self-assembly and characterization of anisotropic ... · and characterizing self-assembled, dense,...
TRANSCRIPT
SELF-ASSEMBLY AND CHARACTERIZATIONOF
ANISOTROPIC METAMATERIALS
A dissertation submitted toKent State University in partial
fulfillment of the requirements for thedegree of Doctor of Philosophy
by
Jacob Paul Fontana
May 2011
Dissertation written by
Jacob Paul Fontana
B.S., California Polytechnic University, 2004
Ph.D., Kent State University, 2011
Approved by
, Chair, Doctoral Dissertation CommitteeDr. Peter Palffy-Muhoray
, Members, Doctoral Dissertation CommitteeDr. Oleg Lavrentovich
Dr. Hiroshi Yokoyama
Dr. Qi-Huo Wei
Dr. Samuel Sprunt
Dr. Robert Tweig
Accepted by
, Interim Director, Department of CPIPDr. Liang-Chy Chien
, Dean, College of Arts and SciencesDr. Timothy Moerland
ii
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Negative index materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Realization of optical negative index materials . . . . . . . . . . . . . . . 8
1.4 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Novel achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Synthesis and surfactants of gold nanorods . . . . . . . . . . . . . . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Gold nanorod synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Gold nanorods dispersed in organic solvents . . . . . . . . . . . . 20
2.3.2 Gold nanorods encased in silica shells . . . . . . . . . . . . . . . . 23
2.3.3 Polyurethane gold nanorod composites . . . . . . . . . . . . . . . 24
iii
3 Field induced orientational order of gold nanorods in organic suspen-
sions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Orientational order parameter . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Dielectric function for a dilute, orientationally ordered, ensemble
of gold nanorods in a vacuum at optical frequencies . . . . . . . . 29
3.3.2 Dielectric function for a dilute, orientationally ordered, ensemble
of gold nanorods in a solvent at optical frequencies . . . . . . . . 36
3.3.3 Field induced orientational order of nanorods . . . . . . . . . . . 42
3.4 Electric field induced alignment of gold nanorods dispersed in toluene . . 46
3.5 Order parameter determination . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 Alignment of gold nanorods in various solvents . . . . . . . . . . . . . . . 56
3.7 Temporal alignment of gold nanorods . . . . . . . . . . . . . . . . . . . . 60
3.7.1 Time averaged phase shifts . . . . . . . . . . . . . . . . . . . . . . 61
4 Measurements of the electric susceptibility of gold nanorods at optical
frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Method for determining the optical susceptibility for gold nanorods . . . 63
iv
4.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Dielectric susceptibility measurements of gold nanorods at optical frequencies 69
4.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Self-assembly and characterization of gold nanorod films . . . . . . . . 89
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Evaporative and field assisted self-assembly of silica encased gold nanorod
films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.3 Characterization of silica encased gold nanorod films . . . . . . . 95
5.2.4 Order parameter determination . . . . . . . . . . . . . . . . . . . 97
5.2.5 Effects of evaporation rate of silica encased gold nanorod films . . 99
5.2.6 Phase shift measurements of silica encased gold nanorod films . . 101
5.2.7 Lift-off technique for silica encased gold nanorod films . . . . . . . 105
5.2.8 Temperature dependence of the absorption spectrum for silica en-
cased gold nanorod films . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 Strain-induced alignment of polyurethane-gold nanorod films . . . . . . . 107
6 Other measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Conjugated molecules as building blocks for optical metamaterials . . . . 111
6.3 Gold-23 films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
v
8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.1 Recipe for gold nanorods . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.2 Organically soluble nanorod coating process . . . . . . . . . . . . . . . . 123
8.3 Improved NR synthesis (monodispersed) . . . . . . . . . . . . . . . . . . 123
8.4 Smin program (Maple) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.5 Correlator program (LabVIEW) . . . . . . . . . . . . . . . . . . . . . . . 129
8.6 NR counter (LabVIEW) . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.7 Dye recipes with mirror sets . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.7.1 R6G (565− 640) . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.7.2 DCM special (610− 710) . . . . . . . . . . . . . . . . . . . . . 132
8.8 Dye laser alignment procedure . . . . . . . . . . . . . . . . . . . . . . . . 132
8.9 MetaMachine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.10 Mechanical drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
vi
LIST OF FIGURES
1 a: positive index material (PIM) b: negative index material (NIM) . . . . 4
2 Schematic of diffracting plane wave. . . . . . . . . . . . . . . . . . . . . . 5
3 Ray diagram for a superlens . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Surface index hyperboloid for light propagation in a uniaxial material. . 8
5 Gold nanorod in an optical field . . . . . . . . . . . . . . . . . . . . . . . 9
6 Cut-wire pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7 Proposed negative index material. . . . . . . . . . . . . . . . . . . . . . . 10
8 Synthesis sequence: a. nanosphere seed with CTAB surfactant. b. the
nanosphere with an anisotropic corona of CTAB(pink) around it allows
for different growth rates on the nanosphere. c. Au NR with CTAB corona. 14
9 Aqueous Au NR suspension . . . . . . . . . . . . . . . . . . . . . . . . . 15
10 Absorption spectrum for an isotropic aqueous suspension of Au NRs . . . 16
11 Calculated absorption cross section at the longitudinal plasmon resonance
absorption peak vs. aspect ratio for Au NRs suspended in water. . . . . . 17
12 Au NRs in suspension for 3 years (undisturbed). . . . . . . . . . . . . . . 18
13 TEM image of Au NRs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
14 a: electrostatic repulsion (ionic surfactants). b: steric repulsion (polymer
surfactants) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
15 a. CTAB stabilized aqueous dispersion of Au NRs. Red spheres indicate
positively charged head groups b. polystyrene stabilized organic Au NR
dispersion. White spheres indicate thiol head groups. . . . . . . . . . . 22
16 Silica encased Au NR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
vii
17 Au NR-polyurethane composite films (25× 125× 1) with various NR
concentrations from pure polyurethane(upper left), increasing by 1012 NRs
per sample, to 1013 NRs(bottom right) . . . . . . . . . . . . . . . . . . . 25
18 a. Schematic of layer-by-layer assembly. 0−: glass substrate. +: pos-
itively charged Au NRs. −: negatively charged polymer. b. Au NR-
polyurethane layer-by-layer composite film. . . . . . . . . . . . . . . . . . 26
19 Schematic of orientational ordered NRs . . . . . . . . . . . . . . . . . . . 29
20 Characteristic resonant behavior for the real and imaginary parts of the
parallel component of the susceptibility tensor as a function of wavelength. 33
21 Experimental setup: a.isotropic suspension of Au NRs in toluene. b.
aligned NR suspension³ bkb´ . . . . . . . . . . . . . . . . . . . . . . 46
22 Experimental absorbance spectrum as a function of applied voltage for Au
NRs suspended in toluene. . . . . . . . . . . . . . . . . . . . . . . . . . 47
23 Left: isotropic suspension of Au NRs in toluene. Right: aligned suspen-
sion of Au NRs in toluene, b k b (out of page) . . . . . . . . . . . . . . 48
24 Absorbance vs. the square of the applied external electric field at the
longitudinal wavelength, k = 659 from Fig.22 . . . . . . . . . . . . . 51
25 Plot of as a function of . . . . . . . . . . . . . . . . . . . . . . . . . 54
26 Order parameter vs. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
27 Absorbance vs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
28 Longitudinal resonance absorption spectrum for Au NRs suspended in
THF, DMF, chloroform and toluene . . . . . . . . . . . . . . . . . . . . . 57
29 Longitudinal resonance absorption shift in various solvents . . . . . . . . 58
viii
30 Absorbance spectrum for Au NRs supended in toluene and chloroformwith
no external field applied(solid lines) and a maximum field applied(dotted
lines.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
31 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
32 Transmitted intensity vs. time: voltage off(black). voltage on(red). . . . 61
33 Experimental setup: Mach-Zehnder interferometer and spectrophotometer 66
34 Schematic of the experimental setup . a: setup for real phase shift mea-
surements using the interfereometer. b: setup for imaginary phase shift
measurements using the spectrometer. . . . . . . . . . . . . . . . . . . . 67
35 Absorbance spectrum for NRs suspended in toluene with superimposed
lines indicating probe wavelengths. . . . . . . . . . . . . . . . . . . . . . 70
36 Imaginary phase shift vs. , ( = 575) . . . . . . . . . . . . . . . . . 71
37 Real phase shift vs. , ( = 575) . . . . . . . . . . . . . . . . . . . . 72
38 Imaginary phase shift vs. , ( = 575) . . . . . . . . . . . . . . . . . 73
39 Real phase shift vs. , ( = 575) . . . . . . . . . . . . . . . . . . . . . 74
40 Imaginary phase shift vs. , . . . . . . . . . . . . . . . . . . . . . . . 75
41 Real phase shift vs. , . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
42 Imaginary phase shift vs. , . . . . . . . . . . . . . . . . . . . . . . . 76
43 Real phase shift vs. , . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
44 Real part of the perpendicular and parallel susceptibility components as a
function of wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
45 Imaginary part of the perpendicular and parallel susceptibility components
as a function of wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . 79
ix
46 Predictions of SHO model for the real part of the perpendicular and par-
allel susceptibility components of a Au NR as a function of wavelength.
The NR dimensions were 20× 29 suspended in toluene similiar to the
measured example above. . . . . . . . . . . . . . . . . . . . . . . . . . . 80
47 Predictions of SHO model for the imaginary part of the perpendicular and
parallel susceptibility components of a Au NR as a function of wavelength.
The NR dimensions were 20× 29 suspended in toluene similiar to the
measured example above. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
48 SHO model fit to the experimental data of the real part of the perpendic-
ular and parallel susceptibility components as function of wavelength . . 83
49 SHO model fit to the experimental data of the imaginary part of the per-
pendicular and parallel susceptibility components as function of wavelength 84
50 Simulated longtidutindal real and imaginary susceptibilities for a Au NR
in vaccuum. (courtesy of L. Greengard, NYU) . . . . . . . . . . . . . . . 86
51 Real refractive index for Au NRs vs. volume fraction . . . . . . . . . . . 87
52 a: absorption spectrum for isotropic Au NR aqueous suspension and film.
b: absorption spectrum for isotropic Si-Au NR suspension and film . . . 91
53 Experimental setup for the Si-Au NR films . . . . . . . . . . . . . . . . . 92
54 a: experimental setup for the Si-Au NR films. b: image of Si-Au NR films. 92
55 NR alignment configuration . . . . . . . . . . . . . . . . . . . . . . . . . 93
56 Possible NR alignment mechanism . . . . . . . . . . . . . . . . . . . . . . 94
57 NRs aligned in the a. planar-parallel and b. planar-perpendicular config-
urations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
58 a: AFM image b: SEM image . . . . . . . . . . . . . . . . . . . . . . . . 95
x
59 a: experimental setup. b: transmission color through Si-Au Nr film for
orthogonal polarization states . . . . . . . . . . . . . . . . . . . . . . . . 96
60 Absorption spectrum vs. polarization for Si-Au NR films. 10 rotation of
the polarizer per absorbance curve. . . . . . . . . . . . . . . . . . . . . . 97
61 Absobance vs polarization at = 662 with a probe beam diameter
∼ 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
62 Images of liquid crystal phases formed by aqueous Au NRs on a TEM grid
under various evaporation rates. . . . . . . . . . . . . . . . . . . . . . . . 99
63 Schematic of evaporating water droplet. . . . . . . . . . . . . . . . . . . . 99
64 Radius vs. time for an evaporating droplet . . . . . . . . . . . . . . . . . 101
65 Si-Au NR film for orthogonal polarization states . . . . . . . . . . . . . . 102
66 Experimental setup to measure the real phase shift of the Si-Au NR films 103
67 Phase shift vs. polarization and wavelength for Si-Au NR films . . . . . . 104
68 Si-Au NR film transferred to an optical adhesive . . . . . . . . . . . . . . 105
69 Experimental setup to measure the temperature dependence of the ab-
sorbance for the Si-Au NR films . . . . . . . . . . . . . . . . . . . . . . . 106
70 Absorbance vs. temperature for the Si-Au NR films . . . . . . . . . . . . 107
71 Experimental setup. a: Image of PU-Au NR film in straining mechanism
b: unstrained sample. c: strained sample . . . . . . . . . . . . . . . . . 108
72 Polarized absorbance spectrum for LBL PU-Au NR films as a function
of strain(∼ 2). The probe polarization was perpendicular to the strain
direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
73 Maximum strained(∼ 10) absorbance spectrum vs. wavelength for PU-Au
NR film for different polarizations. . . . . . . . . . . . . . . . . . . . . . 110
xi
74 Absorption spectrum for gold nanorods and dichroic dyes suspended in
toluene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
75 Real phase shift vs. volume fraction for gold nanorods and dichroic dyes
suspended in toluene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
76 Real refractive index vs. extrapolated volume fraction of gold nanorods
and dichroic dye molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 115
77 a: Image of the sample. b: SEM image of the sample . . . . . . . . . . . 117
78 Absorption spectrum for the sputter-coated gold on a silicon wafer with
the superimposed probe wavelengths. . . . . . . . . . . . . . . . . . . . . 117
79 Real relative phase shift of the gold sputter-coated onto a silicon wafer as
a function of wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
80 LabVIEW correlator program . . . . . . . . . . . . . . . . . . . . . . . . 130
81 LabVIEW NR counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
82 Dye laser head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
83 Entrance and exit holes for the pump beam. . . . . . . . . . . . . . . . . 133
84 Cooling system for the dye laser circulator . . . . . . . . . . . . . . . . . 134
85 Layer-by-Layer deposition . . . . . . . . . . . . . . . . . . . . . . . . . . 136
86 Practical applications of Layer-by-Layer deposition . . . . . . . . . . . . 137
87 MetaMachine schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
88 a: 1st generation. b: screenshot of software.
c: during in situ testing at the University of Michigan . . 139
89 Schematic drawing interferometer . . . . . . . . . . . . . . . . . . . . . . 140
90 Schematic of kinematic mounts . . . . . . . . . . . . . . . . . . . . . . . 140
xii
LIST OF TABLES
1 Solvent vs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
xiii
Acknowledgements
I thank Prof. Peter Palffy-Muhoray for providing a stimulating environment to ex-
plore physics. I thank Dr. Ashish Argawal, Prof. Nicholas Kotov, Dr. Kyoungweon
Park and Dr. Richard Vaia for teaching me the art of gold nanorods and for providing
material samples. I thank Prof. Greengard for providing simulations. I thank Jeanette
Killius for teaching me transmission electron microscopy, Liou Qiu for helping me with
AFM and TEM measurements, Merrill Groom for his help in all things practical, and
the rest of the LCI staff for their help over the years. I thank my family for their support
and guidance, especially my most loving and critical wife, Michele.
xiv
CHAPTER 1
Introduction
1.1 Metamaterials
Metamaterials are a broad class of novel synthetic materials. The fundamental
feature of metamaterials are their unique properties determined from the organization of
the constituents. The novel properties are not simply linear functions of the constituents
from which the metamaterial is built [1]. Specifically we are interested in building
and characterizing self-assembled, dense, parallel, separated, aligned nanorod optical
metamaterials.
1.2 Negative index materials
Negative index materials with roots dating back to 1904 by Lamb and Pocklington
[2—4] pointing out the antiparallel propagation of the velocity and energy of the wave in
a mechanical system. The same year, Schuster [5] showed this behavior could be applied
to optical systems. In 1947 Mandelshatam [6] described negative refraction of optical
systems in detail and in 1967 Veselago [7] fully described the idea of a negative index
material. Nearly a century after its inception Schultz et al [8] in 2001 were able to clearly
experimentally demonstrate negative index of refraction at microwave frequencies.
1
2
The equations governing electromagnetic phenomena are Maxwell’s equations,
∇ ·D = (1)
∇ ·B = 0 (2)
∇×E = −B
(3)
∇×H = J +D
(4)
where E is the electric field, D is the electric displacement, B is the magnetic flux
density, H is the magnetizing field, is the free charge density and J is free current
density. The constitutive relations are,
D = 0E+P = E (5)
B = 0 (H+M) = H (6)
where P is the polarization andM is the magnetization. = 0 and = 0 are
the relative and free space electric permittivity and magnetic permeability, respectively,
and generally are second rank tensors with complex components.
If the material is isotropic, with no free charge and assuming a time independent
solution. The wave equation, from Eq.3 and Eq.4, is,
∇E2 = 2E
2(7)
if,
E = E0(k·r−) (8)
where E0 is the amplitude, is the angular frequency, and wavenumber =2, propa-
gating in the b direction, Eq.7 gives,2 = 2 (9)
3
The phase velocity of the propagating wave follows from Eq.9,
=
=
1√
(10)
If the wave is propagating in a vacuum,
= =1√00
= 2997925× (10)8h
i(11)
if not traveling in a vacuum,
=1√=
1√
1√00
=
(12)
where the refractive index, , of a material is given by,
= ±√ (13)
The electric permittivity, , and magnetic permeability, , are the two macroscopic
quantities that determine the refractive index, , of bulk materials.
The refractive index is, in general, a complex number, = 0 + ”, plugging this
into Eq.8, for a wave propagating in the x-direction,
E = E0−2
”(
20−) (14)
the real part, 0, gives the factor by which the phase velocity of the light decreases
in a material as compared with vacuum and the imaginary part, ”, determines the
attenuation of the wave as it propagates through the medium. As a wave travels through
a medium the intensity exponentially decays,
= 0− (15)
where the absorbance is,
=4”
(16)
4
is the free space wavelength, is the thickness of the sample and 0 is the intensity
at = 0. By measuring the intensity of the light beam before and after the sample, ”
can be determined from the previous two equations.
If the material is isotropic and lossless then the refractive index is,
= ±p00 (17)
If both , 0 , a positive index material (PIM), then is a positive real number
and the wave propagates with velocity . If either or is less than zero, then
becomes an imaginary number and the wave does not propagate. If both , 0,
negative index material (NIM), then is again a real number and the wave propagates.
However, the negative root of Eq.17 must be chosen in order to satisfy the boundary
conditions, kk1 = k
k2, where the subscripts 1, 2 represent two different refractive indices,
this leads to a reversal of Snell’s law, Fig.1b,
kk1 = k
k2 (18)
1k0 sin (1) = −2k0 sin (2) (19)
1 sin (1) = −2 sin (2) (20)
Figure 1: a: positive index material (PIM) b: negative index material (NIM)
5
Moreover the constitutive equation,
k×E = H (21)
E× k×E = (E×H) (22)
k (E ·E)−E (k ·E) = S (23)
where the Poynting vector is S = E×H and if 0 then,
k (E ·E) = −S (24)
In a NIM the direction of energy propagation, S, is antiparallel to the momentum of
the photon, p = ~k.
If the refractive index is negative, this leads to an exponentially increasing evanes-
cent wave inside the NIM. Illustrating this mechanism a plane monochromatic wave of
wavelength traveling in the b direction strikes an object of size ∆ , Fig.2,
Figure 2: Schematic of diffracting plane wave.
where 2 = 2 + 2 = 220 solving for ,
= ±q220 − 2 (25)
6
can be thought of the inverse distance the information is carried. The smaller
is the more high spatial frequency information is transmitted to the far-field.
If 0 then 0. If ∆ = , 20= 2
∆, then ∆ = 0
. This is the diffraction
limit. If 0 , then is imaginary and the high spatial frequency information
(evanescent wave) decays exponentially in the +b direction.However in the NIM case the transport of energy (Poynting vector) is in the +b
direction which requires to have opposite sign,
= −q220 − 2 (26)
The negative sign of the wavevector now causes the evanescent wave to grow exponentially
in the +b direction. Pendry [9] proposed the possibility of a negative index lens, wherethe evanescent waves are exponentially amplified in the NIM leading to sub-wavelength
resolution, a "super-lens".
Figure 3: Ray diagram for a superlens
The superlens is composed of a parallel-sided slab of NIM. If a source is placed 2
away from the lens, where is the thickness of the lens, then a perfect atom-for-atom
image would be formed at 2on the other side of the lens. Noting that the impedance,
=p
, must be positive such that when = −1 the medium is perfectly matched
to free space and there are no reflection at the interfaces.
7
Another approach to achieve subwavelength resolution images is the hyperlens [10—12].
Anisotropic, hyperbolic dispersion, metamaterials, do not require 0: only 0, are
capable of carrying near-field information to the far-field over coming the diffraction
limit and possibly leading to a perfect image of an object. For Eq.25, if becomes
sufficiently large, i.e. 0 , then can become positive, a real traveling wave. To
understand how can become arbitrarily large consider light propagation in a uniaxial
crystal without losses. Solving Maxwell’s equations specifically for monochromatic plane
waves, where the direction of propagation, b, and the dielectric tensor, are known [13],³ − bb´ −1 D =
1
2D (27)
This is a standard eigenvalue equation, where the eigenvalues determine the refractive
index, surface of Fig.4, and the eigenvectors determine the direction D for the three
expected solutions. If E is parallel to b it is an eigenvector with eigenvalue 0 and themode is non-propagating. If D is perpendicular to b and b this is the ordinary solutionwith index =
√⊥. The third solution, , is the extraordinary solution given by,
1
2=
k⊥+
⊥k
(28)
where b is a unit vector along the optical axis and k = b · b. If k 0 , then the
eigenvalue surface is a hyperboloid (hence hyperbolic lens), Fig.4,
8
Figure 4: Surface index hyperboloid for light propagation in a uniaxial material.
If the angle between b and b becomes large , b becomes large³b · b = 0
´and hence
can become arbitrarily large, such that 0 À , leading to subwavelength resolution
lenses .
1.3 Realization of optical negative index materials
The fundamental mechanism driving optical metamaterial phenomena is polarization.
To construct metamaterials with 0 at optical frequencies gold nanorods can be
used for the constituents. If an external electromagnetic field is incident upon a gold
nanorod it can penetrate into the volume of the nanorod, contrary to bulk metals, and
shift the free conduction electrons relative to the ion background. The resulting surface
charges produce a depolarizing field, E, within the nanorod. The resulting equation of
motion for the electrons oscillating in response to the external field can be qualitatively
understood with a simple oscillator model. On the low frequency side of resonance the
electrons and external field oscillate in phase. On the high frequency side of resonance
the electrons and external field are 180 out of phase giving rise to anomalous dispersion,
Fig.5,
9
Figure 5: Gold nanorod in an optical field
For this work, to achieve 0 at optical frequencies two gold nanorods can be
coupled together into a "cut-wire"pair [14]. The two coupled nanorods form a psuedo-
current loop. For incident light with the electric field polarized along the long axis of
the nanorods and the magnetic field perpendicular to the pair, Fig.6, the electric and
magnetic responses can experience a resonate behavior at certain frequencies that depend
on the geometry of the pair. These resonances can be out of phase with respect to each
other, an antisymmetric resonance, giving rise to a current and magnetic moment at
optical frequencies.
Figure 6: Cut-wire pair
The goal of this project is to create the proposed constituents and self-assemble them
into orientationally ordered, high-density, marcoscopic length scale optical metamaterials,
10
Fig.7,
Figure 7: Proposed negative index material.
1.4 Dissertation outline
A critical component of bringing optical metamaterials to fruition is the ability to pro-
duce and characterize self-assembled, highly-ordered, macroscopic ensembles of nanorods.
The bulk of this work can be divided into three sequential tiers. The first is the
synthesis of the constituent nanorods. The second is using these materials to create
macroscopic domains of aligned anisotropic nanorods. The third is the development of
an experimental characterization technique to determine the electric susceptibility tensor
for gold nanorods at optical frequencies, and the characterization of the materials.
First, a brief overview of the synthesis processes used to make the constituent materi-
als used throughout this dissertation is given. This work was done in collaboration with
Prof. Kotov’s group at the University of Michigan. Self-assembly can be thought to ap-
ply to both particle synthesis as well as particle-particle assembly. This work focuses on
the self-assembly of gold nanorods using an aqueous seed-mediated synthesis and growth
procedure. Various novel surfactants were used to disperse the nanorods in both aqueous
and nonaqueous suspensions as well as embedding the nanorods into elastomeric hosts
and encapsulating them in hard shells.
Secondly, the development of novel methods of self-assembly leading to dilute and
high-density films and suspensions of orientationally ordered anisotropic nanorods in
11
macroscopic domains is described. A high degree of alignment was achieved by using
an external low frequency electric field on bulk organic suspensions of small aspect ratio
gold nanorods. Self-assembled films highly loaded with small aspect ratio, orientationally
ordered, silica encased gold nanorods in square millimeter superdomains were obtained
by the combination of solvent evaporation and applied electric field. Thin, free-standing,
gold nanorod-polyurethane composite films were developed by embedding nanorods into
bulk polyurethane as well as assembling bilayers of nanorods and polyurethane using
layer-by-layer deposition. The nanorod-polyurethane composites can then be strained,
and it was found that the strain orients the nanorods. An order parameter extraction
technique was developed to determine the orientational order in our samples from polar-
ized absorbance measurements.
In the third part of this work, a straightforward technique was developed to deter-
mine the principal values of the complex dielectric susceptibility of the gold nanorods
in dilute suspensions as a function of frequency. This was done by measuring the real
and imaginary phase shift of light transmitted through gold nanorod suspensions in or-
ganic solvents. The real and imaginary parts of the phase shift were determined using a
Mach-Zehnder interferometer with a dye laser and a spectrophotometer.
The optical properties of other materials, such as conjugated molecules, were explored.
They are capable of anomalous dispersion at optical frequencies, suggesting the possibility
of using them as novel building blocks for self-assembled optical metamaterials. The real
and imaginary phase shifts of sputtered coated films of gold on silicon wafers were also
measured using a Mach-Zehnder interferometer and spectrophotometer.
12
1.5 Novel achievements
• Alignment of bulk organic suspensions of small aspect ratio gold nanorods by ap-
plying an external electric field .
• Development of an experimental characterization technique to determine the prin-
cipal values of the electric susceptibility tensor for gold nanorods at optical fre-
quencies.
• Self-assembled films of densely-packed, small aspect ratio, orientationally-ordered,
silica-encased gold nanorods in square millimeter domains- via solvent evaporation
and applied electric field.
• Discovered conjugated molecules are capable of anomalous dispersion with figures
of merit comparable to metals.
CHAPTER 2
Synthesis and surfactants of gold nanorods
2.1 Introduction
This chapter gives a brief overview of the synthesis processes used to make the con-
stituent materials used throughout this dissertation.
Traditionally various top-down lithographic techniques have been used to produce
nanorods [15—22]. These processes may yield highly monodispersed nanorods- however
there are significant disadvantages. Typically the processes are expensive, producing
predominantly two-dimensional fixed structures.
Conversely a pragmatic method is bottom-up or self-assembly synthesis. Self-assembly
can be thought to apply to both particle synthesis as well as particle-particle assembly.
We focus on gold nanorod synthesis in this chapter using self-assembly.
There are two main methods of synthesis for gold nanorods. The first, is the hard
template method [23—26]. Gold ions in solution are reduced in nanoporous templates to
produce gold nanorods. This technique is limited by the availability of templates, has
relatively low yields and the nanorods have high polydispersity. The second method is
seed-mediated growth synthesis, which is the focus of this chapter.
2.2 Gold nanorod synthesis
Gold nanorods (Au NRs) were synthesized using the seed-mediated growth method
described in literature [27]. The process involves reducing gold chloride (HAuCl4) in an
aqueous solvent by a strong reducing agent sodium borohydride (NaBH4). Reaction with
13
14
reducing agents such as NaBH4 causes elemental gold to be precipitated from solution
forming nanospheres that act as a seed when added to a growth solution containing gold
ions and the surfactant hexadecyltrimethyl ammonium bromide (CTAB). The lattice
constant of gold is ∼ 04, for the nanosphere seeds with diameter ∼ 4 the lattice
structure is non-negligible. The lattice structure creates a non-uniform affinity for the
ionic surfactant and it attaches around the seed in an anisotropic manner, Fig.8a. When
the seeds are placed into a growth solution containing excess gold ions, the ions precipitate
onto the seeds surface at different rates, Fig.8b,
Figure 8: Synthesis sequence: a. nanosphere seed with CTAB surfactant. b. the
nanosphere with an anisotropic corona of CTAB(pink) around it allows for different
growth rates on the nanosphere. c. Au NR with CTAB corona.
this allows the gold nanospheres to grow into NRs, Fig.8c. For NRs having an aspect
ratio less than ∼ 3 they are ellipsoidal in shape, for larger aspect ratios the NRs become
hemispherical capped cylinders. A seed solution was made by mixing CTAB solution
(5 020) with HAuCl4 (5 of 5× (10)−4) followed by addition of ice-cold NaBH4(06 1 × (10)−2) stirring for 2 minutes. The growth solution was made by adding
AgNO3 (06 4 × (10)−3) to CTAB (10 02) followed by addition of HAuCl4
(10 1 × (10)−3) and ascorbic acid (140 788 × (10)−2). Although the exact
15
mechanism of the AgNO3 is not known, the amount of AgNO3 present is the dominating
factor in determining the aspect ratio of the NRs. To grow NRs 24 of the seed solution
is added to the prepared growth solution and kept on a hot plate with the temperature
set at 30−50 for 3 hours. Temperature is also an important parameter in determining
the length of the NRs- the warmer the reaction temperature leads to an increase in the
reaction rate producing larger aspect ratio NRs. The exact mechanism is not understood
and reverses (smaller aspect ratio NRs with increasing temperature) in the case of binary
surfactant synthesis such as CTAB/BDAC [28].
Synthesized Au NRs have bilayers of positively charged surfactant on the surface
and are dispersed in water. Excess CTAB in the solution is removed by centrifugation
(4 185g for 80 mins), and resuspended in deionized water and the centrifuge process is
repeated two more times. Refer to Chap.8.1 for the complete Au NR synthesis procedure.
Figure 9: Aqueous Au NR suspension
Fig.9 is an example of an aqueous suspension of Au NRs. Au NR suspensions have
16
various colors depending on the aspect ratio of the NRs.
500 600 700 8000.0
0.5
1.0
x
[a.u
.]
[nm]
Figure 10: Absorption spectrum for an isotropic aqueous suspension of Au NRs
Au NRs are strongly absorptive in the visible regime as evidenced by the various colors
for different aspect ratio NRs. Absorbance, , measurements were done with an Ocean
Optics HR4000CG-UV-NIR spectrometer with Mikropack DH-2000 unpolarized white
light source in a cuvette with a = 1 path length unless otherwise stated. Fig.10 is a
characteristic absorbance vs. wavelength spectrum for the Au NR suspensions. The two
absorption peaks are observed, they correspond to the two principal plasmon resonances
on the NR, one along the length of the NR (longitudinal) and one along the diameter
(transverse). In Fig.10 the transverse plasmon absorption peak is centered at 514
and the longitudinal plasmon absorption peak is centered at 680. The length of the
NR can be controlled- tuning the longitudinal plasmon absorption peak from 615 to
850.
17
The NR concentration was determined by using a calculated absorption cross section,
, for Au NRs as a function of aspect ratio at the longitudinal plasmon absorption
peak wavelength in water, Fig.11 [29—31]. The absorption coefficient, , is related to
the absorption cross section as, = , where is the number density. Typical NR
absorbances are ≈ 1 at the longitudinal plasmon absorption peak, aspect ratio2,
Fig.10, this corresponds to NR concentrations of ∼ 10h3
i.
Figure 11: Calculated absorption cross section at the longitudinal plasmon resonance
absorption peak vs. aspect ratio for Au NRs suspended in water.
The ability for NRs to remain in aqueous suspensions for long time periods was
addressed experimentally. In order for the NRs to stay suspended a height, , the
thermal forces, , and buoyancy forces, , must be greater than or equal to
the gravitational force, . The density, , of NRs in suspension goes as, () ∼
−(
−) , this implies that, =
(−) ≈ 4, . The aqueous suspension
synthesized for this work stayed suspended for 3 years, Fig.12, if the NRs do settle out,
18
the suspension can be shaken and the NRs become dispersed again. From the experiment
the NRs remained suspended a height of ≈ 100 (38ml marker).
Figure 12: Au NRs in suspension for 3 years (undisturbed).
Fig.13 is a representative image of aqueous Au NRs taken with a transmission electron
microscope (TEM). (JEOL JEM-100S,40-100kV at NEOUCOM). The samples are not
monodisperse [27]. Typical measured polydispersity for these samples are ∼ ±15% from
the average measured dimension. For the sample in Fig.13, 50 NRs were measured, this
will be discussed in Chap.8.6, diameter= 132± 20 and length= 508± 65.
19
Figure 13: TEM image of Au NRs.
Growing small aspect ratio Au NRs ( 2) sensitively depends on the chemical con-
centrations, reaction temperature and reaction rates; allowing the final solution to sit on
the hot plate for at least an order of magnitude (30 ) longer greatly increased the
monodispersity of the suspensions. In extreme cases the NRs were allowed to sit for 2
years(at room temperature)- greatly increasing monodispersity. Liz-Marzan et al [32]
improved the monodispersity of El-Sayed’s method [27] by slowing down the growth
kinetics to yield highly monodisperse NRs, the procedure will be discussed in Chap.8.3.
Vaia et al [33] were able to achieve near perfect monodispersity (∼ 999%) through
the formation of reversible flocculates by surfactant micelle induced depletion interaction.
Separation is achieved even for the nanoparticles of the same mass with different shape
by tuning the surfactant concentration and extracting flocculates from the sediment by
centrifugation or gravitational sedimentation.
2.3 Surfactants
20
AuNRs synthesized by self-assembly synthesis are solubilized in polar solvents by ionic
surfactants, Fig.15a. These also stabilize the suspension; their dissociation is responsible
for the electrostatic repulsion of particles, Fig.14a. In such systems, the particles are in a
secondary (local) energy minimum due to electrostatic interactions, and will eventually
aggregate since the global energy minimum occurs when the particles are in contact. The
ionic surfactant can be displaced from the surface of the NRs since it is not covalently
bonded to the surface and the NRs will aggregate. By contrast, Au NRs can be solubi-
lized in organic solvents by organic thiolated polymer surfactants which can be covalently
bonded to the surface of the NRs. Such a system is stabilized against aggregation by the
entropic repulsion of the polymer chains(Lennard-Jones like), and the system remains in
the global free energy minimum indefinitely, Fig.14b.
Figure 14: a: electrostatic repulsion (ionic surfactants). b: steric repulsion (polymer
surfactants)
2.3.1 Gold nanorods dispersed in organic solvents
Dispersing Au NRs into organic solvents with thiolated polymer surfactants enables
the production of NR suspensions which remain stable against aggregation indefinitely.
Transfer of Au NRs from aqueous to organic dispersion has been achieved previously using
water-immiscible ionic liquid but is limited in its efficiency of phase transfer [34]. The
21
NRs appear to be aggregated as indicated by the loss of their plasmon absorption peaks
after transfer into organic solvent. Zubarev et al have used a slow and detailed process
for replacing the surfactant CTAB with polystyrene [35]. Chen et al have reported phase
transfer from aqueous into toluene using combination of mercaptosuccinic acid (MSA)
and tetraoctylammonium bromide (TOAB) [36] . We introduce here very easy and generic
way of transferring Au NRs from aqueous dispersion into an organic solvent of choice.
The following method was developed in collaboration with Kotov et al [37]. Or-
ganic dispersion were achieved by using a 20 glass vial in which 1 of concentrated
aqueous solution of Au NRs and CTAB (10−8 ) is added to 1 of polystyrene thiol
(Polymer Source Inc, thiol terminated polystyrene, = 50 = 53) solution in
tetrahydrofuran (THF) (25) and the mixture is vigorously shaken by hand. The
reaction leads to a near instantaneous aggregation and the mixture of aggregates stick
to the wall of the glass vial leaving behind a colorless water and THF. The water and
THF mixture is decanted and the remaining moisture in the glass vial is removed by
drying with a heat gun. The Au NRs inside the aggregates are grafted with polystyrene
molecules Fig.15b and can be dispersed in any organic solvent of choice. Typically 1 of
toluene is added to the vial containing the aggregates and the sample is sonicated to get
excellent dispersion. The complete phase transfer process will be discussed in Chap.8.2.
22
Figure 15: a. CTAB stabilized aqueous dispersion of Au NRs. Red spheres indicate
positively charged head groups b. polystyrene stabilized organic Au NR dispersion.
White spheres indicate thiol head groups.
The choice of THF as a solvent for polystyrene thiol is critical for the reaction because
its solubility in water facilitates reaction between thiol molecules and the gold atoms on
the surface of rods. The nature of CTAB binding on the gold surface is dynamic in nature
and high affinity of sulfur group for the gold causes near instantaneous covalent bonding of
gold and thiolated polystyrene. As the reaction completes the NRs turn from hydrophilic
to hydrophobic and are no longer soluble in the water and THF mixture; this leads to the
aggregation. Drying the aggregates helps remove remaining traces of THF left behind.
The aggregates redisperse into individual NRs when introduced into an organic solvent.
The intermediate step of aggregation of the NRs does not in anyway affect the shape or
size. Once suspended in organic solvents the samples show absorption peaks with the
same amplitudes as in Fig.10, but red-shifted due to change in the dielectric constant of
the solvent, as discussed in [36].
23
2.3.2 Gold nanorods encased in silica shells
If two NRs are in close contact either end-to-end or side-by-side, they are effectively a
new particle with a new shape and new plasmon resonant frequencies. To maintain the
individual properties of densely packed NRs they need to be separated roughly ∼ 25
the diameter of the nanoparticle [38]. By encasing NRs in a silica shell, as shown in
Fig.16, the plasmon resonances of the individual NRs are preserved when the NRs are
packed densely. Moreover, the local dielectric environment around the NRs becomes
fixed. This allows for the NRs to then be transferred to different hosts without shifting
the plasmon resonances.
Figure 16: Silica encased Au NR
Fig.16 shows Au NR encased in silica (Si-Au NR) imaged with a TEM. The dark
ellipsoid is the Au NR and the lighter annulus surrounding the Au NR is the silica shell
(SiO2) with a thickness roughly the diameter of the NR.
Au NRs were coated with silica using a method described in the literature [39]. To
begin the coating process excess CTAB is removed by centrifugation and redispersed
twice in deionized water. The final volume for one batch is 10. In a test tube 1
solution of aqueous NRs is diluted to a final volume of 20 followed by addition of
24
200 of 01solution of NaOH and 6 of tetraethylothoxysilicate (TEOS) under gentle
stirring. The solution is allowed to stand for 2 days. The coated NRs are centrifuged
and redispersed in deionized water. The resulting suspension of Si-Au NRs has similar
absorption spectrum to Fig.10.
2.3.3 Polyurethane gold nanorod composites
An important step for practical "soft" optical nanodevices [20, 40] is to construct
freestanding films containing high densities of NRs that may be used to create novel
optical metamaterials elements such as transformational optics [41]. Liz-Marzan [42]
embedded Au NRs in poly(vinyl alcohol) (PVA) films however they can only undergo
plastic deformations at high temperature limiting their use at room temperatures.
Polyurethane can undergo large elastic deformations at room temperature. Au NRs
are dispersed in a water-soluble polymer suspension. As the water evaporates the poly-
mer cross-links forming a Au NR-polyurethane composite film, Fig.17. The cross-linking
density and NR concentration determines the stiffness, Young’s modulus, = , of the
material, where is the stress and = ∆is the strain. The energy density of the
polymer is, = 12 [43], where is the cross-link density, therefore, =
12 , as
the cross-linking density increases so does . If more NRs are added to the film, less
polymer is present, decreasing and increasing .
This method was developed in collaboration with Kotov et al [37]. The excess CTAB
is removed by centrifugation and redispersed twice in deionized water. Au NRs in varying
amounts are mixed with of cationic polyurethane (07027). The dispersion is
then poured into a petri dish and allowed to dry in an oven for 12 hours. The cross-linked
Au NR composite film is then peeled from the surface of the dish and cut into pieces
with smaller areas for the absorbance vs. strain experiments.
25
Figure 17: Au NR-polyurethane composite films (25 × 125 × 1) with various NR
concentrations from pure polyurethane(upper left), increasing by 1012 NRs per sample,
to 1013 NRs(bottom right)
Qualitatively the shape of the absorbance spectrum is similar to that shown in Fig.10
and has not been significantly altered by the presence of the host polymer.
Layer-by-layer assembly of gold nanorod-polyurethane composite films
A useful self-assembly technique is layer-by-layer (LBL) deposition. A simple binary
LBL system consists of a solution of aqueous positively charged Au NRs, +, and another
solution of negatively charged polymer, −. If a substrate of negatively charged glass,
0−, is dipped into a beaker of the positively charged NRs, then a single layer of the
positively charged NRs adsorbs to the surface of the glass. The substrate is then rinsed
in with ultrapure deionized water, to remove any excess NRs. Then the substrate is
dipped into the negatively charged polymer solution and a single layer of negatively
charged polymer adsorbs to the surface of the positively charged NRs and rinsed again.
This repeats until a film is created, Fig.18a. The film can be removed from the glass
substrate and used for its intended application.
26
a. b.
LBL film
Figure 18: a. Schematic of layer-by-layer assembly. 0−: glass substrate. +: positively
charged Au NRs. −: negatively charged polymer. b. Au NR-polyurethane layer-by-
layer composite film.
Au NR-polyurethane LBL film dimensions are similar to Fig.17 (5× 20) but the
thickness is ∼ 15, Fig.18b. There are many types of devices that may be produced
using this technique such as photovoltaics, energy storage, optical elements and laser
systems. A machine, MetaMachine, was built to automate the dipping algorithm and
to be versatile enough to build current and future LBL metamaterials as discussed in
Chap.8.9.
CHAPTER 3
Field induced orientational order of gold nanorods in organic suspensions
3.1 Introduction
This chapter describes an experimental technique to align small aspect ratio Au NRs
in bulk organic suspensions using an external electric field.
Although the individual response of anisotropic nanoparticles, such as Au NRs, may
be strongly direction dependent, assemblies of randomly oriented NRs are isotropic. To
access the full range of possible materials responses in NR dispersions, control over ori-
entation is required.
To our knowledge this is the first electric field alignment of a bulk organic suspension
of small aspect ratio( 2) Au NRs [37]. Others [44—50] have used electric fields in two
dimensions or aided by the host medium to align NRs. Lekkerkerker et al [46] aligned
an aqueous suspension of large aspect ratio ( 126) NRs with an electric field; only the
transverse absorption peak was measured. Lekkerkerker observed evidence of alignment-
however there were unresolved observations. The entire absorbance spectrum, for both
polarizations parallel and perpendicular to the electric field alignment direction, increased
with increasing field strength and the electric field could only be maintained for seconds.
If the field was applied for 1-2 minutes, they observed the NRs aggregated into strings
of concentrated and less concentrated regions.
3.2 Orientational order parameter
27
28
The bulk properties of NR suspensions and composites depend on the orientational
order of NRs. To specify the degree of orientational order for the NRs, an order para-
meter, is defined. This can be done in a variety ways, we use an approach similar to
liquid crystals. In general the statistical averaging is done with the orientational prob-
ability density function ( ) that determines the probability of finding a NR with a
given orientation within an element of solid angle. The orientational probability density
function is assumed to have a Boltzmann form,
( ) =− Z
− sin ()
(29)
where is the angle between the long axis of each individual NR and the unit vector
along the external field direction, the director b, is the azimuthal angle about b, is
the energy of the external field, is Boltzmann’s constant and is temperature, Fig.19.
We characterize the alignment by a scalar order parameter, , which is a measure of NR
orientation,
() =
¿3
2cos2 ()− 1
2
À=
Z ()
µ3
2cos2 ()− 1
2
¶sin () (30)
where the brackets, hi, indicate the ensemble average. There is cylindrical symmetry
about b, that is ( ) is independent of and b and −b are equivalent. If = 0, forthe NRs (parallel alignment), cos () = ±1and = 1. This corresponds to a nematic
phase where there is perfect orientational order. If the NRs are randomly oriented ( ()
is independent of ), then cos2 () = 13and = 0. This is the isotropic phase.
29
Figure 19: Schematic of orientational ordered NRs
3.3 Background
This work is experimentally based; a minimum amount of theory must be developed
to interpret the experimental results.
3.3.1 Dielectric function for a dilute, orientationally ordered, ensemble of gold nanorods
in a vacuum at optical frequencies
To qualitatively understand the physical mechanisms underlying the behavior of Au
NRs in an optical field a simple harmonic oscillator model is developed. An external
optical electromagnetic field is incident upon a Au NR in a vacuum. The free electrons
in the gold respond in the opposite direction to the applied field, creating a depolariza-
tion field inside the NR. If it is assumed the NRs are ellipsoidal in shape, this is not
unreasonable for small aspect ratio NRs, then the depolarizing field inside a ellipsoid is
uniform [51,52] and has the form,
E = − P
0(31)
30
where P is the polarization in the Au NR, 0 is the permittivity of free space and
is the depolarizing factor which in the diagonal frame is,
=
¯¯¯ 0 0
0 0
0 0
¯¯¯ (32)
The depolarization factor is a measure of how much the internal field in the NR
is affected by the polarization created by the free charges. The components of the
depolarizing factor form a tensor. The principal values, , , and , where , ,
and are the three principal axis of the ellipsoid, are given by,
=
Z ∞
0
2 (2 + )32 (2 + )
12 (2 + )
12
(33)
=
Z ∞
0
2 (2 + )12 (2 + )
32 (2 + )
12
(34)
=
Z ∞
0
2 (2 + )12 (2 + )
12 (2 + )
32
(35)
where , , and are the lengths of the semiaxes for the ellipsoid [52]. The sum of the
principal values of the depolarizing factor, is unity, that is,
+ + = 1 (36)
For prolate spheroids ( = ), where only one depolarizing factor is indepen-
dent, the following analytical expression for as a function of eccentricity, , can be
derived,
=1− 2
23
µln
µ1 +
1−
¶− 2
¶(37)
= =1
2(1−) (38)
31
where =
r1−
³
´2.
Assuming that the local field inside the NR is, E = E + E where E is the
externally applied optical field, the equation of motion for a conduction electron is,
x+ 0
x = (E +E) (39)
where is the position, is the mass of the electron, 0 is the damping constant and
is the charge of the electron. Writing the depolarizing field explicitly,
x+ 0
x =
ÃE − P
0
!(40)
If the applied electric field is of the form E = E0, then P = x = P0
, where
is the number density. Substituting these equations into Eq.40 and assuming a solution
of x = x0 gives,
−2P− 0P = 2
ÃE− P
0
!(41)
rearranging and defining the plasma frequency as 2 =2
0 = 0
, gives,
³−2 − +
2
´P = 0
2E (42)
Solving Eq.42 explicitly for P
P = 0
¯¯¯
22−2− 0 0
02
2−2− 0
0 02
2−2−
¯¯¯E (43)
or P = 0E where is the electric susceptibility. Now we have an explicit
expression for the polarization of a single nanorod in an applied electric field. For a
uniaxial NR = .
32
We can write the susceptibility in a more convenient form,
=
¯¯¯
22−2− 0 0
02
2−2− 0
0 02
2−2−
¯¯¯ =
¯¯¯⊥ 0 0
0 ⊥ 0
0 0 k
¯¯¯ (44)
Noting that Eq.44 only depends on , , and .
We can separate the susceptibility into its real, 0, and imaginary parts,
”
,
0 =
¯¯¯
2(2−2)
(2−2)2+22
0 0
02(
2−2)
(2−2)2+22
0
0 02(
2−2)
(2−2)2+22
¯¯¯ (45)
0 =
¯¯¯0⊥ 0 0
0 0⊥ 0
0 0 0k
¯¯¯ (46)
where 0 is a diagonal element of the real part of the susceptibility tensor.
”
=
¯¯¯
2
(2−2)2+22
0 0
02
(2−2)2+22
0
0 02
(2−2)2+22
¯¯¯ (47)
”
=
¯¯¯”⊥ 0 0
0 ”⊥ 0
0 0 ”k
¯¯¯ (48)
where ” is a diagonal element of the imaginary part of the susceptibility tensor.
Plotting, Eq.44, the real and imaginary parallel component of the susceptibility tensor,
0k = 0k and ”k = ”k, Fig.20,
33
580 600 620 640
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
'
"
[nm]
0
Figure 20: Characteristic resonant behavior for the real and imaginary parts of the
parallel component of the susceptibility tensor as a function of wavelength.
In Fig.20 the plasmon resonance has been shifted from the bulk value by the depo-
larizing factor , where 20 =
2, for Au NRs this shift pushes the resonance into
the visible regime.
If 2 − 2 = 0 for the imaginary part of the susceptibility from Eq.44 then,
” =2¡
2 − 2¢2+ 22
(49)
”max =2
(50)
Therefore as approaches , ” approaches a maximum.
By inspection ” = 12”max when,
¡
2 − 2
¢2= 22 (51)
34
and solving for ,
−2 ± +2 = 0 (52)
with two positive roots,
=1
2
µq2 + 42 ±
¶(53)
FWHM is the difference between the two positive roots, = . If is
negligible compared to 0, then ” = 12”max occurs at 0 − ≈ ±
2.
If the derivative of the real part of the susceptibility from Eq.44 is equal to zero with
respect to the variable = 2 − 2 then the extrema are,
0=
2¡
2 − 2
¢¡2 − 2
¢2+ 22
=2
2 + 22(54)
0
= 2
¡2 + 22
¢−1 − 222 ¡2 + 22¢−2
= 0 (55)
= (56)
0maxmin = ± 2
2= ±”max
2(57)
and occurs at ≈ 0 ±
2. This simple oscillator model can be used, surprisingly
effectively, to understand the behavior of Au NRs.
The real and imaginary parts of the susceptibility are not independent of each other,
as shown in Eq.57, and more generally by the Kramers-Kronig relation [52]. This has
significant consequences, implying that , , and determines the complete behavior
of the system. If , , and are known, then in principle it is straightforward to
determine 0max and ”max.
The figure of merit is,
=
¯0
”
¯(58)
35
and is a measure of the optical performance of a material. Typically for useful optical
applications the should be as large as possible, minimizing losses. The real and
imaginary refractive indices can be related to the real and imaginary susceptibilities. If
Eq.57 is to be believed, then the will always be close to unity for Au NRs at optical
frequencies, with severe consequences for the realization of optical NIMs.
Connecting the susceptibility to the permittivity,
D = E = 0E+P (59)
where,
P = hpi (60)
The average dipole moment for an ensemble of NRs is,
hpi = 0®E (61)
where is the volume of one NR, or
P = 0®E (62)
The susceptibility expression, Eq.44, can be recast as,
= ⊥ +¡k − ⊥
¢bb (63)
where b is a unit vector along the long axis of the NR. Rearranging Eq.63, the averagesusceptibility of an orientationally ordered ensemble of NRs is,
®=
µk + 2⊥
3
¶ +
2
3
¡k − ⊥
¢¿12
³3bb −
´À(64)
36
where the order parameter tensor is given by =D12
³3bb −
´E=
2(3 b b − ).
Simplifying Eq.64 and the notation, the average susceptibility is,
®= +∆
2(3 b b − ) (65)
where b is nematic director in the optical axis direction, =k+2⊥
3and ∆ =
23
¡k − ⊥
¢and is the scalar order parameter.
Plugging Eq.62 into Eq.59,
E = 0E+ 0®E (66)
= + (67)
where is the volume fraction of NRs. Finally substituting Eq.65 into the previous,
= +
µ +∆
2(3 b b − )
¶(68)
Now we have an explicit expression for the electric permittivity of a dilute ensemble
of orientationally ordered NRs.
3.3.2 Dielectric function for a dilute, orientationally ordered, ensemble of gold nanorods
in a solvent at optical frequencies
If a NR is placed into a medium other than a vacuum then the plasmon resonances
will be affected. In general, near a boundary with normal b, the surface charge densityis
= P·b (69)
where P is the polarization. We write
D = E = 0E+P (70)
37
or
P =
µ1− 1
¶D (71)
and
=
µ1− 1
¶D · b (72)
Since the normal component of D is continuous across the interface, we have that,
=
µ1− 1
¶D · b (73)
and
= −µ1− 1
¶D · b (74)
where the subscripts represent the fields inside and outside the surface. The total
surface charge is given by
= + =
µ1
− 1
¶D · b (75)
In terms of the polarization inside,
P =
µ1− 1
¶D (76)
and
D =
− 1P (77)
so
=
µ1
− 1
¶D · b = µ 1
− 1
¶
− 1P · b = 1
− − 1
P · b (78)
and for simplicity,
=1
− − 1
P · b = P · b (79)
where
=1
− − 1
(80)
38
Adjusting Eq.31 for the dielectric effects, we have a new depolarizing field,
E = −P
0(81)
and a new equation of motion
x+ 0
x =
µE − P
0
¶(82)
solving similarly to Eq.42 the induced polarization is
P =20E0
−2 − +2(83)
ignoring the surface depolarizing effects (assuming only a bulk response)
P =2
−2 − 0E0 (84)
and
P = ( − 1) 0E (85)
so
2
−2 − = ( − 1) (86)
where is the dielectric constant of the metal itself. Now we write
P =0E0
− (2+)2
+=
0E01
−1+
(87)
or
P =0E0
1−1
+
³1
−−1
´ = ( − 1) 0E0 ( − ) +
(88)
This is the polarization of the inclusion. It may be the same as that of the surrounding
medium if = . The dipole moment of the inclusion is
p = P =4
3
( − 1) 0E0 ( − ) +
(89)
39
where the volume of the ellipsoid inclusion is = 43, where , , and are the
major axes of the ellipsoid. If there is a mismatch in the dielectric constant, then, outside
there will be a dipole field due to net surface charge on the surface of the ellipsoid.
The field of this surface charge is the same as the field of a polarized ellipsoid in free
space with polarization P.
Recall in that
=1
− − 1
P · b = P · b (90)
this is given by
P = ( − 1) 0E0
( − ) + =1
− − 1
( − 1) 0E0 ( − ) +
=( − ) 0E0
( − ) +
(91)
multiplying by the volume we get the dipole moment
p =4
3
( − ) 0E0
( − ) + (92)
So the field in the dielectric is the same as the field due to the dipole moment p in
vacuum. The polarizability corresponding to this in vacuum is
=4
3
( − )
( − ) + (93)
where the polarizability is defined by
p = E0 (94)
This means the dipole in the dielectric must be times greater to produce such a
field. That is as far as the field around the ellipsoid in concerned, it is (in the far field)
the same as if we had a dipole p in the dielectric, where
p = p = 4
3
( − ) 0E0
( − ) + (95)
40
The above is in agreement with [52]. However, we note, that this is useful in
describing the field around the ellipsoid, but does not give the correct dipole of the
inclusion which is p.
The susceptibility for the nanorod in a solvent can then be written as,
=
⎡⎢⎢⎢⎢⎢⎢⎣1
⊥+
−
0 0
0 1
⊥+
−
0
0 0 1
k+
−
⎤⎥⎥⎥⎥⎥⎥⎦ (96)
If the nanorod is made of gold then the simple harmonic oscillator model can be
applied where,
= 1−2
2 + (97)
or
1
− 1= −
2 +
2= −2 (98)
We now rearrange elements of the susceptibility tensor.
1
+
−=
( − 1)− ( − 1)( − 1) + −( − 1)
=1
−( − 1)( − 1)− ( − 1)
[ −(−1)( − 1) + 1]
(99)
or
1
+
−=
1
−( − 1)1− (−1)
(−1)[ −(−1) +
1(−1)
]=
1
−( − 1)1 + 2( − 1)[ −(−1) − 2]
(100)
substitution for 2 gives
1
+
−=
( − 1) −( − 1)
1(−1)
2 + (
2 + )
[ −(−1)
2 − 2 + ]
(101)
Finally the dielectric susceptibility of a gold nanorod, suspended in a solvent, at
optical frequencies is given by,
41
=
⎡⎢⎢⎢⎢⎣⊥ 0 0
0 ⊥ 0
0 0 k
⎤⎥⎥⎥⎥⎦ (102)
with components,
⊥ =( − 1)
−⊥( − 1)1
(−1)2 + (
2 + )
[ ⊥−⊥(−1)
2 − 2 + ]
(103)
k =( − 1)
−k( − 1)1
(−1)2 + (
2 + )
[k
−k(−1)2 − 2 + ]
(104)
For a dilute (non-interacting) suspension of NRs in a solvent the polarization can be
written as,
P =
Pp +
Pp
(105)
where p is the average dipole moment for the NRs and p is the average dipole
moment for the solvent and is the total volume of the system. The polarization of the
material is,
P = E+ E (106)
where is the polarizability, or,
P = E+ 0E (107)
The permittivity of the colloidal suspension is then,
= + + (108)
= + (1− ) + (109)
where is the volume fraction of a NR and is the susceptibility of the solvent,
or simply,
= + (1− )¡ − 1
¢ + (110)
42
Finally substituting Eq.65 into the previous,
= + (1− )¡ − 1
¢ +
µ +∆
2(3 b b − )
¶(111)
Now we have an explicit expression for the electric permittivity of a dilute ensemble
of orientationally ordered NRs suspended in a host medium. Where the principal values
of the susceptibility for the Au NRs are given by Eqs.103-104.
3.3.3 Field induced orientational order of nanorods
For a dielectric ellipsoid, where the field is uniform inside, the energy of the body in
an external field and consequently the work done by introducing it is [51],
=( − )
2
Z1
E−·E0 (112)
where is the permittivity of the surrounding medium, is that of the dielectric body,
is the volume of the nanorod, E− is the resultant field inside the ellipsoid and E0 is
the initial polarizing field, where,
E−·E0 = 2
1 + 2(−
)1
+2
1 + 2(−
)2
+2
1 + 2(−
)3
(113)
and where now refers to the host and to the immersed body [51]. Now we have
that
=
2 (114)
is the depolarizing factor in vacuum, where the depolarizing factor is defined by,
E− = E − 1P1 (115)
and so our dimensionless (usual) depolarizing factor is
= (116)
43
and
= =
2 (117)
and it follows that
E−·E0 = 2
1 +(−)+
2
1 +(−)+
2
1 +(−)
(118)
and the energy is then [51],
=2
3( − )E
−·E0 (119)
or
=2
3( − )[
2
1 +(−)+
2
1 +(−)+
2
1 +(−)] (120)
or
=2
3( − )[
2
+( − )+
2
+( − )+
2
+( − )] (121)
In the previous notation, this becomes
=2
3(− )[
2
( − ) + +
2
( − ) + +
2
( − ) + ]
(122)
Now note that the total ’excess’ dipole moment of the inclusion is (from above)
p = p = 4( − )E
( − ) + (123)
and if we define the polarizability tensor
=4
3
⎡⎢⎢⎢⎢⎣(−)
(−)+ 0 0
0(−)
(−)+ 0
0 0(−)
(−)+
⎤⎥⎥⎥⎥⎦ (124)
44
in accord with [52], then we have that
p =00E0 (125)
and the energy is
= −12p ·E0 = −
1
200E0E0 (126)
or
= −12
4
3[
( − )2
( − ) + +
( − )2
( − ) + +
( − )2
( − ) + ] (127)
This is the main result for the energy. It is in agreement with [51].
For the case of a Au NR,
−02 − 0= ( − 1) (128)
so at low frequencies, is very large [53].
In this case, the polarizability becomes
=4
3
⎡⎢⎢⎢⎢⎣1
0 0
0 1
0
0 0 1
⎤⎥⎥⎥⎥⎦ (129)
and the energy becomes
= −23[
2
+2
+2
] (130)
For a ellipsoid of revolution, we can write the polarizability as
=4
3
⎡⎢⎢⎢⎢⎣1⊥
0 0
0 1⊥
0
0 0 1k
⎤⎥⎥⎥⎥⎦ = 4
3
1
⊥
⎡⎢⎢⎢⎢⎣1 0 0
0 1 0
0 0 1
⎤⎥⎥⎥⎥⎦+43( 1k− 1
⊥)
⎡⎢⎢⎢⎢⎣0 0 0
0 0 0
0 0 1
⎤⎥⎥⎥⎥⎦(131)
45
The orientational dependent part of the energy of the inclusion in the field is,
= −12pE = −1
2 0EE (132)
= −23(
1
k− 1
⊥)2
(133)
The depolarizing factork is small for prolate ellipsoids (tends to zero as the long axis
goes to infinity), so the anisotropy ( 1k− 1
⊥) is positive. Therefore, at low frequencies,
the rods will line up with the field. If the frequency of the applied field is 60,
is large and negative, and the energy , Eq.133, can be written as,
= −12 0
2
µ1
⊥−µ1
k− 1
⊥
¶cos2 ()
¶(134)
where is the angle between and the long axis of the ellipsoid. Since k ⊥
for prolate ellipsoids, the NRs tend to align with the field. In equilibrium, the degree
of alignment of the inclusions is given by the orientational order parameter, Eq.30. To
achieve alignment, the energy must the comparable to , and writing − = +³
´2cos2 () the field to align the particles must be comparable to
=
vuut 2
0
³1⊥− 1
k
´ (135)
If the length and width of a NR are 40 × 15 and , assuming ellipsoidal shape,
k = 0125 and ⊥ = 0438, and since for toluene ≈ 22, at = 300, Eq.135 gives
= 194× 106. Thus a large field, on the order of 1, is required to align the
NRs.
46
3.4 Electric field induced alignment of gold nanorods dispersed in toluene
The experimental setup is shown in Fig.21. A 035, 1 path length, glass cuvette
containing Au NRs suspended in toluene has 20 × 20 ITO coated glass electrodes
glued to the outside surface of the cuvette. The electrodes are not in contact with
the solution. The cuvette with electrodes is submersed into transformer oil (Clearco
STO-50) to prevent dielectric breakdown. The NR suspension is filled well above the
height of the electrodes. A 15 , 60 transformer (Franceformer 15030P) with a
variable transformer (Variac) is used to apply a voltage to the electrodes. The absorbance
is measured with an unpolarized spectrometer (Oceanoptics HR4000CG-UV-NIR with
Mikropack DH-2000 light source). For the experiment we apply a maximum voltage of
15 across a 3 cuvette (1 path length), =∼ 5 []. For the geometry
shown in Fig.21 (homeotropic alignment) the NRs align along the long axis of the NR
parallel to the b and the applied electric field direction.
Figure 21: Experimental setup: a.isotropic suspension of Au NRs in toluene. b. aligned
NR suspension³ bkb´
Fig. 22 is the absorbance spectrum for Au NRs suspended in toluene. As an illus-
trative example, the nominal NR size is 28 × 45 ± 15% and the volume fraction is,
47
≈ 10−4. The transverse plasmon resonance absorption peak is centered at 520nm and
the longitudinal plasmon resonance absorption peak is centered at 649nm in toluene.
As the strength of the electric field is increased longitudinal plasmon resonance absorp-
tion peak is suppressed and the transverse plasmon resonance absorption peak increases
consistent with similar studies of aligned Au NRs [42,46].
Figure 22: Experimental absorbance spectrum as a function of applied voltage for Au
NRs suspended in toluene.
Fig.23 shows a suspension of Au NRs in toluene. The suspension is isotropic with
no voltage applied and has a uniform blue color (in transmission). Voltage is applied
and the NRs align in the direction of field (out of the page). The longitudinal plasmon
resonance absorption peak is suppressed allowing the red wavelengths to transmit through
the sample.
48
Figure 23: Left: isotropic suspension of Au NRs in toluene. Right: aligned suspension
of Au NRs in toluene, b k b (out of page)
A practical application for this phenomenon would be a display device composed of
electrically addressable pixel elements on suitable substrates. The pixels contain meta-
materials consisting of anisotropic nanoparticles in a host. The nanoparticles are gold
nanorods, possibly coated by silver or other metals, stabilized by surfactants, and are ei-
ther dispersed in a host (as in our experiments) or in neat liquid crystalline phases [54,55].
The susceptibilities of the nanoparticles depend on their shape; controlling their shape
during synthesis enables tailoring their optical properties. Due to their anisotropy, their
optical response depends on the direction of polarization relative to the orientation of the
particles. Since the particles can be individually oriented by an external electric field,
applying a field to a pixel can change the optical properties (absorption, transmission
and scattering spectra). A provisional patent application has been filed [56]
49
3.5 Order parameter determination
To experimentally determine the orientational order of nanorod suspensions the com-
plex index of refraction can be written as
= 2 (136)
= (0 + ”)2= 02 + 20”− ”2 (137)
where 0, ”are the real and imaginary parts of the refractive index. The real and
imaginary part of the permittivity are
0 = 02 − ”2 (138)
” = 20” (139)
The imaginary phase shift can be written as
=4
0” (140)
is the path length through the material, 0 is the wavelength in free space. Substi-
tuting Eq.139 into Eq.140
4
0” =
4
0
µ”
20
¶(141)
where 0 is the real part of the refractive index. The imaginary electric permittivity
for a homeotropic alignment geometry ( b||b) can be written as” =
"Ôk + 2
”⊥
3
!− 13
¡”k − ”⊥
¢
#(142)
where is the volume fraction of NRs, ”k and ”⊥ are the imaginary longitudinal and
transverse principal resonances of the NR susceptibility tensor, respectively. Combining
Eq.141 and Eq.142
4
0
µ”
20
¶=2
00
"Ôk + 2
”⊥
3
!− 13
¡”k − ”⊥
¢
#(143)
50
where the real part of the index of refraction is dominated by the solvent contribution,
0 =
2
00
Ôk + 2
”⊥
3
!+2
300
¡−”k + ”⊥¢ = + (144)
or,
= + (145)
The absorbance is linearly dependent on the NRs orientation. This allows a direct
way to calculate the order parameter from the absorbance spectrum at any wavelength
chosen. It is convenient to measure the absorbance spectrum of Au NRs in suspension
at the longitudinal peak as a function of applied external voltage in the homeotropic
case ( b||b). If a voltage is applied parallel to b(homeotropic) the NRs align with thefield. This corresponds to a suppression of the longitudinal peak absorption as the field
strength is increased and an increase in the transverse peak absorption Fig.22. For the
isotropic case, = 0, the absorbance spectrum is at a maximum and when the NRs are
completely aligned, = 1, the absorbance spectrum is at a minimum. Plotting the
absorbance as a function of squared electric field at the longitudinal absorption peak,
Fig.24,
51
0 5 10 15 20 250.4
0.6
0.8
1.0
1.2
x
[a.u
.]
E2 [V/m]2
Figure 24: Absorbance vs. the square of the applied external electric field at the longi-
tudinal wavelength, k = 659 from Fig.22
To extract the order parameter, , from the data shown in Fig.24 we need to fit the
absorbance data using Eq.145
=
R12(3 cos2 − 1) (
)2cos2 sin R
()2cos2 sin
(146)
where is the critical electric field for alignment, 2 = ≈ 1[]. From
Eq.145 when there is no applied field, = 0, and
0 = (147)
which is directly measured. Defining,
= − 0 = (148)
52
where is known and the applied field . Writing,
= ( ) =
R 1012(32 − 1) 2R 10
2
(149)
where = cos , = 12
and = 2. Then we have,
= ( ) (150)
Given a set of measurements, 0 and 0 we can extract from this and . Once
these are determined then we can get for every . We can do a least-squares fit: that
is, we choose and so that the difference between the experimental measurement
and the functional description, given by Eq.148 is a minimum for all measurements. So
we minimize,
2 =X
( − ( ))2
(151)
with respect to and . Minimization with respect to gives,
2
= 0 =
X2 ( − ( )) ( ) (152)
or X ( )−
X2 ( ) = 0 (153)
or
=
PP2
(154)
this can not be evaluated until is known. Minimizing with respect to gives
2
= 0 =
X
2( − ( )) ( )
(155)
or X
0 ( )−X
( ) 0 ( ) = 0 (156)
where
0 ( ) = ( )
(157)
53
We must have X
0 +X
0 = 0 (158)
and substituting for ,
X
0 +
PP2
X0 = 0 (159)
is given by the condition that
() =X
2X
0 +X
X
0 = 0 (160)
We then evaluate ( ) and 0 ( ) then find the value of for which Eq.160 is
satisfied. Where,
0 ( ) =
R 10122 (32 − 1) 2R 1
0
2
−R 1012(32 − 1) 2 R 1
02
2
hR 10
2i2 (161)
or
0 ( ) =
R 10122 (32 − 1) 2 R 1
0
2
− R 1012(32 − 1) 2 R 1
02
2
hR 10
2i2
(162)
plotting Eq.160 as a function of ,
54
0 2 4 6 8 10-1.0
-0.5
0.0
0.5
1.0
f
C
Figure 25: Plot of as a function of
It is now straightforward to find () = 0. Once is known we can find and
is already known. Solving Eq.145 for and substituting in , , and plotting the
extracted order parameter, as a function of squared electric field in Fig.26.
55
Figure 26: Order parameter vs. 2
The maximum order parameter that can be achieved is max = 091 for this example.
The code to determine the order parameter can be found in Chap.8.
56
Figure 27: Absorbance vs.
The experimental data in Fig.27 is in good agreement with Eq.145.
3.6 Alignment of gold nanorods in various solvents
The degree of alignment that can be achieved for NR suspensions by an external
low frequency electric field is greatly dependent on the solvent. Au NRs (from the
same synthesis batch) were suspended in ultra pure water (182Ω), dimethyformamide
(DMF), tetrahydrofuran (THF), chloroform and toluene, Fig.28.
57
550 600 650 7000.0
0.1
0.2
0.3
0.4
Toluene
Chloroform
DMF
THF
x [a
.u.]
[nm]
Figure 28: Longitudinal resonance absorption spectrum for Au NRs suspended in THF,
DMF, chloroform and toluene
The plasmon resonances of the nanoparticles sensitively depend on the surrounding
dielectric medium, Fig.28. Chen et al [36] developed a simple linear relationship between
the square of the longitudinal plasmon resonant absorption peak of the Au NR and the
square of solvent optical index of refraction,
2 = 2
µ1 +
µ1
k− 1¶2
¶(163)
where is the plasma wavelength. Eq.163 was verified through experiments, Fig.29.
58
1.7 1.8 1.9 2.0 2.1 2.2 2.33.0x10
-13
4.0x10-13
5.0x10-13
2 [m2 ]
n2
s
H20 THF Chloroform Toluene
Figure 29: Longitudinal resonance absorption shift in various solvents
The sensitivity of the plasmon resonances to the surrounding environment may have
promising application in sensor application [57].
The water, DMF and THF suspension showed no evidence of alignment at high fields,
using the measurement techniques described above. Fig.30 is the absorbance spectrum
for the NRs dispersed in toluene and chloroform with no external field applied(solid lines)
and for the maximum field applied, ∼ 5 [], (dotted lines). Toluene suspensions of
NRs, as already demonstrated, produces very good alignment, ≈09 The NRs slightly
align in chloroform suspensions with a maximum order parameter, ≈ 02.
59
450 500 550 600 650 7000.0
0.1
0.2
0.3
toluene: E=0 toluene: E=E
max
chloroform: E=0 chloroform: E=E
max
x
[a.u
.]
[nm]
Figure 30: Absorbance spectrum for Au NRs supended in toluene and chloroform with
no external field applied(solid lines) and a maximum field applied(dotted lines.)
The mechanism of alignment is subtle and not well understood. Eq.134 suggests
that the solvent with the largest dielectric constant should yield the best NR alignment.
Implying water should show the best alignment(for low frequencies), Table.1 [58—61], but
there is no evidence of alignment from the experimental measurements. Experiments
suggest that the degree of alignment should scale inversely proportional to , contrary
to Eq.134.
Water, DMF and THF can be classified as polar solvents and chloroform and toluene
as nonpolar. Since the external field is low frequency the polar molecules may be shielding
the NRs from the external field possibly explaining why only the nonpolar solvents can
allow the alignment of NRs. This hypothesis may be tested by applying high frequency
60
Solvent water 80.10
DMF 38.25
THF 7.58
chloroform 4.81
toluene 2.38
Table 1: Solvent vs.
fields.
3.7 Temporal alignment of gold nanorods
Light from a vertically polarized 100 HeNe laser (0 = 6328) was transmitted
through the cuvette containing the NRs suspended in toluene as described in Chap.3.4.
The maximum absorption for the longitudinal peak for the sample was at = 650.
The transmitted intensity as a function of time was measured with a photodetector and
oscilloscope.
Figure 31: Experimental setup
If no voltage is applied then most of the light is absorbed since the probe wavelength
61
is almost at the maximum absorption peak of the sample, Fig.32(black). If the voltage
is then applied then the transmitted intensity oscillates at 120Hz, Fig.32(red), the NRs
align in phase with the aligning field.
0 10 20 30 400
2
4
6
In
tens
ity [V
]
t [ms]
Figure 32: Transmitted intensity vs. time: voltage off(black). voltage on(red).
3.7.1 Time averaged phase shifts
The orientational order parameter can be written as,
=
2(1 + sin ()) (164)
where is the average order parameter. The change in phase, , that is measured
when orienting the NR suspension, Chap.3.4, is proportional to which is a function of
, 2,
= (165)
62
where is the proportionality constant. Substituting in Eq.164 into Eq.165,
=
2(1 + sin ()) (166)
= 0 + 0 sin () (167)
The real phase shift oscillates as, 0+ 0 sin (), so the maximum phase shift is then,
0 = 200, where the average, measured value, is 00. This implies that the real phase shift
that is measured is really half of the maximum value and the measurements are adjusted
accordingly. The imaginary phase shift is ” = 2, but the actual phase shift is twice
the measured or, ” = . These phase adjustments become relevant in Chap.4.
CHAPTER 4
Measurements of the electric susceptibility of gold nanorods at optical
frequencies
4.1 Introduction
This chapter describes an experimental technique to determine the optical electric
susceptibility of Au NRs as a function of wavelength.
In Chap.2 a method was developed to suspend Au NRs in organic solvents, this
enabled the NRs to be aligned with an external electric field, Chap.3. By aligning the
NRs this enables us to develop a method to measure the real and imaginary phase shifts
of light transmitted through the suspensions and is a key feature of this chapter.
The electric permittivity and magnetic permeability are the macroscopic quantities
that determine the optical properties of bulk materials. They can be related to the
responses of the individual materials constituents. Applying macroscopic properties to
nanoparticles may not be precise [52,53,62—66]. For example, in metallic nanoparticles
the electron mean free path(∼ 50) may be limited by the boundary, therefore the
damping constant, , which is inverse of the collision time for conduction electrons,
is changed because of additional collisions with the boundary of the nanoparticle with
consequences for the optical properties.
4.2 Method for determining the optical susceptibility for gold nanorods
We introduce a straightforward technique to determine the real and imaginary princi-
pal values of the dielectric susceptibility tensor for gold nanorods suspensions at optical
frequencies.
63
64
The average perpendicular component of the susceptibility can be written as,
h⊥i =k − 2⊥
3− 13
¡k − ⊥
¢ (168)
where =k−2⊥
3and ∆ = 1
3
¡k − ⊥
¢. For dilute suspensions, the real and
imaginary parts of the electric permittivity are given by,
0⊥ = 1 + (1− ) (0 − 1) + (0 −∆0) (169)
”⊥ = (”−∆”) (170)
where is the volume fraction of the nanorods and is the permittivity of the solvent.
For dilute suspensions( 1) it is assumed = 1 at optical frequencies [67, 68]. The
index of refraction can be expressed as,
= 2 = 02 + 20”− ”2 (171)
this implies,
0⊥ = 02 − ”2 (172)
”⊥ = 20” (173)
Equating Eq.169, Eq.172 and Eq.170, Eq.173,
1 + (1− ) (0 − 1) + (0 −∆0) = 02 (174)
(”−∆”) = 20” (175)
where ” is assumed to be negligible compared to 0 in Eq.174. Differentiating both
equations with respect to and , and solving for and ∆, yields the following four
equations,
0 =¡2 − 1
¢+
0
µ0
¶(176)
65
∆0 = −0
µ0
¶(177)
” =0
µ”
¶(178)
∆” = −0
µ”
¶(179)
where the real and imaginary parts of the phase are,
0 =2
00 (180)
” =2
0” (181)
respectively, is the index of the solvent, 0 is the free space wavelength and is the
path length in the sample. We solve for the principal values of the electric susceptibility
tensor, i.e. Eq.43,
k = + 2∆ (182)
⊥ = −∆ (183)
4.3 Experimental setup
We measure 0,
0, ”and ”
the unknowns in Eqs.176-179, using a Mach-Zehnder
interferometer and spectrometer as depicted in Fig.33.
66
Figure 33: Experimental setup: Mach-Zehnder interferometer and spectrophotometer
The interferometer, Fig.33, was custom built, Chap.8, to minimize vibrations observed
with standard optical post mounts. The base of the interferometer was milled out of 6061
aluminum into a 19×16×4” base block. The base block had two grooves milled into the
bottom to couple the two support rails 25×25×16” forming a tongue and groove joint.
The entire interferometer was mounted to an optical table with air suspension. Six 3-
point adjustable mounts were built: two to mount 3×3” 5050 unpolarized beam splitters
(12) and four to mount 3× 3” full mirrors (3456), Fig.34. The interferometer can
accommodate a 1”diameter beam. A mounting tower was built over the sample area ().
The tower contains the Teflon mounted female banana plug terminals for the high voltage
67
power supply as well as an electric motor with a plastic stir filament. For the electric
field configuration the leads from the cuvette electrodes are connected, via male banana
plugs, into the high voltage terminals. For the variable concentration configuration a
12VDC electric motor with a plastic stir filament driven by a Vizatek DC power supply is
used to stir the suspension in the cuvette after each concentration change. An acrylic box
(20× 20× 22”) was built (not shown in Fig.33) to enclose the interferometer to prevent
any air circulation that may cause the interference fringes to oscillate. A 2” hole was
drilled into the box at 1 for the beam entrance and another at 2 for the beam exit.
Figure 34: Schematic of the experimental setup . a: setup for real phase shift measure-
ments using the interfereometer. b: setup for imaginary phase shift measurements using
the spectrometer.
Fig.34a is the experimental setup to measure the real phase shift, Eq.180, using a
Mach-Zehnder interferometer. A Coherent Innova 100 CW Argon laser(1) is used
to pump a Coherent 599 Dye laser(2). The dye laser has two mirror/dye sets 700 −
610(DCM special) and 630 − 570(R6G), Chap.8, as well as discrete Argon laser
lines, 514−457. From the dye laser the beam enters a spatial filter( ) consisting of
68
a microscope lens with a 7 focal length focused onto a 10 pin hole plate to expand
the beam. A lens is then used to collimate the beam(1). The beam goes through a
variable attenuator(1), the transmitted beam typically did not exceed 100 . The
beam enters the Mach-Zehnder interferometer(1) and the beam is split into two. The
suspension of NRs() and a liquid crystal cell() was placed in the 1 −3 − 2
arm.
The liquid crystal cell was constructed out of two 254 × 254 pieces of BK7
float glass with an indium tin oxide (ITO) conductive layer coated on one side. An
alignment layer of polyamide 2555 (PI) was spun coated on top of the ITO layers of each
piece. The PI was rubbed with a felt covered bar to align the liquid crystal once it is
filled into the cell. The two pieces of glass were glued together with epoxy with the
ITO/PI layers facing each other and separated by a 20 mylar spacer at the edge of
the glass cell. The cell is then filled with 7 nematic liquid crystal. The liquid crystal
aligns in the rubbing direction of the cell walls creating a planar aligned configuration.
Wire leads were soldered onto each substrate and connected to a Krohn-Hite model 5920
function generator. The liquid crystal cell was oriented so that the rubbing direction was
parallel to the laser polarization. The liquid crystal cell was used to determine the sign
of the phase shift of the NR suspension by applying a voltage across the cell the liquid
crystal molecules switch from a planar to a homeotropic state were the long axes of the
molecules point in the same direction as the applied electric field. When the molecules
switch from the planar state to the homeotropic state the refractive index decreases and
so the interference fringes at the output of the interferometer shift in the direction of
decreasing phase determining the phase shift direction.
Another variable attenuator(2) and phase delay( ), to ensure roughly the same path
length in each arm, were placed in the 1 −4 − 2 leg. Both arms are combined
69
at 2 and a lens(2) expands the interference pattern onto a camera(Viewbits
CMOS Uroria 3MP USB camera f/1.8 6-13 lens). The uncertainty in measured phase
shift was typically ±4.
Fig.34b is the experimental setup to measure the imaginary phase shift, Eq.181, using
a spectrometer to measure the absorbance ( = 2”). Two mirrors(56) are slid into
place on a tongue and groove rail system mounted to the top of the interferometer. A
unpolarized white light() from a (Mikropack DH-2000) is passed through the NR
suspension, and the transmitted light is collected with a 400 core diameter fiber
optic cable( ) connected to a spectrometer(Oceanoptics HR4000CG-UV-NIR ). The
uncertainty in the absorbance measurements was typically ±5%.
As a control, pure toluene was placed into the cuvette. The absorbance(nearly zero
in the optical regime) nor the interference pattern from the interferometer were changed
by applying a high voltage across the cuvette or adding more toluene to the cuvette, as
described above.
4.4 Dielectric susceptibility measurements of gold nanorods at optical frequencies
Measurements were performed at 11 different wavelengths for small aspect ratio Au
NRs suspended in toluene shown in Fig.35.
70
450 500 550 600 650 7000.0
0.5
1.0
1.5
x [a
.u.]
[nm]
Figure 35: Absorbance spectrum for NRs suspended in toluene with superimposed lines
indicating probe wavelengths.
By changing the orientational order parameter, , of the NRs in suspension by varying
the externally applied voltage, as discussed in Chap.3.4 and by changing the volume
fraction of NRs in the suspension, , we can measure the real and imaginary parts of the
phase shifts, Eqs.180, 181 using the Mach-Zehnder and spectrometer, respectively.
For measuring the concentration dependence, a concentrated suspension of NRs is
placed into the cuvette, an image is captured of the interference fringes, then the sus-
pension is diluted with solvent, stirred for 2 minutes, then another image is captured of
the translated interference fringes and the process is repeated.
For any given wavelength there are 4 separate measurements, the real and imaginary
phase shifts (0”) versus the volume fraction() and orientational order parameter()
71
for each measurement 6-10 data points are collected- in total, ∼350 measurements per
spectrum. At a given wavelength for each of the 4 measurements the slopes of the best
fit line, 0, 0, ”
and ”
, are extracted Fig.36-39,
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
" [
rad
]
S
Figure 36: Imaginary phase shift vs. , ( = 575)
72
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
'[
rad
]
S
Figure 37: Real phase shift vs. , ( = 575)
73
0.0 2.0x10-4
4.0x10-4
6.0x10-4
0.0
0.2
0.4
0.6
0.8
1.0
" [
rad
]
[vol. fraction]
Figure 38: Imaginary phase shift vs. , ( = 575)
74
0.0 2.0x10-4 4.0x10-4
-6
-4
-2
0
' [
rad
]
[vol. fraction]
Figure 39: Real phase shift vs. , ( = 575)
The data collected and shown in Fig.36-39 can be well fit with a straight line, within
experimental error, this indicates that real and imaginary phase shifts, 0, ”, are indeed
proportional to , and .
The wavelength-dispersion of all the measurements, 0” vs. , is shown in Fig.40-
43,
75
Figure 40: Imaginary phase shift vs. ,
76
Figure 41: Real phase shift vs. ,
Figure 42: Imaginary phase shift vs. ,
77
Figure 43: Real phase shift vs. ,
From Fig.40-43 and using Eqs.176-179, 0 ∆0 ” and∆”can be calculated at each
wavelength. Once these four quantities are determined the elements of the NR suscep-
tibility tensor, Eq.43, can be determined using Eq.182, 183 as a function of wavelength,
Fig.44-45.
78
450 500 550 600 650 700-3
-2
-1
0
1
2
'
[nm]
'
'
Figure 44: Real part of the perpendicular and parallel susceptibility components as a
function of wavelength
79
450 500 550 600 650 700
0.0
0.1
0.2
"
[nm]
"
"
Figure 45: Imaginary part of the perpendicular and parallel susceptibility components
as a function of wavelength
Figs.44-45 show the real and imaginary susceptibilities spectrum for a Au NR at
optical frequencies.
4.5 Analysis
To interpret these results, we fit the data with a simple harmonic oscillator (SHO)
model, including the solvent contribution, Eq.96. It should be noted that the properties
of Au are more difficult to represent in the optical regime, relative to other metals, with
a free electron (SHO) model. At short optical wavelengths Au is not well modeled by,
Eq.95, because apart from the free conduction electrons, interband transitions of bound
valence electrons occur. Au has at least two interband transitions at = 330 and
= 470 [69] which have non-negligible contributions to the optical response, giving
80
rise to the distinctive yellow color of bulk gold. Due to this mechanism the electron
model is only valid for wavelengths greater than = 550 [36,70—74].
Fig.46-47 show the real and imaginary susceptibility curves the parallel and perpen-
dicular principal resonances in the Au NR. Salient features of the dispersion curves
are when the real part of the susceptibility is zero, 0= 0, then the imaginary part of
the susceptibility is a maximum, ” = ”max, and when the real part of the susceptibil-
ity is a maximum the imaginary part is half its maximum value, 0maxmin = ±”max
2for
both components as discussed in Chap.3.
Figure 46: Predictions of SHO model for the real part of the perpendicular and parallel
susceptibility components of a Au NR as a function of wavelength. The NR dimensions
were 20× 29 suspended in toluene similiar to the measured example above.
81
Figure 47: Predictions of SHO model for the imaginary part of the perpendicular and
parallel susceptibility components of a Au NR as a function of wavelength. The NR
dimensions were 20×29 suspended in toluene similiar to the measured example above.
The data shown in Fig.36-39 can be interpreted using the SHO model, Fig.46-47.
For Fig.36-39 the probe wavelength, = 575, is in the valley between both the
transverse and longitudinal resonant peaks, Fig.35. For Fig.36 as the NRs become aligned
(increasing ) the longitudinal mode is suppressed, slightly decreasing ”. In Fig.37 as
the NRs become aligned, the longitudinal mode is suppressed, increasing 0as a function
of . In Fig.38 as the volume fraction, , of Au is increased,” also increases. In Fig.39
as the volume fraction of Au is increased 0decreases at this wavelength.
Moreover using Fig.46-47 the dependence of the real and imaginary parts of the
susceptibilities per wavelength, Fig.40-43, can be interpreted. In Fig.40, as the NRs are
aligned, ” increases around the the transverse peak, ≈ 514, and the longitudinal,
≈ 650, decreases.
82
In Fig.41 as the NRs are aligned 0increases around ≈ 575 because the negative
longitudinal mode is suppressed, increasing 0. Around ≈ 660 the transverse and
longitudinal susceptibilities are the same, 0= 0 and for 660 the longitudinal
mode is a larger value than the transverse, so there is a sign inversion for 0.
In Fig.42, as the volume fraction of Au increases, ”increases for all wavelengths.
In Fig.43 for ≈ 700, 0 is positive but as the wavelength decreases, 0becomes
negative with a minimum around ≈ 575. Again, the response of 0 for 550
is complicated and not well described by this simple model.
The model, Fig.46-47, is fit to the real and imaginary susceptibility data for the
parallel and perpendicular components and wavelength ranges, Fig.48-49. The real part
of the susceptibility was only plotted for 550 due to the additional interband
transitions in the Au.
83
550 600 650 700-3
-2
-1
0
1
2
'
[nm]
'
'
Figure 48: SHO model fit to the experimental data of the real part of the perpendicular
and parallel susceptibility components as function of wavelength
84
450 500 550 600 650 700
0.00
0.05
0.10
0.15
0.20
0.25
"
[nm]
"
"
Figure 49: SHO model fit to the experimental data of the imaginary part of the perpen-
dicular and parallel susceptibility components as function of wavelength
The model fits the data relatively well overall. When the imaginary longitudinal
component of the susceptibility is a maximum, ”k max, then the real longitudinal com-
ponent of the susceptibility is zero, 0k ≈ 0. The real longitudinal component of the
susceptibility, 0k, is a minimum around ≈ 600. The magnitudes of the real longitu-
dinal and transverse components of the susceptibility, 0k and
0⊥, invert at ≈ 660.
The slope of 0⊥ is relatively constant(not at resonance, ≈ 514). The shapes of ”k
and ”⊥ are as expected from the model.
The SHO model implies that if , , and are known, then the complete behavior
of the system can be determined. However there are difference between the fitted and
accepted values for and [53,62,75]. The model fitting parameters were : = 81015
85
[ ], = 5081014 [
], k = 025, ⊥ = 0375.
The accepted bulk values for Au is = 1371015 [
], = 0411014 [
] [53, 62,
75]. The quality factor is then, =≈ 300. It is clear from the experimental
measurements is ∼ 10, implying the system is far more dissipative than theoretically
predicted.
The fitted damping frequency, , is two orders of magnitude larger than the accepted
value. As discussed in Chap.5 the electron-phonon scattering appears not to be the
principal loss mechanism. There are two other mechanisms that may lead to a broadening
of . First, the NRs are not perfectly monodisperse with two simple plasmon resonances.
The samples are polydispersed with two "effective" plasmon resonances resulting from the
collective response from all the polydispersed nanorods [30]. This size distribution leads
to an effective broadening of the resonance, where the effective resonance is centered
at the most probable NR size distribution. Stoller et al [71] measured the real and
imaginary susceptibilities of a single gold nanosphere as a function of wavelength to test
the role of polydispersity causing the discrepancies. For a single resonance their results
are similar in shape and magnitude to our results for the real susceptibility however our
imaginary susceptibility is an order of magnitude smaller than their values. Another
broadening mechanism is to consider the electron mean free path in metals(∼ 50). For
a Au NR the mean path may be limited by the boundary of the particle, and possibly
anisotropic, therefore the damping constant, , which is inverse of the collision time for
conduction electrons, is increased because of additional collisions with the boundary of
the particle with obvious consequences for the optical properties [52,76,77]. As pointed
out by Bohren et al [52] for measurements of ensembles of small gold nanospheres they
observed that the measured absorption peak is broader and lower than predicted by
theory. The discrepancies can be partly explained due to an additional damping arising
86
from the nanoparticle’s surface, however others [72—74] have corrected for this factor, yet
discrepancies between the measured and calculated value exist.
The fitted plasma frequency, , also varies by appreciably from the accepted value.
Possible reasons may be either bulk or surface defects such as impurities, grain disloca-
tions etc. [52,78]. As already mentioned the interband transitions of the bound electrons
could change the resonances as well as the effects of polydispersity.
The fitted depolarizing factors, , were in reasonable agreement with the measured
value. The fitted aspect ratio value for the NR was 145, the actual measured aspect
ratio of the sample, using TEM, was 157± 012.
Greengard et al [79] preformed advanced simulations modeling the optical response
of Au NRs, Fig.50. Their results are similar in shape and magnitude to our results for
the real longitudinal susceptibility however our imaginary susceptibility is an order of
magnitude smaller than their values and the transverse resonances do no agree, the
simulations did not account for the interband transitions in Au.
Figure 50: Simulated longtidutindal real and imaginary susceptibilities for a Au NR in
vaccuum. (courtesy of L. Greengard, NYU)
Susceptibility measurements by others [52, 71—74] as well as ourselves show there
are significant discrepancies between the accepted bulk material values and measured
87
suggesting a need to carry out more first principle experiments to determine the optical
behavior of metallic nanoparticles.
4.6 Consequences
For the Au NR suspensions measured, as the volume fraction of Au increases the real
phase shift decreases, Fig.39, the real phase contribution from the Au NRs is negative,
at a prescribed wavelength. From these measurements the refractive index vs. volume
fraction can be extrapolated to large volume fractions of Au,
0 = +
2
0
(184)
where is the refractive index of the solvent. Plotting Eq.184,
Figure 51: Real refractive index for Au NRs vs. volume fraction
Fig.51 shows for volume fractions of ≈ 30% the refractive index should be near 0.
Simulations confirm that volume fractions of ≈ 35% are needed for ≈ 0 [79].
88
As discussed in Chap.1 the figure of merit, =¯0”
¯, for metals is order unity.
Moreover from the simple model in Chap.3 it was shown that, 0maxmin = ±”max
2, implying
that the real and imaginary refractive indices will always be comparable. Measurements
taken in this work show that the typical maximum real phase shift was 0 = 20 ≈ 2
and the imaginary phase shift ” = 2” ≈ 1. This implies that to gain a 2 real phase
shift approximately 1of the light is absorbed.
We have determined that the constituent materials are capable of 0 at optical
frequencies, but the losses are large, effectively rendering any realization of bulk optical
metamaterials impractical. A more useful strategy is to produce self-assembled thin
(∼ ), aligned, Au NR films.
CHAPTER 5
Self-assembly and characterization of gold nanorod films
5.1 Introduction
This chapter describes experimental techniques to create and characterize self-assembled,
high-density, anisotropic Au NR films.
Nowwith an understanding of the intrinsic optical properties of the NRs, fromChap.4,
we focus in this chapter on self-assembling, high-density, orientationally ordered, macro-
scopic films. The macroscopic uniform alignment of nanoparticles is critical to the produc-
tion of self-assembled metamaterials with optical properties such as hyperbolic dispersion
metamaterials [10,11,80,81]. We discuss two schemes below.
Self-assembled films highly loaded with small aspect ratio Au NRs encased in silica
are orientationally ordered in square millimeter sized domains where developed by the
combination of solvent evaporation and applied electric field.
Thin, free-standing, gold nanorod-polyurethane composite films were developed by
embedding nanorods into bulk polyurethane as well as assembling bilayers of nanorods
and polyurethane using layer-by-layer deposition. The NR-polyurethane composites can
then be strained orienting the nanorods.
5.2 Evaporative and field assisted self-assembly of silica encased gold nanorod films
5.2.1 Introduction
Controlling the alignment of anisotropic nanoparticles in macroscopic domains is crit-
ical for assembling optical metamaterials. At optical frequencies the permittivity of
89
90
anisotropic metallic nanoparticles can be negative, 0. If the nanoparticles are as-
sembled properly, they may have a negative permeability, 0, at optical frequencies
yielding a negative index of refraction [17,75,82]. The ability to densely pack the Au NRs
while preserving the plasmon resonances is key to developing negative index/refraction
materials.
A practical method to produce these materials is self-assembly. To date, only micro-
scopic, ∼ 2, domains of aligned anisotropic nanoparticles have been achieved using
self-assembly [44,83—89]. These domain sizes are too small for practical optical devices,
such as optical negative index/refraction applications. Jaeger et al achieved macro-
scopic, ∼ 2, domains of Au nanospheres by the controlled evaporation of collodial
nanosphere suspensions [90].
We have used a combination of solvent evaporation and a low frequency electric field
to produce self-assembled films, of orientationally ordered, over length scales, of
small aspect ratio Au NRs. The only factor determining the upper limit of domain size
thus far has been the availability of the constituent material.
Not only is the evaporation rate critical, the choice of surfactant encasing the Au NRs
is paramount in determining not only the mechanical assembly of constituents but the
optical properties for the Au NR film. Au NRs encased in silica shells(Si-Au NRs) were
synthesized as described in Chap.2. These NRs are solubilized in polar solvents (water)
by ionic surfactants (CTAB). These also stabilize the NRs in suspension, if the solvent
is removed, the density of NRs becomes large, the NRs then can aggregate washing
out the individual plasmon resonances, Fig.52a. By encasing the Au NR in a silica
shell the plasmon resonances remain preserved, because the shells prevent the NRs from
physically touching, and unshifted going from a dilute isotropic suspension of Si-Au NRs
to a highly-loaded film of Si-Au NRs in air, Fig.52b.
91
Figure 52: a: absorption spectrum for isotropic Au NR aqueous suspension and film. b:
absorption spectrum for isotropic Si-Au NR suspension and film
The thickness of the silica shell is critical not only to prevent physical aggregation but
to enable tuning of the optical response by changing the separation between particles [66].
Typically the particles must also be separated ∼ 25 the diameter of the particle to
preserve the optical response of the individual particles, ensuring that the induced dipole
field is much less than the stimulus field [38,91].
5.2.2 Experimental setup
Several drops, ∼ 25, of the Si-Au NR suspension, 0 ∼ 10−4, are placed in
between two (50× 50× 1) glass plates separated by a 1 spacer. The glass plates
have ITO coated on the one side. In Fig.53 a voltage (Franceformer 15 ) is applied
across the plates (∼ ). The glass plates with the suspension is in a sealed acrylic
box which is flushed with 2 (5 ), the gas provides a relatively constant humidity
atmosphere.
92
Variac
15 kV transformer
N2 inlet exhaust
Sealedcontainer NR suspension
Figure 53: Experimental setup for the Si-Au NR films
Fig.54a is a vertical cross section of the evaporating Si-Au NR suspension. Fig.54b
is a glass plate with several Si-Au NR films after evaporation.
Figure 54: a: experimental setup for the Si-Au NR films. b: image of Si-Au NR films.
93
There are many subtle parameters which determine the desired self-assembled mor-
phology for the Si-Au NR films. The salient features are the geometry, type and rate of
solvent evaporation, type and thickness of surfactant, substrate, aspect ratio, the presence
of an external electric field and density of NRs in suspension.
There are three scenarios for uniform NR alignment on the substrates for the given
cylindrical geometry, Fig.55. The NRs can align perpendicular to the substrate (homeotropic)
or the NRs can align in the plane of the substrate (planar). If they align in plane the NRs
can align perpendicular to the evaporating interface, parallel to the drop radius direction
(planar-parallel) or they can align parallel to the interface, perpendicular to the radius
direction (planar-perpendicular).
Figure 55: NR alignment configuration
Due to the pinned liquid-vapor interface of the droplet between the substrates, at
the circumference there is a high evaporative flux. As the solvent evaporates, fluid from
the center of the droplet flows to the circumference pushing NRs to the perimeter of the
droplet, Fig.56. This is the so-called coffee stain effect [92—94].
94
Figure 56: Possible NR alignment mechanism
If the NRs align in the planar-parallel configuration, Fig.57a, then the free energy is,
= 2(divn)
2, nonzero, where is the elastic constant and n is the director. However,
if the NRs align out of the page, Fig.57b, then n=constant and = 0, therefore there is
no energy cost to align the NRs in the planar-perpendicular, Fig.57b, configuration. It
was observed that the films were more uniform, having larger domains of aligned NRs,
when produced in the presence of an externally applied low frequency electric field.
Figure 57: NRs aligned in the a. planar-parallel and b. planar-perpendicular configura-
tions.
95
5.2.3 Characterization of silica encased gold nanorod films
The evaporated Si-Au NRs films are microscopically characterized using a SEM and
AFM. The AFM and SEM images in Fig.58 show the preferential alignment, b, of the Si-Au NRs in the plane of the substrate and perpendicular to the radius, b, of the film. Fromthe AFM and TEM images the direction of alignment is clear (planar-perpendicular) and
the domains are uniform over the scanned region (∼ 1002).
Figure 58: a: AFM image b: SEM image
To measure the thickness of the films, a knife was used to cleave and scrap the film
away down to the glass substrate leaving a discrete step from sample to the substrate.
The tip of the AFM was translated just above the surface of the film perpendicular to
the cleaved edge and down to the glass substrate the height difference was measured,
determining the thickness of the film. Over square millimeter domains the thickness
was constant and determined to be ≈ 300 which corresponds to roughly 5 layers of
Si-Au NRs.
The Si-Au NRs films are macroscopically characterized using a polarized white light
source (Mikropack DH-2000 light source) and a spectrometer (Oceanoptics spectrometer
96
HR4000CG-UV-NIR) shown in Fig.59a.
Figure 59: a: experimental setup. b: transmission color through Si-Au Nr film for
orthogonal polarization states
The absorbance of the Si-Au NR films can be measured for the film as a function of
polarization angle and wavelength, Fig.60,
97
Figure 60: Absorption spectrum vs. polarization for Si-Au NR films. 10 rotation of the
polarizer per absorbance curve.
If the polarization direction is parallel to the radial direction of the film then light is
polarized along the small axis of the NR (520) and will be absorbed, Fig.60, and the
film will appear red in transmission, Fig.59b. If the polarization direction is perpendicular
to the radial direction of the film then light is polarized along the long axis of the NR
(660) and will be absorbed, Fig.60, and the film will appear blue in transmission,
Fig.59b.
5.2.4 Order parameter determination
To quantify the amount of orientational order we examine the absorbance as a function
of polarization angle at the longitudinal peak wavelength, Fig.61.
98
Figure 61: Absobance vs polarization at = 662 with a probe beam diameter∼ 5.
The absorbance is linearly proportional to orientational order parameter, this implies
that,
k = + k (185)
⊥ = + ⊥ (186)
where the average absorbance is =k+2⊥
3, k = 2
3
¡k − ⊥
¢, ⊥ = −13
¡k − ⊥
¢The difference divided by the average of the two previous equations yields,
− 1+ 2
= (187)
where =k⊥
[95]. From Fig.61 the absorbance saturates as approaches 90, implying
a high degree of alignment. Using this method we estimate an order parameter, ≈ 08,
over∼ 2 sampling area which implies we have a very high degree of orientational order
verified through this method and using AFM and SEM techniques.
99
5.2.5 Effects of evaporation rate of silica encased gold nanorod films
Figure 62: Images of liquid crystal phases formed by aqueous Au NRs on a TEM grid
under various evaporation rates.
Fig.62, demonstrates the different liquid crystal like phases that can be achieved
by simply controlling the evaporation rate [88, 96—98]. A droplet (∼ 10) of Au NRs
suspended in water was placed onto a TEM grid (300 mesh Cu Formvar/carbon) and the
water was allowed to evaporate at different rates. If the water droplet was evaporated on
the time scale of minutes then an isotropic phase is formed, Fig.62a. If the evaporation
was over hours then a nematic phase is formed Fig.62b. If the evaporation was on the
order of days then a smectic phase can be achieved, Fig.62c.
To study the evaporation rate of a droplet confined between two plates assume the
shape of the experimental droplet to be a cylinder, Fig.63,
Figure 63: Schematic of evaporating water droplet.
100
where is the radius and is the distance between the two glass plates. The surface
area of the evaporating drop is, = (2) . If it is assumed that the change in
volume, with respect to time, , is proportional to the surface area,
= (188)
= (2) (189)
where is the proportionality constant. The volume of the droplet, = 2, then,
= 2
· (190)
where· is the derivative with respect to . Equating Eq.188 and Eq.190,
2· = (2) (191)
or· = , note that the velocity of the interface,
·, is a constant. Integrating with
respect to time,
= + (192)
where is a constant, therefore we expect the radius to decrease linearly in time.
Images of the evaporating droplet were taken as a function of time, normal to the
glass plates, and the radius of the droplet versus time was determined using National
Instruments Vision Assistant 2009, and plotted in Fig.64,
101
0 50 100 1500.0
0.5
1.0
1.5
2.0
2.5
R [
mm
]
t [min]
Figure 64: Radius vs. time for an evaporating droplet
The interface position, , evaporated linearly in time over the course of 3 hours
confirming the evaporation rate is constant and it is possible to control the growth rate
can be controlling the vapor pressure.
5.2.6 Phase shift measurements of silica encased gold nanorod films
The real phase shift of the Si-Au NR film were measured in a Mach-Zehnder inter-
ferometer as described in Chap.4, and were measured as a function of orthogonal polar-
ization and wavelength.. A uniformly align domain (∼ ) was found in a film and
the remaining film was scrapped away with a knife. A aluminum foil frame was tapped
around the rectangular film to form rectangular transmission window, Fig.65 (note: the
sample in the image was heavily used).
102
a. b.
Figure 65: Si-Au NR film for orthogonal polarization states
The Si-Au NR film on the glass substrate, Fig.66a, was fixed to a translational
stage and placed into one arm of the interferometer, Fig.66b and imaged onto a screen.
The light beam from the dye laser was expanded using the spatial filter so the beam
diameter was approximately 10.
103
a. b.
Figure 66: Experimental setup to measure the real phase shift of the Si-Au NR films
The film was translated vertically into the middle of the probe beam, Fig.66a. Above
and below the film is the relative phase shift from the glass substrate. The phase shift
going from the glass substrate onto the Si-Au film, i.e. less than 2.
104
Figure 67: Phase shift vs. polarization and wavelength for Si-Au NR films
The phase shifts were determined using a image analysis program, Chap.8.5. For all
wavelengths and polarizations measured the phase shift of the Si-Au NR was positive.
For the largest measured phase anisotropy, ∆615 = k − ⊥ = −08 [], but still
positive overall and the uncertainty in the measurement was comparable to the measured
difference. Based on the SHO model, at = 615, the longitudinal resonance
should be large and negative and the transverse should be positive so ∆ should indeed
be negative as measured.
To understand the overall positive and small phase shift consider a sphere or radius,
, coated with a shell, radius = 2. The volume of the sphere is =433 and
the volume of the shell is =43 (3 − 3) the ratio of these two is
10%.
From simulations [79] we estimate that there must be approximately a 35% volume filling
fraction of gold before a sign inversion is expected. So there is too much silica and too
105
little gold in the film to drive the net phase shift negative.
5.2.7 Lift-off technique for silica encased gold nanorod films
As a proof of concept experiment, the Si-Au NR film can be transferred onto a flexible
optically clear Teflon FEP silicon adhesive tape by laying the tape over the film on the
glass substrate, applying pressure to the backside of the tape then simply lifting the tape
up, removing the film, Fig.68, this ability to transfer the films to flexible substrates may
have useful applications.
Figure 68: Si-Au NR film transferred to an optical adhesive
5.2.8 Temperature dependence of the absorption spectrum for silica encased gold nanorod
films
The width of the plasmon absorption bands are determined by electron-electron,
electron-phonon and electron-defect scattering process. For bulk metals the electron-
phonon mechanism is the dominant processes. For Au NRs the surface acts as an
additional scatterer, Chap.4. If the electron-phonon scattering process is dominant then
increasing the temperature of the NRs should increase the phonon oscillations which in
turn should increase the width of the plasmon absorption bands.
106
Figure 69: Experimental setup to measure the temperature dependence of the absorbance
for the Si-Au NR films
Fig.69 is the experimental setup to measure the change in absorbance as a function
of temperature. An isotropic Si-Au NR film on a glass substrate is placed into the
beam of an unpolarized spectrometer(Oceanoptics spectrometer HR4000CG-UV-NIR).
A Master heat gun model HG-501A-C is used to heat the film from 292 to 423. The
temperature was measured with a mercury thermometer placed next to the film.
107
500 600 700 8000.0
0.2
0.4
0.6
0.8
1.0
292K 423K
x
[a.u
.]
nm
Figure 70: Absorbance vs. temperature for the Si-Au NR films
Fig.70 is an absorbance spectrum of the isotropic Si-Au film that was heated from
292 to 423 with almost no change in the absorption. This implies that the electron-
phonon loss mechanism is not dominant, consistent with other measurements [65]. The
implications are discussed further in Chap.4.
5.3 Strain-induced alignment of polyurethane-gold nanorod films
Au NR were embedded into bulk polyurethane (PU-Au NR) films as well as thin LBL
PU-Au NR films described in Chap.2. Orientational order of the NRs can be achieved
by straining the samples, Fig.71. Samples were distributed to others for redundant
experiments [99].
108
a. b.a. b. c.
Figure 71: Experimental setup. a: Image of PU-Au NR film in straining mechanism b:
unstrained sample. c: strained sample
The PU-Au NR or LBL PU-Au NR films were placed into a straining device to strain
the PU-Au NR samples up to a factor of 10 and LBL PU-Au NR up to a factor of 3,
Fig.71a. The PU with no NRs embedded into it is optically clear with no absorption in
the visible wavelengths.
The absorbance spectra were measured with a polarized light source and a spectrom-
eter (Oceanoptics HR4000CG-UV-NIR with Mikropack DH-2000 light source). Unlike
the bulk PU-Au NR film where the absorbance spectrum is relatively unaffected by plac-
ing the NRs into the PU, Fig.73, the absorbance for the LBL PU-Au NR films, especially
around the transverse absorption peak, is affected by the LBL PU host, Fig.72. For
Fig.72, the polarized probe beam was perpendicular to the strain direction; as the sam-
ple is strained the overall absorption decreases as well as the changing of the relative
absorption peak ratios. The unstrained spectrum in Fig.72 is the top curve. As the
sample is strained the path length through the sample becomes smaller due to volume
constraints, explaining the overall decrease in absorbance. The change in the relative
absorption peak ratios is assumed to be from the NRs reorienting such that the long
axis of the NRs is along the strain direction. As shown in Fig.72, as the sample is
109
strained, the relative height of the longitudinal peak decreases, which is consistent with
the NRs orienting along the strain direction. The upper limit on the strain for the LBL
PU-Au films was determined from small cuts in the edge of the film resulting from cut-
ting the film to experimental dimensions, as the strain approaches ∼ 2 the cut(fracture)
propagates through the samples cutting the sample in two pieces. Efforts were made to
minimize this effect, but it was a persistent issue with the samples.
500 600 700 800 900 10000
1
2 unstrained max. strain
x
[a.u
.]
[nm]
Figure 72: Polarized absorbance spectrum for LBL PU-Au NR films as a function of
strain(∼ 2). The probe polarization was perpendicular to the strain direction.
To compensate for the overall decrease of spectrum due to sample thinning the ab-
sorbance curves are normalize at a specific wavelength, Fig.73.
110
E F
500 600 700 8000.0
0.5
1.0
1.5
x
[a.u
.]
[nm]
E F
Figure 73: Maximum strained(∼ 10) absorbance spectrum vs. wavelength for PU-Au
NR film for different polarizations.
Fig.73 is a normalized plot of a PU-Au NR film that was strained to a maximum(∼ 10)
and measured for perpendicular and parallel polarizations relative to the strain axis,b . When the probe beam is polarized perpendicular to the strain axis, a decrease in
the longitudinal absorption peak and an increase in the transverse absorption peak is
observed. The converse is true for the orthogonal polarization, consistent with the NRs
aligning in the strain direction. Using Eq.187 the order parameter for both the bulk
PU-Au and LBL NR films was ≈ 04.
CHAPTER 6
Other measurements
6.1 Introduction
This chapter describes other experimental work.
We have demonstrated that conjugated dye molecules are capable of anomalous dis-
persion at optical frequencies with figures of merit comparable to gold nanorods.
The real relative phase shift and absorbance was measured as a function of frequency
for a layered film composed of sputter coated Au in between layers of 23 on a silicon
wafer. The film was of particular interest due to the large filling fraction of Au.
6.2 Conjugated molecules as building blocks for optical metamaterials
Traditionally optical metamaterial constituents have been metallic nanoparticles [10,
14,15,100]. Metallic nanoparticles may be only a subset of a ubiquitous class of soft anom-
alous dispersion materials. Our results show that dichroic dyes are capable of anomalous
dispersion at optical frequencies, resulting in a negative real phase shift contribution
with figures of merit comparable to metals. In addition we consider other materials with
potential for anomalous dispersion.
We carried out real and imaginary phase measurements on Au NRs and dichroic dyes
suspended in toluene using a Mach-Zehnder interferometer and photospectrometer. The
Au NRs, length=30 ± 5, width=20 ± 3, were synthesized using a seed-mediated
growth procedure then phase transferred into toluene using the methods describes in,
Chap.2. The conjugated dye molecules(SLO1215, AlphaMicron, Inc.) have dimensions,
111
112
length=381, width=067. Absorbance, , measurements were done with an
Ocean Optics HR4000CG-UV-NIR spectrometer with Mikropack DH-2000 unpolarized
white light source in a cuvette with a 1 path length. In Fig.74, for the nanorods,
the transverse surface plasmon absorption peak is centered at 522 and the longitu-
dinal surface plasmon absorption peak is centered at 644. Also plotted in Fig.74 is
a characteristic absorbance vs. wavelength spectrum for the dye suspended in toluene.
The dye molecule is similar to the NR but in place of surface plasmons the molecule has
conjugated electrons that can oscillate along the principal axes of the molecule giving
rise to an apparent transverse and longitudinal absorption peaks.
Figure 74: Absorption spectrum for gold nanorods and dichroic dyes suspended in
toluene.
A Coherent 599 dye laser pumped with a Coherent Innova 100 Argon laser was used
113
for the interferometric measurements, Chap.4. The nanorod and dye suspensions were
measured at a wavelength of = 633 and = 514, respectively, in the
anomalous dispersion regime for each material. Initially a 350, 1 path length cu-
vette with either suspension was placed in one leg of the interferometer. The interference
fringes were imaged onto a screen. An image was then captured of the initial position of
the interference fringes with a Viewbits CMOS Uroria 3MP USB camera and f/1.8 6-13
lens. Toluene is then added to the cuvette and mixed ensuring homogeneous distribution
of the particles, another image of the translated interference fringes is captured and the
algorithm is repeated. A liquid crystal phase plate was placed in the same arm as the
cuvette and was used to determine the sign of the phase shift resulting from the dilution
of the nanoparticle suspension. The images were analyzed with a LabVIEW program
and the relative sign and magnitude of the real phase shifts, , were determined, Fig.75.
114
Figure 75: Real phase shift vs. volume fraction for gold nanorods and dichroic dyes
suspended in toluene
The above results, for both nanorod and dye suspensions, have linear dependence and
show a decreasing real phase contribution as the volume fraction of each constituent is
increased. To ensure that the phase shift is negative and not merely a smaller positive
value than the solvent, the effective real permittivity of a dilute suspension, 0 = 1 +
(1− ) ( − 1) + , where is the real permittivity of the solvent, is the volume
fraction and is the susceptibility of the nanoparticles. If → 0 then the effective real
refractive index is, 0 = +12
³ − 1
´, where is the refractive index of the solvent.
The previous equation sets an upper bound on the data; if the data falls below this line
then the nanoparticle susceptibility is negative. If the volume fraction from Fig.75 is
extrapolated for each constituent then the effective real refractive index can be written
115
as 0 = +¡
2
¢ ³
´, for non-interacting particles, and plotted, Fig.76,
Figure 76: Real refractive index vs. extrapolated volume fraction of gold nanorods and
dichroic dye molecules
The phase shift contribution from both the nanorod and dye fall below the = 0
line, clearly demonstrating negative phase contributions for both materials.
The figure of merit (FOM) is given as, =¯0”
¯=¯2
¯, and should be much
greater than unity for the realization of any pragmatic optical metamaterial. The fig-
ure of merit for the nanorods is, =633 = 160, and for the dye molecules,
=514 = 141. The figure of merit for the dye molecules is comparable to that
of the nanorods, noting that the aspect ratios and probe wavelengths were different for
both materials.
116
In addition, other conjugated molecules may be good candidates to exhibit anomalous
dispersion at optical frequencies such as chromonic liquid crystals like Sunset Yellow FCF
and Benzopurpurin 4B. Chromonic liquid crystals are rigid aromatic rings. The −
interaction of the aromatic cores is the primary mechanism that allows the molecular face-
to-face stacking of the mesogen rings into rod-like macromolecules. The macromolecules,
in suspensions, show similar characteristic absorption spectra to that of the nanorods and
dye molecules [101,102]. These macromolecules have delocalized electrons, similar to dye
molecules, which are free to respond to external fields and may give rise to anomalous
dispersion at optical frequencies.
6.3 Gold-23 films
As shown in Chap.5 achieving large volume fractions, 35%, of Au nanoparticle
films while preserving plasmon responses is experimentally challenging. An approach
to assembling such materials, is to use a silicon substrate sputtered coated with 4 layers
of gold islands with 3 spacer layers of 23 in between the gold layers , Fig.77a. The
sample was provided by Dr. Drehman at Hanscom AFB. The sample had a gradient in
the gold coating starting from the right-center of the sample with no gold, to the full gold
coating in the wafer center. The gold islands were random in size distribution with a
nominal size of ∼ 100, Fig.77b. Using the program, described in Chap.8, the volume
fraction of the Au in image, Fig.77b, was determined to be, = 284% (for a single Au
layer), Fig.77b.
117
Figure 77: a: Image of the sample. b: SEM image of the sample
The absorbance spectrum of the sample is shown in Fig.78. The effective absorption
peak of the sample is centered at = 600.
Figure 78: Absorption spectrum for the sputter-coated gold on a silicon wafer with the
superimposed probe wavelengths.
The sample was placed into one arm of the interferometer similarly to Chap.4. The
118
probe beam was expanded to a diameter ∼ 10 and placed in the gold gradient region
shown in Fig.77a, allowing for a continuous shift in the interference fringes. The nega-
tive dispersion region of the sample should be to the left, short wavelength side, of the
absorption peak, as described in Chap.3. The sample was probed for wavelengths from
left to right in Fig.79 at = 496, 514, 580, 590.
Figure 79: Real relative phase shift of the gold sputter-coated onto a silicon wafer as a
function of wavelength.
Fig.79 shows images of the sample region, shown in Fig.77a, with the gold coating
and without. The extremum relative phase shifts were ∆590 2 and as the
wavelength decreases the relative phase shift decreases to ∆496 8, the net phase
shift still remains positive relative to the substrate.
CHAPTER 7
Conclusions
In this dissertation we focused on experimentally synthesizing, assembling and char-
acterizing Au NR composites for optical metamaterial applications. The novel achieve-
ments of this dissertation are:
A method for phase transferring Au NRs from aqueous to organic suspensions was de-
veloped. This enabled the Au NRs to be aligned by a external electric field, achieving high
order parameters. By aligning the NRs this enables us to develop a method to measure
the real and imaginary phase shifts of light transmitted through the suspensions allowing
an experimental characterization technique to be developed to determine the principal
values of the electric susceptibility tensor for gold nanorods at optical frequencies. Once
the optical susceptibilities of the individual NRs were determined self-assembled densely-
packed films of small aspect ratio, orientationally-ordered, silica-encased gold nanorods
in square millimeter domains were created, via solvent evaporation and applied electric
field. We experimentally explored other materials that may be used as novel metamate-
rial constituents such as conjugated dye molecules and discovered conjugated molecules
are capable of anomalous dispersion with figures of merit comparable to metals. This
work provides insights into the design, production, processing and characterization of
optical metamaterials.
119
CHAPTER 8
Appendix
8.1 Recipe for gold nanorods
Overview
A seed solution is made from Au flakes, CTAB and NaBH4 in water. The NaBH4 is
a strong reducing agent and shapes the gold flakes into spheres (~4nm). CTAB has a
nonuniformed affinity for the Au sphere’s surface due to the lattice structure of the gold.
This allows the Au spheres to grow into an anisotropic shape (ellipsoidal). The seed
is injected into a solution of gold flakes, ascorbic acid, AgNO3, CTAB and water. The
ascorbic acid is a mild reducing agent this allows the Au flakes to dissolve onto the spher-
ical seeds. The amount of AgNO3 present is the dominating factor in determining the
aspect ratio of the nanorods, water temperature is also another important factor. CTAB
helps stabilize and control the growth of the nanorods. To make the nanorods soluble in
organic media polystyrene is added to the solution. The polystyrene interdigitates with
the CTAB on the nanorod’s surface. The polystyrene is a much longer molecule than the
CTAB this allows the nanorod to be soluble in organic media.
Component preparation
Purchase highest purity material available
1. CTAB: Dissolve 11.27g of CTAB in 100ml of DI water and place it into a 500ml
flask. Place the flask in a heated sonicator (50C) to dissolve CTAB while continuing
on with the other component preparation process. Scrap off any CTAB that did
not fully dissolve off the bottom of the flask periodically through this preparation
process. (purchase bottle of CTAB 100g)
120
121
2. AgNO3: Dissolve 13.58mg of AgNO3 in 20ml of DI water (water resistivity must
be greater than 18.2 MΩ) and place it into a 20ml vial.
3. Ascorbic acid: Dissolve 138.7mg of ascorbic acid in 10ml of DI water and place it
into another 20ml vial.
4. Gold: Dissolve 54.54mg of Au in 152.5ml of DI water and place it into a 150ml
flask. Measure the gold FAST because it absorbs the moisture from air. If the gold
becomes hydrated before being mixed with the DI water use a pipette with the DI
water in the flask to wash the gold into the flask. (Do not use a metal spatula; it
will oxidize the gold)
5. NaBH4: Dissolve 7.56mg of NaBH4 in 20ml of DI water and place it into another
20ml vial. Work FAST NaBH4 is a very reactive reducing agent and has a very
short lifetime. Place the sample in the freezer after preparation it needs to be cold
before use (5 min).
6. Bring the total volume of the CTAB solution to 155ml using DI water.
Seed and solution mixture
**Once you add the cold NaBH4 the whole process needs to be finished in 5 min.
(seed lifetime)**
1. Mix 2.5ml of DI water and 2.5ml Au solution into a new 20ml vial this is the seed
solution. Mix with a stirring bar. Stir for 5 min.
2. Add 5ml of the CTAB solution into the seed. Stir for 5 min. (Color change: yellow
to orange)
3. ** Add 600 of cold NaBH4 into the seed (all on stir plate). (Color change: orange
to brown). Note time: =0 min. Add 7.5ml of AgNO3 to the CTAB solution. Mix
122
with a stirring bar (2 min). The concentration of AgNO3 determines the aspect
ratio.
4. At =2 min. stop stir plate and hand swirling. Pour all the Au solution into the
CTAB solution (Color: orange-brownish). Mix with a stirring bar (2 min).
5. At =4 min. Add 2.1ml Ascorbic acid into CTAB. (solution becomes colorless,
molarity test). Mix with a stirring bar (1 min).
6. At =5 min. add 360 of the seed into the CTAB solution. Inject the seed solution
under the foam layer in the CTAB solution. Shake violently for 30sec
7. Put the solution on hot plate for 1hr at ~45C. Mix with a stirring bar (10rpm).
Temperature is critical in determine size and aspect ratio of NR. If the solution
immediately becomes pink then the solution is spheres and no good. If the solution
is good, over the course of minutes it will start changing to a purple color and will
gradually become very dark.
After 15 min. you can check the solution in a spectrometer. The characteristic
shape looks like the following[**abs vs spectrum**]
8. At the 1 hour mark reduce heat to 25-30C for 2 hours.
Purification
Add the solution into a centrifuge and spin for 80 minutes at 6000rpm (4185 g’s).
After the first cycle is complete decant the clear solution from the top of the centrifuge
vials. Add a little water in each condensed NR vial and mix them together. Heat up the
condensed solution on a hot 65C hot plate. You want the solution warm when is goes
back into the centrifuge. Then transfer back to centrifuge and repeat this cycle two more
times
123
8.2 Organically soluble nanorod coating process
The aforementioned process yields non-organic soluble nanorods (in water). The
following describes the procedure for coating the nanorods with an organically soluble
coating.
1. Measure 100mg of Thiol terminated polystyrene and mix it into 4ml of THF.
2. Take 2ml of the Au nanorod solution and place it into a new 20ml vial.
3. Add 1ml of the polystyrene/THF solution to the Au nanorod solution and shake
violently (1 min). The nanorods should be clumping and sticking to the sides of
the vial.
4. Add another 1ml of the polystyrene/THF solution to the Au nanorod solution and
shake violently (1 min). The nanorods should mostly be stuck to the sides of the
vial and the solution should be fairly clear. Let the mixture stand for 15minutes.
5. Using a pipette remove the clear solution, being mindful not to remove any aggre-
gated nanorods in the solution.
6. Add 2ml of Chloroform or Toluene or any organic media. Shake and sonicate to
properly disperse nanorods.
7. Centrifuge to clean up if necessary.
8.3 Improved NR synthesis (monodispersed)
Glassware Needed
• Two 500ml flasks
• 100ml graduated cylinder
124
• 6 Vials
Piranha Cleaning
• 3:1 ratio of sulfuric acid : hydrogen peroxide
• Added in the hood
• Careful, it is hot when reacting. . . don’t touch until 3 minutes later with heat
resistant gloves
• Swirl the flasks, but make sure to keep opening facing the inside of the hood
• Move solution from one glassware to the others, and at the end remove the
solution into waste bucket
• Clean the glassware first with deionized water, then with E-pure water to remove
any acid left
• Put glassware in heater for 5 minutes
1. Add 100ml of E-pure water to 2 flask.
2. Weight out 5.649 grams of CTAB for each flask.
3. Use sonic bath to dissolve all the CTAB.
4. Weigh out 33.986 mg of silver nitrate in vial.
5. Weigh out 176.12 mg of ascorbic acid in vial.
6. Weigh out 7.566 mg of sodium borohydride (NaBH4) in vial. (Measure out
accurately)
7. For each CTAB flask, measure out volume of water in graduated cylinder to
100ml, then measure another 55ml for a total of 155ml.
8. Prepare 1M HCl by adding 890ul of 37% HCl and bring total volume to 10ml.
9. Take out 4.7ml from CTAB flask and added to vial marked “seed,” the other
125
0.3ml will be disposed in waste solution.
10. Small stir bar will be added to the “seed” vial.
12. Add 1.5ml of HAuCl4 to CTAB flask. Swirl (Orange-reddish color)
13. Add 2.85ml of HCl to CTAB flask. Swirl (Orange-reddish color)
14. Add 10ml of E-pure water to “Ascorbic Acid” vial. Mix vial.
15. Add 1.2ml of “Ascorbic Acid” to CTAB flask. Swirl (Clear)
16. Add 20ml of E-pure water to “AgNO3” vial. Mix vial.
17. Add 1.8ml of “AgNO3” to CTAB flask. Swirl (Clear)
18. Prepare “Seed” by using a magnetic stirrer at Fastest speed.
19. **Add 25ul of HAuCl4 to “Seed” vial. (Clear to Yellowish)**
20. Add 20ml to “NaBH4” vial. Mix vial.
21. Add 300ul of “NaBH4” to “Seed” vial. (Brownish)
22. Let it mix for 2 minutes.
23. Add 360ul of “Seed” to CTAB flask. Swirl (Clear)
24. Put flask in water bath at 28 degrees Celsius for 2 hours. (in water bath
overnight)
25. During the water bath, the solution will change color multiple times until
the reddish-brownish final color.
Next Day:
1. If solution has no crystals, we can centrifuge in 50ml tubes for 1 hour at
9000rpm. *If crystals are present, use sonic bath or heat plate to dissolve crystals.
2. After centrifuge, take the supernatant out by using a plastic pipette and
getting all clear solution out. Pipette into waste bucket.
3. Use the three 50ml with rods to get a solution of 10ml in E-pure water in 1
126
50ml falcon. You can achieve this by add 2-3ml of water to two 50ml falcons and then
pouring that into one of the 50ml falcon.
4. Pour the 10ml of gold solution into new vial for storage.
8.4 Smin program (Maple)
This program calculates the orientational order parameter, , by measuring the ab-
sorbance spectrum, , vs. applied electric field following the method described in
Chap.3.
restart:
Number of x elements (E2)
X:=15:
Data file location (E2,ax)
file_in:="C:\\Documents and Settings\\user\\Desktop\\E2 vs ax 750nm.txt":
Result file location (E2,ax,S)
file_out:="C:\\Documents and Settings\\user\\Desktop\\Results 750nmE2,ax,S,A,B,C.txt":
Result file location-continuous fit (x,ax,S)
file_continuous:="C:\\Documents and Settings\\user\\Desktop\\Results 750nm con-
tinuous fit x,ax,S.txt":
data:=readdata(file_in,2):
for i from 1 to X do
x[i]:=data[i,1]:
end do:
for i from 1 to X do
y_in[i]:=data[i,2]:
end do:
for i from 1 to X do
127
x[i]:
y[i]:=y_in[i]-y_in[1]:
end do:
g:=C-int((1/2)*(3*z^2-1)*exp(C*x[i]*z^2),z=0..1)/int(exp(C*x[i]*z^2),z=0..1):
dg:=C-x[i]*((int((z^2/2)*(3*z^2-1)*exp(C*x[i]*z^2),z=0..1)*
int(exp(C*x[i]*z^2),z=0..1)-int((1/2)*(3*z^2-1)*exp(C*x[i]*z^2),z=0..1)
*int(z^2*exp(C*x[i]*z^2),z=0..1))/(int(exp(C*x[i]*z^2),z=0..1))^2):
g2sum:=0:
ydgsum:=0:
ygsum:=0:
gdgsum:=0:
for i from 1 to X do
g2:=(g(C))^2:
g2sum:=g2sum+g2:
ydg:=y[i]*dg(C):
ydgsum:=ydgsum+ydg:
yg:=y[i]*g(C):
ygsum:=ygsum+yg:
gdg:=g(C)*dg(C):
gdgsum:=gdgsum+gdg:
end do:
f:=g2sum*ydgsum-ygsum*gdgsum:
plot(f,C=0..10,y=-1..1,labels=[’C’,’f’],title="Find f=0");
C:=fsolve(f=0,C,0..10);
B:=evalf(ygsum/g2sum);
128
A:=y_in[1];
for i from 1 to X do
S[i]:=(y_in[i]-A)/B:
end do:
S[max]:=S[15];
results:=array(1..X,1..6):
for i from 1 to X do
for j from 1 to 6 do
results[i,j]:=0.0:
end do:
end do:
for i from 1 to X do
results[1,4]:=A:
results[1,5]:=B:
results[1,6]:=C:
results[i,1]:=x[i]:
results[i,2]:=y_in[i]:
results[i,3]:=S[i]:
end do:
writedata(file_out,results):
Outputs x,ax, S continuous fit function:
xx:=array(1..250,1..3):
for i from 1 to 250 do
xx[i,1]:=evalf(i/10):
xx[i,2]:=evalf(A+B*int((1/2)*(3*z^2-1)*exp(C*xx[i,1]*z^2),z=0..1)/int(exp(C*xx[i,1]*z^2
129
xx[i,3]:=evalf(int((1/2)*(3*z^2-1)*exp(C*xx[i,1]*z^2),z=0..1)/int(exp(C*xx[i,1]*z^2),z=0..
end do:
writedata(file_continuous,xx):
8.5 Correlator program (LabVIEW)
This program calculates the correlation function between two interference images, 0
and 1, to determine the relative phase shift. The images are first cropped and rotated
such that the interference lines are vertical. The image is composed of (horizontal) and
(vertical) elements. First, the colums are summed for each element, =P
=0
for both images. The direction the image translated, 1, is known from the experiment.
A correlation function is calculated from the two images, 0 = 0 () 1 (). 1 is then
translated, in the opposite direction of the measured translation direction, by a pixel,
∆, and a new correlation function is calculated, 1 = 0 () 1(−∆). The algorithm is
repeated and the relative phase shift is then calculated by determining the distance 1
needed to be translated for the first maxima of .
130
Figure 80: LabVIEW correlator program
8.6 NR counter (LabVIEW)
This program measures the dimensions and standard deviations of NRs imaged with
a TEM. The program works well so long as the NR do not touch each other in the
image and the partial NRs at the borders of the image were manually removed from the
measurements. The image is first converted to a binary image, where below a certain
threshold, the images is black and above the threshold the images is red. The threshold
is manually adjusted for each image. LabVIEW, using the particle analysis function,
fits an ellipsoid to each NRs, and measures the length, width and standard deviations.
131
Figure 81: LabVIEW NR counter
8.7 Dye recipes with mirror sets
8.7.1 R6G (565− 640)
Recipe
premix: 1gr/50ml dye/methonal
add to 950ml Ethylene Glycol (stir 3 hours)
add 1ml COT
Mirror set
M1: 0158-787-07
M2: 0158-788-07
M3: 0500-836-03
132
8.7.2 DCM special (610− 710)
Recipe
premix: 1gr/200ml dye/benzyl alcohol (stir 3 hours)
add to 600ml Ethylene glycol (stir 12 hours)
add another premix: 1gr/100ml dye/benzyl alcohol (to optimize)
Mirror set
M1: 0158-787-08
M2: 0158-788-08
M3: 0500-836-01
8.8 Dye laser alignment procedure
M0M1
M2
M3
Dye Jet
Birefringence plate
Figure 82: Dye laser head
Carefully remove mirrors 0, 1, 3 and the birefringence plate from the laser
head, Fig.82. 2 also needs to be removed, however to remove it, it is necessary to
unscrew the folding mirror holder and slide the mount all the way down the invar finger
close to the dye jet path. Place the coated surface of the mirrors pointing up on an
133
optical lens tissue. Do not touch the coated surfaces. If the mirrors need to be
cleaned- refer to a proper technique for cleaning laser mirrors.
There is a small hole in the front (entrance) and back (exit) side of the laser cavity as
shown in Fig.83. Align the pump beam through both holes. Mark the place where the
pump beam hits the wall after passing the holes. Also, place a white paper 34 from
the end of the laser head, then mark where the pump beam hits the paper.
entrance hole
exit hole
Figure 83: Entrance and exit holes for the pump beam.
Turn the pump laser on( 1 ). Also, turn the dye circulator ON to eliminate bubbles
and mix the dye.
To turn ON the dye circulator, first turn the water for the cooling system ON by
opening the BLUE valve, as indicated on Fig.84. The pressure should be between 50−
60. The little black valve, Fig.84, should be completely opened. There are two dye
circulators. The cooling system for each one is connected to different water pipes, one
with the cw argon laser and the other with the ns Nd:YAG laser. Be sure to turn on the
right one. After, turn the switch on the dye circulator to PUMP position. Adjust the
pressure to 20 by turning the white valve on the dye circulator.
134
blue valve
black valve
Figure 84: Cooling system for the dye laser circulator
Block the pump beam and attach the 0 mirror back in place. The coated surface
should be facing the pump beam. Unblock the pump beam. The beam should be reflected
by 0 into the hole behind the folding mirror holder. Use the knobs located on the top
and side of the laser head to zero the beam into the hole (while the 2 mirror mount is
still next to the jet).
Turn the dye jet ON by switching to the JET position, and then increase the pressure
to 30. The jet should be at an angle of ˜26. DO NOT adjust the angle with the jet
on. If necessary, turn the jet off and use the tip of the jet tube as reference. You may
need to move the orange cover to see the tip. The pump beam should hit the middle of
the dye jet.
Carefully place 2 in the folding mirror holder, the coated surface should face the
dye jet. Move the folding mirror along the Invar finger and observe the fluorescent spot
in the white paper. The folding mirror support should be at 31 from the end of the
invar finger. There were two black marks on the invar finger where the support should
be placed. The spot should be ˜17 above the pump beam mark in the white paper.
135
Use the three silver screws on the support plate of the 2 to adjust the position of the
fluorescent spot.
Carefully insert 1 in place; the coated surface should be facing the dye laser jet.
A second fluorescent spot should appear in the white paper. The spots should have the
same shape and size. Use the three silver screws to align the spot on the top of the first
fluorescent spot from 2.
Carefully insert 3 but before attaching the mirror, rock the mirror by hand to see
where the fluorescent spot is on the 2 support. The spot is very week and has a round
shape. Increase the pump laser intensity in order to see laser from the dye laser (30 in
the Argon laser should be enough). Align the 3 fluorescent spot in the middle of 2.
You also may observe a spot on the white paper. Use the two knobs on the top of the
dye laser to align the spot. As soon the dye laser emitted, decrease the pump intensity
and try to optimize the emission with the two knobs. Then mount the birefringent plate
into the laser head and reoptimize.
8.9 MetaMachine
The is a computer controlled layer-by-layer dip coating machine that
can produce novel metamaterials in relatively large quantities. The primary purpose of
the , , is to enable scientists to efficiently and affordably build novel
metamaterials ranging from, but not limited to, self-assembled monolayer, layer-by-layer,
sol-gel, and photonic band gap material assemblies. Ubiquitous “real world” applications
include photovoltaic devices (solar cells), high-strength composite plastics, energy storage
devices (batteries), optical elements, and conventional or non-conventional laser systems.
As an illustrative example of a metamaterial suppose material has a certain physical
property (i.e. mechanical, electrical, optical, etc.) that is combined with material
136
which has another useful physical property. If the constituent building blocks, materials
and , are sufficiently small relative to the overall size scale then the new material,
, is said to be homogenized, with a novel unique property, which is not simply a linear
combination of and .
A useful type of bottom-up or self-assembly technique is layer-by-layer (LBL) depo-
sition. Consider a solution of positively charged polymer, +, and another solution of
negatively charged polymer, −. If a substrate of negatively charged glass, 0−, is dipped
into a beaker of the positively charged polymer then a single layer of the positively
charged polymer adsorbs to the surface of the glass. The substrate is then rinsed with
ultrapure deionized water, to remove any excess positively charged polymer. Then the
substrate is dipped into the negatively charged polymer solution and single layer of the
negatively charge polymer adsorbs to the surface of the positively charged polymer and
rinsed again. This cycle is repeated until the desired thickness of film is created, Fig.85.
This film can be removed from the glass substrate and used for its intended application.
Figure 85: Layer-by-Layer deposition
A recently developed LBL metamaterial, by Prof Nicolas Kotov at the University
of Michigan, is plastic steel- a clear plastic with the strength of steel but much lighter,
Fig.86 (upper right). By mimicking the brick and mortar molecular structure found in
sea shells Prof. Kotov used nanoclays and a transparent polymer to build these materials
using LBL. There are many different types of materials that may be produced using this
technique, Fig.86,
137
Figure 86: Practical applications of Layer-by-Layer deposition
Typically because these materials are novel they are hand dipped. This is very time
consuming and produces low quantities. There are two types of automated dipping
machine available but both are extremely limited in their functionality. Therefore there
is a need for a dipping machine versatile enough to build current and future composite
metamaterials. To date there are only two types of commercially available dip coating
machines; Nanostrata dipping machine and tinting machines. The Nanostrata machines
are relatively expensive, relatively large, requires a dedicated PCwith specific I/O boards,
may require expensive short lifetime compressed2 cylinders to operate and closed source
software that does not allow for user customization and limits algorithm length and
complexity. The tinting machine suffers from similar shortcomings and is generally less
capable than the Nanostrata machines.
The schematic of the is shown to the below, Fig.87. The structure is built out
of aluminum plate. The consists of four sections.
Section 1 is the carousel a circular plate capable of holding up to four 150ml beakers.
A shaft coupler connects the plate to a stepper motor mounted under the plate. This
motor allows for rotation of the carousel plate thus being able to alternate the solution
where the substrate will be dipped.
Section 2, the substrate stage, allows for vertical translation and rotation of the
138
substrate(s) to and from the carousel and deionized water (DI) water rinse. A stepper
motor rotates a belt, raising and lowering the substrate stage and another stepper motor,
mounted on the substrate stage, rotates the arm, holding the substrates, to the carousel
and water rinse.
Section 3 is the DI water rinse used for rinsing the substrates in between dipping
cycles. Using PC controlled solenoid valves to supply and drain ultrapure DI water into
a 150ml beaker.
Section 4 consists of PC that uses LabVIEW to control a USB-6221 card, solenoid
valves and motor drivers to control all three stepper motors rotation and direction.
Figure 87: MetaMachine schematic
The hardware and software for first generation has been constructed. A key
stage of development is the software algorithms to build various novel metamaterials.
To develop these algorithms the needed to be tested in situ. A collaboration was
started with a leader in the LBL field Prof. Nicolas Kotov.
A first generation was built and loaned to Prof. Kotov so the methods and
algorithms could be developed as well as testing for mechanical reliability. The machine
has been operating continuously for one month with no reported issues.
139
a. b. c.
Figure 88: a: 1st generation . b: screenshot of software.
c: during in situ testing at the University of Michigan
An example of a current exciting material being explored with the 1st generation
is a new algorithm to produce exponential LBL films. Using a combination of gold
nanorods, polyurethane and poly acrylic acid films are constructed that increase in thick-
ness exponentially with every dipping cycle- the physics is exciting and not understood
at this point.
Globally, this collaboration can be thought of as an idea incubator. New materials
and methods can be developed, possibilities for intellectual property, and then these new
ideas can be incorporated into new s.
The , at this stage, is designed to enable scientists and engineers to efficiently
and affordably build novel metamaterials in relatively large quantities. However in the
future once the material algorithms are developed and understood it should be relatively
straightforward to scale up the systems fromR&Dmachines to mass production industrial
machines for general use.
8.10 Mechanical drawings
Mechanical drawing for the Mach-Zehnder interferometer
140
Figure 89: Schematic drawing interferometer
Mechanical drawing for the kinematic mirror/beamsplitter mounts
Figure 90: Schematic of kinematic mounts
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