Consider the system of two nanoparticles of different sizes. The largest nanoparticle consists of a nucleus and a shell. The axis of symmetry of the OZ system connects the centers of the particles. Assume the incidence of external light on the system. To demonstrate the effect of interparticle interaction, we need to calculate the effective susceptibility^{20} of the system under consideration. It is convenient to use the method of the pseudo-vacuum function Green^{21}. Consider the medium consisting of the environment in which the small nanoparticle is integrated. Let the effective susceptibility of the small nanoparticle be \({\rm X}_{ij}^{(p)} ({\mathbf{R}},\omega )\). Note that the effective susceptibility is the characteristic of the object and depends on the material the object is made of and the size and shape of the object. The effective susceptibility relates the linear response (bias) to the external field, i.e. \(P_{i} ({\mathbf{R}},\omega ) = \varepsilon_{0} {\rm X}_{ij}^{(p)} ({\mathbf{R}},\omega )E_{j}^{(0)} ({\mathbf{R}},\omega )\)^{20}. Then, the calculation of the effective susceptibility provides the consideration of self-action processes.

The electrodynamic green function of the environment is \(G_{ij}^{(0)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega )\). Then, using the method proposed in^{16}the pseudo-empty Green’s function \(G_{ij}^{(m)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega )\) of the ‘environment of small nanoparticles’ system can be written as follows (see appendix)

$$\begin{aligned} G_{ij}^{(m)} ({\mathbf{R}},{\mathbf{R^{\prime}}},\omega ) & = G_{ij}^{ (0)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega ) + G_{ij}^{(R)} ({\mathbf{R} },{\mathbf{R}}^{^{\prime}} ,\omega ) \\ & = G_{ij}^{(0)} ({\mathbf{R}},{\mathbf{R} }^{^{\prime}} ,\omega ) – k_{0}^{2} \int\limits_{{V_{p} }} {d{\mathbf{R}}^{^{\prime\ prime}} G_{ik}^{(0)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime\prime}} ,\omega ){\rm X}_{ kl}^{(p)} ({\mathbf{R}}^{^{\prime\prime}} ,\omega )G_{lj}^{(0)} ({\mathbf{R}}^{ ^{\prime\prime}} ,{\mathbf{R}}^{^{\prime}} ,\omega )} , \\ \end{aligned}$$

(1)

where the integration is on the small volume of nanoparticles, \(k_{0}^{{}} = \omega /c\), *vs* is the speed of light, and the subscripts signify x, y, and z in the Cartesian coordinate system. Here and below we use summation of Einstein’s notation, which means that \(A_{ij} B_{jl} = \sum\limits_{j = x,y,z} {A_{ij} B_{jl} = A_{ix} B_{xl} + A_{iy} B_{yl } + A_{iz} B_{zl} }\). In eq. (1) \(G_{ij}^{(R)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega )\) is the impact of the smaller nanoparticle on the Green’s function of the pseudovacuum due to the local redistribution of the field.

Within the framework of the concept of effective susceptibility, developed in^{20}the effective susceptibility of the large nanoparticle embedded in the pseudo-vacuum (the “new” medium consisting of an environment and a small nanoparticle) described by Green’s function \(G_{ij}^{(m)} ({\mathbf{R}},{\mathbf{R^{\prime}}},\omega )\) can be calculated according to

$$\Xi_{ij}^{(b)} ({\mathbf{R}},\omega ) = \left[ {\left( {{\rm X}_{ij}^{(b)} ({\mathbf{R}},\omega )} \right)^{ – 1} – k_{0}^{2} \int\limits_{{V_{v} }} {d{\mathbf{R}}^{^{\prime}} G_{ji}^{(R)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega )} } \right]^{ – 1} ,$$

(2)

where \({\rm X}_{ij}^{(b)} ({\mathbf{R}},\omega )\) is the susceptibility of the large nanoparticle which can be evaluated from the following equation^{22.23}which can only be applied to shelled spherical nanoparticles:

$${\rm X}_{ij}^{(b)} ({\mathbf{R}},\omega ) = \delta_{ij} {\rm X}^{(b)} (\omega ) , \, {\rm X}^{(b)} (\omega ) = 3\;\frac{{(\varepsilon_{2} – \varepsilon_{m} )(\varepsilon_{1} + 2\varepsilon_{ 2} ) + f_{1} (\varepsilon_{1} – \varepsilon_{2} )(\varepsilon_{m} + 2\varepsilon_{2} )}}{{(\varepsilon_{2} + 2\varepsilon_{ m} )(\varepsilon_{1} + 2\varepsilon_{2} ) + f_{1} (2\varepsilon_{2} – 2\varepsilon_{m} )(\varepsilon_{1} – \varepsilon_{2} ) }},$$

(3)

or one_{2} is the outer radius, ε_{1} and ε_{2} are the nucleus (of radius a_{1}) and the shell dielectric functions, respectively. The volume fraction of the core is \(f_{1} = {{a_{1}^{3} } \mathord{\left/ {\vphantom {{a_{1}^{3} } {a_{2}^{3} }}} \right.\kern-\nulldelimiterspace} {a_{2}^{3} }}.\) It should be noted that the idea of effective susceptibility has been expressed for a long time (see for example^{23}, ch. 2) and as a result of the use of this idea can be indicated the obtaining of the Lorenz-Lorentz formula for the polarizability of the sphere in a homogeneous external field. Analogously \({\rm X}_{ij}^{(b)} ({\mathbf{R}},\omega )\) reflects the processes of self-action for the particles located inside the homogeneous isotropic medium. But, \(\Xi_{ij}^{(b)} ({\mathbf{R}},\omega )\) takes into account the processes of self-action via small nanoparticles and, of course, depends on the distance between the nanoparticles (see Fig. 1).

It should be noted that we performed the self-consistency procedure when calculating the ponderomotive forces between the nanospheres. Self-consistency was achieved within the framework of the method developed in^{20}. A similar self-consistent procedure has been demonstrated in^{24}. But in the present work, we performed the self-consistency with the pseudo-vacuum Green’s function method and obtained the self-consistent equations for the effective susceptibilities of nanoparticles taking into account that nanoparticles are not point-like. and have their shape and dimensions, unlike^{24} the same self-consistency was achieved for the nanoparticle dipole moments in the point dipole approximation. Then the interest of the present work is not only the self-consistency but also the taking into account of the inhomogeneities of the local fields inside and at the level of the nanoparticles.

In this respect, it should be noted that the ponderomotive forces were calculated at the points of maximum gradients of the local fields and, of course, the integrations of the density of the ponderomotive forces were made on the volume of the shell of the nanoparticle in the local field hotspot domains.

For the calculation of local field strength (including “hot spots”), the self-consistency equation (called the Lippmann–Schwinger equation)^{25} Should be used

$$\begin{aligned} E_{i} ({\mathbf{R}},\omega ) & = E_{i}^{(0)} ({\mathbf{R}},\omega ) \\ & \,\,\,\, – k_{o}^{2} \int\limits_{{V_{v} }} {d{\mathbf{R}}^{^{\prime}} G_{ij} ^{(m)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega )\Xi_{jl}^{(v)} ({\mathbf{ R}}^{^{\prime}} ,\omega )E_{l}^{(0)} ({\mathbf{R}}^{^{\prime}} ,\omega )} . \\ \end{aligned}$$

(4)

Then the force density (the force per unit volume) acting on the large nanoparticle can be written

$$F_{i} ({\mathbf{R}}) = \left( {P_{j} ({\mathbf{R}}) \cdot \frac{\partial }{{\partial x_{j} } }} \right)E_{i} ({\mathbf{R}}),$$

(5)

where \(P_{j} ({\mathbf{R}}) = \varepsilon_{0} \Xi_{jl}^{(v)} ({\mathbf{R}})E_{l}^{0} ( {\mathbf{R}})\) is the polarization (local dipole moment) of the nanoparticle envelope.

Since in formula (5) the polarization and the field are local characteristics and the polarization is continuously distributed over the shell of a large particle, the force acting on an element of the volume of the shell is calculated as the integral of (5) on this volume. In this sense, we speak of the “density” of the force. This means that the force acting on the small volume of the nanoparticle is calculated for different areas of this nanoparticle. And since the force (5) is determined by the gradient of the local field, when calculating the normal and tangential components of the force acting on an element of the shell volume, such an element is chosen which corresponds to the maximum value of the gradient field (hot spot area).

For example, eq. (5) is equal to the force density equation in Eq. (3.62) of^{26}. If the dipole moment density, i.e. the polarization, is replaced by the dipole moment, then Eq. (5) is a force equal to the first term of Eq. (2.2) of^{27}. Then, using (4) we get

$$\begin{gathered} F_{i} ({\mathbf{R}}) = \left( {\varepsilon_{0} \Xi_{jk}^{(v)} ({\mathbf{R}}) E_{k}^{0} ({\mathbf{R}}) \cdot \frac{\partial }{{\partial x_{j} }}} \right) \cdot \hfill \\ \left[ {E_{i}^{(0)} ({\mathbf{R}},\omega ) – k_{o}^{2} \int\limits_{{V_{v} }} {d{\mathbf{R}}^{^{\prime}} G_{is}^{(m)} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega )\Xi_{sl}^{(v)} ({\mathbf{R}}^{^{\prime}} ,\omega )E_{l}^{(0)} ({\mathbf{R}}^{^{\prime}} ,\omega )} } \right]. \hfill \\ \end{gathered}$$

Taking into account that at distances around the characteristic linear dimension of the nanoparticle, the external field is rather constant, this equation can be reduced to the following

$$\begin{gathered} F_{i} ({\mathbf{R}}) = – k_{o}^{2} \varepsilon_{0} \Xi_{jk}^{(v)} ({\mathbf {R}})E_{k}^{0} ({\mathbf{R}}) \cdot \hfill \\ \int\limits_{{V_{v} }} {d{\mathbf{R}}^ {^{\prime}} \frac{\partial }{{\partial x_{j} }}G_{est}^{(m)} ({\mathbf{R}},{\mathbf{R}}^ {^{\prime}} ,\omega )\Xi_{sl}^{(v)} ({\mathbf{R}}^{^{\prime}} ,\omega )E_{l}^{(0 )} ({\mathbf{R}}^{^{\prime}} ,\omega ).} \hfill \\ \end{gathered}$$

(6)

Note that in (6) the differentiation is about the components of \({\mathbf{R}}\). The action of the force on the entire surface of the shell can obviously modify the properties of the shell until it is destroyed.

In the ellipsoid susceptibility model, homogeneous field tensor \(\Xi_{jk}^{(v)} ({\mathbf{R}})\) does not depend on coordinates^{23.24}that’s to say \(\Xi_{jk}^{(v)} ({\mathbf{R}}) \to \Xi_{jk}^{(v)}\). Then, the dependence of the force of (6) on the coordinate originates in the local distribution of the field on the surface of the large nanoparticle only – via the dependence of the Green’s function \(G_{est}^{(m)} ({\mathbf{R}},{\mathbf{R^{\prime}}},\omega )\) on **R** [see (6)]. Remember that the outer field \(E_{l}^{(0)} ({\mathbf{R}},\omega )\) is the constant in the near-field approximation. Then it can be obtained from (6)

$$F_{i} ({\mathbf{R}}) = – k_{o}^{2} \left( {\varepsilon_{0} \Xi_{jk}^{(v)} \cdot \int\ limits_{{V_{v} }} {d{\mathbf{R}}^{^{\prime}} \frac{\partial }{{\partial x_{j} }}G_{is}^{(m )} ({\mathbf{R}},{\mathbf{R}}^{^{\prime}} ,\omega )\Xi_{sl}^{(v)} } } \right)E_{k} ^{(0)} E_{l}^{(0)} .$$

(seven)

As “hotspots” (strong local field domains) are defined by the dependence of Green’s function on **R**the forces acting on the large nanoparticle (shell) will be concentrated at the hot spots (Fig. 3c).

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