Thickness Effects on Light Absorption and Scattering forNanoparticles in the Shape of Hollow SpheresJeng-Yi Lee,† Min-Chiao Tsai,‡ Po-Chin Chen,‡ Ting-Ting Chen,‡ Kuei-Lin Chan,‡ Chi-Young Lee,‡

and Ray-Kuang Lee*,†,§

†Institute of Photonics Technologies, ‡Department of Materials Science and Engineering, and §Frontier Research Center onFundamental and Applied Sciences of Matters, National Tsing-Hua University, Hsinchu 300, Taiwan

ABSTRACT: We reveal the thickness effects on opticalproperties for nanoparticles in the shape of hollow spherestheoretically and experimentally. Within and beyond theelectrically small limit, hollow spheres are shown to havealmost the same light absorption power as that of solid ones,when the ratio of inner core to whole particle radii is smallerthan 0.4. It means that one can maintain the level of lightabsorption even with a large empty core. In the electricallysmall limit, we expand the exact solution of Mie theory inpower of the thickness parameter and show that the thicknessratio has less influence on light absorption. Moreover, we synthesize highly uniform hollow spheres of TiO2 anatase through aself-sacrificing template method. A variety of particle radii from 94 to 500 nm, with 50 nm in the shell thickness, are performedexperimentally in photocatalytic activity. With experimental demonstrations and theoretical simulations, our results provide aguideline in the design on the thickness for hollow-sphere nanoparticles with an optimized absorption power in light harvesting.

■ INTRODUCTION

Photocatalyst has acted as one of solutions to utilize solar lightinto available energy, with environmental friendly and highlyefficient materials synthesized for water splitting, CO2reduction, and green energy applications.1,2 In particular,TiO2 is the major photocatalyst to decompose water intohydrogen and oxygen and used as a clean and recyclable energysource for hydrogen fuel.3−5 It is known that TiO2 has theenergy gap about 3.2 eV, and when excited by a UV−visiblelight source, a pair of electron and hole is generated.Consequently, this excited electron−hole pair migrates to thesurface to serve as electron donor and acceptor, resulting in thephotocatalytic reaction. Based on this series of electrochemicalreactions, several methods are demonstrated to enhance thephotocatalytic activity with TiO2. For example, to reduce therecombination of separated electron−hole pairs, one can blendvarious TiO2 phases, deposit metal materials on the surface, orintroduce trapping sites to inhibit the recombination.6−8 As theaim to extend absorption regime in TiO2 to visible wavelength,doping specific metal or nonmetal elements are demonstratedfor light harvesting.9−14

Instead of chemical or material approaches mentioned above,intricate structure and morphology can exhibit fascinatingproperties to improve photocatalytic activity.15−19 Inspired bymaterials in nature, like seashells and plant seeds, an alternativeway to enhance photocatalytic performance is utilizingnanosized TiO2 particles.

15 As a rule of thumb, the exposuresurface area grows when the size of particles decreases. For lightabsorption and scattering, it is known that resonance effectemerges as the size of the nanoparticle approaches the

wavelength of incident light, known as Mie’s scattering effect.20

Recently, size-dependent Mie’s scattering effect with TiO2

nanoparticles is reported for a superior photoactivity of H2

evolution.15,16 In addition to synthesizing different geometricalsizes, nanoparticles can also be formed in the shape of hollowspheres, which have been widely used as adsorbents, deliverycarriers, catalysts, and biomedical detectors.17−19,21 Eventhough enhancement in photocatalytic function by a varietyof geometrical structures has been demonstrated, a systematicinvestigation to understand the thickness effect on lightabsorption for nanoparticles in the shape of hollow spheres isstill needed to provide a guideline to design optimized core−shell structures for light harvesting.By taking the complicated photoelectrochemical reaction as a

pure optical problem, in this work, we study optical response ofincoherent hollow spheres with different geometrical andmaterial parameters based on the exact solutions of Mie’sscattering theory. Analytically, we derive the formula for lightabsorption and scattering coefficients by expanding thethickness parameter. Our theoretical results not only givegood agreement to the numerical solutions conducted fromMie theory, but also reveal a counterintuitive scatteringproperty of hollow spheres. We find that as long as thethickness ratio, that is, the ratio of inner core radius to wholeparticle radius is smaller than a critical value, about 0.4, anindividual hollow sphere is shown to have almost the same light

Received: August 30, 2015Revised: October 16, 2015Published: October 23, 2015

Article

pubs.acs.org/JPCC

© 2015 American Chemical Society 25754 DOI: 10.1021/acs.jpcc.5b08435J. Phys. Chem. C 2015, 119, 25754−25760

absorption as a solid one. In contrast to the naive picture, with adifferent thickness parameter, one can also have a higher totalabsorption power for a larger particle size. Moreover, weperform experimentally the photocatalytic activity by synthesiz-ing highly uniform hollow spheres of amorphous TiO2 anatasethrough a self-sacrificing template method. With the measure-ments on photodegradation of methylene blue, thickness effecton a variety of particle radii from 94 to 500 nm is studiedexperimentally for light absorption. With experimentaldemonstrations and theoretical simulations, our results providerecipes to design highly efficient photocatalysis with benefitsfrom reducing volume of material use.

■ MIE THEORY FOR HOLLOW-SPHERENANOPARTICLES

Before giving the theoretical model for nanoparticles in theshape of hollow spheres, we briefly describe our synthesismethod. Typical scanning electron microscope (SEM) imagesof our TiO2 hollow-sphere nanoparticles are shown in Figure1a,b, with different particle radii. These TiO2 nanoparticles aresynthesized using titanium isopropoxide (TTIP, 97%, Aldrich),valeric (99%), and butyric (99.5%) acids in anhydride alcohol(99.5%). Our synthesis process can be viewed as a two-stagetransformation. First, the amorphous TiO2 reacts with F− ionsin the NaF solution to form H2TiF6, resulting in dissolving andshrinking in particle size. Then, in the second stage,nanocrystalline particles grow slowly into the shape of hollowspheres, composed of rhomboid anatase. By controlling theconcentration of NaF solution, we can synthesize TiO2nanoparticles in different sizes, while the concentration of theF− ion modifies the thickness of hollow spheres. As shown inFigure 1, highly uniform TiO2 nanoparticles in the shape ofhollow spheres are demonstrated with the radius varying from

94 to 500 nm. The resulting thickness for the shell region isalso tunable, from 50 to 100 nm.X-ray diffraction (XRD) measurement (Bruker D8-advanced

with Cu Kα radiation λ = 1.5405981 Å) is applied tocharacterize the crystal structure of our hollow-sphere nano-particles, as shown in Figure 1c for diameters in 220, 440, 620,and 800 nm, respectively. By using Scherrer’s formula, our XRDspectra reveal the corresponding grain size with 14.6, 13.2, 13.1,and 12.6 nm, respectively. As shown in XRD patterns, highlyuniform crystallinity of various-sized TiO2 nanoparticles in theshape of hollow spheres is demonstrated. The results show thatthe crystallinity for all samples are the same. More details onthe method to prepare amorphous precursor spheres can befound in our previous work.17,18 Due to its superiorphotocatalytic performance, we use this TiO2 anatase as ourstudying materials.To identify the difference between optical responses from

solid-sphere and hollow-sphere nanoparticles, we consider anincident plane wave on a spherical object, as illustrated inFigure 2. This sphere has a bilayer structure, denoted as thecore and shell regions. The inner radius of core is denoted as ac;while the outer radius of whole particle is denoted as a. Materialproperties are characterized by the permittivity ϵi andpermeability μi, where i = e, s, or c that denotes theenvironment, shell, and core regions, respectively. Withoutloss of generalities, the particle is embedded inside a losslessmaterial. All materials are taken as nonmagnetic ones, that is, μe= μs = μc = 1. In the following, we use prime and double primeto denote the real and imaginary parts of the dielectric function,that is, ϵ = ϵ′ + iϵ″. Moreover, corresponding to thephotocatalytic experiment carried out later, we set ourbackground environment as water, with a constant dielectricfunction in the UV−visible region ϵe = 1.77. We also assume

Figure 1. (a, b) SEM images at various magnifications of TiO2 nanoparticles in the shape of hollow spheres. (c) XRD spectra of our hollow-spherenanoparticles, shown in different diameters.

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that the core region is filled with water. It is known that TiO2anatase is also a birefringence material. However, due to therandom distribution of TiO2 nanoparticles in water, one cansafely ignore such an optical anisotropy property. The opticaldielectric function within UV−visible light spectra for TiO2nanoparticles, in particular, for the wavelength λ = 388 nm(corresponding to 3.2 eV band gap energy), is around 6.25 < ϵ′< 10 for the real part and 0 < ϵ″ < 0.5 for the imaginarypart.22,23 In the following, we also take the complex dielectricconstant into the calculations, with applications in thephotocatalytic process.Now, an x-polarized electric plane wave (incident wave) is

assumed to propagate along the z direction with the timedependence e−iωt, where ω is the angular frequency. Based onMie theory,24 the scattering electric and magnetic fields can begenerated through two auxiliary vector potentials, only withradial components in the spherical coordinate. These twoauxiliary vector potentials are A⃗scat (magnetic vector potential)and F⃗scat (electric vector potential), which can be decomposedinto two infinite series of complex scattering coefficients Sn

TM

and SnTE, with the subscript to denote n-th order spherical

harmonic channel. These complex scattering coefficients, SnTM

and SnTE, can be determined by the boundary conditions of each

layer, due to the continuity of magnetic and electric fields alongthe spherical surface. The complex scattering coefficients, Sn

TM

and SnTE, can be recast in the following compact forms:24

= −+

Si V U

11 ( / )

nn n

TETE TE

(1)

= −+

Si V U

11 ( / )

nn n

TMTM TM

(2)

where

=

ϵ′

ϵ′

ϵ′

ϵ′

ϵ′

ϵ′

U

j k a j k a y k a

k a j k a k a j k a k a y k a

j k a y k a j k a

k aj k a k ay k a k aj k a

( ) ( ) ( ) 0

1[ ( )]

1[ ( )]

1[ ( )] 0

0 ( ) ( ) ( )

01

[ ( )]1

[ ( )]1

[ ( )]

n

n n n

c n n n

n n n

n n n

TM

c c s c s c

cc c c

ss c s c

ss c s c

s s e

ss s

ss s

ee e

(3)

and

=

ϵ′

ϵ′

ϵ′

ϵ′

ϵ′

ϵ′

V

j k a j k a y k a

k a j k a k a j k a k a y k a

j k a y k a y k a

k aj k a k ay k a k ay k a

( ) ( ) ( ) 0

1[ ( )]

1[ ( )]

1[ ( )] 0

0 ( ) ( ) ( )

01

[ ( )]1

[ ( )]1

[ ( )]

n

n n n

n n n

n n n

n n e n

TM

c c s c s c

cc c c c

ss c s c

ss c s c

s s e

ss s

ss s

ee

(4)

Here, both Vn and Un are 4 × 4 determinants, jn(.) is thecorresponding spherical Bessel function, yn(.) is the corre-sponding spherical Neumann function, [xjn(x) ]′ represents

xj x[ ( )]x n

dd

, and [xyn(x) ]′ represents xy x[ ( )]x n

dd

, respectively.

Eqs 1−4) show the exact formulas of Mie’s scattering theory fora core−shell sphere with isotropic and homogeneouspermittivities and permeabilities in each layer. These equationsare also given in a symmetric form, that is, the formulas for Un

TM

(VnTM) and Un

TE (VnTE) have similar mathematical forms but with

slight differences on physical parameters (just replacingdielectric constant ϵ by permeability μ).To give a quantitative analysis, we define the quantity to

measure how much amount of power is absorbed by ananoparticle, that is, the absorption cross section Qabs,

= = −QWI

Q Qabsabs

incext scat

(5)

where Iinc is the intensity of incident wave in the unit of powerper unit area. The corresponding extinction cross section andscattering cross section, dented as Qext and Qscat, have theforms:

∑π= =| |

+ | | + | |=

∞

QWI k

n S S2

(2 1)( )n

n nscatscat

inc e2

1

TE 2 TM 2

(6)

∑π= = −| |

+ +=

∞

QWI k

n S S2

(2 1)Re{ }n

n nextext

inc e2

1

TE TM

(7)

Here, Wabs, Wscat, and Wext correspond to the absorption,scattering, and extinction powers, respectively. Even thoughabove results are a consequence of Mie theory, it is too generalto have a clear physical picture on the thickness effect forhollow spheres. Below we expand the thickness parameter togive analytical formulas.

Electrically Small Limit. First of all, as the size parametera/λ in each layer is small enough, satisfying the condition kea≪ 1 and ksa ≪ 1, we have analytic formulas for thecorresponding cross sections. In this electrically small limit,one can approximate the spherical special functions by a series

of polynomial functions, that is, ≈ !+ !j x x( )nn

nn2

(2 1)

n

and

≈ − !!

− −y xnnn

n(2 )2

1n . With this series expansion, we can expand

the complex scattering coefficients SnTM and Sn

TE shown in eqs 3and 4 into

γ γ

≈ −+

!+ !

+ ϵ ϵ ϵ

× ϵ − ϵ + × ϵ + ϵ + ϵ − ϵ

× ϵ + + ϵ

− − − −

−

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟U

nn

nkk

kk

k a n

n n

n n

12 1

2(2 1)

( ) ( 1)

{( ) [( 1) ] ( )

[ ( 1) ]}

n

n nn

n

TM2

c

e

e

se

2 1s

2e

1c

1

c s2

s e1

s e

c s (8)

Figure 2. Schematic of a hollow sphere in the core−shellconfiguration. The corresponding wavenumbers for each region aredefined as μ= ϵk ke e e 0, μ= ϵk ks s s 0, and μ= ϵk kc c c 0 for the

environment, shell, and core, respectively, where k0 ≡ 2π/λ is thewavenumber in the vacuum with the wavelength denoted by λ.

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γ

γ

≈+

ϵ ϵ ϵ

× + ϵ − ϵ ϵ − ϵ

− + ϵ + ϵ ϵ + + ϵ

− − − −

−

⎛⎝⎜⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟V

nkk

kk

k a

n n

n n n n

1(2 1)

( )

{ ( 1) ( )( )

(( 1) )( ( 1) )}

n

n

n

TM2

c

e

e

se

2s

2e

1c

1

2e s c s

1e s c s (9)

Here, we introduce the thickness ratio, γ ≡ ac/a, which isdefined as the ratio of inner core to whole particle radii. When γ= 0, the nanoparticle is fully filled (a solid-sphere), while whenγ = 1, the nanoparticle is fully empty. A hollow sphere isdescribed for 0 < γ < 1.For nonmagnetic materials, all the values of Un

TE for differentspherical harmonic channels are automatically zero. Thus,within this electrically small regime, the correspondingscattering coefficients for TE mode in every channel do notcontribute to the scattering and absorption cross sections. Onthe other hand, in eq 2, the dominant term in the denominatorof Sn

TM is now VnTM, with the leading term (kea)

−2. In thisscenario, the corresponding cross sections in the electricallysmall limit become

π

γγ

∼ ≈| |

×ϵ + ϵ ϵ − ϵ + ϵ − ϵ ϵ + ϵϵ − ϵ ϵ − ϵ − ϵ + ϵ ϵ + ϵ

⎡⎣⎢

⎤⎦⎥

Q Qk

k a4( )

Im(2 )( ) ( )(2 )

2 ( )( ) ( 2 )( 2 )

abs exte

2 e3

3s e s c e s s c

3e s c s s e c s

(10)

π γγ

≈| |

ϵ + ϵ ϵ − ϵ + ϵ − ϵ ϵ + ϵϵ − ϵ ϵ − ϵ − ϵ + ϵ ϵ + ϵ

Qk

k a83

( )(2 )( ) ( )(2 )

2 ( )( ) ( 2 )( 2 )scate

2 e6

3s e s c e s s c

3e s c s s e c s

2

(11)

Above results are also known as the dynamic dipole results, forone can obtain similar results by solving Laplace equation foran electrostatic dipole.We want to remark that even in such an electrically small

regime, one can have unusual optical properties by using

composited meta-materials. Especially, with plasmonic materi-als, of which the real part of dielectric function is negative,enhancement in light scattering and absorption can happen dueto the localized surface plasmonics.24−29 Nevertheless, withoutthe introduction of meta-materials, we show that counter-intuitive scattering and absorption properties exist for alldielectric materials in the shape of hollow spheres. If we apply

the series expansion = ∑− =

=∞ zz i

i i11 0 , for |z| < 1 by setting

γ≡ ϵ − ϵϵ + ϵ ϵ + ϵz 2 3 ( )

( 2 )( 2 )e s

2

s e e s, then the thickness effect on light

absorption in the electrically small regime can be foundexplicitly:

π γ

γ

π

γ γ γ

=| |

ϵ − ϵϵ + ϵ

−

−

=| |

ϵ − ϵϵ + ϵ

× − −ϵ − ϵ

ϵ + ϵ ϵ + ϵ+

ϵ − ϵϵ + ϵ ϵ + ϵ

⎪

⎪

⎪

⎪

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎧⎨⎩

⎡⎣⎢

⎤⎦⎥⎫⎬⎭

Qk

k a

kk a

4( ) Im

21

1 2

4( ) Im

2

1 2( )

( 2 )( 2 )O( )

abse

2 e3 s e

s e

3

3 ( )( 2 )( 2 )

e2 e

3 s e

s e

3 3 e s2

s e e s

6

e s2

s e e s

(12)

From eq 12, one can see that the thickness, in terms of γ withthe leading term to the cubic power, plays a minor role in lightabsorption. Based on this, we find that there exists a large rangeof thickness ratios to maintain nearly the same absorption crosssection. As long as the ratio between inner core to wholeparticle radii is smaller than γ = ac/a ≤ 0.4, hollow spheresbehave like the solid spheres, even with an empty core.Analytical results based on eqs 10 and 11 are shown in Figure

3 for the absorption and scattering cross sections, Qabs and Qscat,with the dielectric constants ϵs = 7 + 0.1i. It can be seen that theabsorption cross section remains as a constant as long as thethickness ratio, γ = ac/a, is smaller than 0.6. At the same time,the corresponding scattering cross section is not suppressed toomuch when γ < 0.4. However, when γ > 0.4, the scattering cross

Figure 3. Absorption cross section Qabs and scattering cross section Qscat for nanoparticles in the shape of hollow spheres are shown as a function ofthe thickness ratio, γ = a

ac . Analytical equations shown in eqs 10 and 11 based on the dynamic dipole limit and numerical solutions by Mie’s exact

solutions are depicted in solid and dashed curves, respectively. Here, the dielectric constant in the shell region is set as ϵs = 7 + 0.1i, and the dielectricconstants for environment and core regions are ϵe = ϵc = 1.77 for water, respectively. Two different sets of size parameters are calculated for the shell

region composited by TiO2 nanoparticles: (a, b) = λa30

and (c, d) = λa18.

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section decreases to zero very quickly. Compared to numericalresults conducted from exact solutions of Mie theory, shown inthe dashed curves in Figure 3, our theoretical formulas based onthe electrically small limit give the same tendency on thethickness effect. Even though a discrepancy arises when thethickness ratio γ approaches zero, it is due to the electricallysmall limit used. Nevertheless, we want to emphasize that thisthickness-independent result can be applied to all compositedsystems, as long as off-resonance and electrically smallconditions are satisfied.Beyond Electrically Small Limit. The electrically small

limit becomes invalid when the radius of TiO2 nanoparticle islarger than 50 nm, that is, ksa ≈ 1. In this scenario, we can notapproximate related spherical functions by a series ofpolynomial functions. To go beyond the electrically smalllimit, we look for the characteristic features in the originalscattering spectrum for solid spheres. From Mie theory, it iswell-known that interference between the incident andscattered fields occurs when the phase difference betweenthem is fixed as a constant. As a result, the corresponding phasedifference along the forward direction in a hollow sphere can befound as

ϕ πλ

πλ

πλ

γ

γ

Δ = × ϵ − + ϵ − × ϵ

= ϵ − ϵ − ϵ − ϵ

≡ × −

a a a a

a

a

22{ ( ) }

22

4{( ) ( )}

const (1 )

s c e c e

s e s e

(13)

where const stands for the constant related to the givenmaterial parameters. In eq 13, we reveal the relation betweenphase shift to particle radius a and thickness ratio γ. That is,there exists a trajectory for the constant scattering cross sectionin the phase space defined by the particle radius a and thicknessratio γ, and this trajectory can be described by a(1 − γ) = const,as shown in Figure 4.Based on the exact solutions conducted from Mie theory, in

Figure 4, we plot the values of normalized absorption andscattering cross sections Qabs/πa

2 and Qscat/πa2 as a function of

the particle size a (in unit of λ) and thickness ratio γ, for threedifferent sets of dielectric constants for TiO2 nanoparticles: (a,b) ϵs = 7 + 0.02i, (c, d) ϵs = 8 + 0.1i, and (e, f) ϵs = 9 + 0.3i. Asthe dashed curves in white color shown in Figure 4b,d,e, theformula a(1 − γ) = const, shown in eq 13, can provide the roleof thickness ratio on the scattering cross section for hollowspheres. Moreover, as shown in Figure 4a,c,e, the trajectoriesfor a constant normalized absorption coefficient Qabs/πa

2 canbe represented in straight lines along the vertical direction. Itmeans that when the thickness ratio γ is smaller than 0.4, thenormalized absorption cross section is almost independent ofthe thickness ratio. For the normalized absorption cross section,one can see that its maximum value approaches to 1 when theimaginary part ϵs″ and outer radius a increase. However, thisinsensitive thickness effect survives for all three different sets ofdielectric constants, as well as for different outer radii. It impliesthat we can have almost the same light absorption performancefor a given particle radius, but with an additional degree offreedom in the thickness ratio.

■ EXPERIMENTS ON PHOTOCATALYTIC ACTIVITYTo verify our theoretical results for the absorption ofnanoparticles in shape of hollow spheres, we perform

experimentally the test of photocatalytic activity with thesynthesized TiO2 hollow spheres through the photodegradationof methylene blue. Here, uniform samples of hollow-spherenanoparticles are tested with different radii, that is, but all withthe same thickness in the shell region. By mixing 0.01 g of ourTiO2 nanoparticles with 50 mL of 25 ppm methylene bluesolution, the solution is illuminated on top with a Xe bulb, fromOsram Inc., operated at the output power 180 W and kept thetemperature to 20 °C with a water cooling jacket. Samples of0.5 mL are taken out every 5 min, filtered with a 0.2 μm filter,and diluted with 2 mL of deionized water. The filtrate isanalyzed to determine the dye concentration, by a Hitachi UV-vis 3010 spectrophotometer.Since the solution is dilute, we can safely assume that our

photocatalytic behavior has a pseudo-first-order degradationrate constant, κ-value, which can be calculated by

κ = − Δ⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

CC

tln /0 (14)

where C and C0 are the corresponding final and initialconcentrations of methylene blue, and Δt is the time period ofinteraction.To have a comparison, same weight of hollow-sphere

nanoparticles is performed under all the same conditions, but

Figure 4. Normalized absorption and scattering cross sectionsπ

Q

aabs

2 and

π

Q

ascat

2 shown as a function of the particle size a (in unit of λ) and

thickness ratio γ. Three different sets of dielectric constants for TiO2nanoparticles are used: (a, b) ϵs = 7 + 0.02i, (c, d) ϵs = 8 + 0.1i, and (e,f) ϵs = 9 + 0.3i. The dashed curves in white color shown in thenormalized scattering cross sections (b, d, e) are depicted from theformula a(1 − γ) = const, shown in eq 13.

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with different particle radii. We want to emphasize that ourhollow spheres are composed with the shells in 50 nm, which ismuch larger than that of the average migration length for thephoton-generated carriers (holes and electrons). In thisscenario, the effects from the migration to surface can also beruled out. Experimentally measured data on these hollowspheres is shown in Figure 5, marked by Δ, for the degradation

constant κ. From Figure 5, one can see that the κ-value tomeasure photocatalytic activity is enhanced for hollow sphereswhen the particle radius is smaller than 250 nm, or equivalent γ= 0.8 or a = 0.644 in the unit of λ = 388 nm. Then, this valuedrops gradually from 0.05 to 0.02 as the radius increases to a =0.9 (350 nm) or the thickness parameter γ = 0.9.To link our experimental measurement on photocatalytic

performance for hollow spheres, we take the weight ofphotoncatalyst into consideration. With the condition ofsame weight, we calculate the total absorption power of lightfor nanoparticles, Wtot, by including the total number ofparticles Nparticle, that is,

= ×W NWItot particle

abs

inc (15)

Here, due to its dynamic randomness and the dilute watersolution, scattering from individual particles is assumed to beincoherent. Therefore, we can safely sum over all the totalabsorption power from each individual nanoparticle. Forparticles in shape of a fully filled solid-sphere, the total numberof particles is Nparticle

solid = 3M/4πρa3, for a given mass M, theknown material density ρ, and the particle radius a. But forparticles in the shape of a hollow sphere, with the outer andinner radii defined as a and ac, the corresponding total numberof particles becomes Nparticle

hollow = 3M/4πρ(a3 − ac3). It can be

understood that, for the same weight, we have more particlenumbers in the shape of hollow spheres.Numerical simulation of the total absorption power for

hollow-sphere nanoparticles using the experimental conditions,Wtot, is also shown in Figure 5, marked by ● for differentparticle radii a in unit of λ. Again, a clear transition from high-absorption to low-absorption region can be seen as the particleradius increases. Our theoretical calculations based on eq 15

not only give good agreement to the experimental measure-ments on the photocatalytic activity, but also confirm thethickness effect of TiO2 nanoparticles in the shape of hollowspheres. Moreover, a reference sample made by crushing theoriginal hollow-sphere particles into powders with size of 10nm, is also performed for the comparison. Here, for themeasurements on crushed ones, the samples are prepared froma variety of hollow-sphere nanoparticles with different sizes. Ineach crushed samples, all of them reveal almost the samephotocatalytic activity. In other words, the data is not takenfrom some specific hollow samples. The comparison to thecrushed samples strongly supports our conclusion that thestructural effect plays the dominant role upon photocatalyticactivity. Experimental data from the measured degradationconstant for this crushed sample is shown in Figure 5, too.Theoretical calculation by taking the particle radius as small aspossible (taking a = 10 nm) also gives the same result for asmaller total absorption power.By defining the parameter space through the particle size a

(in unit of λ) and thickness ratio γ, in Figure 6 we show the

total absorption power Wtot for different geometrical variables.In this diagram, several ways to achieve higher values in thetotal absorption power can be found with a suitable set of thesize parameter and thickness ratio. As a comparison, the sameexperimental condition, in terms of (a,γ), from the photo-catalytic activity shown in Figure 5 is also depicted in thisdiagram by using the same marker, Δ. Again, we can see thatthere are two groups for the data points: one for the hollowspheres located in the up-right quadrature, and the other onefor crushed powders located near the origin. For the crushedpowders, the corresponding total absorption power is prettylow Wtot < 0.02. But for the hollow spheres, one can see that allthe data points with the particle size a > 0.9 have a low totalabsorption power Wtot ∼ 0.02, while with the particle size a <0.8, we have Wtot ≥ 0.04. Moreover, there exists more delicateregions to support the total absorption power up toWtot = 0.07.Based on this phase diagram, one can have an optimizedabsorption power for hollow-sphere nanoparticles by adjustingparticle radius and its thickness.

■ CONCLUSIONIn conclusion, we reveal the thickness effect on light absorptionand scattering for nanoparticles in shape of hollow spherestheoretically and experimentally. Theoretically, based on theexact solution of Mie theory, we derive analytical formulas with

Figure 5. Total absorption power Wtot and related degradationconstant κ for hollow-sphere nanoparticles in different particle radius,a (in unit of λ). Experimental hollow spheres have a thickness in theshell region about 50 nm, and λ = 388 nm is used as the incident lightwavelength. Our measured particle radii differs from 94 to 500 nm.The Δ and ● markers correspond to the measured data points in thetest of photocatalytic activity and theoretical calculations based on eq15, respectively. A reference sample (on the left bottom side), made bycrushing the original hollow-sphere particles into powders with thesame weight, is also shown for the comparison, that is, a = 0.02λ. Here,the dielectric constant used is ϵs = 6.25 + 0.4i.

Figure 6. Same experimentally geometrical conditions as those shownin Figure 5, marked in Δ, are represented in the plot for the totalabsorption power,Wtot, as a function of the particle size a (in unit of λ)and thickness ratio γ.

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respect to the thickness ratio. Based on the theoretical formula,we find that, within and beyond the electrically small limits, onecan maintain light absorption of hollow spheres to those ofsolid-spheres as long as the thickness ratio is smaller than 0.4.Moreover, with a different thickness parameter, we also revealthe possibility to have a higher total absorption power evenwith a larger particle size. Experimentally, we synthesize highlyuniform hollow spheres of TiO2 anatase through a self-sacrificing template method and perform related photocatalyticactivity for light absorption. A variety of particle radii from 94to 500 nm, with 50 nm in the shell thickness, are tested andgive a good agreement with our theoretical results. Withexperimental demonstrations and theoretical simulations, thesenanoparticles in the shape of hollow spheres would serve as theplatform for green energy applications from light harvesting,solar cell, and water splitting to CO2 reduction.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: rklee@ee.nthu.edu.tw.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe research was supported by the National Center forTheoretical Science and the Ministry of Science andTechnology in Taiwan.

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