Thermal Behavior – IChemical Processes & Transition State Theory

[based 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

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

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.

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

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.)

Potential energy surface

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

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!

1-dimensional PES

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

1-dimensional transition state theory

€

kA → B =ν exp −E t − EA

kT

⎛

⎝ ⎜

⎞

⎠ ⎟

Rate of transition from A to B

Vibrational frequency at A~ 1012-1013 Hz

Probability of system being at transition state

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?

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

€

kA → B =ν1 ×ν 2 × ...×ν 3N

ν1t ×ν 2

t × ...×ν 3N −1t exp −

E t − EA

kT

⎛

⎝ ⎜

⎞

⎠ ⎟

• Still, the first fraction works out to ~ 1012-1013 Hz, and hence the rate of the process is dominated by E

• 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

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?

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

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

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

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

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:

O Interstitial MigrationO Interstitial MigrationSi:HfOSi:HfO22 Interfaces Interfaces

Interfacial segregation:Thermodynamic driving force (implied by decreasing formation energy as interface is approached)

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)

Si HfO2

Hf

O

Point Defect MigrationAmorphous HfO2

HfO

O vacancy

O interstitial

Hf vacancy

VO2+ most mobile in a-HfO2

C. Tang & R. Ramprasad, Phys. Rev. B (in print)

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

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 system’s time evolution can be simulated

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

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”

H2 + D2 2HD

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

H

D

H

D

H

D

H

D

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

H

D

H

D

Orbital Symmetry ConsiderationsH

D

H

D

H

D

H

D

Ground state of reactants/products correlates with excited state of products/reactants, and hence, high barrier!

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

2NO N2 + O2

(Symmetry Forbidden)2NO N2O + O N2 + O2

(Symmetry Allowed)

Ramprasad, et al, J. Phys. Chem. B (1997)Ramprasad, PhD Thesis

Gas Phase Reactions

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 ThesisSchneider et al, J. Phys. Chem. B (1998)

Catalytic NO DecompositionCorrelation Diagram

Catalytic CO Oxidation

• One of the most studied catalytic reactions on metal and metal oxide surfaces

• Spawned careful surface science work (cf. Ertl’s 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)

• 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

CO Oxidation on RuO2 surfaces• Surface phase diagram and turn over frequency (TOF) for CO2

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 θ(1-θ), where θ is the coverage of O on the cus sites; but this was not the case

• kMC simulation clarified this …

Catalyst Design from First Principles