1.2 Surface Dynamics
2.1.2 Surface Plasmon Resonance–SPR
2.1.2.3 Surface Plasmon Resonance Spectroscopy–SPR 66
PS (nP S≈ 1.59,λ = 632.8 nm).
3 0 4 5 6 0 7 5
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8
P S
h
0 n m 5 0 n m 1 0 0 n m
R p (a.u.)
ψ1 ( ° ) 1 0 n m
Figure 2.11: By increasing the thickness or refractive index of a polymer thin film on top of gold, the resonance angle shifts towards higher values. This case shows the evolution of resonance angle with film thickness for a fix refractive index (nP S = 1.59). The resonance broadens as the film becomes thicker.
The intensity of the total internal reflection value decreases by increasing film thickness.
The intensity of the electric field of waveguide modes throughout the film thickness depends on which mode is excited (T Mi orT Ei). The distribution of intensities is shown in Figure 2.13.
Each of these modes, (except TM0 for thick films) has a “tail” that leaks into the dielectric medium above. The higher the order of the mode,i.e., the greater the value of i, the greater the penetration depth of the leaky mode into the dielectric add-layer:
ξT M1<ξT M2<ξT M3 <... (2.11) and
ξT E1<ξT E2<ξT E3<... (2.12) These concepts presented about surface plasmon resonance spectroscopy and waveguide spectroscopy will be necessary for understanding the working mechanism of the REDLS and WEDLS techniques.
3 0 4 5 6 0 7 5 9 0 0 . 0
0 . 2 0 . 4 0 . 6 0 . 8
T M 2 T M 1
T M 0
6 0 0 n m
P S
h
3 0 0 n m 1 0 0 n m
R p (a.u.)
ψ1 ( ° )
Figure 2.12: If the film thickness is continuously increased, the surface plas-mon resonance angle reaches a maximum value (this just happens for the half cylindrical lens, while triangular prisms do not show an upper resonance plateau). Waveguide modes appear by increasing the film thickness. In this case, as the incident beam is p-polarized, TM modes are excited using a half cylinder lens. The thicker the film, the higher the number of modes it supports.
I R
n
1> n
2n 1 n 2
y
1Polymer Gold
TM0 TM1
TE0 TM2
Leaky-modes
Figure 2.13: Leaky Modes and waveguide modes: according to the selected mode chosen, a different optical field distribution is found inside and outside the waveguide. By using p-polarized light, TM modes can be excited, while s-polarized light just allows TE modes. The tail of each mode inside the add-layer is called leaky-mode. The higher order modes have leaky-modes with higher penetration depth inside the add-layer. Surface plasmons (TM0) are sensitive to the polymer-solid contact in this example, and does not feel the influence of the free-surface.
2.1.2.4 Kinetic–SPR
Kinetic-SPR4is a powerful technique applied to detect small variations of the surface properties with time. As shown in section 2.1.2.3, an increase in the dielectric film thickness or in refractive index let the resonance minima shift towards higher angles, cf. Figure 2.11. The study of Tg from supported poly-mer has been experimentally addressed using such techniques [117]. When the dielectric material on top of the gold layer is heated, the thickness in-creases due to thermal expansion, and the refractive index dein-creases due to the decrease of density. The overall result can be estimated from the Lorentz-Lorenz relation (or Clausius-Mosotti model), so that the effect of expansion is greater than the refractive index decrease, and thus a shift to higher res-onance angles happens, and reflectivity increases with temperature [117], Figure 2.14. In the glassy state (T <Tg), the expansion coefficient has an absolute value smaller than in the liquid or rubbery state (T > Tg), thus different slopes in the reflectivity curve with temperature can be observed in these ranges, Figure 2.14–right. Tg is defined as the temperature at which an interception of both curves occurs, Figure 2.14–right. More about thermal expansion of polymer thin films is discussed in Chapter 3.1.2.
Lorentz-Lorenz equation. The Lorentz-Lorenz equation, also known as Clausius-Mossotti equation or Maxwell’s formula (equation 2.13), relates the refractive index,n, of a substance (or the real part of the dielectric constant
5, ′), to its polarizability, α, and N/V is the bulk density of polarizable monomer units.
N α 30V =
n2−1 n2+2 =
′−1
′+2 (2.13)
0 is the vacuum permittivity. Assuming that for a supported polymer film standing on a solid substrate with expansion coefficient much smaller than that of the polymer, the areal number density of monomer units,NA=N/A, remains constant during thermal expansion, one can write:
N α 30A ⋅1
h = NAα
30 ⋅ 1 h =
′−1
′+2 (2.14)
with V =A⋅h. Therefore:
4Wave guides are also used for this purpose, due to their higher resolution originated from sharp resonance minima [117].
5The refractive indexnrelates to the dielectric constant through: n2 =′
T g
R ef le ct iv ity ( a. u. )
T e m p e r a tu r e ( ° C )
L iq u i d
G l a s s y
∆R
R 2
R 1
Ψ1 , k ΨS P R
R 1R 2 E x p a n s io n
∆R
R e fl e c ti v it y ( a .u .)
Ψ 1 ( º )
R e fl e c ti v it y ( a .u .)
L iq u id
o r R ub b e ry
Reflectivity (a.u.)
Ψ1 ( º ) G la s s y
Figure 2.14: When the polymer films are heated, thermal expansion leads the SPR curve to shift towards higher resonance angles (left figure). By keeping the laser and SPR detector at a constant angle,ψ1,k, the reflectivity increases from R1 to R2 during heating. As the thermal expansion coefficient in the liquid state is greater than at the glassy state, a change in slope atTg takes place (right figure).
h= ( NAα
30 ) ⋅ ′+2
′−1 (2.15)
The derivative of thickness, h, in respect to the dielectric constant, ′, gives:
Bh B′ = (
NAα 30 ) ⋅
(′−1) − (′+2) (′−1)2
= − ( NAα
0 ) ⋅ 1
(′−1)2 (2.16) From equation 2.14, one can write:
(′−1) = (′+2) ⋅ ( NAα
30
) ⋅1
h (2.17)
Substituting the value of (′−1) in equation 2.16 (still keeping one term (′−1)free):
Bh B′ = − (
NAα 0
) ⋅ 1 (′−1)
⋅ 30h
(′+2)NAα (2.18)
Thus:
Bh B′ =
−3h
(′−1)(′+2) (2.19)
or
Bh h =
−3B′
(′−1)(′+2) (2.20)
By dividing both sides of the equation 2.20 by ′, it can be rewritten as:
(′−1)(′+2)
3 ⋅ 1
′ ⋅Bh h =
−B′
′ (2.21)
substituting the dielectric constant,′, of a polymer (e.g., PS with′ =2.528) in the left-hand side of equation 2.21, one gets as a good approximation:
0.91⋅Bh h ≈
−B′
′ (2.22)
meaning that, for the case of PS, as the angular shift of a surface plasmon resonance induced by a thin dielectric coating always depends on its optical thickness,n⋅h[117], the negative contribution due to the decrease of dielectric constant (or refractive index) while heating is just about 91 % of the positive increase in thickness due to thermal expansion. Therefore, the SPR curve shifts towards higher resonance angles, and the reflectivity value increases at a constant angle, ψ1,k, Figure 2.14. Naturally, the opposite affirmation is valid for cooling,i.e., the reflectivity decreases while decreasing temperature.
For a polymer as PMMA, in which the dielectric constant is ′ = 2.16, the negative contribution of the dielectric constant is just 75 % of the positive contribution of thermal expansion. The lower the refractive index of the polymer film, the higher will be the difference between both contributions.
Testing the validity of the Lorentz-Lorenz equation for PS thin films. The validity of the result obtained from the Lorentz-Lorenz relation for polystyrene is not trivial, and has to be tested. To check whether or not equation 2.22 describes well the physical behavior of PS, real values of thickness and dielectric constant, based on experiments will be next used.
Assuming a PS 1821 g/mol film, with initial thickness, hi = 5 nm, at the initial temperature, Ti = 20 ○C, is heated to Tf = 28.85 ○C, then the final thickness is, hf = 5.01 nm, as the linear expansion coefficient for this
polymer at T < Tg is αL ≈ 2.03×10−4/K, as calculated from ref. [118]. One can calculate the dielectric constant, by using the experimentally determined relation, that can be calculated from ref. [119]:
n= { 1.5920−1.4×10−4 T if T <Tg
1.5864−3.6×10−4 (T −50 ○C) if T >Tg (2.23) AsTg ≈ 50○C for PS 1821 g/mol, at the initial and final temperatures ones has:
′i=n2i = (1.5892)2 =2.5256 (2.24)
′f =n2f = (1.5878)2 =2.5212 (2.25) therefore:
∆′
′i = 2.5212−2.5256
2.5256 = −1.7422×10−3 (2.26)
and ∆h
hi
=5.01−5
5 =2×10−3 (2.27)
resulting:
∆′/′i
∆h/hi =
−1.7422×10−3
2×10−3 = −0.871 (2.28) which is in good agreement with the theoretical prediction given by equa-tion 2.22 (error≈4.3 %), confirming the validity of the Lorentz-Lorenz equa-tion for PS 1821 g/mol.