Materials Science and Technology of Optical Fabrication

By Tayyab I. Suratwala

Covers the fundamental science of grinding and polishing by examining the chemical and mechanical interactions over many scale lengths

Manufacturing next generation optics has been, and will continue to be, enablers for enhancing the performance of advanced laser, imaging, and spectroscopy systems. This book reexamines the age-old field of optical fabrication from a materials-science perspective, specifically the multiple, complex interactions between the workpiece (optic), slurry, and lap. It also describes novel characterization and fabrication techniques to improve and better understand the optical fabrication process, ultimately leading to higher quality optics with higher yield.

Materials Science and Technology of Optical Fabrication is divided into two major parts. The first part describes the phenomena and corresponding process parameters affecting both the grinding and polishing processes during optical fabrication. It then relates them to the critical resulting properties of the optic (surface quality, surface figure, surface roughness, and material removal rate). The second part of the book covers a number of related topics including: developed forensic tools used to increase yield of optics with respect to surface quality (scratch/dig) and fracture loss; novel characterization and fabrication techniques used to understand/quantify the fundamental phenomena described in the first part of the book; novel and recent optical fabrication processes and their connection with the fundamental interactions; and finally, special techniques utilized to fabricate optics with high damage resistance.

• Focuses on the fundamentals of grinding and polishing, from a materials science viewpoint, by studying the chemical and mechanical interactions/phenomena over many scale lengths between the workpiece, slurry, and lap
• Explains how these phenomena affect the major characteristics of the optic workpiece—namely surface figure, surface quality, surface roughness, and material removal rate
• Describes methods to improve the major characteristics of the workpiece as well as improve process yield, such as through fractography and scratch forensics
• Covers novel characterization and fabrication techniques used to understand and quantify the fundamental phenomena of various aspects of the workpiece or fabrication process
• Details novel and recent optical fabrication processes and their connection with the fundamental interactions

Materials Science and Technology of Optical Fabrication is an excellent guidebook for process engineers, fabrication engineers, manufacturing engineers, optical scientists, and opticians in the optical fabrication industry. It will also be helpful for students studying material science and applied optics/photonics.

• Language: English
• Publisher: Wiley
• Release date: Jul 30, 2018
• ISBN: 9781119423782

Book preview

Preface

The objective of this book is to re‐examine the age‐old field of optical fabrication from a materials science perspective. Optical fabrication is the manufacture of optical elements such as passive optics – for example, lenses, transmission flats, mirrors, and prisms – and active optics – for example, laser gain media, frequency converters, polarizers, and adaptive optics. These are crafted in a variety of shapes, sizes, and materials. The improved manufacturing of next‐generation optics has been instrumental in boosting the performance of advanced lasers and imaging and spectroscopy systems. The interdisciplinary field of materials science and engineering explores how fabrication processes influence material structures and chemistry, and hence their properties and performance.

Optical fabrication dates back to Roman times, with the first use of metal convex mirrors and spherical glass optics [1]. Owing to the vast possibilities in workpiece materials, shapes, polishing materials, techniques, and process variables, the number of feasible fabrication recipes is essentially infinite. Historically, this field has been primarily an art in which a master optician developed a unique set of effective processing techniques through years of apprenticeship and trial and error. While there have been significant advances in the scientific understanding of optical fabrication over the past decades, this field has yet to reach the maturity required for full control, via a truly deterministic process, of the surface characteristics of workpieces made of various optical materials.

A wealth of optical fabrication books provide excellent guidance and references as to materials, grinding and polishing, processing methods, tools, optical‐metrology techniques, and optical design [1–19]. Despite the seminal work of researchers in the past 40 years; however, notably Brown in the 1970s [10], Izumitani in the 1980s [16], Jacobs, Lambropoulos, and Cook in the 1990s [20–45], and Dornfeld and coworkers in the 2000s [46–50], a recent comprehensive review of optical fabrication has been lacking. The time is ripe for a single‐source reference on this topic.

In the related field of chemical–mechanical planarization (CMP), which concerns the fabricating of integrated circuits, many reference works are available – some of which cover the materials science aspects of the optical‐fabrication process [13–15, 17, 51–53]. However, because CMP has important differences from optical fabrication with respect to materials, amount of material removed, and specifications, the topic warrants a separate treatment.

Optical‐fabrication processes, all of which involve grinding and polishing, share a basic set of interactions or phenomena among their three component parts, which are

the workpiece – that is, the optic to be manufactured

the lap or tool, which determines the time‐ and spatially dependent mechanical loading of the workpiece

the slurry or lubricant, which may contain particles to remove material from the workpiece

These fabrication components interact as to their mechanics, tribology, physics, and chemistry to affect the four characteristics of optical fabrication (surface figure, surface quality, surface roughness, and material removal rate). We present these fundamental interactions in an organized, systematic, quantitative way, with illustrations of proposed models using experimental data. We also discuss the research and development in these basic elements that has led to novel characterization methods, fabrication techniques, and processing recipes not previously collected.

This book contains two parts. Part I: Fundamental Interactions – Materials Science begins with an introduction in Chapter 1. Chapters 2-5 discuss the mechanical and chemical phenomena in play during optical fabrication and their quantitative effects on surface figure, surface quality, surface roughness, and material removal rate. We begin with macroscopic interactions, such as those affecting surface figure, and move to the microscopic and molecular interactions that affect surface quality, surface roughness, and material removal rate. Numerous mechanisms and models have been proposed for the fundamental interactions, many of which are covered in this book. However, the aim of the book is to illustrate these concepts and to show details of some of the models (some of them from our own research group) in a standardized and organized fashion, as opposed to comprehensively describing the details of all the existing models.

Part II: Applications – Materials Technology introduces related topics that are tightly linked to the fundamentals in Part I. Chapter 6 discusses diagnosis and prevention of fracture‐related failures, both catastrophic and those affecting surface quality, to improve optical fabrication yield. Chapter 7 describes novel supporting fabrication and characterization techniques to improve and better understand optical‐fabrication processes. Novel optical fabrication processes are examined in Chapter 8. Chapter 9 covers how these techniques have been applied toward the fabrication of high‐energy laser optics with high damage resistance.

This book is both an educational reference for students and scientists and a practical handbook for opticians, process engineers, and technicians in industry. Thus, Part I is in written textbook‐style, describing basic physical principles and theoretical backgrounds. Many of these principles are discussed vis‐à‐vis brittle glass and ceramic workpieces. For fused silica and other glasses, quantitative examples are emphasized, as these materials have been the model systems in much of the research and development. Part II is a handbook for improving optical‐fabrication processes. For example, the scratch forensics section in Chapter 6 is a useful tool set for the diagnosis of defects (i.e. scratches) and increasing yield.

Despite much recent progress in the ancient field of optical fabrication, it is still a young science. This book is offered as an overdue technological summary, reference, and compilation of advancements. It is hoped that the information in this volume will spur further scientific progress, pushing the envelope of optic performance and applications.

Lawrence Livermore National Laboratory, 2018

Tayyab I. Suratwala

References

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22 Lambropoulos, J.C., Fang, T., Funkenbusch, P.D. et al. (1996). Surface microroughness of optical glasses under deterministic microgrinding. Appl. Opt. 35 (22): 4448–4462.

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25 Shafrir, S.N., Lambropoulos, J.C., and Jacobs, S.D. (2007). Subsurface damage and microstructure development in precision microground hard ceramics using magnetorheological finishing spots. Appl. Opt. 46 (22): 5500–5515.

26 Cerqua, K.A., Lindquist, A., Jacobs, S.D., and Lambropoulos, J. (1987). Strengthened glass for high average power laser applications. SPIE: Conference on New Slab and Solid‐State Laser Technologies and Applications, Volume 0736, pp. 13–21.

27 Guo, S., Arwin, H., Jacobsen, S.N. et al. (1995). A spectroscopic ellipsometry study of cerium dioxide thin films. J. Appl. Phys. 77 (10): 5369–5376.

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29 DeGroote, J.E., Marino, A.E., Wilson, J.P. et al. (2007). Removal rate model for magnetorheological finishing of glass. Appl. Opt. 46 (32): 7927–7941.

30 DeGroote, J.E., Jacobs, S.D., Gregg, L.L. et al. ed. (2001). Quantitative characterization of optical polishing pitch. International Symposium on Optical Science and Technology. International Society for Optics and Photonics.

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32 Golini, D., Jacobs, S.D., Kordonski, V., and Dumas, P. ed. (1997). Precision optics fabrication using magnetorheological finishing. Proceedings Volume 10289, Advanced Materials for optics and precision structures: A Critical Review.

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36 Kordonski, W.I. and Jacobs, S. (1996). Magnetorheological finishing. Int. J. Mod. Phys. B 10 (23–24): 2837–2848.

37 Lambropoulos, J.C., Miao, C., and Jacobs, S.D. (2010). Magnetic field effects on shear and normal stresses in magnetorheological finishing. Opt. Express 18 (19): 19713–19723.

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44 Arrasmith, S.R., Kozhinova, I.A., Gregg, L.L. et al. ed. (1999). Details of the polishing spot in magnetorheological finishing (MRF). SPIE's International Symposium on Optical Science, Engineering, and Instrumentation. International Society for Optics and Photonics.

45 Jacobs, R.R. and Weber, M.J. (1976). Dependence of the ⁴F3/2‐⁴I11/2 induced‐emission cross section for Nd³+ on glass composition. IEEE J. Quantum Electron. QE‐12 (2): 102–111.

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48 Luo, J. and Dornfeld, D.A. (2003). Effects of abrasive size distribution in chemical mechanical planarization: modeling and verification. IEEE Trans. Semicond. Manuf. 16 (3): 469–476.

49 Wang, C., Sherman, P., Chandra, A., and Dornfeld, D. (2005). Pad surface roughness and slurry particle size distribution effects on material removal rate in chemical mechanical planarization. CIRP Ann. Manuf. Technol. 54 (1): 309–312.

50 Moon, Y. (1999). Mechanical aspects of the material removal mechanism in chemical mechanical polishing (CMP). ProQuest Dissertations and Theses, thesis PhD. University of California, Berkeley, CA.

51 Li, Y. (2007). Microelectronic Applications of Chemical Mechanical Planarization. Wiley.

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Acknowledgments

This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory (LLNL) under Contract DE‐AC52‐07NA27344. Special thanks to the LLNL midcareer award program for providing the funding and bandwidth needed to complete this book, and the Laboratory Directed Research and Development (LDRD) program for funding much of the research presented.

Thanks to Wiley and the American Ceramic Society for considering the important topic of optical fabrication for publishing and to the editors, graphic artists, and reviewers (especially Margaret Davis, Brian Chavez, and James Wickboldt).

Special thanks to the many coresearchers, collaborators and mentors, especially the LLNL coresearchers Rusty Steele, Mike Feit, Phil Miller, and Lana Wong, who joined this 15‐year journey to unwind the mysteries and complexities of the fascinating field of optical fabrication. Their contributions greatly enriched this book. Finally, and most important, a debt of gratitude to my parents, who set the foundation of what I am today, and to my family: Maleka and my kids (Fatima, Maryam, and Aamina), who provided endless support and tolerated the endless hours.

Glossary of Symbols and Abbreviations

Symbols (with most common units and/or values in parenthesis).

Physical Constants

e el charge of electron (−1.602 × 10 −19 C) g gravitational acceleration (9.80 m s −2 ) h Planck's constant (6.626 × 10 −34 J s) k b Boltzmann's constant (1.38 × 10 −23 J K −1 ) R gas constant (8.314 J mol −1  K −1 ) v e electron absorption frequency (3.1 × 10 ¹⁵ s −1 ) ε o permittivity in vacuum (8.854 × 10 −12 F m −1 )

Variables

a Hertzian contact radius (nm) a ij Hertzian contact radius between two bodies ( i and j ) (1 for workpiece, 2 for lap, and 3 for particle) (nm) a j Hertzian contact radius for particle j (nm) a 0 vertical displacement of workpiece (μm) a 2 workpiece curvature (μm) a c crack flaw size (μm) average contact zone radius due to plastic removal (nm) average contact zone radius due to chemical removal (nm) a KDP lattice parameter for KDP crystal (Å) a DKDP lattice parameter for DKDP crystal (Å) A Auerbach's constant for Hertzian crack initiation (N m −1 ) A i slurry island area; subscript i is a descriptor for each individual island (μm ² ) A s area of a single surface site (nm ² ) A H Hamaker constant (J) A o pre‐exponent constant (multiple units) A p cross‐sectional area of the plastic nanoscratch (nm ² ) A f area fraction of pitch buttons A sf fracture surface area of a fractured workpiece (cm ² ) A pad contact area between workpiece and lap (mm ² ) A m fracture mirror marking constant for a given material (MPa m ¹/² ) b empirical constant for single bond strength correction in MRF removal equation B friction constant for trailing indent cracks B i index of brittleness of a material (MPa m −1/2 ) B nd nanodiamond constant (μm ⁴/³ ) B CI carbonyl ion constant (μm ⁴/³ ) c crack depth (μm) c i crack depth ( i = h for Hertzian fracture, i = ℓ for lateral fracture, i = r radial fracture, i = t trailing indent fracture) (μm) mean crack depth (μm) c 90 crack depth for 90% probability of removing (μm) c max maximum crack depth or SSD depth (μm) C L lap wear constant (μm h −1 ) C concentration of impurities (cm −3 ) C o pre‐exponential constant (m s −1 ) C K K concentration (atoms cm −3 ) [Ce]s steady state Ce concentration at the surface (atoms cm −3 ) C p1 heat capacity of workpiece (J kg −1  K −1 ) C p2 heat capacity of lap (J kg −1  K −1 ) C pb linear rate of increase in deflection with area fraction for PBB (μm) C nd nanodiamond concentration (cm −3 ) C CI carbonyl‐iron concentration (cm −3 ) d particle diameter or effective abrasive or polishing particle diameter (μm) d a pre‐exponential constant for workpiece roughness – slope of PSD relationship (μm) mean abrasive or polishing particle diameter (μm) d b spacing between pitch buttons (mm) d c critical depth of cut (μm) d e effective distance from edge of workpiece (μm) d g spacing between grooves on pad (mm) d M moment arm distance from spindle pivot to workpiece–lap interface plane (mm) d m chemical removal depth on workpiece by a polishing particle (nm) average chemical removal depth on workpiece by all polishing particles (nm) d p nanoplastic removal depth on workpiece by a polishing particle (nm) average nanoplastic removal depth on workpiece by all polishing particles (nm) d s stroke amplitude distance (cm) d set settling distance (cm) d t total depth of penetration of particle into the workpiece and lap (nm) d ij depth of penetration between two bodies ( i and j ) (1 for workpiece, 2 for lap, and 3 for particle) (nm) d j depth of removal of a given particle j d PSD inverse exponential slope of the tail end in the slurry PSD (μm) d rem depth of removal from a workpiece from slurry islands (nm) D c diameter of etch cusp (μm) D r ribbon penetration depth (μm) D co initial cusp or crack size (μm) D impurity cation diffusivity (cm ²  s −1 ) D s(pHMRF) percent weight loss of the glass in the MR fluid supernatant as a function of MRF fluid pH E a, E b modulus components for viscoelastic lap (GPa) E eff effective modulus of three‐body system (GPa) E ij composite modulus between two bodies ( i and j ) (1 for workpiece, 2 for lap, and 3 for particle) (GPa) E i elastic modulus ( i = 1 for workpiece, i = 2 for lap, i = 3 for particle) (GPa) E rel stress relaxation function for the viscoelastic lap material (GPa) E s elastic modulus of surface layer (GPa) E sbs single bond strength (kJ mol −1 ) E r activation energy of the relative rates hydrolysis ( r Ce:Si ) (kJ mol −1 ) ΔE activation energy for temperature dependence on removal rate (kJ mol −1 ) f fill fraction of particles between workpiece and lap f s fraction of the lap circumference loaded by the septum f p fraction of particles contributing to nanoplastic removal f m fraction of particles contributing to chemical removal f r fraction of active particles at interface f L fraction of applied load carried by particles (rather than by pad directly) f ∞ fill fraction of particles f(r) fractional slurry PSD f A fraction of pad area making contact with workpiece f redep fraction of the glass products removed from the workpiece that deposits on a particular position of the lap f island(h) slurry island height distribution f load(c) fraction of particles being loaded for a given fracture depth f pad(h) pad height distribution of the pad topography F pad(h) cumulative height distribution of the pad topography f o(c) instantaneous or fundamental crack depth distribution f c(c) final incremental distribution of cracks F c(c) final cumulative crack depth distribution f o(r) fraction of the lap circumference loaded by the workpiece as a function of radial distance ( r ) from lap center f s(r) fraction of the lap circumference loaded by the septum as a function of radial distance ( r ) from lap center F ij force between two bodies ( i and j ) (1 for workpiece, 2 for lap, and 3 for particle) (N) F a applied force during cleaning (N) F vdw van der Waals force (N) F electrostatic electrostatic image force (N) F double electrical double layer force (N) F capillary capillary force (N) F gravity gravitation force (N) F, F x , F y friction force (N) g p, g pi interface gap or interface gap at pad asperity height i (μm) g b(d) PSD of the base slurry abrasive g r(d) PSD of the rogue particles h ij heat transfer coefficient from various surfaces ( i = 1 for workpiece, i = 2 for lap, j = t for top, j = b for bottom, and j = s for side) h L(x) relative height of the lap normalized to zero at the ends (μm) h PV peak‐to‐valley height of the full lap (μm) h o height of the workpiece or height of dome asperity (μm) h c height at the center of the workpiece (μm) h f pad compression or slurry island height upon workpiece loading (μm) h p particle penetration into lap (μm) h m maximum height of dome surface feature (nm) h, h i height of pad asperity, slurry island or dome (μm) h t heat transfer coefficient (W m −2  K −1 ) Δh oL(x) workpiece–lap mismatch gap (μm) Δh oL* gap at the point of maximum pressure on the workpiece (μm) constant describing the rate at which pressure drops with increase in workpiece–lap mismatch (μm) h vdw van der Waals constant (eV) dh/dt average thickness material removal rate on workpiece (μm h −1 ) dh i/dt instantaneous thickness material removal rate on workpiece (μm h −1 ) dh lap/dt thickness material removal rate on lap (μm h −1 ) ΔH optic tilt height (μm) H i hardness of workpiece (GPa) ( i = 1 for workpiece, i = 3 for particle) H k Knoop hardness of the workpiece (MPa) [H]s H concentration at surface (atoms cm −3 ) IEPs isoelectric point of slurry particle J z flux of particles contacting surface (m −2  s −1 ) J gp glass reaction product flux into workpiece–lap interface (mol m −2  s −1 ) J diffusion flux of impurities into workpiece (m −2  s −1 ) J material removal flux of impurity removal during polishing (m −2  s −1 ) J s(r) evaporative flux of a drying fluid droplet (mol s −1  cm −2 ) J(t) creep compliance function for viscoelastic lap material (GPa −1 ) k, k′ constant relating material constants of two different materials (unitless or GPa −1 ) ²⁵ K a equilibrium reaction constant for HF dissociation (mol l −1 ) ²⁵ K equilibrium reaction constant for HF ²− formation (mol l −1 ) K I stress intensity (MPa m ¹/² ) K Ic fracture toughness (MPa m ¹/² ) K s1, K s2 equilibrium constants for change in surface hydroxyl surface charge (mol l −1 ) K f constant for fail‐safe design of pressure differential windows (MPa m ¹/² ) k c growth constant for etch cusps (μm ²  s −1 ) k i thermal conductivity ( i = 1 for workpiece, i = 2 for lap) (W m −1  K −1 ) k p Preston coefficient for workpiece (m ²  N −1 ) k lap Preston coefficient for lap (m ²  N −1 ) k olap Preston coefficient for lap at initial use of pad (m ² N −1 ) k max proportionality constant between surface roughness and crack depth K sp spring constant of AFM tip [K]s K concentration at surface (atoms cm −3 ) ℓ o characteristic dimension (diameter or side length) of workpiece (m) L characteristic length for dome convergence during tumble finishing (μm) L max maximum crack length (μm) L t trailing indent fracture crack length (same as a scratch width) (μm) mean crack length (μm) mean length of scratch (μm) LH lapping hardness (MPa −2 m −1/2 ) m, n nonnegative integers used for describing Zernike polynomials with n ≥ m m s constant in exponent describing influence of material properties to SSD M, M x , M y moment force (N m) Ms surface metal atom M wp, M p surface metal atom of workpiece (wp) or slurry particle (p) M gp molecular concentration of glass reaction products at the workpiece–lap interface (mol l −1 ) MWgp molecular weight of glass reaction products (gm mol −1 ) n s areal number density of cracks at surface (cm −2 ) n ps number of nanoscratching passes n o counter ion concentration (cm −3 ) n i refractive index ( i = 3 for colloidal particle, i = 4 for liquid medium) N number of discrete, equally spaced measured points along a surface N b number of pitch buttons on workpiece N c power constant for empirical slow crack growth relationship N L number of abrasive particles being loaded (i.e. active particles) N t areal number density of particles at the interface (cm −2 ) N T total number of particles between workpiece and lap O(c) SSD depth distribution represented as obscuration versus crack depth ( c ) p water vapor partial pressure (Pa) p o saturation water vapor pressure (Pa) p ad probability of an incident particle adsorbing to the surface p si stoichiometry of atom i P applied load (N) P b applied load for blunt impact (N) P s applied load for sharp impact (N) P ci critical load to initiate indentation fracture ( i = h for Hertzian fracture, i = ℓ for lateral fracture, i = r for radial fracture, i = t for trailing indent fracture, and i = e for edge fracture) (N) Pe Peclet number PV peak‐to‐valley height (μm) P tw Twyman stress (N m −1 ) q d surface charge density of particle (C m −2 ) q(r) heat generation rate per unit area (W m −2 ) Q frictional heat flux (J m −2  s −1 ) Q I activation energy for hydrolysis reaction (kJ mol −1 ) Q II activation energy for H 2 O transport (kJ mol −1 ) Q r constant used in calculating SSD distribution with rogue particles r particle, island, or surface radius (μm) or radial distance (μm) average particle radius (nm) r i , r j radius of curvature of body ( i and j ) (1 for workpiece, 2 for lap, and 3 for particle) r A pre‐exponential constant r arc arc radius; distance from lap center to a given leading edge point ( x L , y L ) on workpiece (mm) r b bulk thickness etch rate on a flat surface (μm h −1 ) r L radius of lap (cm) r m mirror marking radius on fracture surface (μm) r o radius of workpiece (cm) r p radius of pitch button (mm) r Ce:Si hydrolysis reaction rate ratio between Ce–O–Ce and Si–O–Si R p radius of a pinned droplet on a drying surface (mm) R L, lap rotation rate (rpm) R o, optic rotation rate (rpm) R s stroke rotation rate (rpm) R ts thermal shock FOM (K) terms for describing Zernike polynomials s, separation distance from workpiece center and lap center (cm) s c crack separation distance (μm) S p areal density of Si atoms in fused‐silica glass (nm −2 ) t time (min) t f time‐to‐failure due to slow crack growth (h) t i thickness ( i = 1 for workpiece, i = 2 for lap) (mm) t s thickness of surface layer (mm) t j characteristic etch times ( j = a, b, c) (min) t Beilby effective thickness of the Beilby layer (nm) t Lap removal rate decay constant for lap material removal rate (h) t L(x,y) time of lap exposure at point x , y on the workpiece for the corresponding point on the lap (s) T temperature (K) T i initial temperature (K) T f final temperature (K) T ave average temperature in the bulk (K) T surf surface temperature (K) ΔT temperature difference, T ave − T surf (K) T g glass transition temperature (K) T e edge toughness (N mm −1 ) U work function difference between particle and workpiece (mV) v c slow crack growth velocity (m s −1 ) v I slow crack growth velocity in Region I (m s −1 ) v II slow crack growth velocity in Region II (m s −1 ) v 3 impacting particle velocity (m s −1 ) v 3bc critical velocity for fracture initiation for blunt projectile (m s −1 ) v 3sc critical velocity for fracture initiation for sharp projectile (m s −1 ) v r local relative velocity (m s −1 ) V a activation volume (m ³  mol −1 ) V L volume of workpiece (cm ³ ) average relative velocity (m s −1 ) relative velocity of septum (m s −1 ) w o initial width of Gaussian dome (FWHM) (μm) w width of Gaussian dome (μm) w c width of crack (μm) w 1 workpiece deflection (μm) w max maximum workpiece deflection (μm) w p width of the plastic nanoscratch (nm) W t interaction energy between colloidal particles (J) x r number fraction of rogue particles x L,y L point on the leading edge of the workpiece Y geometric constant for cracks z separation distance (nm) z c valence of counter ion z o local surface height above or below the mean height of the surface (nm) z g thickness removed from workpiece during etching (μm) z p Zernike polynomials z s scaling factor for impact damage as function of projectile and workpiece hardness dz g/dt thickness removal rate during etching (μm h −1 ) Z interaction constant used in DLVO model (J m −1 ) Z sf constant relating fracture surface area to peak stress and workpiece volume (cm ²  psi −2  L −1 ) α stress parameter to determine between finite and infinite scratch length α 1 thermal diffusivity of the workpiece (m ²  s −1 ) α2–3 combined thermal diffusivity of the lap and slurry (m ²  s −1 ) α t1 thermal expansion coefficient of the workpiece (K −1 ) α Al thermal expansion coefficient of aluminum (K −1 ) α K numerical factor in the range 0.03–0.04 used to determine k max α r radial crack initiation constant α e edge angle on workpiece (°) α s fraction of the kinetic energy that is used to plastically deform surface α g groove height (μm) β geometric constant γ liquid media surface tension (J m −2 ) γ SG solid–gas interface surface energy (J m −2 ) γ SL solid–liquid interface surface energy (J m −2 ) γ LG liquid–gas interface surface energy (J m −2 ) γ f fracture surface energy (J m −2 ) Δ removal increment amount (μm) ΔPV workpiece deflection due to PBB (nm) Δz AFM tip displacement (nm) δ RMS surface roughness (nm) δ i partial charge of atom i δ s–wp partial charge difference between slurry particle and workpiece δ s partial charge of slurry particle δ wp partial charge of workpiece material δ PV PV surface roughness (μm) δ b baseline PV roughness used in 2D etch model (μm) δ o roughness constant for workpiece roughness – slope of PSD relationship (nm) δ e strain generated upon cooling a glass bilayer with different compositions ε strain on lap ε c elastic strain of lap at workpiece center lap strain rate due to loading by workpiece (s −1 ) ε i dielectric constant ( i = 3 for colloidal particle, i = 4 for liquid medium; i = r for interface media) η s fluid viscosity (Pa s) η 2 viscosity of viscoelastic lap material (Pa s) θ angle in polar coordinate system (°) θ a angle of applied force on particle (°) θ c contact angle of solvent droplet on a workpiece surface (°) θ x , θ y slopes of the workpiece in the x and y directions relative to the lap plane (°) θ L solid angle of the lap covered by the workpiece (°) ν i Poisson's ratio ( i = 1 for workpiece, i = 2 for lap, i = 3 for polishing or abrasive particle) ν rate of desorption of an adsorbed particle (s −1 ) κ inverse Debye length (nm −1 ) μ interfacial friction coefficient ρ radial distance for Zernike polynomials (mm) ρ c crack tip radius (nm) ρ i mass density ( i = 1 for workpiece, i = 2 for lap, i = 3 for polishing or abrasive particle) (gm cm −3 ) ρ f mass density of fluid (gm cm −3 ) ρ L radius of curvature of the lap surface (m) ρ o x , y position coordinate on the workpiece relative to workpiece center (mm, mm) σ, σ i local stress or pressure at workpiece–lap interface or local stress at pad asperity height i (Pa) σ f critical failure stress for fracture (Pa) σ o average applied stress on workpiece (Pa) σ part standard deviation in PSD (nm) σ p peak stress in workpiece (Pa) σ s surface stress in workpiece (Pa) σ sep applied pressure on septum (Pa) σ t thermal stress on workpiece (Pa) τ shear stress (Pa) τ c creep compliance time constant for viscoelastic lap (s) τ s stress relaxation time constant for viscoelastic lap (s) ϕ azimuthalangle (°) ϕ nd nanodiamond particle size (nm) ϕ CI carbonyl iron particle size (nm) χ i crack growth constant ( i = h for Hertzian fracture, i = ℓ for lateral fracture, i = r for radial fracture) (N) χ v term related to strain of a viscoelastic lap (unitless) χ i ο electronegativity of neutral atom i mean electronegativity of the compound Ψa abrasive angle (°) Ψ, Ψ i , Ψ j Stern potential, Zeta potential, or surface potential (1 for workpiece, 2 for lap, and 3 for particle) (mV) Ω factor relating crack length to crack depth ξ geometric constant for Vickers