An Improved Synthesis of 2'-O-(13C) Methylguanosine

Article abstract of DOI:10.1135/cccc19960059

Using methyl iodide as a source of 13C label, an efficient and regioselective synthesis of 2'-O-(13C)methylguanosine has been elaborated.The synthesis of the fully protected 2'-O-(13C)methylguanosine-3'-O-phosphoramidite (9), the building unit for the oli

Full text of DOI:10.1135/cccc19960059

Linear Aggregate of Ag Colloidal Particles
59
THE MODEL OF LINEAR AGGREGATE OF Ag COLLOIDAL PARTICLES
WITH VARIABLE INTER-PARTICLE DISTANCES
Ondrej SESTAK, Pavel MATEJKA* and Blanka VLCKOVA**
Department of Physical and Macromolecular Chemistry,
Charles University, 128 40 Prague 2, Czech Republic
Received March 1, 1995
Accepted July 15, 1995
A simplified method of calculation of the surface plasmon energy states of the Ag colloidal
aggregates characterized by varying inter-particle (inter-sphere) distance has been developed. Ag
colloidal aggregate is approximated by a linear (one-dimensional) assembly of N silver spheres (of
identical radii r and identical inter-sphere distances D) mutually interacting by a dipole–dipole
interaction. The calculations use the following parameters: N from 1 to 25, r = 2, 5 and 10 nm, D = 0,
0.5, 1 and 2 nm, water and/or vacuum embedding media. The perturbation energies Vmin (stabilization
energy) and Vmax (destabilization energy) of the excited plasmon state of a linear aggregate of N
spheres interacting by the dipole–dipole interaction were calculated as the eigenvalues of perturbation
matrix using the above-mentioned parameters. The stabilization energy Vmin increases with increasing
number of spheres in the aggregate and with increasing sphere radius, while it decreases with
increasing inter-particle (inter-sphere) distance. Calculations of the square values of the eigenvector
coefficients show that the contribution of a particular single sphere to the total energy of the
aggregate is the highest for the central sphere in the odd-N aggregates and for the two central spheres
in the even-N aggregates. The results of the model calculations support the hypothesis that the
differences between the surface plasmon absorption curves of the Ag colloid/monomeric adsorbate
and of the Ag colloid/polymeric (oligomeric) adsorbate systems have their origin in the difference in
the inter-particle distance distributions.
Key words: Ag-colloid; Linear aggregates; Aggregation theory, SERS; Surface plasmon states.
Ag colloids are suitable model systems for investigation of the electromagnetic mecha-
nism of SERS (Surface Enhanced Raman Scattering) chiefly because of the possibility
to follow the surface plasmon absorption (SPA) of the Ag colloid/adsorbate system by
UV/VIS absorption spectroscopy. The changes in the SPA of Ag colloid upon aggrega-
tion were explained theoretically by the Fornasiero–Grieser’s model of linear aggregate
of Ag colloidal particles1. For a linear aggregate of Ag particles interacting by a di-
* Present address: Department of Analytical Chemistry, Prague Institute of Chemical Technology, 166 28
Prague 6, Czech Republic.
**The author to whom correspondence should be addressed.
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

60
Sestak, Matejka, Vlckova:
pole–dipole interaction, two excited surface plasmon states originate and two absorp-
tion bands, one at lower energy than the monomer (single particle) absorption band, the
other at a higher energy, are observed. Differences in the SPA spectra of SERS active
systems with different adsorbates are ascribed to a varying degree of the Ag colloid
aggregation2. In our previous studies3–6, we found systematical differences in the SPA
spectra of SERS-active systems containing polymeric (i.e. proteins4) and/or oligomeric
(i.e. polypeptide3, Triton X-100 (refs5,6)) species in comparison to the spectra of sys-
tems formed by monomeric species. In this paper, we first introduce the hypothesis
which provides a possible explanation of the differences between the SPA curves of the
Ag colloid/monomeric adsorbate systems and those of Ag colloid/polymeric (oli-
gomeric) systems. Furthermore, in order to verify our hypothesis, we present here the
results of the calculations of the energy states of a linear colloidal aggregate.
HYPOTHESIS
The second maximum of plasmon absorption, which is observed in systems containing
species of monomeric nature in the 540–620 nm range, is ascribed to SPA of the aggre-
gates of spherical colloidal particles. In contrast to that, for SERS-active systems con-
taining species of polymeric character, only an asymmetric broadening of the original
monomer (single particle) peak towards higher wavelengths is typical (without appear-
ance of a pronounced second maximum). Up to now, no explanation concerning the
origin of a different character of the plasmon absorption curves for systems formed by
polymeric (oligomeric) and monomeric species has been published. On the basis of our
previous results3–6, we introduce here a hypothesis, that differences between plasmon
absorption curves are given by different distances of Ag colloidal particles in the aggre-
gates. We assume that for the monomeric adsorbates, the colloidal particles in aggre-
gates are almost in touch with each other (D = 0 nm), while for aggregates formed by
polymeric or oligomeric species, non-zero distances between individual spheres (Ag
colloidal particles) have to be considered. To verify this hypothesis, we have developed
a new model of colloidal aggregate which combines Aravind’s and Fornasiero’s theore-
tical approaches1,7. The model describes a linear aggregate of spheres (Ag colloidal
particles) mutually interacting by a dipole–dipole interaction in which the inter-sphere
distances are generally non-zero and are involved as an adjustable parameter in the
calculations of the energy states of the aggregate. In our calculation, the dipole–dipole
interactions of the spheres in the linear aggregate are considered as a perturbation af-
fecting the energy of the excited plasmon state of an assembly of N non-interacting
^
spheres and is expressed in terms of the perturbation operator V. The perturbation energies
Vmin (stabilization energy) and Vmax (destabilization energy) of the excited plasmon
state of a linear aggregate of N spheres interacting by the dipole–dipole interactions
will be calculated as eigenvalues of the perturbation matrix (the matrix representation
^
of the perturbation operator V). The Vmin values will thus represent the stabilization
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

Linear Aggregate of Ag Colloidal Particles
61
energy of the excited plasmon state of the longitudinal plasmon mode of the linear
aggregate of N interacting spheres with respect to the degenerate excited plasmon state
of an assembly of N non-interacting spheres.
THEORETICAL
The model of linear colloidal aggregate is based on the following assumptions:
1. only dipolar interactions between colloidal particles in the aggregate are con-
sidered;
2. the radii of all colloidal particles are identical;
3. the distances between adjacent spheres (colloidal particles) are identical;
4. the aggregate axis is oriented along the x-axis.
→
The vector of distances Rij between the centers of individual spheres (i and j) can
thus be described by Eq. (1):
→
→
Rij = (2r + D) (j − i) k ,
(1)
→
where r is the sphere radius, D is the distance between adjacent spheres and k is the
→
unitary vector (|k| = 1) oriented in direction of the aggregate axis.
5. The longitudinal plasmon mode1 is oriented along x axis and the transversal modes
are oriented along the y and z axis, respectively.
The individual dipolar interaction energies Vij (between each two individual spheres
i and j, i ≠ j) are described by Eq. (2):
→
→
→
→
→
3(Rij µi) (Rij µj)
1
4πεe
1
→→
µi µj −
Vi,j =
,
(2)
→
(|Rij|)3
|Rij |2
→
→
where Rij is defined as above and µ is the induced dipole moment for a single
sphere1,8,14, where εe, the dielectric function of environment, is the product of the per-
mittivity of vacuum ε0 and relative permittivity of environment εr. The induced dipole
→
moment for a single sphere µ can be described by Eq. (3):
→
ε(ω) − 1
ε(ω) + 2
→
µ = 4πεe
r3 E(ω) ,
(3)
→
where E is a vector of the intensity of the incident electric field and ε(ω) is the dielec-
tric function of silver8.
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

62
Sestak, Matejka, Vlckova:
With these equations we have developed a new formula for dipolar interaction ener-
gy of a single sphere (Eq. (4)): (The details of formulation of Eq. (4) are given in
Appendix).
8π
c
(n2 − k2) (n2 − k2 − 2) + 1
r6
Vi,j =
εr
(n2 − k2) (n2 − k2 + 4) + 4 |(2r + D) (j − i)|3
2π
λ
cos
(2r + D) (i + j − 2) (sin (θ) cos (ϕ) − 1) [1 − 3 sin2 (θ) cos2 (ϕ)] I ,
(4)
where c is the velocity of light in vacuum, n is the refractive index, k is the absorption
coefficient (n and k are referred to as “bulk constants” (ref.8)) for Ag and depend on the
wavelength of incident radiation), λ is the wavelength of incident radiation, θ, ϕ are the
angles of incidence (in spherical coordinates), I is the intensity of the incident radiation.
The individual values Vi,j form a perturbation matrix V of the N-th order for the
particular N-membered colloidal aggregate. This perturbation matrix is the matrix-repre-
^
sentation of perturbation operator V of the overall energy of aggregate. The eigen-
values of this perturbation matrix have to be calculated9 to determine the energy states
of aggregate. The least square method10 was used for the calculation of the largest
eigenvalue of the perturbation matrix. Furthermore, to obtain the contributions of the
individual spheres in the aggregate to its total energy, we calculate the normalized
eigenvectors of the perturbation operator belonging to the particular eigenvalues of
energy. Squares of eigenvector coefficients determine the contribution of each individual
sphere to the total energy of aggregate9. For angles of incidence θ, ϕ = 90°, 0° the
solution describes the energy level of the longitudinal plasmon mode (Vmin), while for
angles of incidence θ, ϕ = 0°, 90° and 0°, 0° the solution gives the energy levels of the
transversal plasmon modes Vmax (the linear colloidal aggregate is oriented along x axis).
A necessary step in our calculation was evaluation of the square values of induced
→
dipole moment |µ|2 in dependence on the wavelength of incident radiation Eq. (5):
32π2
c
(n2 − k2) (n2 − k2 − 2) + 1
(n2 − k2) (n2 − k2 + 4) + 4
→
|µ|2 =
εeεr
r6I .
(5)
RESULTS OF CALCULATIONS
The square values of the induced dipole moment of a silver sphere as a function of the
wavelengths of the incident radiation are shown in Figs 1 and 2. The maximum of the
square values of the induced dipole moment for a 5 nm sphere embedded in water is
located at 375 nm (Fig. 1), while for the sphere of the same radius embedded in va-
cuum, the corresponding maximum is at 355 nm (Fig. 2). The shift of the maximum of
the square values of the induced dipole moment for a sphere embedded in water in
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

Linear Aggregate of Ag Colloidal Particles
63
comparison to that embedded in vacuum has its origin in the different values of permit-
tivity ε0 and permeability µ0 for water and vacuum, and is in a good agreement with the
results published11. The same maxima of the square values of the induced dipole mo-
ments were obtained also for the 2 nm and 10 nm spheres. The deviation of the calcu-
lated value (375 nm) from the real value11 (390 nm) is most probably caused by the
fact, that particles of the real Ag colloid are embedded in an “electrolyte” with a higher
ionic strength than that of pure water. The presence of BH−4 and BO3− ions in SERS-ac-
tive system results from the procedure of preparation of Ag colloid12. The existence and
stability of isolated Ag particles is conditioned by the existence of an electric bilayer on
the surface of each colloidal particle13. The calculated values of wavelengths (375 nm
for water, 355 nm for vacuum) at which the squares of induced dipole moment reach
their maxima were used for selection of input parameters (“bulk constants” for Ag
(ref.8)) to the following model calculations.
8
2
51
|µ| . 10
2
2
C
m
4
FIG. 1
Square values of the induced dipole mo-
ment µ as a function of the excitation
wavelength λ for 5 nm sphere em-
bedded in water
0
340
375
410
λ, nm
0.12
51
2
|µ| . 10
2
2
C
m
0.08
0.04
0
FIG. 2
Square values of the induced dipole mo-
ment µ as a function of the excitation
wavelength λ for 5 nm sphere em-
bedded in vacuum
340
375
410
λ, nm
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

64
Sestak, Matejka, Vlckova:
The square values of eigenvector coefficients were calculated to determine the con-
tribution of the particular single sphere to the energy states of the colloidal aggregate9.
The dependence of the squares of eigenvector coefficients on the position number of
the sphere in the aggregate (N) is shown in Figs 3 and 4. The Figs 3 and 4 show the
0.4
2
V
0.2
0
0
5
10
15
20
N
FIG. 3
The dependence of the square values of the eigenvector coefficients (V2) on the position numbers of
the spheres (N) in the even membered aggregate of twenty spheres (5 nm sphere radius, 2 nm inter-
sphere distance, aggregates embedded in water and excited by 375 nm radiation). The same depen-
dence is depicted also for other even membered aggregates: ❐ N = 20, × N = 10, ✶ N = 6, ❍ N = 4.
For the sake of comparison, the centres of 4, 6, 10 membered aggregates are located into the central
positions of the spheres of the 20 membered aggregate (position numbers N = 10, 11)
0.6
2
V
0.4
0.2
0
0
5
10
15
20
N
FIG. 4
The dependence of the square values of the eigenvector coefficients (V2) on the position numbers of
the spheres (N) in the odd membered aggregate of nineteen spheres (5 nm sphere radius, 2 nm inter-
sphere distance, aggregates embedded in water and excited by 375 nm radiation). The same depen-
dence is depicted also for other even membered aggregates: ❐ N = 19, × N = 9, ✶ N = 5, ❍ N = 3.
For the sake of comparison, the centres of 3, 5, 9 membered aggregates are located into the central
positions of the spheres of the 19 membered aggregate (position numbers N = 10)
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

Linear Aggregate of Ag Colloidal Particles
65
dependences for the even/odd membered aggregates. Our calculations show that the
highest contribution from a single sphere to the total energy states of the aggregate
arises: (i) in the case of even sphere-numbers from the two spheres which are located
in the centre of the aggregate, and (ii) in the case of odd sphere-numbers from the
sphere located in the center of the aggregate. The values of squares of the eigenvector
coefficients are independent of the wavelength of incident radiation, the radii of
6
17
V
J
. 10
max
2
–2
–6
17
V
J
. 10
min
–10
0
5
10
15
20
N
FIG. 5
Dependence of the energies Vmin and Vmax on the position number of the spheres (N) in aggregate at
varying inter-sphere distances D (in nm) of the 5 nm spheres embedded in water and excited by 375 nm
radiation. ❐ D = 0, Vmin; + D = 0.5, Vmin; ❍ D = 1, Vmin; ■D = 0, Vmax; ✶ D = 0.5, Vmax; ● D = 1, Vmax
3
18
V
J
. 10
max
1
–1
18
. 10
V
min
J
–3
–5
0
5
10
15
20
N
FIG. 6
Dependence of the energies Vmin and Vmax on the position number of the spheres (N) in aggregate at
varying inter-sphere distances D (in nm) of the 5 nm spheres embedded in vacuum and excited by 355 nm
radiation. ❐ D = 0, Vmin; + D = 0.5, Vmin; ❍ D = 1, Vmin; ■ D = 0, Vmax; ✶ D = 0.5, Vmax; ● D = 1, Vmax
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

66
Sestak, Matejka, Vlckova:
spheres, and the distances between adjacent spheres. Their values depend only on either
the odd or the even numbers of spheres in the aggregate. The nondependence of squares
of eigenvector coefficients on the varying distances between adjacent spheres is an
important result which indicates formal correctness of the approach adopted in our
model of colloidal aggregate.
Finally, we have calculated the eigenvalues of energy for the longitudinal plasmon
mode (Vmin) and for the transversal plasmon modes (Vmax). The values Vmin represent
the stabilization energies of the aggregate. The dependence of the energies (Vmin, Vmax
on the following factors has been calculated:
)
1. the number of spheres in the aggregate (N)
2. the radius of a single sphere (r)
3. the distance between adjacent spheres in the aggregate (D).
Figures 5 and 6 show the dependences of Vmin, Vmax on the number of spheres in the
aggregate for varying inter-sphere distances D at constant sphere radii r (r = 5 nm)
using n, k values of Ag (ref.8) (at 375 nm wavelength of incident radiation for the
aggregates embedded in water, and at 355 nm for those embedded in vacuum). Further-
more, Figs 5 and 6 show that an increase in the distance between the adjacent spheres
in the aggregate leads to a decrease in the absolute values of energies Vmin, Vmax. This
result can readily be explained by the decrease in the dipole–dipole interaction between
the spheres in the aggregate with the increasing inter-sphere distance. Analogous calcu-
lations were made for spheres of 2 and 10 nm radii.
CONCLUSIONS
For a linear aggregate consisting of N identical Ag colloidal particles (spheres) mu-
tually interacting by a dipole–dipole interaction, the stabilization energy Vmin of the
longitudinal plasmon mode and the “destabilization” energy of the two degenerate
transversal plasmon modes Vmax were calculated as functions of (i) number of spheres
in the aggregate; (ii) radius of the spheres; (iii) inter-sphere distance.
The absolute values of Vmin and Vmax (i) increase with increasing number of spheres
in the aggregate N for N ≤ 15, but become virtually independent of N for N ≥ 15; (ii)
increase with increasing radius of the spheres; (iii) decrease with increasing inter-
sphere distance.
From the results of this model calculation, important conclusions concerning the real
Ag colloid/adsorbate systems can be drawn:
1. As Ag colloidal aggregates in real systems consist of more than a hundred Ag
particles15, the number of the spheres in the aggregate cannot significantly influence the
shapes of SPA curves of real Ag colloid/adsorbate systems.
2. The size of the Ag colloidal particles significantly influences the energies of the
surface plasmon states of the aggregate. The shapes of the SPA curves of real Ag
colloid/adsorbate systems thus reflect particle size distributions in these systems.
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

Linear Aggregate of Ag Colloidal Particles
67
3. The inter-particle distances significantly influence the energies of the surface plas-
mon state of the aggregate. For systems with non-zero inter-particle distances, the dis-
tribution of the inter-particle distances can substantially influence the shape of the SPA
curve of the SERS-active system.
The above conclusions provide a key to explanation of the systematic differences
between SPA curves of the SERS-active systems formed by the polymeric (oligomeric)
and/or the monomeric adsorbates. For Ag colloidal aggregates formed by monomeric
species, the individual spheres are nearly in touch and the inter-sphere distances can be
considered to be close to D = 0 nm. On the other hand, in the case of polymeric (oli-
gomeric) species, one polymer chain can be adsorbed simultaneously on two or more
Ag particles thus preventing them from touching. Therefore, in the Ag colloidal aggre-
gates formed by polymeric species, the distances between the adjacent spheres are non-
zero (D > 0 nm). Their distribution can be described by a distribution function.
Consequently, the shapes of SPA curves of the Ag colloid/monomer systems are gov-
erned by only one distribution function describing the distribution of particle sizes, and
a well defined maximum in the red spectral region is usually observed. By contrast, the
shapes of the SPA curves of Ag colloid/polymer (oligomer) systems are governed by
two distribution functions (i) distribution of particle sizes and (ii) distribution of inter-
particle distances. This is the reason why the SPA curves of these systems do not show
a single, well pronounced maximum, but a broad absorption feature extending from the
absorption band of the residual isolated particles towards the red spectral region.
APPENDIX
The dielectric function of the metal ε(ω) consists of a real ε1 and imaginary iε2 part
which are related to the refractive index n and absorption coefficient k of the metal by
the following equations8:
ε(ω) = ε1 + iε2
ε1 = n2 − k2
ε2 = 2nk .
By inserting ε1, the real part of ε(ω) into Eq. (3) (ref.11), we obtain Eq. (A-1):
(n2 − k2) − 1
(n2 − k2 + 2)
→
→
µi,j = 4πεe
r3 Ei,j(ω) ,
(A-1)
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

68
Sestak, Matejka, Vlckova:
where εe is the dielectric function of environment. In our calculation, the aggregate is
fixed along the x axis. This assumption means, that the components Ryij and Rizj of the
→
vector Rij vanish. Therefore a simplification of Eq. (2) gives Eq. (A-2):
1
4πεe
1
Vi,j =
(µyi µyj + µzi µjz − 2µix µjx) .
(A-2)
|(2r + d) (j − i)|3
Subsequent expansion of this equation with the use of Eq. (A-1) results in Eq. (A-3):
r6 (n2 − k2) (n2 − k2 − 2) + 1
Vi,j = 4πεe
|(2r + d) (j − i)|3 (n2 − k2) (n2 − k2 + 4) + 4
(EyiEyj + EziEjz − 2EixEjx) .
(A-3)
Furthermore we assume that incident plane electromagnetic wave creates the incident
→
electric field intensity E(ω):
→
→
i,j
→→
Ei,j(ω) = E0 exp [ik r − iωt] .
We take only the real part of this expansion16, Eq. (A-4):
→
→
i,j
→→
Ei,j = E0 cos [(k r − ωt] .
(A-4)
Using spherical coordinates for description of the aggregate position, we obtain Eq.
(A-5):
→
→
Ei,j(ω) ≡ Exi,j,Eyi,j,Ezi,j
;
E0 ≡ Ex0 ,Ey0 ,Ez0
i,j i,j i,j i,j
→
k ≡ kx,ky,kz
Ex0 = |E0| sin (θ) cos (ϕ)
i,j
Ey0 = |E0| sin (θ) sin (ϕ)
i,j
Ez0 = |E0| cos (θ) .
(A-5)
i,j
→
→
The aggregate is fixed along the x axis, the vector k has the form: k ≡ {kx,0,0}, where
→
kx = |k| cos (θ) sin (ϕ); |k| = 2π/λ. The vector r = {rx,0,0}; rx = (2r + D) (i + j – 2). Thus
the first term in Eq. (A-4) reads as follows:
Collect. Czech. Chem. Commun. (Vol. 61) (1996)

Linear Aggregate of Ag Colloidal Particles
69
2π
λ
→ →
cos (k r) =
(2r + D) (i + j − 2) sin θ cos ϕ .
For description of the term ωt, we assume that ωt = ω(ti + tj) where ti and tj are times
when the incident wave is in contact with sphere i or j, respectively. The circular fre-
quency is ω = 2πc/λ, the time tj = ti + [(2r + D) (j − i)]/c. We assume the initial time t1 =
0. Thus we obtain ti = t1 + [(2r + D) (i − 1)]/c. We use these expressions in the term ω(ti
+ tj) and, consequently, in the equation for the product of plane electromagnetic waves.
Thus we obtain Eq. (A-6):
→
→
2π
λ
Ei(ω) Ej(ω) = cos
(2r + D) (i + j − 2) (sin (θ) cos (ϕ) − 1)
((E0y)2 + (E0z)2 − 2(E0x)2) .
(A-6)
We use equations Eq. (A-5) for further expansion of Eq. (A-6) and introduce the result
into Eq. (A-3). By expressing |E20| as |E02| = 2cµ0I, where I is the intensity of the incident
radiation, we directly obtain the Eq. (4).
The authors gratefully acknowledge the financial support by Grant No. 203/93/0106 awarded by the
Grant Agency of the Czech Republic and Grant GAUK awarded by the Grant Agency of Charles
University (Grant No. 179 UK/1993).
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