D-AiSI 195 GAS-SOLID DYNAMICS AT DISORDERED RND ADSORBATE COVERED iSLJRFRCES(U) PRINCETON UNIV NJ DEPT OF CHEMISTRYH RABITZ 15 MAY 87 AFOSR-TR-87-0753 AFOSR-84-086

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4. PERFORMING ORGANIZATION R NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBER(S)

I'D AFOSR-TI 87- 0 7536a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION

(if applicable)Princeton University AFOSR/NC

6c. ADDRESS (City, State, and ZIP Code) 7b. ADDRESS (City, State, and ZIP Code)Department of Chemistry Building 410Princeton, NJ 08544 Bolling AFB, DC 20332-6448

&a. NAME OF FUNDING/SPONSORING I8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION (If applicable)AFOSR NC AFOSR-84-0106

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Building 410 ELEMENT NO. NO. NO ACCESSION NO.Bolling AFB, DC 20332-6448 61102F 2303 BI

11. TITLE (Include Security Classification)

Gas-Solid Dynamics at Disordered and Adsorbate Covered Surfaces (U)

12. PERSONAL AUTHOR(S)

Herschel Rabitz13a. TYPE OF REPORT 13b. TIME COVERED 114. DATE OF REPORT (Year, Month, Day) 15. PAGE COUNT

FINAL I FROM8 Z4 TO 8.20 281 1987/14av/15 10

16. SUPPLEMENTARY NOTATION

Published by Princeton University, May 15, 1987.

17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)FIELD GROUP SUB-GROUP 1> chemical dynamics wavefunction

chemical kinetics

gas-surface g.19. ABSTRACT (Continue on reverse if necessary and identify by block number)§The research under this grant was aimed at several closely related objectives involving

studies in chemical dynamics and chemical kinetics. In the case of chemical dynamics,both gas-phase and gas-surface molecular collisions were studied with a specialemphasis on the development and application of new techniques. In particular, anaverage wavefunction procedure was developed especially designed for treating the

scattering from complex targets. Applications were made to gas-phase scattering aswell as gas-surface scattering in the presence of defects. In the latter case, thefocus was on the role that natural defects or adsorbed species have with regard totheir influence on the chemical and physical properties of surfaces.

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22a. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE (Include Area Code) 22c OFFICE SYMBOLMaior Larry. Davisa (2021 767-4961 TAynq/mr

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FINAL TECHNICAL REPORT

GAS-SOLID DYNAMICS AT DISORDERED ANDADSORBATE COVERED SURFACES

Sponsored by AFOSR-84-0106B

Herschel RabfzDepartment of Chontty

Princeton University

F.TI 7 CRAR-IDTIC TAB

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ABSTRACT

Cme research under this grant was aimed at several closely relatedobjectives involvng studies In chemical dynamics and chemical kinetics. Inthe case of chemical dynamics, both gas-phase and gas-surface molecularcollisions were studied with a special emphasis on the development andapplication of new techniques. In particular, an average wavefunctionprocedure was developed especially designed for treating the scattering fromcomplex targets. Applications were made to gas-phase scattering as well asgas-surface scattering in the presence of defects. In the latter case, thefocus was on the role that natural defects or adsorbed species have withregard to their Influence on the chemical and physical properties ofsurfaces.. An enhanced degree of momentum transfer was found in thepresence of surface defects. Scattering of helium atoms off of surface bounddiatomic molecules was studied with the underlying lattice being in a finitetemperature to include phonon effects. In this case, the strong phononcoupling was found to have a significant effect on the gas-surface energytransfer processes. In addition to this dynamics work, continuing developmentand applications of sensitivity analysis techniques were pursued for problems inchemical kinetics. Special applications of concern involve problems incombustion and atmospheric chemistry. Finally, research was also carried outin several other related areas stemming from these primary studies. Inparticular, applications of filtering theory to problems in quantum mechanicsand chemical kinetics were investigated. This resulted in a new means forinvestigating these phenomena as well as providing insight into the degree thatlaboratory data may be inverted back to more fundamental Information.Finally, a variety of variational techniques and their generalizations wereexplored for application to problems arising in quantum mechanics as well asother areas. Given below is a summary of these various research projects.

1. Semiclassical Perturbation Theory for Atom Scattering from Surfaceswith Defects 1

Presented here is an extension of semiclassical perturbation theory(SCP) to gas-surface scattering incorporating surface defect sites with therestriction that the defects be diffuse enough such that they have no directinteraction. The formalism is applied to a corrugated surface with atomsmissing, modelling a nearly complete layer of atoms adsorbed on a metalsurface. The results show the redistribution of the Bragg intensities to acontinuum of momentum transfers about the former peaks.

2. Atom Molecule Scattering with the Average Wavefunction Method 2

The average wavefunction method (AWM) is applied to atom moleculescattering. In Its simplest form the labor Involved in solving the AWMequations in equivalent to that Involved for elastic scattering in the sameformulation. As an initial Illustration, expUclt expressions for the T-matrzlx aredertved for the scattering of an atom and a rigid rotor. Results arepresented for low energy scattertng and corrections to the Born approximationare clearly evident. In general, the AWM is particularly suited to polyatomscattering due to its reduction of the potential in terms of a separableatom-atom potential.

3. Applications at Filtering Theory tar the Inversion at Temporal ChemicalSystems 3

This paper presents a method tar Inverting temparal experimental datafrom chemical models ta abtain estimates at unknawn parameters. Most atthe madels under consideration are deterministic and we assume that themeasurements obtained fram experimental abservations are represented as thesolutian at a differential equatian cantaining the variables at the madel. ToIncarparate any extraneous laboratory effects that are not included In themodel, we assume that these equations are perturbed by a white noiseprocess so that the measurements became time dependent Stochasticvariables. A particular measurement is then equivalent to a realization atthese variables and applying stochastic estimation theory this realization can beused to obtain estimates at the unknown parameters In the model. As anexample of this estimation method, we consider chemical kinetics models withvarious observational equations and construct an estimator tar the unknownreaction rate constants. We also show the estimators tar the structuralconstants in a laser model depending on the representation of theexperimental data. In some cases the observations are simulated numericallyand we present the parameter estimates as a tunction at time. The efficiencyat the estimation process is calculated as the ratio at the a-posteriorivariance at the parameter estimator and the Rao-Cramer lower bound. SomeIssues in the numerical implementation at the filtering equations are discussedand a comparison is made between the minimum least square estimationmethod and the filtering method.

4. An Application at Filtering Theory to Parameter Identification inQuantum Mechanics 4

In this paper we present a method tar inverting observations onquantum mechanical systems to obtain estimates at unknown parametersresiding in the Hamiltonian. The quantal system Is represented in matrix formwith respect to a chosen basis and it Is assumed that the associatedexpansion coefficients are truncated to a finite dimension. The uncontrollablelaboratory noise will be modelled by means at an inhomogeneous white noiseprocess so that the experimental observations are represented as stochasticvariables satisfying a stochastic differential equation. We will assume thatmeasurements obtained from an experiment are now equivalent to a realizationat these stochastic variables. It is known from filtering theory that theminimum variance estimate at the unknown parameters in the quantal model isnow given by the expectation at the unknowns conditional on this realization.This estimator can be calculated analytically from the associated a-posterioriprobability density IN the original quantal system does not contain any randomelements. We calculate this probability density tar the unknown matrixelements and we demonstrate that tar a full Hamiltonian matrix the asymptoticvariance at the parameter estimator decreases as a third power In time and afourth power in the initial conditions. Some differences with the minimumleast square method are mentioned and a few issues at numericalimplementation are discussed.

5. Time-dependent Resonant Fluorescence Spectrum of Two-level Atomsin Strong Short Pulsed Fields: A Nonperturbative Treatment s

A study Is presented for the tre-dependent resonance fluorescencespectrum of two-level atoms Irradiating by hyperbolic secant pulsed lasers,allowing for an arbitrary spontaneous emission rate from the excited level tothe ground one. Novel features of the temporally transient spectrum havebeen observed. The spectrum is dominated by a central peak whichoscillates in time and Is accompanied by multiple side peaks. Asymptotically(i.e., after the pulse has passed) the side peaks disappear leaving a tail atthe line center which displays a Lorentizlan shape of width equal to the sumof the spontaneous emission rate and the width of the spectrometer consideredand of magnitude proportional to the tnstaneous intensity of the pulse. Theexact behavior of the spectrum is dictated by the area of the engaged pulseand by how the atom is prepared initially. In particular, it is found that thespectrum can be asymmetric even when the laser is tuned to exact resonanceif the atoms are initially prepared in a certain class of proper coherentstates, Irrespective of how the field amplitude may vary in time. An analysisof the spectral symmetry without resorting to any explicit evaluation of thespectrum itself Is also presented. Furthermore, the associated optical Blochequations are solved analytically in terms of hypergeometrtc functions in asimilar manner to that done previously for the undamped two-level Schrodingerequation. Specific numerical results are presented for pulses of area equalto v, 21r, 31r, 4r and 51r for both symmetric and asymmetric cases.

6. Sensitivity Analysis of Combustion Systemss

This paper presented an updated review of how sensitivity analysistechniques may be used to analyze chemically reacting flows with a particularemphasis on combustion phenomena. The paper included mathematical detailsas well as a particular emphasis on anticipated new directions for the field.Selected numerical examples were included to illustrate the material.

-' 7. Vibrational Energy Transfer at the Gas-solid Interface: The Role of-:' Collective and of Localized Vibrational Mods 7

We present a study of energy transfer (kinetic to vibrational) incollisions of atoms with diatomic molecules adsorbed on the surface of ametal substrate, for hyperthermal coilislon energies (0.1 to 1. 0eV). In orderto make the many-body problem computationally tractable, atomic motions arerestricted to one spatial dimension and the combined diatomic-metal target ismodeled by a linear chain of coupled harmonic oscillators, so that vibrationsof the target can be solved analytically for any arbitrary number of atoms.The collision Is described in the semiclassical lnimt appropriate forhyperthermal velocities: translation of the projectile is obtained from a

qclassical trajectory, while vibration of the target Is treated quantummechanically. The intensity of scattered atoms ts obtained from thetime-correlation function of the semiclassical transition operator. As a result,the Intensity is evaluated analytically without need of tnternal-state expansions,and it includes the quantum-statistical average over the distribution of initIalphonon states at nonzero temperature. The theory is applied to Heprojectiles scattered from OCPtx , OCNIx, N 2 Wx, and from the pure metals.

k2.

The results are presented in the form of energy-lose spectral simulated for atypical experimental detector of finite resolution. The calculations are donewith realistic values of force constants and He-target potentials. Hence, theone-dimensional model contains the basic vibrational features of the gas-solidinterface, namely: a very large number of low-frequency modes Involvingcollective vibrations of the target atoms and a few high-frequency modeswhose atomic displacements are localzed near the surface (the latter roughlycorrespond to the vibration of the free diatomic and to stretching of thediatom-substrate bond). The simulated spectra show rich structure due tomany-quantum excitations of collective and localized modes. We show howthe structure Is related to the eftenfrequencem and eigenvectors of the target,and we examine how the contributions of each mode vary with collision energyand target temperature. We find that excitation of localized modes follow aPoisson distribution and only the first few transitions are significantly excited.In contrast, collective modes undergo many-quantum transitions characterizedby a quasicontinuous and Gaussian distribution of energy transfer. We derivesimple expressions that explain the amounts of energy transferred to eachmode In terms of the physical parameters (force constants, atomic masses,temperature, thickness of the target, and strength of the potential). Thesimulations indicate that vibrational transitions of the adsorbed molecule canbe resolved from the background of substrate phonons using presenttechnology. They also suggest that lattice phonon structure could beobserved using a thin substrate film weakly coupled to an underlying support.We also find that the spectrum Is quite sensitive to small variations of theprojectlle-admorbate potential, so that scattering experiments can yieldinformation about how Intermolecular forces are changed by chemlsorption.

S. Autocorrelations In the Center Manifold of Dissipative Systems'

The phase-space contraction for the systems under considerationcorresponds to an asymptotic-center manifold functional dependence offast-reLxting variables Xt,j's on the slowty-relaxing ones, the X8 's. Thepropagation of perturbations along the center manifold In obtaine from theGreen-function sensitivity matrix. Scaling and self-similarty relations amongthe matrix elements are found. This fact allows for a simplification nt thecomputation of the Xg,j-XC,j and the X&,-X#, autocorrelations. The validityof the results is confirmed in two different contexts: (a) An analyticalderivation of power spectra at the onset of periodic instabilities is performed.We demonstrate that n the infinite relaxation time limit for slow variables, thepower spectra for all the Xf,j's converge to the same distribution. This resultIs In accord with previous computations. (b) The power spectrum for arandomly driven anharmonic damped oscillator is computed at theasymptotic-center manifold regime and tested vis-a-vis previous plots exhibitinga very good agreement.

9. Fundamental Sensitivity Propagators in Dissipative Systemswith a Statistical Enslaving of Fast-relaxing Variables9

We consider dissipative systems where the contraction In phase spacecorresponds asymptotically to the center-manifold-functional dependence of thefast-relaxing variables X on the order parameters given by the slow variablesXs. We derive analyticaly the Green-function sensitivity matrix to calculate.the response on the subordinated variables to nstantaneous perturbations onXf or on Xs . We thus show how these perturbations propagate along thecenter manifold. This information Is obtained from sensitivity measurementson the experimentally relevant variables Xs's using also the center-manifoldexpansion. The results ar applied to a randomly driven damped anharmonicoscillator. The spectral density 18 found by Fourier transforming the sum ofthe different contributions to the equilibrium correlation: The X&Xg,XAJg, andXJ44 terms. The average local attractvity of the center manifold Is calculatedand used In this computation. The sensitivity functions involving only thesubordinated degrees of freedom are used as fundamental propagators tocalculate the X.,g term. The results are tested against an analog computersimulation and via-a-via plots obtained from statistical linearization methods.A very good agreement Is observed.

10. Atom Scattering from Disordered Surfaces: Randomly corrugated HardWalls and the Sudden Approximationl °

We used a randomly corrugated bard well model and the suddenapproximation to analyze two experiments on atom scattering from disorderedsurfaces. In one, the structural surfce disorder was caused by ionbombardment. In the other, the disorder was due to an incomplete ovedayerof adsorbed atoms. We also present a study of the scattering of a rigidrotor by a randomly corrugated hard wall using the sudden approximation.

11. The Sudden Approximation for Scattering from NoncrystallineSurfaces: Applications to Models of Adsorbed Impurities and toMixed OverlayersI L

It was recently proposed that the sudden approximation should be apowerful tool for the calculation of the angular Intensity distribution Inhigh-energy aton scattering from one- and two-dimensional models of: (1)Isolated adsorbed impurities on crystalline surfaces (Ar on Cu); (2) Mixedoverlayers on an underlying surface (Xe+Ar mixtures on a smooth surface).The results are tested against numerically exact quantum-mechanioal wavepacket calculations. Except for very low collision energies, the suddenapproximation gives results of excellent quantitative accuracy for both types ofnonorystalline surfaces. At low energies, several features of the intensitydistribution are not produced correctly by the sudden: These are found to bedue mainly to double collision effects. The accuracy and validity range of themethod are discussed in the light of the results obtained in the testcalculations.

12. Variational Methods for Approximating Solutions of V2 u(x) f(x) +k(x) u(x)and Generalizations

1 2

The Itarattve procedure of Guy, Sales, Brami-Depaux, andJoly-Cabaret (GSBJ) for approximately solving equations of the formV2 u--f+ku is analyzed and generalized from the viewpoint of thecalculus of variations. The GSBJ method involves a functional of thesolution and either one or two auxiliary functions which are chosen empiricallyso an to make the Iterations converge suMciently fast, even for problems forwhich the corresponding Neumann series diverges or converges too slowly. Inthe case in which one of the auxiliary functions Is identically unity, thisfunctional is seen to be the scale-invariant form of the affine functionalstudied by Kahan, Rideau, Roussopoulos, and Pomraning (KRR)), and asystematic way of seeking appropriate auxiliary functions Is developed bydemanding that the functional be stationary about the true solution of theequation in question. The variational approach is tested on three simpleexample problems, and the numerical results are seen to compare favorablywith those of 05B3 and with the exact solution. In addition, two newvariational functlonals are derived, various iteration schemes are discussed,the method is extended to more general equations, and further connectionswith other work are established.

13. Average Wave Function Method for Gas-surface Scattering 1 3

The average wave function method (AWM) is applied to scattering of agas oft a solid surface. The formalism is developed for both periodic as wellas disordered surfaces. For an ordered lattice an explicit relation is derivedfor the Bragg peaks along with a numerical lustration. Numerical results arepresented for atomic clusters on a flat hard wall with a Gaussian-ike potentialat each atomic scattering site. The effect of relative lateral displacement oftwo clusters upon the scattering pattern is shown. The ability of AWM toaccommodate disorder through statistical averaging over cluster configurationsis illustrated. Enhanced uniform backscattering is observed with increasingroughness on the surface.

14. Local and Global Parametric Analysis of Reacting Flows 1 4

This paper presents a selected overview of current techniquesavailable for analyzing parametric behavior of dynamical systems includingreactive flows. The paper is based on a series of lectures presented atCornen University, and the discussion is broadly divided into the realms oflocal and global techniques. Local procedures are capable of addressing arich variety of questions associated with the behavior of dynamical systemsnear a chosen operating point In parameter space. On the other hand,global procedures are specifically designed to map out solution behavior overwide ranges of parameter space for both statistical analysis as well asobjective function mapping. Techniques for accomplishing all of these tasksare in various stages of development and comments on current researchdirections are mentioned when appropriate.

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15. Dynamics and Kinetics on Surfaces Exhibiting Defects's

This paper considers dynamics and idnetic phenomena on surfaces withdefect structures. Quantum mechanical scattering as well as Idnetics anddiffusion processes are treated as examples. In the case of quantummechanics, a theoretical formulation is presented capable of handling atomicdisorder on surfaces. The approach is based on an approximation which is-best In the quantum long wave-length regime. Some simple illustrationsconsidering scattering off ordered and disordered lattices will be presented.The second part of the paper considers the steady state non-linearreaction-diffusion equations used to describe adsorption, desorption, diffusionand reaction on surfaces with macroscopic scale defects. These latterdefects may arise due to inherent faulting of the lattice or foreign material onthe surface. Defects within a mean diffusion length of each other are shownto exhibit cooperativity in their chemical properties. Finally, in the lastportion of the paper reaction-diffusion models are again considered from adifferent perspective. In this case, all of the non-linear chemical ordesorptive aspects of the problem are restricted to the edges of the defectsites and the intervening surface is assumed to be characterized byadsorption, desorption and simple diffusion. These physically realistic modelsclearly show the capability of multiple steady states existing on active chemicalsurfaces. A variety of non-linear surface phenomena could possibly beimportant in appropriate gas-surface systems.

Publications Resulting from AFOSR Support:

1. Semiclassical Perturbation Theory for Atom Scattering fromSurfaces with Defects, L.M. Hubbard and H. Rabltz, J. Chem.Phys. (to be published).

2. Atom Molecule Scattering with the Average Wavefunction Method,H. Singh, D.K. Dacol, and H. Rabitz, Chem. Phys., 1987 (in press).

3. Applications of Filtering Theory for the Inversion of Temporal ChemicalSystems, J.G.B. Beumee and H. Rabitz, J. Chem. Phys. 1987(in press)

4. An Application of Filtering Theory to Parameter Identification toQuantum Mechanics, J.G.B. Beumee and H. Rabltz, J. Math. Phys.1987 (in press).

5. Time-dependent Resonant Fluorescence Spectrum of Two-level Atoms inStrong Short Pulsed Fields: A Nonperturbative Treatment, T-S. Hoand H. Rabitz, Phys. Rev. A, 1987 (submitted).

6. Sensitivity Analysis of Combustion Systems, H. Rabitz, TheMathematics of Combustion, J. Buckmater, editor, (SAM,Philadelphia, 1985).

7. Vibrational Energy Transfer at the Gas-solid Interface: The Role ofCollective and of Localized Vibrational Modes, E. Vllallonga andH. Rabitz, J. Chem. Phys., 85, 2300 (1986).

8. Autocorrelations in the Center Manifold of Dissipative Systems,A. Fernandez and H. Rabitz, Phys. Rev. A, 33, 3314 (1986).

9. Fundamental Sensitivity Propagators in Dissipative Systems with aStatistical Enslaving of Fast-relaxing Variables, A. Fernandez andH. Rabttz, Phys. Rev. A, 33, 1913 (1986).

10. Atom Scattering from Disordered Surfaces: Randomly Corrugated HardWalls and the Sudden Approximation, D.K. Dacol, H. Rabitz, andR.B. Gerber, J. Chem. Phys., 86, 1616 (1987).

11. The Sudden Approximation for Scattering from Noncrystalline Surfaces:Applications to Models of Adsorbed Impurities and to Mixed Overlayers,A.T. Yinnon, R.B. Gerber, D.K. Dacol and H. Rabitz, J. Chem.Phys., 84, 5955 (1986).

12. Variational Methods for Approximating Solutions of V 2 u(x) =fx) +k(x) u(x)and Generalizations, M. McClendon and H. Rabitz, SIAM J. AppL.Math., 46, 525 (1986).

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13. Average Wave Function Method for Gas-surface Scattering, H. Singh,D.K. Dacol and H. Rabitz, J. Chem. Phys., 84, 1852 (1986).

14. Local and Global Parametric Analysis of Reacting Flows, H. Rabitz,Physica 20D, 67-81 (1986).

15. Dynamics and Kinetics on Sirfaces Exhibiting Defects", H. Rabitz,Dynamics on Surfaces, B. Pullman et al, Editors (D. Reidel PublishingCompany, 77-88, 1984).

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