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The new model of linear colloidal aggregate
Journal of
MOLECULAR STRUCTURE
ELSEVIER
Journal of Molecular Structure 348 (1995) 297-300
THE NEW MODEL OF LINEAR COLLOIDAL AGGREGATE O.Sest&a, P.Matejkab ,B.VlZkoviia, aDept. of Physical and Macromolecular Chemistry, Charles University, Hlavova 2030, 12840 Prague 2, Czech Republic bDept. of Analytical Chemistry, ICI, Technicka 5, 16628, Prague 6, Czech Republic
1. Introduction Ag colloids are suitable model systems for investigation of the electromagnetic
mechanism
of SERS (Surface Enhanced Raman Scattering) chiefly due to the possibility to follow the surface plasmon absorption (SPA) of the Ag colloid/ adsorbate system by UV-vis. absorption spectroscopy.
For a linear aggregate of Ag particles interacting by a dipol- dipol interaction,
two excited surface plasmon state originate and two absorption bands, one at lower energy In our
than the monomer (single particle) absorption band, the other at a higher energy.
previous studies, we have found systematical differences in SPA spectra of SERS-active systems containing polymeric (i.e. proteins) and/or oligomeric (i.e. polypeptide, 100) species in comparison to the spectra of systems formed by monomeric second maximum of plasmon absorption
Triton X-
species. The
appears for species of monomeric nature in range
540-620 run. In contrast to that, for SERS-active systems containing species of polymeric character , only an extensive broadening of original peak (with maximum at 400 nm) towards higher wavelengths
is typical. We have formed a hypothesis , that differences
between
plasmon absorption
curves are given by different distances of Ag colloidal particles in the
aggregates. For verification of this hypothesis we have developed a new model of linear colloidal aggregate with parametric inter-sphere distances. The new model of linear colloidal aggregate is based on following approximations: 1) -only dipolar interactions between colloidal particles in aggregate are considered 2) - the radii of all colloidal particles are identical 3) - the distances between adjacent spheres (colloidal particles) are identical 4) -the longitudinal
plasmon mode [l]
is oriented along x axis and the transverse
modes are oriented along the y and z axis, respectively. With these assumption, we have developed a new formula for the dipolar interaction energy of single spheres : 0022-2860/95/$09.50 0 I995 Elsevier Science B.V. SSDI 0022-2860(95)08647-l
All rights reserved
298
n2 v iJ
I
=&f -
cos [9(2r
+ d)(i+
j-
2)(sin
(8)cos
r6
1(2r + d)(j
(p)-
I
- i)13
’
I)].
[I - 3 sin 2(e)c0s ‘(p)]4?,2
(
Er- relative
permeability of vacua (for water: 6,. =Sl,l) , n,k - are the “bulk constants” for
Ag, r -is the sphere radius ,d -is the distance between adjacent radiation,
spheres, 1 -is wavelength of
0, Q, -are angles of incidence, ,!?i - is the intensity of the electric field of the
incident radiation) The details of the calculation are published elsewhere [2] . For angles of incidence 0, ~=90,0; the solution describes the energy level of a longitudinal plasmon mode (Vmin) while for angles of incidence 0, q.~=0,O; and 0,90; the solution gives energy the levels of transversal plasmon modes (Vmax; the linear colloidal aggregate is oriented along x axis). These model calculations thus provide the following results:
2. Results The dependence of Vmin,Vmax on the number of spheres in the aggregate were calculated for varying inter-sphere distances. The results for the spheres of 5 nm radii are shown in Figure 1, for 2 nm spheres in Figure 2. The values of Vmin correspond to the stabilisation energy of the aggregate. As in real systems with aggregated Ag colloids,
the SPA curves are
routinely measured in the range 300-700 nm, we prefer these values to the Vmax values. The absolute value of Vmin is directly proportional to the shift of the maximum
of the SPA
curves of non aggregated Ag colloid. The absolute values of Vmin and Vmax
a) increase
with the increasing number of spheres in the aggregate N for NI 15 , but became independent on N for N2 15 ; b) increase with the increasing radius of the spheres ; c) decrease with the increasing inter - sphere distance, however, sufficient stabilization occurs even for the inter sphere distances of 2 nm (Fig. 1).
299
3. Conclusions 1) As real Ag colloidal clusters consist of more than 100 Ag particles, 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 the particle size distributions of the SERS - active systems. 3) The inter - particle distances significantly influence the energies of the surface plasmon state of the aggregate. For systems with non - zero inter - particle distances, the distribution of the inter - particle distances can substantially influences the shape of the SPA curve of the SERS - active system. Thus the shapes of SPA curves of the Ag colloid/monomer
systems are governed 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 SPA curves of Ag colloid
/ polymer(oligomer)
overlapping)
distribution
systems are influenced
by two (most propably
mutualy
function: a) distribution of particle sizes and 2) distribution of
interparticle 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 isolated particles towards the red spectral region.
Figure 1
THE DEPENDENCE OF ENERGY VMIN,VMAX ON THE NUMBER OF SPHERES AT VARYING INTERSPHERE DISTANCES >de FOR 6 NM SPHERES IN WATER AT 375 NM WAVELENGTH
3,00E-19 2,00E-19
--+---
l,OOE-19
60 nrqVtin
-d=On~Vmx _f_
d=OS nm W-tin
-
d=OS nr-q Vmax
---+---
d=l ntq Vn-in
~d=lnrtjVmax
NUMBER
OF SPHERES
IN AGGREGATE
__t__
d=2 nrfjVnin
-
d=2 nrqVrri3x
300
Figure 2
THE DEPENDENCE OF ENERGY WIN, VMAX ON THE NUMBER OF SPHERES AT VARYING INTERSPHERE DISTANCES >d< FOR 2 NM SPHERES IN WATER AT 375 NM WAVELENGTH
8
4,00E-21
-
2,00E-21
--
O,OOE+OO 5
-8,OOE-21
7
9
11
13
15
17
19
d=O nm, Vmin
-
d=O nm, Vmax
-
d=OS
-
d=O.S,
--+----
d=l
nm, Vmin
--+---
d=l
nm, Vmax
U
d=2
nm, Vmin
+
d=2
nm, Vmax
nm, Vmin Vmax
NUMBER OF SPHERES AGGREGATE
IN
References 1. Fomasiero D., Grieser F., J.Chem. Phys. 1987,5,3213 2. in preparation
-
MOLECULAR STRUCTURE
ELSEVIER
Journal of Molecular Structure 348 (1995) 297-300
THE NEW MODEL OF LINEAR COLLOIDAL AGGREGATE O.Sest&a, P.Matejkab ,B.VlZkoviia, aDept. of Physical and Macromolecular Chemistry, Charles University, Hlavova 2030, 12840 Prague 2, Czech Republic bDept. of Analytical Chemistry, ICI, Technicka 5, 16628, Prague 6, Czech Republic
1. Introduction Ag colloids are suitable model systems for investigation of the electromagnetic
mechanism
of SERS (Surface Enhanced Raman Scattering) chiefly due to the possibility to follow the surface plasmon absorption (SPA) of the Ag colloid/ adsorbate system by UV-vis. absorption spectroscopy.
For a linear aggregate of Ag particles interacting by a dipol- dipol interaction,
two excited surface plasmon state originate and two absorption bands, one at lower energy In our
than the monomer (single particle) absorption band, the other at a higher energy.
previous studies, we have found systematical differences in SPA spectra of SERS-active systems containing polymeric (i.e. proteins) and/or oligomeric (i.e. polypeptide, 100) species in comparison to the spectra of systems formed by monomeric second maximum of plasmon absorption
Triton X-
species. The
appears for species of monomeric nature in range
540-620 run. In contrast to that, for SERS-active systems containing species of polymeric character , only an extensive broadening of original peak (with maximum at 400 nm) towards higher wavelengths
is typical. We have formed a hypothesis , that differences
between
plasmon absorption
curves are given by different distances of Ag colloidal particles in the
aggregates. For verification of this hypothesis we have developed a new model of linear colloidal aggregate with parametric inter-sphere distances. The new model of linear colloidal aggregate is based on following approximations: 1) -only dipolar interactions between colloidal particles in aggregate are considered 2) - the radii of all colloidal particles are identical 3) - the distances between adjacent spheres (colloidal particles) are identical 4) -the longitudinal
plasmon mode [l]
is oriented along x axis and the transverse
modes are oriented along the y and z axis, respectively. With these assumption, we have developed a new formula for the dipolar interaction energy of single spheres : 0022-2860/95/$09.50 0 I995 Elsevier Science B.V. SSDI 0022-2860(95)08647-l
All rights reserved
298
n2 v iJ
I
=&f -
cos [9(2r
+ d)(i+
j-
2)(sin
(8)cos
r6
1(2r + d)(j
(p)-
I
- i)13
’
I)].
[I - 3 sin 2(e)c0s ‘(p)]4?,2
(
Er- relative
permeability of vacua (for water: 6,. =Sl,l) , n,k - are the “bulk constants” for
Ag, r -is the sphere radius ,d -is the distance between adjacent radiation,
spheres, 1 -is wavelength of
0, Q, -are angles of incidence, ,!?i - is the intensity of the electric field of the
incident radiation) The details of the calculation are published elsewhere [2] . For angles of incidence 0, ~=90,0; the solution describes the energy level of a longitudinal plasmon mode (Vmin) while for angles of incidence 0, q.~=0,O; and 0,90; the solution gives energy the levels of transversal plasmon modes (Vmax; the linear colloidal aggregate is oriented along x axis). These model calculations thus provide the following results:
2. Results The dependence of Vmin,Vmax on the number of spheres in the aggregate were calculated for varying inter-sphere distances. The results for the spheres of 5 nm radii are shown in Figure 1, for 2 nm spheres in Figure 2. The values of Vmin correspond to the stabilisation energy of the aggregate. As in real systems with aggregated Ag colloids,
the SPA curves are
routinely measured in the range 300-700 nm, we prefer these values to the Vmax values. The absolute value of Vmin is directly proportional to the shift of the maximum
of the SPA
curves of non aggregated Ag colloid. The absolute values of Vmin and Vmax
a) increase
with the increasing number of spheres in the aggregate N for NI 15 , but became independent on N for N2 15 ; b) increase with the increasing radius of the spheres ; c) decrease with the increasing inter - sphere distance, however, sufficient stabilization occurs even for the inter sphere distances of 2 nm (Fig. 1).
299
3. Conclusions 1) As real Ag colloidal clusters consist of more than 100 Ag particles, 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 the particle size distributions of the SERS - active systems. 3) The inter - particle distances significantly influence the energies of the surface plasmon state of the aggregate. For systems with non - zero inter - particle distances, the distribution of the inter - particle distances can substantially influences the shape of the SPA curve of the SERS - active system. Thus the shapes of SPA curves of the Ag colloid/monomer
systems are governed 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 SPA curves of Ag colloid
/ polymer(oligomer)
overlapping)
distribution
systems are influenced
by two (most propably
mutualy
function: a) distribution of particle sizes and 2) distribution of
interparticle 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 isolated particles towards the red spectral region.
Figure 1
THE DEPENDENCE OF ENERGY VMIN,VMAX ON THE NUMBER OF SPHERES AT VARYING INTERSPHERE DISTANCES >de FOR 6 NM SPHERES IN WATER AT 375 NM WAVELENGTH
3,00E-19 2,00E-19
--+---
l,OOE-19
60 nrqVtin
-d=On~Vmx _f_
d=OS nm W-tin
-
d=OS nr-q Vmax
---+---
d=l ntq Vn-in
~d=lnrtjVmax
NUMBER
OF SPHERES
IN AGGREGATE
__t__
d=2 nrfjVnin
-
d=2 nrqVrri3x
300
Figure 2
THE DEPENDENCE OF ENERGY WIN, VMAX ON THE NUMBER OF SPHERES AT VARYING INTERSPHERE DISTANCES >d< FOR 2 NM SPHERES IN WATER AT 375 NM WAVELENGTH
8
4,00E-21
-
2,00E-21
--
O,OOE+OO 5
-8,OOE-21
7
9
11
13
15
17
19
d=O nm, Vmin
-
d=O nm, Vmax
-
d=OS
-
d=O.S,
--+----
d=l
nm, Vmin
--+---
d=l
nm, Vmax
U
d=2
nm, Vmin
+
d=2
nm, Vmax
nm, Vmin Vmax
NUMBER OF SPHERES AGGREGATE
IN
References 1. Fomasiero D., Grieser F., J.Chem. Phys. 1987,5,3213 2. in preparation
-