to
  determine the normal mode frequencies of the
  equilibrium and the transition states
• What are normal modes?
• A system with N atoms will display 3N real
  frequencies at equilibrium and 3N-1 real
  frequencies at a 1st order saddle point
• The A ? B rate is given by

• Still, the first fraction works out to
  1012-1013 Hz, and hence the rate of the process
  is dominated by DE
• The above expression applies to any process (not
  just site-to-site hopping of adsorbates) as long
  as starting and ending equilibrium situations,
  and transition states can be defined

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Two questions

• To calculate rates of elementary processes, we
  need the activation barrier (i.e., energy
  difference between transition state and initial
  equilibrium situation). How do we determine the
  barriers?
• Even after the barriers for all (or most)
  possible elementary steps are determined, how do
  we assemble all this information to determine
  overall macroscopic experimental quantities such
  as turn over frequency (TOF) or conversion
  efficiency?

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Determining Extrema

• Transition state is a maximum along one
  direction in phase space, but a minimum along
  other orthogonal directions
• First, let us review how minima are found
  numerically consider a function (Energy, E,
  which is a function of several coordinates) that
  needs to be minimized let us suppose that
  methods are available to compute the function
  value (DFT energy) and its first derivatives
  (Hellmann-Feynman forces) at a chosen set of
  coordinates
• The most obvious choice for minimizing the
  function numerically is the steepest descent
  method
• A much better choice is some flavor of the
  conjugate gradient method

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Steepest Descent (SD)

• Move along the negative gradient of the function
  till you reach a minimum (the line search)
• Then find the gradient again, and commence
  another line search, etc.
• This can take any number of line searches even if
  we are in the quadratic region

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Conjugate Gradient (CG)

• First search direction is identical to the
  steepest descent (SD) method
• Subsequent search directions are linear
  combinations of the new gradient direction and
  the previous gradient direction (called
  conjugate directions)
• This is done to account for the fact that the new
  SD gradient direction contains an already
  searched direction component
• For a N-dimensional function in the quadratic
  region, the CG algorithm takes exactly N line
  minimizations to locate the minimum

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Transition State

• Finding transition states is an entirely new ball
  game, as the point of interest is a maximum and
  minima simultaneously
• Starting with an initial guess (as is always the
  case), attempting to find a point with with zero
  (or negligible) gradient will inevitably take us
  to one of the local minima, rather than the
  transition state!
• Remember criterion satisfied by transition
  state first derivative is zero, and second
  derivative is negative along one dimension and
  positive along all other dimensions
• A few methods are available, the most popular one
  in electronic structure calculations being the
  nudged elastic band method

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Nudged Elastic Band Method

• Much more intensive than conjugate gradient, and
  works very differently
• Involves multiple configurations along the
  reaction coordinate separated by fictitious
  springs to keep the configurations from
  falling into a local minimum
• A snapshot

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O Interstitial MigrationSiHfO2 Interfaces
Si
HfO2
Interfacial segregation Thermodynamic driving
force (implied by decreasing formation energy as
interface is approached)
Hf
O
Kinetic driving force, and O penetration into Si
(implied by decreasing migration barriers as
interface is approached)
Excess O interstitials lead to the formation of
SiOx at the interface
C. Tang R. Ramprasad, Phys. Rev. B 75, 241302
(2007)
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Point Defect MigrationAmorphous HfO2
Hf
O
O vacancy
O interstitial
Hf vacancy
VO2 most mobile in a-HfO2
C. Tang R. Ramprasad, Phys. Rev. B (in print)
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Now What .

• We have the frequencies, we have the barriers,
  and hence we have the rates
• How do we put it all together?
• We roll the dice!
• The kinetic Monte Carlo (kMC) method
• kinetic because rates and temperatures are
  involved, and Monte Carlo because elementary
  processes are stochastic

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The kinetic Monte Carlo (kMC) Method

• The idea behind this method is straightforward
  If we know the rates for all processes that can
  occur given the current configuration of our
  atoms, we can choose an event in a random way
  that is consistent with these rates. By repeating
  this process, the systems time evolution can be
  simulated

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The kMC Algorithm

• Consider a Pd-alloyed Cu surface, with a
  predetermined number of Ag atoms randomly
  adsorbed on this surface
• The dynamical evolution of this system may be
  modeled using the following algorithm

• The output of a kMC simulation is typically
  surface coverage at given (T, P) conditions rate
  of formation of various competing products via
  competing mechanisms
• kMC will be contrasted with Molecular Dynamics
  (MD) in the last lecture

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Case Studies of Catalysis

• Although a reaction/process may be allowed by
  thermodynamics, it may be slow at low
  temperatures due to large barriers
• Catalyst A magical substance that speeds up
  chemical reactions without (of course) altering
  the overall thermodynamics
• Large barriers may be due to steric or bond
  breakage reasons, or because the process may be
  forbidden

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H2 D2 ? 2HD

• Intuitively, we may expect this reaction to occur
  as follows

H
H
H
H
H
H
D
D
D
D
D
D

• But this almost never happens, and H and D atomic
  intermediates are generally found during the
  course of the reaction. Why?

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Orbital Symmetry Considerations
H
H
H
H
D
D
D
D
Ground state of reactants/products correlates
with excited state of products/reactants, and
hence, high barrier!
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Catalytic decomposition of NO

• NO is one of the harmful effluents of automotive
  exhaust
• Need to accomplish 2NO ? N2 O2 although
  reaction is thermodynamically downhill, it has a
  huge barrier (because it is symmetry-forbidden
  next slide)
• Mechanism of catalytic decomposition of NO using
  a Cu-exchanged zeolite catalyst

2NO
N2 O2
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2NO ? N2 O2(Symmetry Forbidden)
Gas Phase Reactions
2NO ? N2O O ? N2 O2(Symmetry Allowed)
Ramprasad, et al, J. Phys. Chem. B
(1997) Ramprasad, PhD Thesis
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Catalytic decomposition of NO

• Various modes of interaction of NO with
  Cu-exchanged zeolites
• Mechanism of NO decomposition is many step process

2NO
N2 O2
Ramprasad, PhD Thesis Schneider et al, J. Phys.
Chem. B (1998)
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Catalytic NO DecompositionCorrelation Diagram
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Catalytic CO Oxidation

• One of the most studied catalytic reactions on
  metal and metal oxide surfaces
• Spawned careful surface science work (cf. Ertls
  work from the 1960s)
• Essential steps
• Adsorption of CO (generally unactivated, meaning
  negligible activation barrier)
• Dissociative adsorption of O2 (may be activated)
• Surface diffusion of CO and O (generally with a
  large barrier)
• Reaction of CO and O to form weakly bound CO2
  (may be activated)
• CO2 desorption (may be activated)

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• RuO2(110) surface has 2 types of adsorption
  sites coordinatively unsaturated site (CUS)
  bridge site, forming alternating rows
• Energies and barriers for all elementary steps
  (previous slide) were computed, and used in a kMC
  simulation
• Results Surface phase diagram of RuO2, and CO2
  conversion efficiency

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CO Oxidation on RuO2 surfaces

• Surface phase diagram and turn over frequency
  (TOF) for CO2

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kMC Simulation

• Barrier for COcus Ocus ? CO2 was lowest
• If this was the only operative reaction, it
  should have resulted in a rate of CO2 production
  proportional to ?(1-?), where ? is the coverage
  of O on the cus sites but this was not the case
• kMC simulation clarified this

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Catalyst Design from First Principles