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Theoretical Background of SPR Method According to Green's FunctionTable of Contents:
Surface molecular layer susceptibility and reflection coefficientAs mentioned before, the main principle behind usingtheSPRmethod for studying the physical properties of coatings on solid surfaces is the measurement of the shifts in resonant angle if a molecular covering is present on the sensitive surface of a sensor [26]. The angle shift is, of course, dependent on molecular concentration as well as the type of the molecules. The physical models usually used for the description of this shift are based mainly on the concept of an additional layer on the surface of SPR transducer, which is characterized by the effective thickness and refractive index, analogous to the similar idea of ellipsometry of thin films [2629]. Another approach is based on the idea of ultrathin film representation [30]. The main point of this approach is the representation of the molecular layer as an effective ultrathin homogeneous film characterized by any susceptibility, which was calculated with selfconsistent equations for local field using molecular polarization and effective film thickness. Similar approaches do not allow one to obtain information about the concentration or individual dielectric properties of molecules at the surface. To describe the optical properties of molecular coverings at surfaces, one needs to take into account the individual properties of the adsorbed molecules, their interaction with the surface, and intermolecular (lateral) interactions. According to Bobbert and Vlieger [31], one solution to the problem of light reflection from a substrate covered with spherical particles can be obtained by defining the reflected electromagnetic wave as the sum of Fresnel's plane wave and the number of spherical waves, which are raised at the scattering on the spherical particles in accordance with the Mie theory. Another method of calculating the reflection coefficient for a surface covered by a molecular layer is based on Green's function [32] . This subsection covers an approach based on the concept of linear response for the monolayer of nonpointlike protein molecules, which have the shape of oblate or prolate ellipsoids. As it was mentioned in Subsection 2.2.1, the use of the light scattering matrix for a multilayer system is the most common approach for calculating the angular dependence of the reflected light intensity on the excitation of SPR in the Kretschmann configuration [33] . The determination of effective optical constants in this approach yields an approximate estimate of the thickness and complex refractive index of the molecular layer, while the use of such layer parameters as polarization and surface concentration of molecules in the application of Green's function significantly increases the informative value and reliability of the calculations. In this case, in order to calculate the reflection coefficient, it is necessary to know the effective susceptibility of the molecular layer. Let us consider a dilute thin layer of oblate or prolate ellipsoid organic molecules that are homogeneously distributed on the surface. According to the LippmannSchwinger equation, the field at an arbitrary point in the system obeys the equation [34]:
where E,^{0>}(R,co) is the external longrange field, a is the coefficient defined by the system of units (for SI, a = o9/c^{2}e_{0}, where со is the angular light velocity, c is the speed of light, £_{0} is the permittivity of vacuum], Q is the number of molecules on the surface, V_{a} is the molecular volume, Xjii®) is the molecular susceptibility, G^R.R'.co) is a photon propagator that describes the propagation of light of frequency to from point R' to point R [35]. Summation is made over all positions that are occupied by molecules. Because molecular linear dimensions are much less than light wavelength and average distances between the molecules are considered larger than molecular linear dimension (submonolayer cover], one can make the next approximation:
where X/iU^{0}) = V_{a}Xji{^{(}°)r, r_{a}, z, z_{a} are vector and scalar coordinates of the observation point and the center of ath molecule. Here Xjiico) is the response on the local (total) field, which connects the polarization of the molecule and the local field via
The averaging over molecular coordinates if molecules are homogeneously distributed along the surface is performed using the equation:
where S is the area of the surface at which the Q molecules are situated; N_{s} = Q/S is molecular surface concentration; and к , к' are the wave vectors for different points of a field. Then, an equation of selfconsistent field in the Weil representation can be written as
Making Fourier transformation in the plane of the surface, one obtains from Eq. (2.16)
Then, Eq. (2.18) can be rewritten in the form: The solution to this equation is Then, the effective susceptibility of submonolayer of the molecules at the surface, which connects the Fouriertransformants of layer polarization and external field, has the form For SPR simulation, one needs to know the reflection coefficient of the molecular layer (see Fig. 2.6). For calculation of the reflection coefficient, let us consider the planar layered medium, the electrodynamical properties of which are characterized by photon propagator G_{/7}(/c ,z ,z' ,(o). Let the light propagation from semispace z > 0 to the same semispace be described by the photon propagator G^^{+,+}k,z,z',(o), the light propagation from semispace z < 0 to semispace z > 0 by the photon propagator G^{+}’^{_)}(/c,z,z',co), and the light propagation from semispace z > 0 to semispace z < 0 by the photon propagator G~‘^{+}k,z,z',co). Then, an effective susceptibility of the molecular layer situated at the surface of semispace z < 0 is defined by Eq. (2.22) with the photon propagator GjT‘~k,z,z',(o). If the field E^°k,z,(o) acts at the molecular layer, the field reflected by the layer will be written as Then, the reflection coefficient, which connects the amplitudes of reflected by the molecular layer and incident ppolarized fields £p^{R)} = R_{p}Ep^{0)}, can be written in the form where в is the incident angle. Because light is reflected by both the molecular layer and the surface, the total reflection coefficient should be written as the sum
where Rj,^{0)} is the Fresnel reflection coefficient of the surface. Figure 2.6 (a) Reflection of the light by molecular layer situated at a surface, (b) Schematic presentation of the system under investigation. Reprinted from Ref. [36], Copyright 2008, with permission from Elsevier. Nanoparticles shape influence on the dispersion dependences of SPRGreen's function allows considering both molecular surface concentration and shape of molecules. Let us analyze the molecular coating of a metal surface where the molecules are represented as homogeneous particles with the shape of ellipsoids uniformly distributed along the plane of the metal surface. To calculate the effective susceptibility, one can use an approach described in Refs. [34, 37]. As a result, the effective susceptibility of the molecular layer can be obtained as
where СДk, /, /, to) is the electro dynamic Green's function of the environment where the molecular layer is located, N_{s} is the concentration of surface particles, and Xijfa) is the single surface molecule susceptibility. Obviously, in this case, one should use Green's function for two semispaces with a flat boundary. Since we assume that molecules can be represented as homogeneous ellipsoidal particles, the polarizability of an ellipsoidal particle on a surface can be used for molecular susceptibility: where x_{±}{(0) and ^ц(ю) are normal and lateral parts of the linear response tensor, respectively. To see the influence of the shape and concentration of dielectric nanoparticles on the dispersion dependence ofSPPs,letusinvestigate the following system. Assume that nanoparticles are located on the metal surface, whose optical properties are described by a dielectric function £_{m}(a)) = l(0p/(0^{2} with a plasma frequency co_{p} = 2 x 10^{s} cm"^{1}, which is a typical value for metals (for example, (O_{p} = 1.2 x 10^{s} cm'^{1} for aluminum and co_{p} = 4 x 10^{s} cm"^{1} for gold). The geometric parameters of the nanoparticles were selected in the range of 110 nm in such a way that their volume remained unchanged and was equal to V_{p} = 8.38 x 10"^{21} cm^{3}, and the surface concentration of the monolayer coating was set to be equal to N_{s} = 0.25 x 10^{13} cm"^{2}. It is worth noting that exactly such sizes of nanostructures are common for biomolecules. The dielectric permittivity of the model nanoparticles was chosen equal to e_{p} = 5. In the case of ptype waves (Fig. 2.7), the presence of a layer of nanoparticles on the metal surface leads to the splitting of the dispersion curve of the typical SPP into four branches. It should be noted that this effect is common for both prolate and oblate nanoparticles. However, for spherical nanoparticles, only two dispersion curves can be observed. The obtained result can be explained by the fact that the appearance of the four modes is the result of the interaction of the x and zcomponents of the electric field of the SPP with the longitudinal and lateral oscillations of ellipsoidal nanoparticles. In the case of spherical nanoparticles, the equality of polarizability values along the axes of OX and OZ leads to the appearance of only two dispersion curves. It should be noted that this fact is not universal for consideration of the interaction of nanoparticles with the surface of solid. In the case discussed in this subsection, the degree of interaction between the surface and the spherical nanoparticle does not cause significant differences between the various components of polarizability. Figure 2.7 Dispersion curves for ppolarized surface wave upon interaction with monolayer of nanoparticles with oblate ellipsoid shape (blue lines) and spherical shape (red lines). The concentration of nanoparticles is n = N_{s}. Reprinted from Ref. [21], Copyright 2008, with permission from Elsevier. Figure 2.8 Dispersion curves for ppolarized surface wave upon interaction with monolayer of nanoparticles with oblate ellipsoid shapes. The concentration of nanoparticles is n = 0.1A/_{S}. The dashed line represents the dispersion curve of the surface wave, which is typical for the free surface. Reprinted from Ref. [21], Copyright 2008, with permission from Elsevier. Figure 2.8 shows the influence of lateral interactions, which occur due to the nanoparticles' concentration dependence, on the form of the dispersion curves of the ppolarized surface wave. Comparing these results with the ones mentioned before in Fig. 2.7, it can be noticed that the change in nanoparticles' concentration toward a sparser monolayer causes the displacement of the dispersion curves to the region of higher frequencies with the simultaneous narrowing of the occupied bandwidth. Calculations show that the decrease in the concentration of surface particles causes the dispersion curves grouping near the surface wave dispersion curve, which is typical for the free surface (Fig. 2.8). It should also be noted that the dispersion curves disappeared after some critical particle concentration point is reached, since there were no solutions to the dispersion equation in this case. Figure 2.9 Influence of prolate ellipsoidal nanoparticles on ppolarized wave dispersion (concentration of particles in the monolayer is n = N_{s}). Semiaxis of the nanoparticles are (1) /?_{x} = 0.71 nm, h_{z} = 4 nm; (2) /)„ = 1 nm, h_{z} = 2 nm; (3) h_{x} = 1.26 nm, h_{z} = 1.26 nm. Reprinted from Ref. [21], Copyright 2008, with permission from Elsevier. Let us analyze the influence of the prolate shape of nanoparticles on the lower branches of the ppolarized surface wave dispersion curves (Fig. 2.9). One can see that the highfrequency branch, common for prolate nanoparticles, is higher than the corresponding branch for spherical nanoparticles, and when the shape of the nanoparticle approaches the spherical one, it converges from higher frequency values, whereas the lowfrequency branch behaves vice versa, approaching the corresponding curve from lower frequency values. In the case of oblate nanoparticles, the dispersion curves shift in other direction than the spherical nanoparticles. In Figs. 2.7 and 2.9, an interesting effect of the deviation of the dispersion curves is observed with the growth of the wave vector toward the decrease in frequencies. This shows that such waves are characterized by a negative dispersion, which physically signifies the opposite directions of the phase and the surface wave energy propagation. It is known that waves of this type can exist in a solid [38]. The obtained results show that the laws of SPP dispersion, in the case of nanostructured coating on the metal surface, essentially depend on the nanoparticles' shape, type, and surface concentration. In particular, the dispersion curve shifts toward the lower frequencies when the shape of the nanoparticles changes from the prolate ellipsoids (h_{±}/h^ = 2) to the spherical nanoparticles at constant concentration (Fig. 2.9, upper curves 2 and 3). Peculiarities of SPR study to account the 3D polarization factor of the moleculesAs was mentioned before, molecular surface concentration N_{s} and components of molecular susceptibility can be determined from the SPR experiments instead of effective layer thickness and refractive index, which are usually estimated. For calculating the molecular layer reflection coefficient, it is needed to know the initial molecules' susceptibility value ;fy(ft)), which describes linear response on the local field by a single molecule located on the surface. Since the SPR simulation will be taken out for immunoglobulin biomolecules, one can assume that the molecule on the surface has a shape close to the ellipsoidal one. The linear response of the ellipsoidal particle on the surface was calculated earlier [39]. If h_{x}, h_{r} h_{z} are the semiaxes of the molecule and V_{p} is the volume of the molecule, then the plane component of the molecular susceptibility is equal to where the plane local field factor is
where e_{p} is the dielectric constant of particle; e_{s} and e_{a} are dielectric constants of the substrate and the environment, respectively; m, is the depolarization factor; # = h_{x}h_{y}hj(2z_{p})^{3} is the local field factor; and z_{p} is the zcoordinate of the center of the molecule. The normal part of the molecular susceptibility is
where
Depolarization factors for a molecule with prolate shape, where h_{z}> h_{x} = h_{y} (see Fig. 2.10a), are as follows:
where ц = (1  <^)^{1/2}, £ = h_{x}/h_{z}. Depolarization factors for a molecule with oblate shape, where h_{x }= h_{y}>h_{z} (see Fig. 2.10b), are as follows:
where v= (^  1)^{1/2}. It should be noted that the interaction between the molecule and the surface can lead to the phenomenon of the local field amplification, which causes a significant increase in molecular polarizability [40]. Figure 2.10 Protein molecule on the surface: (a) as a prolate ellipsoid; (b) as an oblate ellipsoid. Reprinted from Ref. [36], Copyright 2008, with permission from Elsevier. To simulate the SPR experiment schematically shown in Fig. 2.6b, it is necessary to calculate the reflection coefficient for the system consisting of an ATR glass prism, a thin gold film, and a liquid with an adsorbed molecular layer. If one uses the described approach, then it is obvious that the molecular layer is located on the surface of a metallic film, which is located on the ATR glass prism. Using Eq. (2.25), one can calculate the SPR curves related to different molecule shapes. From Eqs. (2.22) and (2.24), it is seen that the reflection coefficient, which defines the SPR curve, depends on the concentration and shape of the molecules. The reflection coefficient was calculated using a specially designed software, according to Eq. (2.25) for different values of the parameter £ = h/hj_ (where hц and h_{±} are the semiaxes of ellipsoid, parallel () and perpendicular to (_L) to the substrate surface plane), which defines the shape of the molecule of similar mass. It turned out that SPR curves corresponding to molecules that have the shape of a prolate ellipsoid are characterized by a rather strong shift with the change in the parameter £ (see Fig. 2.11a). Molecules characterized by the shape of an oblate ellipsoid show very small shifts for different values of the parameter £ For example, a change in the value of <" from 1.1 to 10 results in a change in the angle of the minimum of the SPR curve from £?_{min} = 64.121° to 0_{min} = 64.262°. Figure 2.11 (a) Calculated SPR curves dependent on the shape of molecules. (1) Free surface, 0_{mln} = 62.747°; (2) oblate molecules £ = 2.0, 9_{min} = 64.262°; (3) prolate molecules ( = 0.12, 9_{min} = 66.585°; (4) prolate molecules ( = 0.11, 0_{min} = 68.302°. (b) Calculated dependences of SPR curves on the composition of the molecular film consisting of prolate (£ = 0.12) and oblate (£ = 2.0) molecules. Part of prolate molecules: f = 1 (curve 1, 0_{min} = 66.282°) and f = 0.5 (curve 2, 0_{min} = 65.777°). Reprinted from Ref. [36], Copyright 2008, with permission from Elsevier. Calculations show that the shift in the minimum of the SPR curve with increasing concentration of molecules is obvious. In particular, an increase in molecular concentration (for molecules that are characterized by £ = 0.15) by 5% leads to an angular shift of Дв = +0.1°. This means that the proposed approach may be useful for expanding the informativity of experimental results obtained with SPR measurements for the evaluation of molecular coatings or layers, because this approach may determine the surface concentration of molecules having their own molecular characteristics, such as the polarizability of one molecule or its shape. It should be noted that this approach allows the possibility of considering the twocomponent coating on the surface of the SPR sensor. In particular, it provides an option to calculate the SPR curves for molecular layers consisting of molecules with prolate and oblate ellipsoid shapes. The dependences of the SPR curves on the molecular film composition consisting of prolate (£ = 0.12) and oblate (£ = 2.0) molecules are shown in Fig. 2.11b. It is seen that the change in the percentage of the prolate molecules from/= 1 (corresponding to the molecular layer, which consists only of prolate molecules) to /= 0.5 leads to the minimum shift in SPR curve of about 0.505°. This result shows strong dependence of the molecular layer dispersion properties on the shape of molecules. 
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