1

Optical effects of charges in colloidal solutions

Railing Chang1*, Hung-Yi Chung2, Chih-Wei Chen3, Hai-Pang Chiang1**, and P. T. Leung4†

1Institute of Optoelectronic Sciences, National Taiwan Ocean University, Keelung, Taiwan, R.O.C.

2No. 129, Hangzhou Rd, Zhongli City, Taoyuan County 320, Taiwan, R.O.C. 3National Synchrotron Radiation Research Center, 101, Xin'an Rd, Hsinchu, Taiwan 300, R.O.C.

4Department of Physics, Portland State University, P. O. Box 751,Oregon 97207-0751, U.S.A.

Abstract

The optical response of charged polymeric and metallic colloids is investigated using

effective medium theories for composite systems of nanoparticles. Based on the Bohren-Hunt

theory for generalized Mie scattering from charged particles, an effective quasi-static dielectric

function previously obtained is applied to the present study to characterize the response from the

various colloidal particles. It is found that such effects are more prominent for polymeric and

nonmetallic colloidal solutions in general. In addition, the effects of clustering among the

colloidal particles are also studied via a fractal model available from the literature. Detailed

numerical studies of the dependence of these effects on the amount of extraneous charge, as well

as on the geometry and volume fraction of the colloidal particles are presented.

Key Words: extraneous charges, composites, polymeric and metallic colloids

Corresponding authors: * rlchang@mail.ntou.edu.tw ; ** hpchiang@mail.ntou.edu.tw;

† hopl@pdx.edu

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Introduction

Among the various colloidal systems, polymer [1] and metal [2] colloids are the two most

intensively studied due to their distinct properties leading to various important applications. These

include, for example, various biomedical and pharmaceutical applications of the former and

spectroscopic enhancement applications of the latter [3]. Of the many physical and chemical

properties, the optical properties are significant since light scattering has remained one of the

powerful probes for the understanding of the structure and behavior of these colloidal solutions.

While there have been many studies in the literature on these properties [4, 5], most of them have

been limited to neutral colloidal particles leaving the effects due to possible presence of

extraneous charges [1, 2] unaccounted for in such studies. Though the first study on light

scattering from colloidal electrolytes including charge effects dates back to the 1950’s, it was

limited to a semi-empirical statistical approach accounting for the decrease in optical

effectiveness of fluctuations due to the presence of charge in the electrolyte [6]. To our

knowledge, the only study based on fundamental optical theories was limited to a single metallic

colloidal particle and the charge-induced optical effects were found to be rather insignificant [7].

The more recent study has indeed considered low (RF) frequency scattering from charged colloids

as a composite of metal particles, but its focus has been on the effects due to the counterions and

the motions of the particles and was limited only to metallic colloids [8].

It is the purpose of our present work to study exclusively these charge-induced optical

effects in both nonmetallic (polymeric) and metallic colloids via the application of

electromagnetic scattering and effective medium theories. We focus on higher frequency

scattering and the metallic surface plasmons can be excited while the particle motion can be

ignored in this case. Our interest is mainly in the optical response of the excess (free) charges on

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the colloidal particles in both the polymeric and metallic cases. While in the literature, there have

been many well-established effective medium theories (e.g., the Maxwell-Garnett; the

Bruggeman theory and their extensions) for the optical properties of composites [9], most of them

have considered only neutral particles, and composites of charged particles have not been

systematically studied previously. Moreover, synthesis of various colloids often lead to charged

ingredient particles [1, 2]. Hence it will be of interest to study these “charge-induced” optical

effects in a composite of nanoparticles and the results will provide an extra dimension for the

control of the optical response of these materials besides manipulations over the shape, size,

material, concentration, … of the nanoparticles.

It is of interest to note that the charge state of single emitting quantum dots has been

recognized to be of high significance in some recent studies on the fluorescence intermittency [10,

11] of these emitters; as well as in higher order fluorescence from single atomic emitters doped

within a crystalline host [12-15]. However, our present focus here is mainly on the optical

response from the plasmonic motion of these extraneous surface charges on various polymeric

and metallic colloidal particles, rather on the atomic emitters. Nevertheless, these plamonic

effects could indeed modify the fluorescence properties of nearby atomic emitters as

demonstrated in one of our previous works [16].

In the literature, Bohren and Hunt (BH) [17] were among the first to have studied the

optical response of charged particles with their generalization of the Lorenz-Mie scattering theory

to apply to a charged sphere. By characterizing the extraneous surface charges with an effective

surface conductivity, BH were able to derive modified Mie coefficients via an implementation of

a modified boundary condition for the discontinuity of the tangential components of the magnetic

field across the surface of the particle. The problem of light scattering from charged spherical

particles has since been studied by many people, including some very recent works in the

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literature (see, e.g., [18-22]). In particular, effective modified dielectric functions have been

derived from the BH theory in the long wavelength limit [19, 20] which highly simplified the

incorporation of such charge effects into the optical response from small particles. The BH

theory in the lowest (dipole) order has been previously applied to a single metallic colloidal

particle [7]. Our previous formulation [19] allows one to extend to include higher order

multipoles in conjunction with the extended effective medium theories available in the literature

[23]. In the following, we shall apply these effective dielectric functions [19] to study such

“charge-induced” optical response from different colloidal systems, and we shall start with a brief

review of the previous theories.

Theoretical models

To account for the surface charge effects, we recall the generalized Mie coefficients

obtained in the BH theory [17] for the electromagnetic scattering from a charged sphere (in

vacuum):

aℓ = −

′ψ ℓ(x)ψ ℓ(nx)−ψ ℓ(x) n ′ψ ℓ(nx)− iτψ ℓ(nx)⎡⎣ ⎤⎦′ξℓ(x)ψ ℓ(nx)−ξℓ(x) n ′ψ ℓ(nx)− iτψ ℓ(nx)⎡⎣ ⎤⎦

, (1)

bℓ = −

ψ ℓ(x) ′ψ ℓ(nx)− ′ψ ℓ(x) nψ ℓ(nx)+ iτ ′ψ ℓ(nx)⎡⎣ ⎤⎦ξℓ(x) ′ψ ℓ(nx)− ′ξℓ(x) nψ ℓ(nx)+ iτ ′ψ ℓ(nx)⎡⎣ ⎤⎦

, (2)

where ψ ℓ ,ξℓ are the Riccati-Bessel functions, x ka= is the size parameter of the sphere, n = ε

is the refraction index of the (nonmagnetic) sphere (assumed to be placed in vacuum), and the

charge parameter τ is proportional to the surface conductivity [19].

In the limit when we have both x≪1 and | n | x≪1, the small argument limits of the

various Bessel functions lead to the following results for the Mie coefficients:

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aℓ → O(x2ℓ+3) , (3)

bℓ →

i(ℓ+1) / ℓ(2ℓ−1)!!(2ℓ+1)!!

×ε + i(ℓ+1)τ / x⎡⎣ ⎤⎦ −1

ε + i(ℓ+1)τ / x⎡⎣ ⎤⎦ + (ℓ+1) / ℓx2ℓ+1 . (4)

When these results are compared to the response of a neutral sphere in the quasistatic

approximation where it is completely governed by the multipolar polarizability of the sphere:

α ℓ =

ε −1ε + (ℓ+1) / ℓ

a2ℓ+1 , (5)

we conclude that the surface charge effects can be accounted for in this limit by simply

introducing the following effective dielectric function [19]:

βℓ = ε + i(ℓ+1)τ / x . (6)

Hence to apply the following well-known Maxwell-Garnett (MG) theory to the charged colloid as

a composite [8, 9]:

2 2

j iMG ij

MG i j i

fε εε ε

ε ε ε ε−− =

+ +, (7)

where phase i represents the host and phase j the particle, and 1i jf f+ = with f the volume

fraction of each of the two phases, one simply replaces jε by β ≡ β1 = ε j + 2iτ / x , since the MG

theory has considered only the dipole response of the particles. With this replacement, Eq. (7)

will then yield the following effective dielectric function for the charged colloid:

εeff =

(1+ 2 f )β + 2(1− f )εh

(1− f )β + (2+ f )εh

εh (8)

where f ≡ f j and h stands for the host medium of the colloidal solution. In the literature, this

MG model has been found to be adequate for the homogenization of dilute colloids with particles

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of small sizes [see, e.g., 24-26].

For the case when the particles coalesce to form clusters, we shall adopt a fractal cluster

(FC) model available from the literature [27]. Assuming the particles of radius a to form local

fractal clusters of radius R≫ a in the host, this model leads to the following cubic equation for

the effective dielectric function ( )Rε of one of the clusters:

( )3

3( )( ) '( ) ( )

i

i

aR fa R

ε εεε ε ε

−⎡ ⎤− =⎢ ⎥−⎣ ⎦ , (9)

where ε(a)→β1 and 3

'fdRf

a

−⎡ ⎤= ⎢ ⎥⎣ ⎦

is the volume fraction of the particles in the cluster of fractal

dimension fd . The dielectric function FCε of the composite which contains these clusters is

finally obtained by a further application of the MG theory to the host-cluster system with a

concentration of the clusters Cf given by / 'jf f . Hence we have FCε to be obtained from the

following equation:

ε FC − ε i

ε FC + 2ε i

= fC

ε(R)− ε i

ε(R)+ 2ε i

. (10)

Note that within the above formalism, it is possible to extend the MG theory to include

higher order multipoles which can be significant for colloidal particles of large size. One

approach will be to follow the recent theory of Malasi et al [23] to replace the polarizability of the

colloidal particle used in the MG theory in (7) by the Mie coefficients as modified in the BH

theory via Eqs. (1) and (2). An alternative approach will be just to apply Ref. [23] with the

particle dielectric function replaced by βℓ in (6). The second approach turns out to be more

accurate for metallic colloid than for nonmetallic ones as discussed in our previous work [28].

We would like to remark that in the only such study in the literature [7], the authors have

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essentially applied β1 to a single metallic colloid particle and concluded the charge –induced

optical effects are negligible in light-colloid interaction in general. In the following numerical

study, we shall extend such study to both polymer and metal colloids, modeling them as

composites using the effective medium theories as described above. We shall also simplify our

simulation to include only dipolar response of the particles for the small particle sizes we are

going to consider for which the charge effects are prominent, and shall see that while the

composite modeling also confirms the insignificance of such effects for metal colloids, these

effects will be manifested in the case of polymer colloids and should be observable in

specifically-designed experiments.

Numerical Results

We first make a comparison between the charge-induced optical effects from a metallic

and a polymeric colloid, respectively, by considering silver and polystyrene colloidal particles in

a solution which we just take to be water. We shall assume a constant refractive index of 2.4 for

polystyrene while a Drude model applies to silver [19]. Both colloids have been reported to be

likely highly charged during their fabrication processes [29, 30]. We shall illustrate the charge

effect on the imaginary part of the effective dielectric function as a function of frequency for

different volume fractions of the colloidal particles. As for the particle size, we shall consider 20

nm radius polystyrene particles but much smaller metallic particles (1 nm radius) since charge

effects are expected to be rather insignificant for the latter except for ultra-fine particles. We first

assume random dispersion of the particles in the colloid so that the MG theory in Eq. (8) can be

applied.

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Figure 1(a) shows the charge effects on the polymeric colloid for surface potentials φ =

0.1 V, 1 V, 10 V (note: 1 V corresponds to a uniform surface charge of 2.2 x 10-18 C) and for two

volume fraction f = 0.02 and 0.2, respectively. It is observed that for such range of charge

values, the blue-shifts of the free-charge (“plasmon”) resonance can extend over two order of

magnitudes with noticeable more intense “peaks” (note that such “peak” does not exist for

polymeric colloid in the absence of extraneous free charges). Furthermore, it is also noted the

increase in particle concentration will lead to slightly lower resonance frequencies due to inter-

particle electrostatic interactions.

Figure 1(b) shows the corresponding results for a metallic colloid with Ag particles of 1

nm radius. We first note that the effect from increasing volume fraction is much more significant

here compared to that in the polymeric case, with the peak for the f = 0.2 case 10 times that for

the f = 0.02 case. However, the charge effects are much insignificant in this case: amounting to

~ 1% blue-shifts within the surface potential range and completely unnoticeable change in peak

values (see the separately-magnified results in Fig. 2(b)). Thus our conclusion here for the

charge-induced optical effects for metallic colloids is consistent with the previous investigation

(based on the study of a single colloidal particle), which concluded such effects are completely

insignificant.

To illustrate some observable effects from this charge-induced optical response, we show

in Fig. 3 the optical absorption of a polystyrene colloid layer (of 1 micron thickness) with a

particle volume fraction of 0.2 and different charge amount (i.e. different surface potential). The

figure shows the spectral sum of reflection and transmission R +T = 1− A for different surface

potentials from which a resonant absorption of more than 10% at a frequency ~ 3.5 x 1013 Hz can

take place at a surface potential of 10 V. The blue-shifts in resonance with increasing surface

charge is also noted and one expects this can enter the optical regime for even greater amount of

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surface charges. Note that the absorption resonance is not exactly Lorentzian but with a longer

tail at the high frequency end as shown clearly for the 0.1 V case. When we compare with the

results in Fig. 1 (a), it is noted that while greater imaginary part of the dielectric function does

correlate to greater absorption, the resonance positions are not exactly aligned with each other at

the same frequencies.

To proceed further, we shall investigate the effect of coalescence among the particles on

the optical response of these colloids. In order to simplify the solution of the cubic equation in (9),

we have found that it is desirable to use a smaller damping frequency [19] in the charge parameter

τ which appeared in Eq. (6). Hence the damping used in τ in the following FC model

calculations is only 10% that used in Figs. 1-3. With this set, we show first in Fig. 4 the

imaginary part of the dielectric function for a fractal cluster of silver particles (assuming a

background of vacuum) at a fixed fractal dimension of 2.5 and a surface potential of 1.0 V. Note

that we have ignored the redistribution of the surface charges on each particle upon clustering

since in a real colloid, the particles are likely isolated from each other by the background solvent

even upon coalescence. The results show that clustering will split and shift the resonance with

one towards the low frequency end and one at the volume plasmon frequency of the metal as

observed previously in the literature [27]. The low frequency shift in resonance becomes more

prominent as the cluster radius increases.

To illustrate any observable optical effects in the FC model, we show in Fig. 5 the

imaginary part of the dielectric function along with the absorption ( A = 1− R −T ) of the

polystyrene colloid layer in Fig. 3 allowing the particles to coalesce. Two very distinct

characteristics from coalescing of (charged) colloidal particles are noted: (1) a decrease in

absorption of the colloid with red-shifted resonances (Fig. 5(b)); and (2) an “anti-correlation”

between the absorption peak and the magnitude of the “plasmon peak” (Fig. 5(a)) rather than a

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direct correlation as seen in Fig. 3. Note that as the fractal dimension decreases which amounts to

stronger coalescence, absorption also decreases. This thus provides a signature for particle-

clustering within the colloid in contrast to having them dispersed randomly through the host

solvent, and likely has its origin from enhanced light-scattering from clustered particles in the

colloid.

Conclusion

We have thus carried out a comprehensive study on the optical effects due to the surface

charges on the colloidal particles in both polymeric and metallic colloids using various effective

medium theories, and have come to the conclusion that such effects are only significant for

nonmetallic (polymeric) colloids. For such colloids, these charges will lead to new plasmon and

absorption peaks which are enhanced and blue-shifted with the increase in the amount of charges.

However, when coalescence of particles takes place within the colloid, such peaks will be red-

shifted and absorption of the colloid will be weaken while the plasmon peaks will be enhanced.

These novel phenomena thus provide a means to optically characterize the charge state of these

polymeric colloids as well as the dispersion pattern of the particles in these colloids.

Acknowledgments

The authors are grateful to the Ministry of Science and Technology of Taiwan, Republic of China

for supporting this research with contract number MOST 103-2112-M-019-003-MY3.

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Figure captions

Fig. 1 (a) Imaginary parts of the dielectric function of polymeric colloid evaluated from Maxwell-

Garnett model under different conditions: f=0.02, φ =10.0V (solid line), f=0.02, φ =1.0V (dash

line), f=0.02, φ =0.1V (dot line); f=0.2, φ =10.0V (dash-dot line) f=0.2, φ =1.0V (dash-dot-dot

line) f=0.2, φ =0.1V (short dash line). (b) Imaginary parts of the dielectric function of metallic

colloid evaluated from Maxwell-Garnett model for the same conditions as in (a) and indicated by

the same lines, respectively. In addition the case of zero potential for (b) with f=0.02 (0.2) is

shown in thin solid (thin dash dot) line.

Fig. 2 (a) Magnified curves of the portion in Fig. 1 (b) for f=0.02, and (b) for f=0.2.

Fig. 3 (a) Sum of the reflection and transmission of normally incident light beam onto a polymeric

colloid thin film of thickness 1.0 mµ , under conditions of f=0.2, φ =10.0V (dash dot line); f=0.2,

φ =1.0V (dash dot dot line); and f=0.2, φ =0.1V (short dash line), respectively. (b) Enlargement

of portion of the figure in (a).

Fig. 4 The imaginary part of the dielectric function of a metallic colloid evaluated from fractal

cluster model for different radius of cluster. The fractal dimension is fixed at d f = 2.5 , the overall

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volume fraction of metal is 0.1, the radius of metallic sphere is 1.0 nm, and the background host is

assumed to be vacuum.

Fig. 5 (a) Imaginary part of the dielectric function for polymeric colloid modeled by fractal

cluster with different fractal dimension d f . The radius of polymeric sphere is set at a = 20nm ,

that of the cluster is at R = 10a , the dielectric constant of polymer is taken to be 2.4 and that of

the solvent, assumed to be water, is 1.78. The electrostatic potential due to the added surface

charge on polymeric sphere is at 1.0V. (b) The sum of reflection and transmission of normally

incident light beam onto a thin film of polymeric colloid model by fractal cluster. The conditions

of calculation are the same as (a) with thickness of thin film fixed at 1.0µm.

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Fig. 1

1010 1011 1012 1013 1014

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Im(ε

Μ)

ω

X10

(a)

5.6x1015 5.8x1015 6.0x1015 6.2x1015 6.4x10150

50

100

150

200

250

300

Im(ε

Μ)

ω

X5

(b)

17

Fig. 2

6.24 6.28 6.32 6.36 6.40

5

10

15

20

25

30

Φ=1.0V

Φ=0.1V

Φ=0.0V

Im(εM)

ω/1015Hz

Φ=10.0V

f=0.02

(a)

5.60 5.64 5.68

50

100

150

200

250

300

Φ=1.0V

Φ=0.1V

Φ=0.0V

Φ=10.0V

Im(εM)

ω/1015Hz

f=0.2

(b)

18

1010 1011 1012 1013 1014 1015

0.90

0.92

0.94

0.96

0.98

1.00

R+T

ω

(a)

1010 1011 1012 1013 1014 1015

0.988

0.990

0.992

0.994

0.996

0.998

1.000

R+T

ω

(b)

Fig. 3

19

Fig. 4

0.0 0.5 1.0 1.5 2.0

1E-4

1E-3

0.01

0.1

1

10

Im(ε

)

ω/ωp

radius_c/radius=(2i-1);i=1-15

20

Fig. 5

1010 1011 1012 1013 1014 1015-2

0

2

4

6

8

10

12

14

df=2.4-2.9Im(ε)

ω

(a)

1010 1011 1012 1013 1014 1015

0.90

0.92

0.94

0.96

0.98

1.00

df=2.4-2.9

R+T

ω

(b)