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Journal of Non-Crystalline Solids 355 (2009) 797–810
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Journal of Non-Crystalline Solids
journal homepage: www.elsevier .com/ locate/ jnoncrysol
The effect of HF/NH4F etching on the morphology of surface fractures on fused silica
L. Wong *, T. Suratwala, M.D. Feit, P.E. Miller, R. SteeleLawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USA
a r t i c l e i n f o
Article history:Received 17 April 2008Received in revised form 20 January 2009Available online 10 April 2009
PACS:42.86.+b81.65.�b81.65.Cf
Keywords:GlassesFractureIndentation, microindentationOptical microscopyLasersSilica
0022-3093/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.jnoncrysol.2009.01.037
* Corresponding author. Tel.: +1 925 422 8178; faxE-mail address: [email protected] (L. Wong).
a b s t r a c t
The effects of HF/NH4F, wet chemical etching on the morphology of individual surface fractures (inden-tations, scratches) and of an ensemble of surface fractures (ground surfaces) on fused silica glass has beencharacterized. For the individual surface fractures, a series of static or dynamic (sliding) Vickers and Brin-nell indenters were used to create radial, lateral, Hertzian cone and trailing indentation fractures on a setof polished fused silica substrates which were subsequently etched. After short etch times, the visibilityof both surface and subsurface cracks is significantly enhanced when observed by optical microscopy.This is attributed to the increased width of the cracks following etching, allowing for greater optical scat-ter at the fracture interface. The removal of material during etching was found to be isotropic except inareas where the etchant has difficulty penetrating or in areas that exhibit significant plastic deformation/densification. Isolated fractures continue to etch, but will never be completely removed since the bottomand top of the crack both etch at the same rate. The etching behavior of ensembles of closely spacedcracks, such as those produced during grinding, has also been characterized. This was done using a sec-ond set of fused silica samples that were ground using either fixed or loose abrasives. The resulting sam-ples were etched and both the etch rate and the morphology of the surfaces were monitored as a functionof time. Etching results in the formation of a series of open cracks or cusps, each corresponding to theindividual fractures originally on the surface of the substrate. During extended etching, the individualcusps coalesce with one another, providing a means of reducing the depth of subsurface damage andthe peak-to-valley roughness. In addition, the material removal rate of the ground surfaces was foundto scale with the surface area of the cracks as a function of etch time. The initial removal rate for theground surface was typically 3.5� the bulk etch rate. The evolving morphology of ground surfaces duringetching was simulated using an isotropic finite difference model. This model illustrates the importancethat the initial distributions of fracture sizes and spatial locations have on the evolution of roughnessand the rate at which material is removed during the etching process. The etching of ground surfacescan be used during optical fabrication to convert subsurface damage into surface roughness therebyreducing the time required to produce polished surfaces that are free of subsurface damage.
� 2009 Elsevier B.V. All rights reserved.
1. Introduction
Fluoride based wet chemical etching can be accomplished byusing a variety of reagents including hydrofluoric acid (HF) or, un-der suitably acidic conditions, fluoride or bifluoride salts. The dis-solution of silicate glasses results in the formation of the stablehexafluorosilicate ðSiF2�
6 Þ anion [1–11]. When hydrofluoric acid isused, the overall reaction can be summarized as:
SiO2 þ 6HF! SiF2�6 þ 2H2Oþ 2Hþ: ð1Þ
The underlying mechanism of such reactions involves a number ofsteps and intermediate species. For example, in aqueous solution,
ll rights reserved.
: +1 925 423 0792.
HF acid acts as a weak acid where an equilibrium exists betweenthe un-dissociated acid, H+ ions and the F� anions:
HF! Hþ þ F� 25Ka ¼ 6:7 � 10�4 mol=l; ð2Þ
where 25Ka is the equilibrium constant at 25 �C [12].In the presence of un-dissociated HF, the fluoride ion reacts to
form the bifluoride anion ðHF�2 Þ which is thought to be a primaryspecies responsible for the attack of the silica matrix [2,11,12]:
HFþ F� ! HF�225K ¼ 3:96 mol=l: ð3Þ
A variety of reagents, so long as they produce both fluoride (F�) andhydrogen (H+) ions, can form the bifluoride ions in situ, and thus canbe expected to etch silicate glasses.
In the present study, solutions of buffered oxide etch (BOE)were used as the etchant. However, qualitatively similar resultswould be expected using other fluoride based etchants. BOE refers
798 L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810
to a series of commercially available mixtures of high purity, aque-ous-phase hydrofluoric acid (HF) and ammonium fluoride (NH4F)which were developed for use in the semiconductor industry[1,2,9,10]. In such solutions, NH4F dissociates completely providingan abundant source of F� ions which are free to react with the un-dissociated HF to form HF�2 ions (Eq. (3)). In addition, the smallfraction of dissociated HF (Eq. (2)) provides the H+ ions whichare known to catalyze the overall reaction [13].
Most contemporary studies of fluoride etching on glass surfaceshave been performed in the context of the semiconductor industry.Such studies most often concentrate on masking and dielectric filmapplications, and describe the evolution of masked silica surfaceswith etch time [1,2,14,15]. Significantly less work has been re-ported on the use of fluoride etching during optical fabrication.Preston [3] was among the first to examine how HF etching onground surfaces of glass changes the surface topology of glassand recognized the utility of acid etching as an inspection aid. Spi-erings [1] combined the results from the glass science and inte-grated circuit (IC) fabrication literature in a comprehensivereview that describes the etching mechanism, morphologicalchanges of glass surfaces with etching, and the effects of etchantand glass composition [1]. More recently, Zhou et al. [4] and thepresent authors [5–7], applied HF etching to reveal subsurfacecracks.
In the present work, we report on the effect that fluoride etch-ing has on the morphology and observability of three classes offractures: static indentation fractures (digs), sliding indentationfractures (scratches), and ensembles of fractures typical of groundfused silica surfaces. The present work suggests the use of etchingboth as an aid in the inspection process and as a means of remov-ing subsurface fractures/damage (SSD) introduced during the fab-rication and finishing of silicate based glasses or ceramics,particularly in those applications where SSD must be minimizedin the final product. Examples include the finishing of SSD free op-tics for use in high-peak-power laser systems [16] and applicationswhere material strength is of paramount importance, such as forwindows or barriers used in aerospace or deep-sea applications.
2. Experimental procedures
2.1. Etching of isolated fractures (static and sliding indentations)
A series of static and sliding indentations were created, underambient conditions, on the surface of polished fused silica (Corning7980) substrates and the morphology of these cracks was moni-tored as a function of BOE etch time. Static Hertzian (blunt) frac-tures were created on a polished, fused silica substrate (Corning7980; 7 � 7 � 0.7 cm3) using a 1 mm diameter stainless steel Brin-nell indenter using a Zwick microhardness tester (Model 3212)using loads of 39 and 49 N. A loading rate of 0.3 mm/s and a dwelltime of 60 s were used in each case. Static Vickers (sharp) indenta-tions were also created on the same substrate using a standardVickers diamond indenter at loads of 1, 2, 59, 78 and 98 N. The 1and 2 N loads were applied using a Shimadzu microhardness tester(Model HMV2) while the 59, 78 and 98 N loads were applied usingthe Zwick microhardness tester. In all cases, the loading rate wasnominally 0.2 mm/s and the dwell time was 60 s. Finally, a seriesof sliding blunt indentations [5,17]) were created by sliding an11 mm diameter stainless steel ball across the substrate surfaceat �1 cm/s with a nominal load of �90 N.
All etching was performed by affixing the substrate to a Teflon�kinematic mount. The entire assembly was then immersed in 20:1BOE (Air Products, Allentown, PA) for various etch times and sub-sequently removed from the etchant for characterization. The kine-matic mount provided a means of reproducibly mounting the
sample during subsequent characterization and metrology cyclesso that the same location of the substrate could be measured andcharacterized. This characterization included weighing of the sam-ple to the nearest 1 mg, recording the surface morphology, in bothreflection and transmission using optical microscopy (Nikon Optip-hot 300), and recording the surface topology using a stylus profi-lometer (KLA Tencor P-10) which has a vertical resolution of 1 Åand horizontal resolution of 1 lm. Based on the scale resolution,the uncertainty in the measured mass is ±1.4 mg. All measure-ments from the photomicrographs were made using Image J soft-ware (National Institutes of Health) and the uncertaintycalculated for each length measurement is ±0.42 lm. For theroughness measurements, the peak heights (and valleys) are 3–4orders of magnitude greater than the instrument resolution andthus, no formal error analysis was performed. The accuracy ofthe profilometer is dominated by the ability of the needle to pen-etrate into the narrow and deep cracks. This is discussed in detailin Section 4.1. For clarity, the error bars are not shown on the fig-ures in the results section.
2.2. Etching of ground surfaces
One of the faces of each round fused silica samples (Corning7980; 10 cm diameter � 1.0 cm thick; polished on all surfaces)was ground using either fixed or loose abrasive processes. The firstsample was ground using a Model II Blanchard (S/N 7769)equipped with a 150 grit (100 lm) diamond in a metal matrix tool(downward feed rate = 230 lm/min, rotation rate = 41 rpm,time = 20 s). The second sample was loose-abrasive ground on an8” borosilicate glass lap (load = 0.3 psi, rotation rate = 15 rpm,time = 1 h) using a 30 lm Al2O3 abrasive particle slurry (MicrogritWCA30T). The SSD distributions of the two ground surfaces (150grit, fine Blanchard and 30 lm, loose abrasive) were determinedby: (1) creating a shallow (�40–50 lm) wedge/taper on the sur-face by magneto-rheological finishing (MRF); (2) exposing theSSD by acid etching; and (3) performing image analysis of the ob-served cracks from optical micrographs taken along the surface ta-per. Details of this characterization technique are providedelsewhere, including the error analysis associated with the obscu-ration measurement [5–7]. A second set of the ground samples de-scribed above were etched for various times in 20:1 BOE andcharacterized in the same manner as described in Section 2.1.Based on the scale resolution and time measured to the nearestminute, the uncertainty in the measured mass removal rate andthe thickness removal rate are ±0.012 g/h and ±0.2 lm/h, respec-tively. Again, for clarity, the error bars are not shown on thefigures.
3. Results
3.1. Etching of static and sliding indentations
Fig. 1 shows optical micrographs and the surface profilometryof the static Brinnell and Vickers indentations before etching andas each indentation fracture evolved during a total of 18 h of etch-ing. The morphology of the surface before etching was typical ofthose expected following Brinnell (Hertizan cone cracks) and Vick-ers (radial and lateral cracks) indentation (see Fig. 2 and top ofFig. 1). In the absence of etching, the surface ring crack at 39 N isbarely visible (Fig. 1(a)). At a higher applied load of 49 N, the ringcrack is more discernible although the extent of the surface frac-ture in either case is not readily apparent (Fig. 1(b)). After even ashort etch time (30 min), corresponding to the removal of�500 nm of material, both the surface cracks and the overall depthand extent of the subsurface fractures resulting from each indenta-
(a) (b)
(c) (d)
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-60
-50
-40
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-10
0
Surf
ace
Prof
ile ( μ
m)
Position (μm)
100 μm
18 hrs
12 hrs8 hrs4 hrs2 hrs 0.5 hr0 hrs
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0
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ace
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ile (μ
m)
Position (μm)
100 μm
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ace
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ile (μ
m)
100 μm
0 100 200 300 400 0 100 200 300 400
0 100 200 300 400 0 100 200 300 400
-50
-40
-30
-20
-10
0
Surf
ace
Prof
ile ( μ
m)
100 μm
Position (μm) Position (μm)
Fig. 1. Optical micrographs and surface profilometry (bottom) of the fused silica surface indented by Brinnell or Vickers indentation, before (top) and after (middle) etching:(a) Brinnell indent at 39 N, (b) Brinnell indent at 49 N, (c) Vickers indent at 1 N and (d) Vickers indent at 2 N. Profilometry was performed for each indent at the same,corresponding etch times shown in (a). Note that the optical micrograph for each indent is shown at the same scale as the lineout in the surface profile. Alignment of eachindent to its surface profile is aided by the dashed lines.
L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810 799
(a) (b) (c) (d)
P
ch
r
Trailing Hertzian Indents
r
Movement
cth
PT
Lateral Cracks
P
b
c
P
cr2a
Radial Cracks
P
ch
r
2a
Hertzian Cracks
Fig. 2. Schematic illustration of the fracture geometry of the idealized fractures created by indentation: (a) Hertzian cone crack from a blunt indenter, (b) radial or mediancracks from a sharp indenter, (c) lateral cracks from a sharp indenter and (d) trailing Hertzian indents (scratches) from a sliding blunt indenter.
0.5 hr
2.37 mm
800 L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810
tion are much more apparent. It should also be noted that evenafter long etch times (18 h), each of the isolated fractures areclearly discernible. This is a result of the isotropic nature of theetching process where the bottom of the crack etches at nominallythe same rate as the sides of the crack.
Fig. 1(c) and (d) show the Vickers indentations at 1 and 2 N,respectively. Prior to etching, the presence of radial and lateralcracks, as well as a plastically deformed or densified zone, is dis-cernible. However, following etching in addition to the expectedradial and lateral cracks, Hertzian (cone) [17] cracks are also pres-ent in each case. This was similarly observed for the Vickers inden-tations created at the higher loads (Fig. 3). The anomalous presenceof such cone cracks, induced by a sharp indenter in fused silica, haspreviously been described by Roesler [18] and Hagan [19].
Finally, Fig. 4 shows the morphological changes of a series oftrailing indentation cracks (i.e. a scratch) as a function of etch time.At short etch times, the individual fractures remained isolatedfrom one another. At longer etch times, these cracks etch open toform cusps which can be described as rounded pits on the surfacehaving depths equivalent to the crack depth. With sufficient etch-
59N 78 N 98 N
½ hr
4 hrs
12 hrs
20 hrs
32 hrs
60 hrs
84 hrs
594 μm
Fig. 3. Optical micrographs of the Vickers indents as a function of etch time.
ing, the cusps, corresponding to each of the individual cracks, coa-lesce with one another to form a nearly continuous trench on thesurface of the optic. However, because this coalescence occurs onlyalong a single dimension, along the surface of the optic, the scratchremains prominent even after 120 h of etching.
40 hrs
120 hrs
0 400 800 1200 1600 2000-240
-180
-120
-60
0
Surf
ace
Prof
ile (μ
m)
Position (μm)
0.5 hr
120 hrs
96 hrs80 hrs64 hrs56 hrs40 hrs
Fig. 4. Optical micrographs and surface profilometry of a scratch (trailing indentcracks) after etching with BOE at various times. Note that the optical micrographsare shown at the same scale as the lineout of the surface profile. Points of referencealong the open scratch to the surface profile are given by the dashed lines.
(a) (b)
Surf
ace
Prof
ile
Position (μm) Position (μm)
306.5 hrs
240 hrs
152.5 hrs
84 hrs
60 hrs
38.5 hrs
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15 min
20 μm
237 μm
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15 min
0 200 400 600 800 1000 0 200 400 600 800 1000
Surf
ace
Prof
ile
20 μm
237 μm
4 hrs
16 hrs
24 hrs
32 hrs
40 hrs
48 hrs
Fig. 5. Optical micrographs and surface profilometry of two ground fused silica surfaces after etching with BOE for various times: (a) 150 grit fixed abrasive grind and (b)30 lm loose abrasive grind.
0
20
40
60
80
100
120
140
160150 grit generator grind 30 μm loose abrasive grind
Ave
rage
Cus
p D
iam
eter
, D (μ
m)
Etch Time, t1/2 (sec1/2)
200 400 600 800 1000 1200
Fig. 6. Measured cusp diameter vs. etch time1/2 for the ground fused silica surfacesetched in 20:1 BOE.
L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810 801
3.2. Etching of ground surfaces
Fig. 5 shows a series of optical micrographs and the surface pro-filometry scans that were taken at approximately the same loca-tion after various etch times for the two ground fused silicasurfaces. Fig. 5(a) describes the results for the etched fused silicasubstrate which was ground using the 150 grit fixed abrasive,while Fig. 5(b) describes the results for the sample which wasground using the 30 lm loose abrasive. Etching results in the for-mation of a series of individual cusps which are created as silicais etched from the sides and bottom of each fracture site. As etch-ing progresses, individual cusps coalesce with one another result-ing in the formation of successively larger cusps. As shown inFig. 6, one finds that the average cusp diameter scales with thesquare root of time. Unlike a scratch, ground surfaces consist ofan ensemble of closely spaced fractures that are randomly distrib-uted on the surface. As material is etched from the sides of individ-ual fractures, individual cusps intersect and coalesce with oneanother. Such coalescence increases both the spatial period of thesurface roughness and removes SSD.
The SSD depth distributions (cumulative crack obscuration,O(c), as a function of depth) for both the fixed and loose abrasiveground samples are shown in Fig. 7. The maximum SSD depthwas found to be �42 lm for the Blanchard-ground sample and�41 lm for the loose abrasive ground sample (see Table 1).
4. Discussion
4.1. Etching of isolated fractures
During the polishing of high quality optical components or highstrength glass or ceramic components, it is often desirable to deter-
0 10 20 30 40 5010-6
10-5
10-4
10-3
10-2
10-1
100
150 grit generator grind30 μm loose abrasive grind
Obs
cura
tion,
O(c
)
Crack Depth, c (μm)
Fig. 7. The subsurface damage (SSD) depth distributions for the 150 grit fixedabrasive ground and 30 lm loose abrasive ground samples.
Table 1Measured parameters for each grinding process used to model the etch behavior.
Indentation parameters Variable 150 Grit fineblanchard
30 lm Looseabrasive
SSD depth (lm)* c 42 41Maximum crack length (lm)* Lmax 94 134Average crack length (lm)* <L> 15 14Average crack width (lm)* w 4.4 4.4Average crack spacing (lm) s 22 17Pre-exponent for crack
distribution*
A 1.01 0.96
Annihilation time (h) t2 8 6Measured P–V roughness after
stabilization (lm)dPV 30 20
* Data from Ref. [5].
(a) (b)
Before Etching
Light scattering due to surface fractures.
After Etching
Cracksvisible
Bielby or redeposition layer, 50-500
Cracks not
visible
w = 2 r t
Fig. 8. (Top) edge or side-lit images of a polished fused silica surface, (a) before and(b) after etching. (Bottom) schematic showing the cross-section of the glass surfacecontaining hidden cracks prior to etching which open up after etching.
802 L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810
mine the origin of, or the extent of, scratches (sliding indentation)or digs (static indentation), which can occur during fabrication orthe handling process. The size and morphology of the indentationcracks combined with a knowledge of the fundamental aspects oftribology [20] and indentation fracture [17] can be utilized as a toolfor determining the nature of the indenter (sharp vs. blunt), thephysical size of the indenter, and the load which must have beenpresent during the indentation event [5–7]. Such information,when combined with specific process knowledge, can often beused to identify the process or handling steps during which specificflaws were introduced.
In this context the fluoride etching of fused silica has two ben-efits: (1) exposing surface cracks (Fig. 8) and (2) revealing the truemorphology of the cracks (Fig. 9(a)–(c)). For example, a short etchof a glass surface typically reveals all of the hidden subsurfacedamage (i.e. surface fractures) by two means. First, etching in-creases the size and separation of those fracture surfaces whichwere too narrow to scatter or reflect light prior to etching (seeFig. 8(a) and (b)). Second, etching may possibly remove the in-dex-matched layer of re-deposited or rehydrated silica (the Beilbylayer) which typically covers the first �100 nm of polished glasssurfaces [21–24].
The crack morphologies for the Brinnell (Hertzian) indentations,Vickers indentations and trailing Hertzian indentations created inthis study are well known and have been previously described[5–7,17,20]. An idealized schematic representation of each cracktype is shown in Fig. 9.
Using elastic Hertzian contact mechanics [17], the size of thecontact zone (2a) between a spherical indenter and a planar sur-face is given by:
2a ¼ 24P r k
3E
� �13
; ð4Þ
where P is the applied load, r is the radius of the spherical indenter,E is the Young’s modulus of the substrate, and k is the contactparameter which is defined as:
k ¼ 916
1� m2� �þ E
Ep1� m2
p
� �� ; ð5Þ
where m and mp are Poisson’s ratio for the substrate and the indenter,and Ep is the modulus of the indenter. Substituting known materialconstants for fused silica (substrate) and stainless steel (indenter)[25], the contact zone sizes (2a) for a Brinnell indentation at 39and 49 N are found to be 126 and 136 lm, respectively. As shownin Fig. 1(a) and (b), these values are in reasonable agreement withthe measured size of the surface ring cracks which were found tobe 131 and 135 lm, respectively.
The profilometry results for the Brinnell indentations revealedthat the true depth for these cracks can not be measured using con-tact profilometry due to the narrow width of crack opening [18–20] relative to the size of the profilometry stylus. In comparingthe fracture profiles shown in Fig. 1(a) and (b) to the predictedetching behavior shown in Fig. 9(a), the combination of the narrowwidth of the cracks, the angle of the cone cracks, the width of thestylus (10 lm), and the angle of the stylus, prevents the stylus fromproviding a true representation of the crack depth at short etchtimes. However, the width of the opening of the Hertzian cracksafter each etch time could still be accurately measured by profil-ometry and/or microscopy (Fig. 1(a) and (b)). By simple geometry,the width of the fracture opening (wh) with etch time (t), assumingisotropic etching, is given by:
wh ¼2rbtsin h
; ð6Þ
where rb is the bulk etch rate expressed in units of thickness/unittime, and h is the angle of the crack relative to surface of the glass(see Fig. 10). By measuring the depth of the Hertzian cone crackand the distance away from the contact ring, h is found to be�26�; which is similar to that cited by Hagan [19]. Using a bulk etchrate of 1.6 lm/h for 20:1 BOE [26,27], one can compare the size ofthe cone crack determined experimentally to the dimensions of
θθrbt
cone crack before etching
at t = 0
rbt
wh: crack opening after time, t
Fig. 10. Schematic illustration of the opening of Hertzian cone cracks with etching;the solid lines represent surfaces before etching and the dashed lines representsurfaces after etching for time, t.
0
20
40
60
80
100
120
140
Her
tzia
n C
one
Cra
ck W
idth
, wh ( μ
m)
Etch Time, t (hrs)
39 N Model 39 N Brinnell Indent 49 N Model 49 N Brinnell Indent
0
20
40
60
80
100
120 59 N 78 N 98 N
Etch Time (hrs)
Rad
ial c
rack
hal
f wid
th, 0
.5w
r
or th
ickn
ess
rem
oved
(μm
)
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radial crack opening
0 4 8 12 16 20
(a)
0 12 24 36 48 60 72 84 96
(b)
Fig. 11. (a) Comparison of the measured crack widths of the Hertzian cracks (wh)from Brinnell indentation on the fused silica surface after etching to that predictedby Eq. (6). (b) Comparison of half of the measured radial crack width (wr) to theamount of the base thickness removed as a function of etch time.
(a) (b) (c)
Vickers Indent
material modification
radial crack
lateral crack
Hertziancrack
plastic deformation
Hertzian crack
2a
Brinnell Indent
Bef
ore
Etch
ing
Afte
r Et
chin
g
Sliding Indent
trailing indent crack
Fig. 9. Schematic cross-sectional view of: (a) Hertzian indent, (b) Vickers indent and (c) sliding indent, before and after etching.
L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810 803
the crack that can be calculated assuming isotropic etching (see Eq.(6)). Fig. 11(a) compares the measured value of wh, from the profil-ometry results in Fig. 1(a) and (b), with that calculated using Eq. (6).The good agreement between the two suggests that etching occursnearly isotropically for Hertzian cone cracks.
Unlike ideal blunt indentations, which take place under purelyelastic conditions, sharp indentations involve irreversible or plasticdeformation. In particular, sharp indentations typically result inthe formation of radial and lateral cracks, together with a plasti-cally deformed and/or densified zone directly below where the in-denter contacts the surface (see Fig. 9(b)). In contrast to Hertziancracks, the radial cracks from the Vickers indentations were notfound to etch isotropically. This effect is illustrated in Fig. 11(b)where the half-width of the radial crack ðwr
2 Þ is plotted versus theetch time. As shown in this figure, the radial cracks were foundto etch �20–25% slower than would be expected assuming an iso-tropic etch rate. One would assume that the plastic deformationand/or densification associated with the sharp indentation(s) re-sult in the reduced etch rate.
In the absence of fluoride etching, either direct or microscopicexamination of indentation fractures can lead one to misinterpretthe morphology or extent of a fracture or fracture system. Forexample, when one compares the apparent size of fractures formedas the result of a Vickers indentation prior to (top of Figs. 1(c) and(d)) and after etching (middle of Fig. 1(c) and (d)) one finds that thetrue extent of damage differs by a factor of approximately ten. Thisdifficulty results from the partial healing [28] of indentation frac-tures that may occur following removal of the tension impartedby the indenter. Both of these effects reduce the light reflectionat the crack surface, thus making these cracks invisible beforeetching.
4.2. Etching of ground surfaces
4.2.1. Cusp growth ratesEtching ground surfaces results in the formation of cusps. As de-
scribed by Spierings [1], each cusp originates from the etching ofan individual surface fracture. As shown in Fig. 6, the cusp diame-ter (D) scales with the square root of etch time (t1/2). Although themechanisms differ significantly, the formalisms describing thegrowth rate of such cusps and grain growth in polycrystallineceramics [29] are similar. Specifically, the diameter of each cusp(D) grows at a rate that is inversely proportional to its size:
804 L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810
dDdt¼ kc
D; ð7Þ
where kc is a growth constant. Integration of Eq. (7) gives:
D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kct þ D2
o
q; ð8Þ
where Do is the initial cusp or crack size. When Do� D, thenD ¼
ffiffiffiffiffiffiffiffiffi2kct
por D / t1/2. As shown in Fig. 6, experimental data from
both the fixed and loose abrasive grinding processes are consistentwith the stated dependence and result in similar growth constants(0.01 lm2/s for the 150 grit fixed abrasive sample and 0.006 lm2/sfor the 30 lm loose abrasive sample). One would expect the cuspgrowth constant to be a function of crack distribution (depth,length, spacing and orientation) (see Section 4.3).
4.2.2. Material removalThe number density of cracks on a ground surface is typically so
high that cracks overlap each other, obscuring the identification ordepth of individual fractures. Previous work [5–7] has shown thatas several micrometers of material are polished from a ground sur-face, one finds that the number density of cracks drops exponen-tially until a maximum depth of subsurface damage is reached[5–7]. The rate at which material is removed during etching isdependent on the surface area exposed to the etchant. Given this,one would expect the rate of material removal during etching todecrease exponentially as the number of fractures in the near sur-face region of the substrate decreases.
Consider a disk of radius, rs, and thickness, hs, which has beenground on one surface and polished on the opposite face and alongthe edge. The time dependent mass of material removed duringetching (mt) is the sum of the masses that are removed from boththe ground (mg) portion of the sample and the polished (mp) sur-faces of the disk. Thus, one can express mg as a function of the bulketch rate (rb), the dimensions of the disk (rs and hs), the etch time(t), and the density of fused silica (q = 2.2 gms/cm3):
mg ¼ mt �mp ¼ mt � rbðpr2s þ 2prshsÞqt: ð9Þ
Similarly, the thickness removed from the ground surface, zg, is gi-ven by:
zg ¼mg
p r2s q; ð10Þ
where pr2s is the geometric area of the ground surface.
Fig. 12 shows the material removal rate (Dzg/Dt) of the groundfaces using Eqs. (9) and (10). The rate at which material is etchedfrom the ground surface drops exponentially and reaches anasymptotic value that is equivalent to the etching rate of the bulkfused silica.
0 10 20 30 40 500.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.0
1.1
2.2
3.4
4.5
5.6
6.7
7.8
9.0
10.1
Mas
s R
emov
al R
ate,
dm
g/dt (
g/hr
)
Etch Time, t (hrs)
150 grit generator grind 30 μm loose abrasive grind
Thickness Rem
oval Rate, dz
g /dt (μ m/hr)
Fig. 12. Mass and thickness removal rate as a function of etch time for the groundfused silica surfaces.
The behavior illustrated in Fig. 12 suggests that the surface areainitially available to the etchant on the ground faces is about 3.5�larger than that of the similarly sized polished face. As the cuspsassociated with each fracture coalesce with one another duringthe etch process, the true surface area (e.g the area available tothe etchant) diminishes until it becomes equivalent to the nominalgeometric surface area at which time the rate at which material isetched from each face of the disk becomes equivalent to the bulketch rate.
4.2.3. Roughness evolutionThe formation, growth, and coalescence of cusps, which result
from the local dissolution of silica in the immediate vicinity of indi-vidual fracture sites, implies a change in the roughness of a groundsurface as it is etched. As shown in Fig. 5, the surface profiles forthe samples ground with the 150 grit fixed abrasive and the30 lm loose abrasive, respectively, show that the cusp size notonly grows with etch time, but the surfaces appear to becomesmoother as well.
Characterizing the surface roughness in terms of the powerspectral density (PSD(kz)) as a function of spatial frequency kz pro-vides one with a means of examining how the surface roughnessevolves as a function of etch time. The function PSD(kz) was calcu-lated from the Fourier transform of the profilometry data hz(xi)using the expression:
PSDðkzÞ ¼PN
i¼1hzðxiÞe�ikzx��� ���2
L; ð11Þ
where hz(xi) is the raw profilometry data (height vs. position alongsurface), L is the length of profilometry scan performed and kz is thespatial frequency. The spatial frequency range for which the PSD isvalid is bound by the length of the scan and the spatial length overwhich each data point is taken. The bounds of the spatial frequencyrange which can be represented range between 3/L for the longscale length features and N/8L (i.e. Nyquist number/4) for shortscale length features where N is the number of data points in theprofilometry scan. For the current data set, N = 10,000 andL = 5000 lm, resulting in a spatial frequency range from 0.0006–0.25 lm�1, or spatial periods between 4 and 1700 lm.
The calculated PSD from the measured profilometry data, atvarious etch times, for both the fixed and loose abrasive groundsamples is shown in Fig. 13. The behavior of the PSD for each ofthe ground surfaces appears to differ for features which have spa-tial frequencies that are greater than and less than �0.02 lm�1. Forfeatures having spatial frequencies less than 0.02 lm�1 (>50 lmspatial period) the PSD (apparent roughness) appears to increasewith etch time and reaches a rough plateau. In contrast, the PSDof features having a spatial frequency greater than about0.02 lm�1 (<50 lm spatial period) appears to reach a maximumvalue which then falls during extended etching. These effects aremore clearly illustrated in Fig. 14, which shows the peak-to-valley(P–V) roughness and root-mean-square (RMS) roughness calcu-lated for both samples as a function of etch time over the spatialranges described above. The apparent increase in roughness ineach case is simply an artifact due to the increased ability of the fi-nite-sized profilometer stylus to penetrate the surface cracks asthey etch open (see Section 4.1). The subsequent drop in small-scale (<50 lm) roughness is due to the coalescence of intersectingcusps.
4.2.4. Cusp coalescenceThe coalescence of neighboring cusps during etching is of cen-
tral importance in understanding the changes in etch rate, surfaceroughness and SSD that occur during the etching process. A simplegeometric model can be used to describe the key features of cusp
1E-3 0.01 0.11E-3
0.01
0.1
1
10
100
1000150 grit
PSD
(A2 /μ
m)
Spatial Frequency (μm-1)
0 hrs2 hrs4 hrs6 hrs8 hrs
16 hrs24 hrs32 hrs40 hrs48 hrs
0 - 48 hrs
1E-3 0.01 0.11E-3
0.01
0.1
1
10
100
100030T
PSD
(A2 / μ
m)
Spatial Frequency (μm-1)
0 hrs2 hrs4 hrs6 hrs8 hrs
16 hrs24 hrs32 hrs40 hrs48 hrs
0 - 48 hrs
(a) (b)
Fig. 13. Power spectral density (PSD) for roughness for the (a) 150 grit fixed abrasive ground surface and (b) 30 lm loose abrasive ground surface, as a function of etch time.The behavior of the PSD for each of the ground surfaces differs for features which have spatial frequencies that are greater than and less than �0.02 lm�1 (50 lm spatialperiod) which is shown by the dashed line.
0
5
10
15
20
25
30
35
40
45
δ PV (μ
m)
Etch Time, t (hrs)
0.0006-0.02 μm-1
0.02-0.25 μm-1
> 50 μm features
< 50 μm features
0
5
10
15
20
25
30δ PV
( μm
)
Etch Time, t (hrs)
0.0006-0.02 μm-1
0.02-0.25 μm-1
0
1
2
3
4
5
6
δ RM
S (μm
)
Etch Time, t (hrs)
0.0006-0.02 μm-1
0.02-0.25 μm-1
0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 500
1
2
3
4
δ RM
S (μm
)
Etch Time, t (hrs)
0.0006-0.02 μm-1
0.02-0.25 μm-1
(a) (b)
Fig. 14. Peak-to-valley (top) and RMS (bottom) roughness for the (a) 150 grit fixed abrasive ground surface and (b) 30 lm loose abrasive ground surface, for different spatialranges.
L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810 805
coalescence and to provide an understanding of how the surfacetopography would be expected to evolve as a function of etch time.
For the present, let us assume that the surface consists of aninfinite series of equally spaced cracks of uniform depth (c), sepa-
rated by a distance (s) (see Fig. 15). Fractures on real surfaces, ofcourse, are characterized by a distribution of crack depths andspacings. A refined model reflecting this will be described in Sec-tion 4.3 and in Appendix A.
Fig. 15. Simple, 2-D, geometric model describing the evolution of the surfaceprofile of cracks with etch time.
806 L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810
As shown in Fig. 15, at t = 0, etching has not yet started and eachcrack remains closed. At time t1, the cracks begin to etch resultingin the formation of a series of cusps each of which remain isolatedfrom one another. At time, t2, the cusps intersect and coalesce withone another. For a series of uniformly spaced fractures, the time re-quired for intersection (t2), and thus coalescence, is given by:
t2 ¼s
2rb: ð12Þ
Before cusp coalescence occurs, i.e. t < t2, the peak-to-valley rough-ness is defined by the crack depth (dPV = c). The surface area is alsoinitially large owing to the large number of surface fractures. At t2,
0
2
4
6
8
10
41
42
432D Model
Experiment
δ PV
(μm
)
Etch Time, t (hrs)
150 grit
0.00
0.04
0.08
0.12
0.16
0.20
0.00
2.24
4.49
6.73
8.97
11.21
Mas
s R
emov
al R
ate,
dm
g/dt (
g/hr
)
Etch Time, t (hrs)
2D Model Experiment
150 grit
Thickness Rem
oval Rate, dz
g /dt (μm/hr)
0 10 20 30 40 50
0 10 20 30 40 50
(a)
Fig. 16. Comparison of the experimental roughness (top) and mass removal rate (bottomsurface and (b) 30 lm loose abrasive ground surface.
dPV decreases instantaneously as the cracks are assumed to havevertical sides in the present model. The surface area also decreasessignificantly. After coalescence, i.e. t P t2, the surface roughness be-comes independent of crack depth and scales as:
dPV ¼ rb t �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrbtÞ2 � 1
4s2
rþ do; ð13Þ
where do is the baseline roughness.One can compare the time evolution of the peak-to-valley (P–V)
roughness of ground surfaces generated experimentally with pre-dictions generated using the present simplified geometric model.Such a comparison is shown in Fig. 16. Specifically, Fig. 16(a) com-pares the time evolution of the P–V surface roughness, in the spa-tial range of 0.05–0.25 lm�1 (�4–20 lm scale lengths), of thesample which was ground using a 150 grit fixed abrasive with pre-dictions which were generated using the present model. A similarcomparison for the sample ground using a 30 lm loose abrasive isshown in Fig. 16(b).
The modeled results, in each case, were generated by substitut-ing the average crack spacing and maximum subsurface damagedepths that were determined experimentally (see Section 2.2) intoEq. (12) which was used to estimate the initial coalescence time,and Eq. (13) which was used to estimate the time dependent evo-lution of the P–V roughness. As shown in Table 1, the average crackspacings (s) of the 150 grit fixed abrasive and the 30 lm looseabrasive ground surfaces were found to be 22 and 17 lm, respec-tively. The maximum SSD depths (c) as determined using the taperpolish technique [5–7] were 42 and 41 lm, respectively (see Table1). In each case, a bulk etch rate of 1.6 lm/hour, corresponding tothat of 20:1 BOE, was used in calculating the model results.
0
2
4
6
8
40.5
41.0
41.52D Model
Experiment
δ PV
(μm
)
Etch Time, t (hrs)
30T
0.00
0.04
0.08
0.12
0.16
0.20
0.00
2.24
4.49
6.73
8.97
11.21
Mas
s R
emov
al R
ate,
dm
g/dt (
g/hr
)
Etch Time, t (hrs)
2D Model Experiment
30T
Thickness Rem
oval Rate, dz
g /dt (μ m/hr)
0 10 20 30 40 50
0 10 20 30 40 50
(b)
) to the 2-D etch model (Eqs. (13)–(15)) for the (a) 150 grit fixed abrasive ground
L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810 807
As shown in Fig. 16(a) and (b), the model predicts a crack inter-section time (see Eq. (12)) of 6.9 h for the 150 grit fixed abrasiveground surface and 5.3 h for the 30 lm loose abrasive ground sur-face. For the experimental data, the initial rise in the measuredroughness is an artifact due to the finite (�10 lm) width of theprofilometer stylus. A probe of such dimension is incapable of pe-netrating the narrow, high aspect ratio (�1 � 40 lm) cracks pres-ent prior to etching. For full penetration, the cracks would haveto be etched to a width exceeding �10 lm. This would require al-most 4 h of etching (4 h � 1.6 lm/h � 2�13 lm) and for a uniformdistribution of cracks, would correspond to the maximum P–Vroughness. Moreover, if the probe was sufficiently narrow to pen-etrate all cracks, the initial P–V roughness would equal the subsur-face damage (SSD) depth, or �40 lm, as described by the model.Because of the physical limitations of the profilometer stylus, thetrue P–V roughness cannot be measured at the onset of etching un-til the openings of the etched cracks exceed the width of the pro-filometer stylus. None-the-less, one finds that there is generallygood agreement between the experimental data and the modeledresults. For example, once the time required to etch beyond thephysical dimensions of the profilometer probe is exceeded, thepresent 2-D geometric model provides a reasonable prediction ofthe time dependent evolution of the surface roughness.
The present model can also be used to describe the material re-moval rate as the surface area exposed to the etchant evolves withtime. The effective removal rate before the coalescence of adjacentsurface fractures (t < t2) is given by the bulk etch rate multiplied bythe ratio of the surface area at time t, to the initial flat surface area:
dzg
dt¼ rb
ð2p� 8Þrbt þ 4c þ 2s2s
: ð14Þ
After coalescence (t P t2) the removal rate is given:
dzg
dt¼ 2r2
bts
sin�1 s2rbt
� �: ð15Þ
As shown in Fig. 16(a) and (b), there is qualitative agreement be-tween the model and the experimental material removal rates. Asone would expect, the material removal rate is maximized at theonset of the etching process when the surface area exposed to theetchant is also at a maximum owing to the large number of fracturesurfaces. After the cusps begin to coalesce with one another, thesurface area decreases and consequently, the removal rate de-creases until it converges to the etch rate of the bulk material (rb).The shortcoming of the present model is also evident in Fig. 16(a)and (b). Specifically, whereas the mass removal rate during theetching of each of the ground samples decreases exponentially,
-140
-120
-100
-80
-60
-40
64 hrs
80 hrs
56 hrs
48 hrs
Peak
Hei
ght,
h z(x) (
μm)
Position, x (μm)
40 hrs
0 20 40 60 80 100
(a)
Fig. 17. Comparison of the 3-D etch model profile results (dotted lines) to the experimwithout enhancement at the peak and (b) with enhancement at the peak (given by Eqs
the modeled results experience a precipitous drop at a single frac-ture coalescence time (t2). This difference is due to the distributionof crack depths and spacings which exists on each of the real groundsurfaces [5,7] rather than the single values of depth (c) and spacing(s) which have been assumed in the present model.
The present results, together with the results discussed in Sec-tion 4.2.2, provide an illustration of the utility of etching groundsurfaces during the fabrication of SSD free substrates. As shownin Fig. 14, after extended etching, a P–V roughness of �30 lm isachieved for the 150 grit fixed abrasive ground sample while aP–V roughness of �20 lm is achieved for the 30 lm loose abrasiveground surface. Recall that the maximum subsurface damage wassimilar (�42 lm) for each of these samples. By etching the sub-strate immediately after grinding, one can significantly reducethe amount of material that must be polished away to achieve aSSD-free substrate. The effectiveness of etching in reducing SSD,and thus reducing polishing time, is promoted by grinding pro-cesses which yield a high density of fractures that have relativelyuniform dimensions, which can readily coalesce with one anotherduring the etching process.
4.3. Finite difference etch model
The etch model discussed in the previous section assumes thateach of the fractures are of uniform dimensions and are equallyspaced. In the present section, we consider a three-dimensional(3-D) finite difference model that describes the etching behaviorof a surface which contains a distribution of cracks which vary withrespect to length, depth, orientation and spacing.
In this model, the initial surface is represented as a height on atwo dimensional rectangular grid. This surface is determined sta-tistically from the relevant crack distributions to set positions,crack length and depth. Since only a finite area can be modeled,the resulting surface can be considered as one member of anensemble of surface patches corresponding to the crack distribu-tions. At a given time, each surface point is assumed to etch alongthe surface normal. In practice, the surface is represented by thepyramid whose vertices are the heights at a specified grid locationplus its four nearest neighbors. The etch direction is then taken asthe average of the normals to the four sides of the pyramid. A moredetailed quantitative description of the 3-D finite difference etchmodel, as well as its features, are provided in Appendix A. Thethree-dimensional (3-D) finite difference etch model was first eval-uated by comparing the coalescence data shown for cracks etchingin a scratch (Fig. 4) to values calculated from the model. Fig. 17(a)shows a comparison of the etch model and experimental data for
-140
-120
-100
-80
-60
-40
Position, x (μm)
Peak
Hei
ght,
h z(x) (
μ m)
40 hrs
48 hrs
56 hrs
64 hrs
80 hrs
0 20 40 60 80 100
(b)
ental results (solid lines) for a single convex surface of a trailing indent crack (a). (A7) and (A8)).
Fig. 18. The 3-D, finite difference etch model simulation results for three different distributions of cracks on the surface.
808 L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810
that of a single peak, and illustrates that the convex surface hassome enhancement in removal. The mechanism for this enhance-ment is not currently understood; however, the model has beenmathematically calibrated to the experimental data using anenhancement in the etch rate (see Appendix A for details).Fig. 17(b) shows the resulting etch simulation with this enhance-ment and illustrates the improved agreement between the modeland the experiment.
The 3-D etch model was then used to explore the effect thatcrack distributions have on the evolution of surface topology dur-ing etching. Three scenarios were simulated: (1) randomly spacedand oriented cracks of the same length and depth, (2) orderedcracks of the same length and depth, and (3) randomly spacedand oriented cracks of different lengths and different depths. Ineach case, the same number of cracks was evaluated using thesame grid size. The results of this simulation are shown in Fig. 18where the top (planar) view of these distributions and the contourmaps of these crack distributions, before and after etching for 16 h,are presented.
1
2
3
8
10
12
14
16A: random cracks, same LB: ordered cracks, same LC: random cracks, L dist'n
δ PV (
μm)
Etch Time (hrs)
C
B
A
2
0 5 10 15 20
(a)
Fig. 19. The 3-D, finite difference etch model results for (a) surface roughness and (b) maon the surface.
As shown by the first scenario, the etch model reveals that inareas where the cracks are spaced far apart, crack intersectionand coalescence do not take place and the surface remains highand fairly flat. In such areas, the P–V roughness also changes lessthan in areas of high crack density. In the areas where the cracksare spaced closer together, the cracks are able to intersect and coa-lesce with one another leading to a reduction in surface roughness.As shown by the second scenario, the reduction in P–V roughnessis most efficient when the majority of the cracks are closely spaced.This reinforces the idea that etching most efficiently converts SSDto surface roughness for surfaces which are characterized by a uni-form distribution of closely spaced cracks. However, because realground surfaces typically consist of a distribution of crack lengths,depths, orientations, and spacings, a third scenario was evaluatedby the 3-D etch code. The roughness and mass removal rates as afunction of etch time for each of the three simulations are com-pared in Fig. 19. These results are qualitatively similar to thosefound experimentally using the 2-D code; both the roughnessand the removal rate were initially high but decreased with etch
3.5x10-4
4.0x10-4
4.5x10-4
5.0x10-4
5.5x10-4
A
B
C
A: random cracks, same LB: ordered cracks, same LC: random cracks, L dist'n
Mas
s R
emo
val R
ate
(g/c
m h
r)
Etch Time (hrs)
0 5 10 15 20
(b)
ss removal rate, as a function of etch time, for three different distributions of cracks
x
y
A
B
C
D
E 12
3 4
Fig. 20. x–y Projection of point A on surface S and its nearest neighbors points B, C,D and E at to. The dashed lines outline the four different planes (1, 2, 3 and 4) thatcan be described containing point A.
L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810 809
time. The rate of surface fracture transformation to roughness isstrongly governed by the crack number density and the spatial dis-tribution of cracks.
5. Conclusions
The effects of HF/NH4F (BOE) wet chemical etching on the mor-phology of isolated surface defects (indentations and scratches)and of ground surfaces have been characterized for fused silicaglass. For polished surfaces, fluoride etching has proven to be asimple and significant aid in assessing the presence of surface de-fects (scratches and digs) during the fabrication process. Moreover,etching has proven to be helpful in revealing the true size and mor-phology of fractures which might otherwise be obscured by thefractures which are of an insufficient width to reflect or scatterlight and/or by the presence of the Beilby layer for fractures thathave been post-polished but removed. Three types of indentationfractures have been examined: blunt indentations (Hertziancracks), sharp (Vickers) indentations (radial and lateral cracks),and scratches (trailing indentation cracks). The crack morphologiesobserved were consistent to those that had previously been re-ported elsewhere. Similarly, there was also good agreement be-tween the measured crack dimensions and those which arecalculated from standard fracture indentation relationships.
When isolated fractures are etched, the bottom and the sides ofthe fractures etch at nominally the same rate, making the removalof such defects by etching impractical. In contrast, when groundsurfaces are etched, as each of the individual cracks open laterally,they may intersect and coalesce with numerous adjacent cracks.This behavior results in the formation of a cusp-like morphologywhose diameter increases with the square root of etch time. Partic-ularly in cases where there is a uniform distribution of closelyspaced cracks, this phenomenon can be used to reduce both the ex-tent of SSD and surface roughness. Similarly, the rate of etching fora ground surface is always elevated relative to the bulk due to thelarge surface area presented by the numerous fracture surfaces. Asetching proceeds, the material removal (etch) rate decreases as thesurface area associated with fracture surfaces is reduced, until thebulk etch rate is reached.
For ground surfaces, etching provides a means of reducing oreliminating the SSD introduced by the grinding process. This pro-vides an economical way of reducing the amount of material thatmust be removed by subsequent grinding or polishing processesin order to yield SSD-free surfaces. Etching also ensures that most,if not all, of the SSD has been effectively removed prior to the nextprocessing step as SSD can be readily seen as cusps after etching.Simple, geometric and isotropic etch models that predict the mate-rial removal and surface evolution of ground fused silica surfaceshave been developed and can be used as guidelines to optimizethe optical fabrication process.
Acknowledgements
The authors would like to thank James Embree for his assistancein performing the grinding processes on the samples. Work per-formed under the auspices of the US Department of Energy by Law-rence Livermore National Laboratory under Contract No. DE-AC52-07NA27344, with the LDRD program.
Appendix A. Finite difference etch model
This appendix describes a finite difference model of the isotro-pic etching of a surface. At time t0, the surface is represented by anarray of points in Cartesian coordinates (x,y,S) in which the z com-ponent S(x,y,to) is the height of the surface at position x,y. The val-
ues of x,y are uniformly spaced (dx and dy) in the base plane usingan array index of io, jo where x = iodx and y = jody. The basic idea isthat the surface etches along the local surface normal. Numerically,the surface at time to + Dt is determined by moving each surfacepoint along the direction which is the average normal of the sur-face facets surrounding the given point (see Fig. 20). This vectordisplacement ~dTðx; yÞ of the point (x,y,S) is defined as:
~dTðx; yÞ ¼ rb Dt n̂ðx; yÞ; ðA1Þ
where rb is the base etch rate (thickness/time) and n̂ðx; yÞ is the unitvector defined by the average of the normal unit vectors of each ofthe nearest neighbor planes containing the given point. This aver-age normal unit vector is given by:
n̂ðx; yÞ ¼ ðnxðx; yÞ;nyðx; yÞ;nzðx; yÞÞ ¼n̂1 þ n̂2 þ n̂3 þ n̂4
jn̂1 þ n̂2 þ n̂3 þ n̂4j; ðA2Þ
where nx, ny and nz are the components of the average normal vec-tor and where
n̂1 ¼AC�!� AB�!
jAC�!� AB�!j
n̂2 ¼AE�!� AB�!
j AE�!� AB�!j
n̂3 ¼AD�!� AE�!
jAD�!� AE�!j
n̂4 ¼AC�!� AD�!
jAC�!� AD�!j: ðA3Þ
Here AB�!
; AC�!
; AD�!
and AE�!
are the vectors described by the points A–E (see Fig. 20). The numerical subscripts represent the four planarfacets connecting point A with its nearest neighbors. The new sur-face is then calculated for every original point (io,jo) on the surfaceusing Eq. (A1), (A2), (A3) for a time increment Dt using:
i ¼ io � rbDt nx
j ¼ jo � rbDt ny
Sði; j; to þ DtÞ ¼ Sðio; jo; toÞ � rbDtnz: ðA4Þ
After time increment Dt, the basal points corresponding to theheights S(x,y,to + Dt) are most likely not uniformly spaced by dx
and dy. For ease of continuing the surface calculation for many timeincrements, it is convenient to interpolate the surface array S(x,y,-to + Dt) back to the original uniform grid of x, y points. To illustratethis process, consider the x, y projection of point a and its nearestneighbors (points b–e) at time t + Dt onto original point A and itsnearest neighbors (points B–E). To interpolate back to the originalx, y grid, we want to calculate the equivalent depth ze for point a atpoint A (see Fig. 21). This can be done by first determining which ofthe nearest neighbor planes point A resides in, which is determined
y
x
(io,jo) A
B
C
D
E
(i,j) a
b
c
d
e
12
3 4
Fig. 21. x–y Projection of point A on surface S and its nearest neighbors points B, C,D and E at time zero and time Dt (shown by equivalent point with lower caseletters).
810 L. Wong et al. / Journal of Non-Crystalline Solids 355 (2009) 797–810
by the minimum average distance to any of the nearest neighborpoints. The desired value can then be found by interpolation be-tween the three values at the (i,j,S) points that define this plane.For example, as shown in Fig. 21, the nearest neighbor plane forpoint A contains points a, b and e. The normal direction for thisplane is along:
~n2 ¼ ab�!� ae�! ¼ ðn2x;n2y; n2zÞ; ðA5Þ
where n2x, n2y and n2z are the components of the vector. By definingthe components of point A as (io,jo) and those of point a as (i,j), theinterpolated height corresponding to point a at point A is given by:
Sðio; jo; to þ DtÞ ¼ �n2x
n2zði0 � iÞ � n2y
n2zðjo � jÞ þ Sði; j; to þ DtÞ: ðA6Þ
This interpolation process is repeated for every point after eachtime increment.
As discussed in Section 4.3 and Fig. 17(a), experimentally thereis an enhancement in removal on convex surfaces. To account forthis effect in our model, the etch rate is enhanced by locallyincreasing the magnitude of the surface displacement vector to:
~dTðx; yÞ ¼ rb Dt n̂ðx; yÞ ð1þ Qðx; yÞÞ; ðA7Þ
where Q(x,y) is the local relative enhancement and depends on thelocal curvature. Fig. 22 shows an x, z projection where point A is on aconvex surface. Note the magnitude of the normal vector is en-hanced by a factor of 1 + Q. The mechanism of the enhancementis not understood; however, we found good correspondence withthe experimental data when Q was empirically defined for a givendx, dy as:
Qði; jÞ ¼ 0:4ð1� e�ð4�Sði;jÞ�Sðiþ1;jÞ�Sði�1;jÞ�Sði;jþ1Þ�Sði;j�1ÞÞ2 Þ; ðA8Þ
where the value in the exponent is related to the curvature aboutpoint A(i,j).
Because the numerical method described above is explicit, un-less the time step is very small, it is possible to induce numericalinstabilities in which high spatial frequency features grow expo-nentially fast. One way to avoid these instabilities and allow amore generous time step is to smooth away the highest spatial fre-quencies. We have chosen to do this by adding a pseudo-diffusionstep to the algorithm described above. To see how to implementthis, consider the diffusion equation
Z
X
dT
dT
dT (1+Q)
A E
c
e a
c
a
C
Fig. 22. x, z Projection of points A, C, and E at to and points a, c, and e at to + Dt.
@ S@ t¼ De
@2S@ x2 þ
@2S@ y2
!; ðA9Þ
where De is the effective diffusion constant. This equation is easilysolved if the surface height S is written as a Fourier transform~sðkx; kyÞ where kx and ky are the spatial frequencies in the x and ydirection. If we take
Sðx; y; tÞ ¼Xkx ;ky
~sðkx; kyÞeiðkxxþkyyÞ; ðA10Þ
the solution to Eq. (A9) is given by
Sðx; y; t þ DtÞ ¼Xkx ;ky
~sðkx; kyÞe�Deðk2xþk2
y ÞDteiðkxxþkyyÞ: ðA11Þ
In practice, for each time step integration, we first transform tothe spectral domain, i.e. Fourier transform the present surface S,thus finding ~sðkx; kyÞ. We then multiply each ~sðkx; kyÞ by the terme�Deðk2
xþk2y ÞDt and inverse Fourier transform back to the spatial do-
main. This accomplishes the smoothing after which we evaluateEq. (A7) at each point. The exponential term in Eq. (A11) amountsto a low pass filter, i.e. it preferentially filters out high spatial fre-quencies. The important parameter in this process is
ffiffiffiffiffiffiffiDeDtdx2
qwhich is
the ratio of the smoothing length to the grid size. When thisparameter is of order one, the highest spatial frequencies sup-ported by the grid (kmax
x ¼ pdx) are attenuated at each time step
while slightly lower frequencies are hardly affected because ofthe exponential dependence of the filtering function on k2
x þ k2y .
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