Influence of Gold Metallodielectric Partial-shell Geometrical Irregularities on Dark

Plasmon Resonances

⇤

Janina Wirth, Hans Hallen and Shuang Fang Lim†

Department of Physics, North Carolina State University, Raleigh, NC 27695, USA(Dated: June 29, 2017)

ABSTRACT: The geometric asymmetry of real, fabricated gold partial-shells leads to orientationand gold partial coverage dependent local fields and scattering. We illustrate this with single particlemeasurements and finite element calculations. In particular, we show that the position and numberof well-defined protrusions on the edge of a partial-shell qualitatively changes the spectra. Themetallic protrusions result in geometrical asymmetry, which leads to excitation of the opticallydark quadrupole mode as a function of incident light excitation and polarization. The far fieldscattering peaks result from the bright dipole resonance with contribution from the dark resonancein the presence of the partial-shell surface protrusions. With more dark modes, the overall scatteredintensity decreases, reflecting the energy trapped in the local electric fields of the dark modes, untilit ultimately dissipates in the metal.

:

Introduction

Plasmonic nanoparticles absorb and scatter light atresonant frequencies, the intensity of which is depen-dent on the size, shape, symmetry and orientation ofthe nanoparticles 1–4. The excitation of a localized sur-face plasmon resonance (LSPR) on these nanoparticlesresults in enhancement of the local electric field. Due toenergy conservation, scattering must be reduced as en-ergy flows into the LSPR fields, where eventually, it willbe locally absorbed by metal or nearby materials. Hence,such nanoplasmonic structures enable light to be concen-trated into nanoscale volumes and actively manipulatedin new ways. Current intense interest and research maybring about applications in optical and temperature sens-ing 5–7, photothermal treatment 8–12, and in metamateri-als 13. Metallic partial-shells present a three-dimensionalreduced-symmetry structure that interacts with the elec-tric and magnetic fields of the incoming excitation. Thesestructures di↵er from closed nanoshells, whose plasmonicresonances are less complex, and are not dependent ontheir orientation 14–16. Hence, partial-shell structures,which include nanocaps, nanobowls, half-shells and nano-cups 3,17–27, demonstrate a 3D orientation dependence,which produces new scattering features such as magneticdipole resonances that can be exploited for metamate-rial design. It has been shown that symmetry breakingof a plasmonic system via metallic nanofeatures such asin gold nanorings 28, single nanorods 29, nanorod trim-mers 30, nanodisks 31, interelectrode nanoscale junctions32, nanocrosses 33, pillar array [34, nanocube array 35

and nanoparticle chains 36, lead to coupling of dipolemodes with dark plasmon modes. Dark plasmon modescan be quadrupolar, multipolar, or propagating 37–39.These modes are considered as dark because they only

⇤A footnote to the article title

†sflim@ncsu.edu

weakly couple with incident light and hence propagatingfields do not radiate 30,36,40. Therefore, these dark modesdisplay an increased lifetime, which enhances near fieldinteractions 36 that lead to potential applications includ-ing waveguides in nanophotonic devices 41, nanolenses42 and far-field color imaging 43. A loss of symmetryof the nanostructures, whether through intentional de-sign, change in orientation, or simply unintentional de-fects, modify the oscillator strengths of both the brightand dark modes, which enables coupling between bothmodes. In this work, we will show, with both measure-ments and calculations, how geometric irregularities in-troduced by standard nanofabrication processes lead tosymmetry breaking, and consequently to the appearanceof dark modes and qualitative changes to the spectra.Typical fabrication of semi-shells includes deposition ofa hemi-spherical shell around a dielectric core, such as sil-ica, immobilized on a substrate via thermal evaporation22,26 or electroless plating 3,24. Our approach uses singleparticle scattering measurement, which can reveal the ex-act influence of the parameters such as particle geometry,orientation, and surface irregularities on the correlatedmeasured spectra. This method is favored over conven-tional bulk measurements where spectra are broadeneddue to size distribution, non-uniform geometries or inter-acting/aggregated particles. We explore the influence ofthe semi-shell orientation, gold fractional coverage, andthe position and number of gold protrusions on the scat-tered light. Our individual particle imaging and opticalmeasurements allow us to present clearly the e↵ect ofeach parameter on the scattering spectra and compareto models. Other researchers have shown the influence offixed orientation partial shells immobilized on a substrate3,17–27. In this work, we report, for the first time, experi-mental evidence of the influence on the spectra of defectson typical realistic, fabricated particles, of the partialshell gold fractional coverage and number and positionof gold protrusions on the edge of the partial shell. The-oretical predictions of the gold fractional height and goldsurface roughness have been carried out by Cortie et al 17.

2

FIG. 1: Schematic of gold partial shell on silicananoparticle fabrication, showing a) immobilization ofsilica nanoparticles on a silicon substrate, b) thermal

evaporation of gold partial shell, and c) transferred goldpartial shell coated nanoparticles onto a gold coatedsilicon substrate. The panels above show SEM images

for the corresponding preparation step, and thecorresponding particle size histogram obtained by AFM

measurements and SEM images of bare particles.

However, their roughness surface models did not intro-duce enough asymmetry to observe the dramatic spectralchanges seen in our results. Our roughening mimics thatobserved in real particles and helps us to understand howthe number and exact position of each gold protrusion,with or without the presence of a substrate surface, givesrise to new spectral features. We demonstrate that thegold protrusions result in geometrical asymmetry, whichleads to excitation of the optically dark mode as a func-tion of incident light excitation and polarization. Thespectral position, number and intensity of the dark plas-mon modes, are shown to be strongly influenced by thenumber and position of the protrusions, as shown both inmeasured and calculated far field scattering. We performfinite element modeling in order to explain the e↵ects ofeach gold nanofeature on the scattering.

Methods

Gold semi-shells were prepared by thermal evaporationof gold onto silica nanoparticles (Fig.1). As a first step,silica nanoparticles were synthesized after the methodfrom Stoeber et al 44 with a diameter of about 89±6.4nm(Fig.1a), and immobilized on poly-4-vinylpyridine (PVP,molecular weight, 160k) functionalized gold coated sil-icon substrates. Afterwards, these silica nanoparticleswere used as a template to evaporate gold partial shells.Following that, the nanoparticles are transferred onto agold coated silicon substrate via a polydimethylsiloxane(PDMS) transfer stamp 45. This process left randomlyoriented gold partial shells on a gold surface layer. Foroptical correlation, a wide field fluorescence image of thesemi-shell nanoparticles was taken, with their positionsrecorded on indexed grids, using an Andor NEO sCMOS

camera. Afterwards, the recorded positions were char-acterized with a MFP-3D-BIO Atomic Force Microscope(AFM, Asylum Research) in tapping mode to find singlenanoparticles. Dark Field optical microscopy was carriedout using an Axio Imager Z1.m (Carl Zeiss MicroImagingGmbH, Jena, Germany). For comparison to the calcu-lations, the fractional coverage and angle of the parti-cles is needed. This is retrieved from two SEM imagesat two tilt angles for each particle, as described in thesupplementary data. We use a commercially availablesoftware package, Comsol, based on the Finite Elementmethod, to quantify plasmonic response and the enhance-ment of our nanostructures during excitation. The goldsemi-shells were modeled on silica nanoparticles, on goldsubstrates, for four di↵erent relative configurations be-tween the incident light direction, excitation polarizationand half-shell orientation. The modeled silica nanoparti-cle diameter and gold shell thickness was 90 nm and 15nm,respectively. The fractional height of the gold semi-shell was varied from 0.2-0.5. Surface geometrical irreg-ularities that resemble some of those on measured par-ticles, were modeled by placing small gold hemispheresnear the rim of the semi-shell edges. We modeled a per-fect continuous, but thinner, gold shell of about 15 nm,in place of a real granular gold shell. The gold shell inthe present modeling is assumed as a perfect continuousshell with bulk dielectric function adopted from previ-ous experiment results 46. We used Bruggemans e↵ec-tive medium theory 47 to calculate the e↵ective index ofthe real film, n

r,eff

+ni,eff

, that would accurately repre-sent the imperfect experimental gold shell thickness. Wecalculate the expected measured thickness t

exp

by requir-ing the same propagation loss for light passing throughthe perfect model film t

Au

as through the imperfect filmby setting the Beer’s Law exponents to be the same,ni,Au

.tmodel

= ni,eff

.texp

, with gold dielectric constantat 960 nm 46, (n

r,Au

+ni,Au

)2 = -46.797 + 3.0 i. We usethe gold dielectric constant, the silica index of refractionof 1.52, and the gold volume fraction estimated from theelectron micrographs to be 0.66. The calculated resultfor t

model

corresponds to the expected measured thick-ness t

exp

= 21.0 nm, which is in reasonable agreementwith the measured 21 nm given the uncertainty in thevolume fraction measurement. In our experiment, the in-clusion of the gold-coated substrate resulted in reflectionfrom the substrate surface that adds a broad backgroundto the measured far field scattered spectra. We subtractthis gold-film background scattering from the measuredspectra in order to clearly show the plasmonic modes.Correspondingly, the gold substrate scattering was alsosubtracted from our calculated spectra by subtractingthe far field spectra of the gold layer with no particlespresent. Both far and near field spectra were calculatedwith a spectral range from 400 to 1000 nm.

3

FIG. 2: Measured far field spectra and correspondingSEM images, comparing a) control: no protrusion, b)top protrusion, c) side protrusion, and d) crown ofprotrusions configurations. The silica cores shown in

the SEM images are of 90 nm in diameter.

Results and Discussions

Fig. 2 shows the measured far field spectra and thecorresponding SEM images of a partial shell a) withoutany protrusions, b) with a top protrusion, c) with a sideprotrusion, and d) with a crown of protrusions. The par-tial shell without any protrusions, displays a peak at 680nm attributed primarily to the dipole mode of the partialshell. The single protrusion configurations b) and c) bothshow an additional prominent peak from 560 to 590 nm,where this peak dominates the multi-peaked scatteredspectrum of the partial shell with a crown of protrusionsin (d). Hence, with the exception of a) the partial shellwithout a protrusion, all other configurations show atleast an additional prominent peak between 560 to 590nm. This led us to speculate that, 1) this additional peakmay arise from the presence of the protrusions, and 2)that there is dependence on the intensities of the 560 to590 nm, with respect to the 680 nm peaks, on the num-ber of protrusions. Moreover, we observe that the totalintensities of the measured peaks are highly dependenton the partial shell orientation 22, as shown in Fig.S8.The peak positions are only weakly dependent upon ori-entation, and are the primary emphasis of this work. Inorder to understand the origin of the 560 to 590 nm peak,and its relation to the protrusion position and number,finite element modeling was performed to obtain bothnear field and far field scattering spectra (See Fig. S1).Fig. 3 shows the calculated near field spectra of a partialshell of fractional gold coverage of 0.2, and with its axisof symmetry oriented 6 degrees from the horizontal. Inthe near field spectra in Fig. 3, the partial shell only con-figuration, without a protrusion, shows a single peak only

400 500 600 700 800 9000

40

80

120

Wavelength(nm)

Elec

tric

Fiel

d St

reng

th (V

/m) Control

TopSide

Crown

FIG. 3: Calculated near field spectra comparing a) noprotrusion, b) top protrusion, c) side protrusion, and d)

crown of protrusions configurations.

at 700 nm, where the low peak intensity is attributed toscattering of most of the energy to the far field. Whena single protrusion is present, as seen for both configura-tions with top and side protrusions, peaks are obtained at720 nm, and at 580 nm. The partial shell with side pro-trusion shows a slightly lower total intensity compared tothe partial shell with a top protrusion. This is becausewe have modeled an incident excitation angle of about60 degrees from the vertical to match the oblique illumi-nation of the dark field objective. Hence, any incomingexcitation more e�ciently scatters from a protrusion lo-cated closer to the direction of incidence, which is thetop of the partial shell, as compared to on the right sideof the shell. In comparison with the partial shell onlyconfiguration, the e↵ect of the added single protrusion is2-fold. First, the protrusions increase the gold surfacecoverage, which shifts the initial peak at 700 nm to 720nm, and second, it introduces an additional peak at 580nm. Furthermore, as shown in Fig. 3, multiple peakscan be seen as more protrusions are added to the partialshell. As shown in Fig. 3 (and Figs. S3 to S7), as ex-pected, the crown configurations increased gold surfacecoverage shifts the 700 nm peak further to 730 nm, andintroduces two additional strong peaks at 580 nm and670 nm, and other weaker peaks and shoulders. The ad-ditional peak at 670 nm can be attributed to the positionsof the additional protrusion positions in the crown con-figuration (See Fig. S1(d) and S3 to S7). Qualitatively,each protrusion creates an additional possible magneticdipole (in this wavelength range), and these dipoles mixto form the same number of normal modes. Similar ef-fects have been observed in nanoclusters 50,51, nanoshells52 and in spiky nanoshells 53. The corresponding far fieldcalculations for scattered light collected within a 60 de-gree cone, were carried out for the same configurations,and are shown in Fig. 4. The 560 to 590 nm peak is no-tably absent. The absence of the 560 to 590 nm plasmonmode implies that it is non-radiative, which is indicativeof a dark plasmon mode. The reason we observe it in

4

500 600 700 800 9000.0.100

2.0.10–29

4.0.10–29

Wavelength (nm)

Elec

tric

Fiel

d St

reng

th (V

/m)

ControlTopSideCrown

FIG. 4: Calculated far field spectra comparing a) noprotrusion, b) top protrusion, c) side protrusion, and d)

crown of protrusions configurations.

our scattering measurements is likely due to the near-field energy being scattered by grains and tiny inhomo-geneities on the real particles and the thermally evapo-rated gold film, which exhibits small scattering inhomo-geneities due to its granular or columnar nanostructures48,49. Additionally the main dipole mode at 720 nm is thestrongest intensity for the clean partial shell only. Theintensity of this 720 nm peak is slightly reduced in thepresence of protrusions, when located either on the sideor on top. Since dark modes are dissipative, the over-all scattered intensity is expected to decrease when thesedark modes are excited. Further evidence for attributionof the dark plasmon mode from 560 to 590 nm, can beseen in the crown configuration. As shown in Fig.4, theoverall scattered intensity is dramatically decreased, asthe number of protrusions increase. The asymmetric ap-pearance of the peaks, seen in the crown configuration,can be explained by the interference between bright anddark modes, giving rise to Fano resonances. Therefore,from both near and far field calculations, the presence ofthe gold protrusions disrupts the nanostructure symme-try, which then enables coupling between the bright anddark modes. Comparisons between experimental and cal-culated spectra show that the experimental spectra area combination of both the near and far field calculatedspectra. In order to understand the nature and origin ofthis 560 to 590 nm peak, 3D electric field and currentdensity plots are calculated for all the di↵erent configu-rations. As seen in Fig. 5, the localization of the electricfield enhancement (Fig. 5a and 5b) and current densities(Fig. 5c and 5d) at 590 nm are largely at the protru-sions, while that at 720 nm are mostly localized on thepartial shell itself (See Fig. S2). This is evident in boththe single and multiple protrusions configurations (Fig.S3 to S6). In addition, the position of the protrusionstrongly influences the electric field and current densityintensities, where the top protrusion results in a highercurrent density than the side protrusion. This observa-tion is also correlated with the near field spectra shownin Fig. 3, where peak intensities at 590 nm and 720 nm

!

!

!Fig.%5!Electric!field!magnitude!E2!plots!of!(a)!top!protrusion,!and!(b)!side!protrusion,!and!current!density!plots!of!(c)!top!protrusion,!and!(d)!side!protrusion,!on!the!edge!of!partial!shells!at!590!nm.!

(a)! (b)!

(c)! (d)!

FIG. 5: Electric field intensity E2 plots, at 590 nm,with 1 V/m far field excitation at 60 degrees towardsthe right from ’z’ (vertical) and polarized along the xdirection (out of the page), to simulate a dark field

illumination from above, of partial shells, with (a) topprotrusion, with color range 50 V 2/m2, and (b) sideprotrusion, with color range 80 V 2/m2, and currentdensity plots with color range 106 A/m2 of (c) topprotrusion model, and (d) side protrusion model.

are lower for the side position, in comparison to the topposition. We also conclude from these 3D plots that the590 nm mode is quadrupolar in nature. The top andcrown scattered spectra in Fig. 4 clearly show a splittingof the 720 nm peak. The top protrusions peak splits intotwo, which can be understood heuristically by the con-sideration that this magnetic dipole peak corresponds tocurrent oscillation around the bowl of the partial shell,and the protrusion versus clear areas of the rim representdi↵erent physical distances around the bowl, hence di↵er-ent resonant wavelengths. The longer distance includingthe protrusion corresponds to a longer wavelength: thenew peak. The quantitative understanding must also in-clude the slight shift to shorter wavelengths of the 720nm peak. This is due to a mixing of the wave resonancesthat causes the energy levels to repel, as is often observedwith electron waves in clusters using perturbation theory53,54. Essentially, the normal modes of the system includea superposition with energy shifts. The superposition isalso evident in where the current flows. The peaks in thecrown configuration, as shown in the near field spectrain Fig. 3, at 590 nm, 670 nm, 740 nm (shoulder at 790nm), and 880 nm (shoulder at 900 nm), are attributed tothe interaction of both the partial shell and protrusions(See also Figs. S3 to S6), and include many more pos-sible modes, with mixing driving the 720 nm peak anda derivative to even shorter wavelengths. The ring ofprotrusions strongly disrupts the partial shell symmetry,leading to multiple peaks as shown in Fig. 3 (See Fig.S7), and the resultant drop in scattering intensity in thefar field. We also observe in Fig. S7 that the positionof the pair protrusions contributes to splitting of the pri-mary dipole mode into spectral pairs. A further analysisof the e↵ect of the gold fractional coverage was carriedout, in Fig.6, showing the calculated scattering spectra

5

FIG. 6: Dependence of calculated far field scattering onthe partial shell fraction.

of partial-shells, with fractional gold coverage, from 0.20.5, with no protrusions. In our calculations, an addi-tional quadrupole peak at 590 nm is observed as goldfractional coverage is increased to 0.4 and 0.5, even withno protrusion. The metallic structure has become largeenough to support a quadrupole for wavelengths allowedby the gold dielectric constant. The position of the 720nm peak is also seen to shift to longer wavelengths withincreasing gold fractional coverage. This is expected asthe size of the metallic fraction or the particle size is in-creased. Note that the 590 nm feature also shifts to theright of the gold dielectric constant cuto↵ with increasinggold coverage. When we compare the influence of goldfractional coverage with that of the protrusions, we seesimilar trends in the emergence of the 590 nm peak andalso the shifts in the peak positions. The e↵ect of theprotrusions is to 1) increase the gold fractional coverage,and 2) disrupt the geometry of the nanostructure, whereboth e↵ects lead to increased coupling between brightand dark modes. One might expect that the crown ofprotrusions should di↵er from the clean, protrusion freepartial shell only by having a larger fractional coverage,

since both the size of and spacing between the protru-sions is small compared to the wavelength over the entirepartial shell perimeter. Our results show that this is notthe case and hints that the capacitive viewpoint ratherthan a wave viewpoint is more predictive for a qualitativeunderstanding of the local fields.

Conclusions

We show that realistic processing-induced particle sur-face inhomogeneities give rise to nanoparticle asymme-try and the subsequent generation of dark plasmons withqualitative changes to the spectra. This e↵ect is not typi-cally observed in ensemble measurements. Through care-ful measurements at the single particle level, we correlatethe metallic protrusion number and position to the ap-pearance of the dark plasmon modes.

Associated Content

Supporting Information. Additional current densityplots for all sample configurations (S1-S9). This ma-terial is available free of charge via the Internet athttp://pubs.acs.org.

Author Information

Corresponding Author *(S.F. Lim) Email:sflim@ncsu.edu.

Acknowledgements

This work was financially supported by the NationalScience Foundation CBET 1067508 grant and the NCState Undergraduate Research Grant.

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!!

!

! !

TOC Graphic

Influence of gold metallodielectric partial-shell

geometrical irregularities on dark plasmon

resonances

Janina Wirth, Hans Hallen, Shuang F. Lim*

Department of Physics, North Carolina State University, Raleigh, NC 27695, USA

*corresponding author, E-mail: sflim@ncsu.edu

S1!

Supporting*Information*

Fig.*S1!The!4!different!partial!shell!configurations!computed!in!COMSOL,!a)!partial!shell!only,!b)!Top!protrusion,!c)!side!protrusion,!and!d)!crown,!on!gold!substrates.!

(a)! (b)!

(c)! (d)!

S2!

Fig.*S2!3D!current!density!plots,!at!720!nm,!sliced!across!the!(a)!top!and!(b)!side!protrusion,!and!(c)!sliced!across!the!partial!shell!for!side!protrusion!configuration.!

(a)! (b)!

(c)!

S3!

Fig.*S3!3D!current!density!plots,!at!590!nm!for!a!crown!of!protrusions,!sliced!through,!(a)!top!protrusion!(b)!2!upper!protrusions,!(c)!2!center!protrusions,!and!(d)!2!lower!protrusions.!!

S4!

Fig.*S4!3D!current!density!plots,!at!670!nm!for!a!crown!of!protrusions,!sliced!through,!(a)!top!protrusion!(b)!2!upper!protrusions,!(c)!2!center!protrusions,!and!(d)!2!lower!protrusions.!!!

(a)! (b)!

(c)! (d)!

S5!

Fig.*S5!3D!current!density!plots,!at!740!nm!for!a!crown!of!protrusions,!sliced!through,!(a)!top!protrusion!(b)!2!upper!protrusions,!(c)!2!center!protrusions,!and!(d)!2!lower!protrusions.!!

(a)! (b)!

(c)! (d)!

S6!

Fig.*S6!3D!current!density!plots,!at!880!nm!for!a!crown!of!protrusions,!sliced!through,!(a)!top!protrusion!(b)!2!upper!protrusions,!(c)!2!center!protrusions,!and!(d)!2!lower!protrusions.!!

(a)! (b)!

(c)! (d!

S7!

Fig.* S7! (a)!Unnormalized! calculated! far! field! spectra! comparing! pairs! of!protrusions!b)!on!top,!c)!in!the!center,!and!d)!at!the!bottom.!

(b)! (c)! (d)!

(a)!

S8!

Fig.*S8!Measured!far!field!spectra!of!two!partial!shell!only!samples!with!SEM!images!showing! one,! with! gold! shell! oriented! sideways! (higher! scattering! intensity),! and!another,!on!the!substrate!side!(lower!scattering! intensity).!The!silica!cores!shown!in!the!SEM!images!are!of!90!nm!in!diameter.!

S9-1!

S9.!Determination!of!angle!of!tilt!of!sample!

The!calculation!of!the!partial!coverage!and!tilt!of!the!partial!shell!based!upon!SEM!images!is!difficult!due!to!the!flattening!effect!resulting!from!the!long!focal!length!of!the!SEM.!To!solve!this,!we!rotate!the!SEM!stage!until!the!plane!containing!the!edge!of!the!cap!is!has!a!horizontal!intercept!(cap!is!towards!top!or!bottom!of!image).!We!then!take!a!first!SEM!image.!Next,!we!tilt!the!SEM!stage!with!a!horizontal!rotation!axis!by!θ,!typically!55!degrees,!and!take!another!image.!For!both!images,!we!measure!the!distances!l,!the!vertical!(in!the!image)!distance!from!the!edge!of!the!coating!in!the!center!of!the!particle!to!the!edge!of!the!bare!side!of!the!particle,!and!d,!the!vertical!distance!from!where!the!inner!coating!just!begins!to!be!shadowed!by!the!bare!particle!to!the!horizontal!line!that!is!tangent!to!the!base!of!the!bare!side!of!the!particle.!For!the!coordinate!system!x!vertical!in!the!image,!y!horizontal!to!the!right,!and!z!into!the!image,!the!l!and!d!are!measured!along!parallel!lines!in!the!x!dimension.!Note!that!d!is!not!measured!from!where!the!outer!edge!of!the!coating!disappears,!but!rather!is!the!equivalent!to!l!if!the!particle!were!swiveled!ninety!degrees!about!a!vertical!(x)!axis.!The!fractional!coating!of!the!particle,!δ,!can!be!related!to!the!halfVangle!of!the!coated!part!referenced!from!the!particle!center,!φ!as!

.!We!take!the!(untilted)!angle,!ψ,!!to!be!the!angle!between!the!radius!to!the!center!of!the!coated!part!and!the!z!direction,!so!ψ=0!would!place!the!coated!portion!directly!behind!the!particle!core!(of!radius!R)!in!the!SEM!image.!

We!determine!φ!and!ψ!in!terms!of!the!known!l1,!l2,!d1,!d2,!and!θ.!We!have!the!relation!!with!θ1=0!and!θ2=55,!typically.!We!use!a!vector!approach!

to!obtain!d.!The!vector! !points!to!the!front!of!the!coating,!where!l!is!measured!from,!and!the!unit!vector! !towards!the!center!of!the!coating.!We!then!rotate! !by!α!about!!until!the!resultant,! ,!has!no!component!along!z,!at!which!point!the!xVcomponent!of! ,!vx,!will!be!dVR.!Specifically:!

,

,

. The rotation of can be determined by noting that the part parallel to the axis ,

, is not changed by the rotation, whereas the part perpendicular, , becomes . The rotated vector is the sum of its

parts, . Setting vz = 0 gives α as

� = (1–cos(�))/2

lj = R(1 + sin(�j + j + ✓j))

~sq̂ ~s

q̂ ~v

~v

q̂ =

0

@rotation

about y

by ✓ +

1

A

0

@00R

1

A =

0

@sin(✓ + )

0cos(✓ + )

1

A

~s =

0

@rotation

about y

by ✓ + + �

1

A

0

@00R

1

A =

0

@sin(✓ + + �)

0cos(✓ + + �)

1

A

~v =

0

@rotation

about q̂

by ↵

1

A~s

~s q̂

~sk = (~s · q̂)q̂~s? = ~s� ~sk

cos(↵)~s? + sin(↵)(q̂ ⇥ ~s?)

~v = (1–cos(↵))(~s · q̂)q̂ + cos(↵)~s+ sin(↵)(q̂ ⇥ ~s)

cos(↵) = cos(�)cos(✓+ )cos(�)cos(✓+�)–cos(✓+ +�)

S9-2!

and , where the d and θ take onsubscripts 1 and 2 as above. These formulae need to be inverted to obtain the φ!and!ψ, so Octave or MatLab are used with the fminunc method on the function(lmeas,1 – lcalc,1)2+( dmeas,1 – dcalc,1)2+(lmeas,2 – lcalc,2)2+( dmeas,2 – dcalc,2)2 to obtain the angles ψ and φ, and from them the fractional coverage. There is sometimes morethan one minimum, but comparison to the SEM images eliminates the extraneous ones.

d = R(1 + cos(�)(sin(✓ + �) + cos(✓ + �)cot(✓ + �))