1. Introduction

Nanocomposite materials consisting of polymeric matrices with embedded inorganic nanoparticles are promising for numerous applications in photonics, electronics and biomedicine due to their unique properties [1–5].

The nanoparticles can be prepared separately ex-situ and then introduced within the polymeric materials by different ways [6]. In-situ technique suggests growing of nanoparticles just within the matrix due to either thermal/photothermal or photochemical destruction of precursor molecules initially occurring within the bulk.

Laser initiated growth of inorganic nanoparticles directly within polymer matrices provides an opportunity for laser recording of micro structures possessing the nanoparticles, e.g., by direct writing. This offers a new promising approach towards laser printing of optoelectronic devices, as has been discussed in detail in a recent review [7]. Laser printing of microstructures by means of a mask of colloidal microparticles, which was demonstrated in [8], implies additional opportunities in this field.

The laser-induced nanoparticles can be of different nature. They can be metallic nanoparticles demonstrating plasmon resonance response. They can be semiconductor nanoparticles with optical absorption and luminescence spectra significantly dependent on their sizes.

Review [7] mainly discusses the approach where the laser heating is the only reason for initiation of nanoparticle growth. The considered precursors are decomposed by laser radiation purely photothermally, that is due to thermally activated reaction.

The precursor decomposition could also result from the photochemical effect of a UV photon. It is interesting that in some cases the quantum yield of such photochemical reaction can significantly depend on the ambient temperature [9], thus, laser radiation can simultaneously provide both photochemical and photothermal effects [8].

The current trend in this field is towards discovering new precursors for different semiconductor nanoparticles, elucidating the features of nanoparticle growth at different conditions and laser micropatterning of these materials. It is recognized that for the laser patterning purposes the precursors should be soluble within the polymer matrix [10–12].

Another promising development of this activity mentioned in [13] relies on irradiation-induced growth of several kinds of nanoparticles within the same sample. It suggests incorporating several kinds of precursors simultaneously within the same media. Laser irradiation of such media could provide nanoparticles of different nature at different domains, different nanoparticles within the same domain and, closer to the subject of this paper, laser-induced core-shell nanoparticles. The latter will open up the possibilities for the laser-induced growth of semiconductor/semiconductor and metal/semiconductor nanoparticles. The former, e.g., CdS/ZnS, CdSe/CdS, etc., core-shells are nanoparticles with enhanced luminescent quantum yields [14–16]. The metal/semiconductor core-shells are promising for exciton-plasmon nano systems, including the “spaser” [17–21].

Below we consider one particular theoretical problem within the aforementioned trend. We assume that due to previous manipulations there exist some metallic (e.g., Au) nanospheres within a polymer matrix. The polymer medium around these nanoparticles contains precursor molecules. The metal nanoparticles are heated by picosecond laser pulses with a frequency close to that of the plasmon resonance. This heating provides generation of semiconductor elementary species (e.g., CdS) in the vicinity of the nanosphere due to the photothermal precursor destruction. These species diffuse towards the sphere, forming the shell layer. The idea that the light-to-heat conversion by metallic nanoparticles can be used for photothermal chemical reaction was recognized recently and presented in some review papers [22–25] and recent publications [26,27]. In [28,29] the authors address quite a close problem of temperature regimes of optically pumped spasers and nanolasers, however, the problem of laser-induced core-shell nanoparticle growth within the polymer matrix has not yet been considered. We will use our previous results on the kinetics of reaction initiated by laser heating of a metallic nanoparticle [30]. We show that because of sharp localization of laser heating provided by the light-to-heat conversion of metal nanoparticles emphasized by the Arrhenius dependence of thermally activated precursor decomposition the diffusion processes appear to be effective even in such a matrix as polymers, despite the relatively small diffusion coefficients. This offers new opportunities for building effective diffusion-supplied nanoreactors within the bulk of condensed materials.

The paper is organized as follows. In the next sections, we present a mathematical formulation of the problem and its approximate solution. This solution relies on the large parameter, the ratio of activation temperature to the actual temperature of the laser-heated metal nanoparticle. The solution is also based on considerations of two time scales, fast picosecond-nanosecond time of afterpulse cooling of the nanoparticles and heat propagation within the surrounding matrix with precursor molecules, and the slow time of multi-pulse irradiation. Within this slow time scale, the millisecond interval between the pulses can be considered as infinitesimally small. Multipulse heating of the nanoparticle and the surrounded matrix can result in the thermal destruction of the latter. We consider the model of thermal decomposition of the polymer matrix actual for its ultrafast laser heating and cooling mediated by the metallic nanoparticle. We also elucidate the restrictions on the shell growth process imposed by the matrix stability.

2. Problem formulation and the approximate solution

Photoinduced nanocomposites originate from light-induced destruction of special compounds, precursors, embedded in the polymer matrix. The products of such destruction, elementary species, interact with each other leading to the formation of inorganic nanoparticles.

We study the possibility of employing these elementary species to build core-shell nanoparticles by depositing them onto existing metal nanoparticles. For this sake, we consider a metal nanoparticle embedded in a polymer matrix that contains a dissolved molecular precursor of another inorganic compound.

Irradiating the spherical metal (e.g., gold) nanoparticle by a short (picosecond) pulse with the wavelength close to the maximum of the plasmon resonance results in heating of the nanoparticle. Just after the pulse, the heat diffuses through the surrounding material initiating the thermally activated reaction of precursor destruction and thereby generation of the elementary species constituting the semiconductor nanostructures.

The first (thermal) part of the problem can be described by the spherically symmetric heat diffusion equation (D_T is the heat diffusivity of the matrix)

This boundary condition addresses the cooling of the metallic nanoparticle by heat diffusion. Here

Here c and p are specific heat and density, respectively, while subscript p is related to the particle. For a gold nanoparticle in poly(methyl methacrylate) matrix $\beta = 1.821.$ Initial conditions are

Here we have introduced the characteristic time

In what follows, we will consider the precursor destruction as a thermally activated reaction

Here A is the number density of precursor molecules, B is the number density of the elementary species extracted from the precursor molecules, T_A is the activation temperature of precursor destruction, and K_A is the corresponding preexponential factor. It is assumed that each precursor molecule generates only one elementary product species.

We consider multi-pulse irradiation of the material, suggesting that only a small fraction of precursor is decomposed by a single pulse. In this case, as has been shown in [30], the relation holds

Here, the decrement of precursor number density (the left hand side) at the point r is proportional to the value of the number density at this point before the pulse and r-dependent factor

Within the considered problem, we assume that the material is stable at the thermostat temperature, that is

Here N_p is the number of pulses, and ${N_\textrm{p}}\nu _\textrm{p}^{ - 1}$ is the irradiation time. This condition will be considered in detail below.

In Eq. (9) we formally take the integration up to the time between pulses, however, as it is seen from solution Eq. (5), the temperature relaxes to the thermostat temperature for time about 10t_fast (see Eq. (6)).

For typical values ${r_\textrm{p}} \approx 10\textrm{nm}$ and ${D_T} \approx {10^{ - 3}}{{\textrm{c}{\textrm{m}^\textrm{2}}} / \textrm{s}}$ this time is about 10⁻⁸s. Typical pulse repetition rate ${\nu _\textrm{p}} = \textrm{ }1\textrm{KHz}$, so the time between pulses is 10⁻³s. It means that practically all the time between the pulses all the parts of the sample have the temperature of the thermostat. The average heating of the material by the laser pulses and its contribution to the thermostat temperature is considered below.

There are different time scales in the considered problem differing in orders of magnitude: the pulse duration (several picoseconds), the cooling time just after the pulse (10 ns), the time interval between the pulses (millisecond) and irradiation time (tenths of minutes). The set Eqs. (1–4) and solution Eq. (5) are relevant for the fast time scale of the problem. considering the time just after the pulse. Below we consider slow time processes corresponding to thousands of time intervals between the pulses.

Between the pulses, the monomers diffuse up to the metal nanoparticle forming the semiconductor shell. Because the cooling time is 4 to 5 orders of magnitude smaller than the time interval between the pulses, the diffusion occurs predominantly at the constant thermostat temperature, T_therm. On the overall irradiation time scale, the time between the pulses is very small. The corresponding diffusion problem reads

It was shown in our previous publication [32] that the diffusion of precursor molecules is very important for the effective formation of local structures in photoinduced nanocomposites. Indeed, when the precursor destruction occurs only within a small volume, the amount of monomer generated within this volume is small and a flux of precursor molecules from the surroundings is required to provide a significant amount of elementary species forming the shell structure. The equation for the precursor number density reads

The schematic of the considered process is seen in Fig. 1.

 

Fig. 1. Schematic of laser-induced shell growth. The energy of the laser pulse comes to a gold nanoparticle within the polymer matrix. The matrix contains the precursor molecules (top left). The neighborhood of the particle is heated due to heat diffusion. The elementary species are formed. This is the fast time scale, just after the pulse (top right). The elementary species diffuse toward the nanoparticle surface and are deposited on it (bottom left). The precursor molecules diffuse toward the neighborhood of the nanoparticle and replace the broken ones (bottom right). The bottom pictures are of slow time scale.

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We are interested in the flux

The approximate solution of the above problem can be found in Appendix A. It reads

Here $\gamma = {r_\textrm{p}}{\nu _\textrm{p}}F/{D_A}, F = \int\limits_{{r_\textrm{p}}}^\infty {{G_A}(r)dr} ,$ and the coordinate scale Δr is introduced in such a way that for $r > {r_\textrm{p}} + \Delta r$, ${G_A} \approx 0.$ The solution is obtained with first-order accuracy with respect to the value of parameter Δr/r_p. When obtaining the approximate solution, we take into account that the problem contains a big parameter ${T_A}/({T_{\textrm{therm}}} + \Delta {T_\textrm{p}}) > > 1.$ Typically, activation energy of thermally activated reaction is larger than 1 eV, i.e., ${T_A} > 10000\textrm{K,}$ whereas the maximal value of temperature T is at least an order of magnitude smaller. Employing the saddle point method [33] with respect to this parameter allows one to obtain the approximate expressions for F and Δr (see Appendix B)

The physical meaning of the term proportional to Δr/r_p in Eq. (17) is the fraction of created elementary species that diffuse from the nanoparticle surface and are not deposited on it. These species are lost for the growth process.

3. Results and discussion

According to Eq. (18), F is an increasing function of ΔT_p (see Fig. 2(a)). In its turn, ΔT_p is an increasing function of laser fluence. If pulse duration is compared with the time of electron-phonon relaxation, the temperature increment is approximately proportional to laser fluence, Ф, and can crudely be approximated as $\Delta {T_\textrm{p}} \approx \frac{3}{{4\mathrm{\pi }r_\textrm{p}^3}}\frac{{{\sigma _\textrm{p}}\Phi }}{{{c_\textrm{p}}{\rho _\textrm{p}}}}.$ Here ${\sigma _\textrm{p}}$ is the nanoparticle absorption cross-section that can be calculated using Mie theory. Thus, increasing laser fluence increases F. As it is seen from Eq. (17), if $\gamma < < 1$ the deposition rate, flux (typically $\Delta r/{r_\textrm{p}} < < 1$, see Eq. (19))

 

Fig. 2. Typical temperature dependences of a - F, see Eq. (18), b – flux J_B, see Eq. (17), and c – A(r_p)/A_∞, see Eq. (16). Calculations are performed for T_A=16000 K, K_A=10¹³s⁻¹, and for different values of precursor diffusion coefficients. Here and in all the following calculations the nanoparticle radius r_p, A_∞ = 6·10¹⁹cm⁻³, v_p = 10³s⁻¹.

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Irradiation of absorbing materials such as the polymer film with gold nanoparticles inevitably provides the overall heating of the material. With the gold mass fraction of about 1%, the concentration of gold nanoparticles of 20 nm radius within PMMA film is ${N_{\textrm{Au}{\kern 1pt} \textrm{p}}} = 1.8 \cdot {10^{13}}\textrm{c}{\textrm{m}^{\textrm{ - 3}}}.$ The absorption cross-section of such a nanoparticle ${\sigma _\textrm{p}},$ according to Mie theory, is 4.7·10⁻¹¹cm² (gold nanoparticle with the optical properties from Ref. [35], wavelength 532 nm, matrix refractive index 1.49). Thus, the effective absorption coefficient is $\alpha = {N_{\textrm{Au}{\kern 1pt} \textrm{p}}}{\sigma _\textrm{p}} = \textrm{ }8.46 \cdot {10^2}\textrm{c}{\textrm{m}^{\textrm{ - 1}}}.$ For pulse heating of such a nanoparticle up to $\Delta {T_\textrm{p}} = 1000\textrm{K}$ one needs a fluence of approximately $\Phi = 1.77 \cdot {10^{ - 3}}\textrm{J/c}{\textrm{m}^\textrm{2}}.$ Taking in mind that in our consideration the repetition rate is 1KHz, the average power density is $I = 1.77\textrm{W/c}{\textrm{m}^\textrm{2}}.$ For PMMA films, the density is $\rho = 1.27\textrm{g/c}{\textrm{m}^\textrm{3}}$ and specific heat is $c = 1.27\textrm{J/gK}\textrm{.}$ This gives the heating rate $\frac{{\alpha I}}{{c\rho }} \approx {10^3}\textrm{K/s}\textrm{.}$ It means that the free-standing film will be destroyed immediately. In order to prevent this, we can place the film between two massive quartz plates, as it was done in [36]. Suggesting that the plates are kept at constant temperature, T_plate, and the reflectivity of the system is small, we can estimate the stationary temperature increment relative to this temperature within the film with thickness $l = 1\mathrm{\mu }\textrm{m}$ to be

In all our following considerations, we put ${T_{\textrm{therm}}} = 370\textrm{K}\textrm{.}$ In order to maintain this temperature, the quartz plates should be kept at a temperature close to 0°C. If $\Delta {T_\textrm{p}} = 650\textrm{K,}$ then the quartz plates should be kept at about 35°C.

When employing the precursor which can thermally decomposed at some elevated background temperature, one faces the problem that during the irradiation time the precursor can provide growth of parasitic nanoparticles. For example, in addition to the desired core/shell Au/CdS nanoparticles one can obtain parasitic pure CdS nanoparticles away from the metal cores. Avoiding this process imposes some restrictions on the kinetic parameters of precursor destruction reaction, T_A, K_A.

The fraction of destructed precursor molecules during irradiation time t_irrad should be smaller than some threshold value $\Delta {A_{\textrm{thresh}}}/{A_\infty }.$ For CdS nanoparticle growth at ${A_\infty } = 6 \cdot {10^{19}}\textrm{c}{\textrm{m}^{\textrm{ - 3}}},$ $\Delta {A_{\textrm{thresh}}}/{A_\infty } \approx {10^{ - 2}}.$ If the amount of broken precursor molecules exceeds this value, the growing CdS nanoparticles start absorbing within the near UV range, affecting the optical properties of the sample [37]. Substituting in Eq. (7) $T = {T_{\textrm{therm}}}$ we find the rate of the precursor destruction at thermostat temperature at $A \approx {A_\infty }$ and find the limiting irradiation time by dividing $\Delta {A_{\textrm{thresh}}}$ by this rate to obtain the first limiting time ${t_{\textrm{irrad}}} \le {t_{\lim 1}}$,

Another restriction is imposed by the stability of the matrix under the effect of laser pulses. When a plasmonic nanoparticle is irradiated at a wavelength close to the plasmon resonance, the strong nearfield can provide ionization of the matrix. In order to avoid plasma effects, we consider the laser pulses of picosecond-ten picosecond duration. Because the time of electron-phonon relaxation in metals is about 10ps, from the point of view of heating of the nanoparticle, the effects of femtosecond and picosecond pulses are similar, whereas the nearfield is much weaker for picosecond pulses. Therefore, we consider only the effect of laser heating of the matrix. This aspect should be specifically clarified for each matrix. Typically, at high enough temperatures, if oxidation is not involved, the polymer destruction proceeds through random bond breaking. In addition polymers, which can be synthesized by the radical polymerization, the chain breaking is followed by depolymerization or unzipping [38,39]. This destruction process goes through cleavage of either C-C or C-N bonds in the main chain or in the side groups. For example, in PMMA, the side group is ester group. The ester group cleavage reaction is practically irreversible. It results in production of gaseous species. It can start the unzipping process [38]. Damage of adjacent sites could lead to carbonization up to charring. The typical activation energy of breaking C-C bonds is about 3 eV. The reported data on activation energy of the random chain breaking in the most studied PMMA matrix scatters around this value. The data on preexponential factor in kinetic constant of bond cleavage shows a discrepancy of several orders of magnitude. It can hardly be smaller than the C-C oscillation frequency measured by IR spectroscopy (approximately 3.3·10¹³s⁻¹). The experimentally estimated factor is at least an order of magnitude larger. Below we consider two models for the matrix response to the effect of laser heating of the metallic nanoparticles. For relatively stable matrix 1, we take the activation energy of matrix destruction to be somewhat larger than 3 eV, namely 300kJ/mol $({T_{\textrm{C1}}} = 36101\textrm{K),}$ see [39] and the preexponential factor ${K_{\textrm{C1}}} = 3.3 \cdot {10^{13}}{\textrm{s}^{\textrm{ - 1}}},$ as discussed above. This is the strong matrix. For a less stable matrix 2, we assign the activation energy of destruction somewhat smaller than 3 eV, namely, 62kcal/mol [38] (activation temperature ${T_{\textrm{C2}}} = 31221\textrm{K}$) and ${K_{\textrm{C2}}} = 2 \cdot {10^{14}}{\textrm{s}^{\textrm{ - 1}}}.$ The latter is the lower limit for the value of this parameter in [38]. This is the weak matrix.

In order to address the bond breaking process, we introduce C, the number density of virgin bonds. Its evolution just after the pulse is addressed by the following equation

Here T(r, t) is described by Eq. (5). Consideration similar to Eqs. (7–9) yields that after N laser pulses

Here C₀ is the initial value of C and

Depolymerization or unzipping is the process when the free radical occurring at the end of a polymer chain provides splitting off of the monomeric unit from the chain.

The activation energy of PMMA depolymerization is known to be 23.4kcal/mol $({T_M} = 11783\textrm{K)}$ [40]. Taking into account the data on polymerization constant (reverse process), activation energy 4.7kcal/mol and preexponential factor 0.705·10³litre/mol, and the temperature 220°C at which the depolymerization and polymerization have the same reaction rates [40] yields estimate for the preexponential factor of depolymerization ${K_M} = \textrm{ }0.85 \cdot {10^{15}}{\textrm{s}^{\textrm{ - 1}}}.$

The number of monomers unzipped from the existing radical per pulse can be found from the equation

The solution of this equation reads

Here

All the G_X(r) functions (here X = A, C, M, see Eq. (9), Eq. (25), Eq. (28)) have the same form and have their maximum values at $r = {r_p}.$ We will designate ${G_X}({r_p}) \equiv {G_X}_p.$

For fixed values of all the parameters T_therm, T_X, K_X, G_Xp is a function of ΔT_p. This function within the considered approximations can be found as (see [30] and Appendix B):

Fraction G_Cp(ΔT_p), see Eq. (25), Eq. (29), is shown in Fig. 3(a). Figure 3(b) shows the number of unzipped monomers per one radical per pulse G_Mp(ΔT_p) according to formulas Eqs. (27–29).

 

Fig. 3. a - The fraction of broken bonds per laser pulse at the surface of the nanoparticle as a function of temperature increment ΔT_p. b - The number of monomers unzipped per radical for a single pulse at the surface of the nanoparticle as a function of temperature increment ΔT_p.

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It is seen that for relevant values of ΔT_p the number of unzipped monomers is close to unity. Taking in mind that at thermostat temperature the polymerization process dominates over the depolymerization and that the lifetime of the free monomer just near the radical at such temperature is about 10⁻⁴−10⁻⁵s, it is possible to neglect the depolymerization when considering the matrix destruction at a useful working ΔT_p. Thus, even in addition polymers the most dangerous destruction process here is the bond cleavage. Below we consider the material response within the bond cleavage model addressed by Eqs. (23–25), Eq. (29) for two sets of kinetic parameters, the strong and weak matrices discussed above.

Let us require that the allowed fraction of broken bonds for the irradiation time should not exceed $\Delta {C_{thresh}}/{C_0} = 0.01$ at $r = {r_p}$. According to Eq. (24) and bearing in mind that ${G_C}({r_\textrm{p}}) = {G_{C\textrm{p}}}(\Delta {T_\textrm{p}})$

Here N is the number of laser pulses. For the considered small values of $\Delta C/{C_0}$, the latter equation reads

Taking into account that $N = {t_{\textrm{irrad}}}{\nu _\textrm{p}}$, where ${\nu _\textrm{p}}$ is the pulse repetition rate, we find from the condition that $\Delta C \le {C_{\textrm{thresh}}}$ the second limit for irradiation time, ${t_{\textrm{irrad}}} < {t_{\lim 2}}$,

Below we consider the maximum grown shell volume, and correspondingly the maximum shell thickness that can be obtained taking in mind the time irradiation restrictions Eq. (22) and Eq. (30) for given precursor (T_A, K_A) and matrix (T_C, K_C) parameters by varying the temperature increment ΔT_p. Here we will use the upper estimation of flux Eq. (20). In any case, ${t_{\textrm{irrad}}} = \min ({t_{\lim 1}},{t_{\lim 2}})$ (see Eq. (22) and Eq. (30)). The shell volume is

Here ${v_B}$ is the molecular volume of the B component within the shell. For example, for CdS element ${v_B} \approx 0.5 \cdot {10^{ - 22}}\textrm{c}{\textrm{m}^\textrm{3}}$ (see [37]).

We start with considering a relatively large temperature increment ΔT_p at which ${t_{\lim 2}} < {t_{\lim 1}},$ that is ${t_{\textrm{irrad}}} = {t_{\lim 2}}.$ We substitute in Eq. (31) the flux J_B in the form of Eq. (20) with F in the form of Eq. (18). We take irradiation time in the form of Eq. (30) with G_Cp(ΔT_p) found in Eq. (29). It reads

Because typically ${T_C} > {T_A},$ it follows from Eq. (32) that $\frac{{d{V_{\textrm{shell}}}}}{{d(\Delta {T_\textrm{p}})}} < 0.$ When increasing ΔT_p the flux increases; however, the lifetime of the matrix decreases stronger, thereby decreasing the net shell volume. Decreasing ΔT_p results in a decrease in the flux but allowed irradiation time t_lim2 increases in such a way that the net shell volume increases. However, one cannot increase the irradiation time above t_lim1. It means that the maximum value of shell volume is reached when

In Eq. (33) we put t_lim2 in the form of Eq. (30), and t_lim1 in the form of Eq. (22). After simple algebraic transformations Eq. (33) yields

For the fixed kinetic constants of precursor (T_A, K_A) and matrix, (T_C, K_C), Eq. (34) is an equation with respect to ΔT_p. By solving this equation using relation Eq. (29) at $X = C$ or using Fig. 3, one can find the optimum value ΔT_{p max} at which the maximal shell volume can be obtained taking in mind irradiation time limitations t_lim1 and t_lim2 (see Eq. (22) and Eq. (30)). Substituting this value into Eq. (18), Eq. (20) one finds the maximum value for flux ${J_B}_{\max } = {J_B}(\Delta {T_\textrm{p}}{}_{\max })$. Taking into account that here ${t_{\textrm{irrad}}} = {t_{\lim 1}} = {t_{\lim 2}}$ (see Eq. (33)), Eq. (31) yields ${V_{\textrm{shell}}}_{\max } = {v_B}{J_B}_{\max }{t_{\lim 1}}$ . Expression for ${t_{\lim 1}}$ comes from Eq. (22). The corresponding maps presenting dependences of the shell thickness ${d_{\textrm{shell}\max }}$ on the set (T_A, K_A) for two sets of matrix constants - (T_C1, K_C1), the strong matrix, and (T_C2, K_C2), the weak matrix - are shown in Figs. 4(a) and 5(a). Here the shell thickness is obtained from the relation

 

Fig. 4. Strong matrix. T_C = 36101 K, K_C = 3·10¹³ s⁻¹. Dependence of maximum shell thickness on kinetic parameters T_A and K_A (a), corresponding value of temperature increment (b), needed irradiation time (c), and characteristic value of diffusion coefficient D_lim (d). If D_A >> D_lim, the precursor diffusion cannot affect the shell growth process. The dotted line in Fig. 5(a) shows the border below which t_irrad > 10⁵s.

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Fig. 5. Weak matrix. T_C = 31283 K, K_C = 2·10¹⁴s⁻¹. Dependence of maximum shell thickness on kinetic parameters T_A and K_A (a), corresponding value of temperature increment (b), needed irradiation time (c), and characteristic value of diffusion coefficient D_lim (d). If D_A >> D_lim, the precursor diffusion cannot affect the shell growth process. The dotted line in Fig. 5(a) shows the border below which t_irrad > 10⁵s.

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At small ${d_{\textrm{shell}}} < < {r_\textrm{p}}$, ${V_{\textrm{shell}}} \approx 4\mathrm{\pi }r_\textrm{p}^2{d_{\textrm{shell}}}.$

Comparing the dependences of the maximal shell thickness on kinetic parameters of precursor destruction one can see that the strong matrix allows the shell growth to significantly larger thicknesses than the weak matrix. Besides the values of shell thickness (Fig. 4(a) and Fig. 5(a)), the figures also present corresponding values of optimal temperature increment (Fig. 4(b) and Fig. 5(b)) as well as needed irradiation time ((Fig. 4(c) and Fig. 5(c)). Figures 4(d) and 5(d) shows the corresponding values of D_lim (see Eq. (21)). It is seen that this value typically does not exceed 10⁻¹²cm²/s. The estimation of the diffusion coefficient of elementary species constituting the CdS nanoparticles at temperatures close to the glass transition point of the matrix [37] yields the values of ${D_B} \approx {10^{ - 9}} - {10^{ - 10}}{{\textrm{c}{\textrm{m}^\textrm{2}}} / \textrm{s}}.$ Taking in mind the size dependence of the diffusion coefficient in soft polymer matrices (see, e.g., the discussion in [41] and references therein), the value of ${D_A} \approx {10^{ - 12}}{{\textrm{c}{\textrm{m}^\textrm{2}}} / \textrm{s}}$ looks to be a reliable value for the precursor diffusion coefficient. Figures 4(b) and 5(b) shows that within a significant area of the map the optimal ΔT_p is higher than 1000 K. At a temperature increment close and above 1000 K other processes that are less significant at lower temperatures should be taken into account, e.g., unzipping, see Fig. 3(b). This would make our model of matrix destruction not relevant. At higher temperatures, the effect of metallic core particle melting and evaporation also should be taken into account [42,43]. Figures 4(c) and 5(c) show that very long irradiation times are sometimes needed to gain the maximum shell thickness. The dotted line in Figs. 4(a) and 5(a) indicates the border of the area (bottom right part) where the maximal shell thickness is obtained for time longer than 10⁵s.

For reasons of practical applicability, we can introduce additional restrictions on irradiation time, t_lim3, i.e., ${t_{\textrm{irrad}}} \le {t_{\lim 3}},$ and on the value of temperature increment, ΔT_lim, i.e., $\Delta {T_p} \le \Delta {T_{\lim }}.$ The results of corresponding calculations of the shell thickness for two matrices are presented in Figs. 6(a) and 7(a). We employ the values ${t_{\lim 3}} = {10^5}\textrm{s,} \Delta {T_{\lim }} = 1000\textrm{K}\textrm{.}$ The algorithm of obtaining Figs. 6 and 7 can be found in Appendix C.

 

Fig. 6. Strong matrix. T_C = 36101 K, K_C = 3·10¹³ s⁻¹. Dependence of maximum shell thickness on kinetic parameters T_A and K_A with restrictions t_irrad < 10⁵s, ΔT_p < 1000 K. (a) Approximation when D_A >> D_lim. Dotted line shows the border below which D_lim < 10⁻¹²cm²/s. (b) Results of calculations with D_A = 10⁻¹²cm²/s.

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Fig. 7. Weak matrix. T_C = 31283 K, K_C = 2·10¹⁴s⁻¹. Dependence of maximum shell thickness on kinetic parameters T_A and K_A with restrictions t_irrad < 10⁵s, ΔT_p < 1000 K. (a) Approximation when D_A >> D_lim. Dotted line shows the border below which D_lim < 10⁻¹²cm²/s. (b) Results of calculations with D_A = 10⁻¹²cm²/s.

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In Figs. 6(a) and 7(a), the dotted line distinguishes the region of parameters where the value of D_lim is less than 10⁻¹³cm²/s (bottom right part from the dotted line). Here, there is no saturation related to the diffusion restrictions at ${D_A} \ge {10^{ - 12}}\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s}\textrm{.}$ Figs. 6(b) and 7(b) demonstrate the results of shell thickness calculations when the diffusion is taken into account. Here relation Eq. (17) for flux J_B is used in the calculation instead of Eq. (20) and ${D_A} = {10^{ - 12}}\textrm{c}{\textrm{m}^\textrm{2}}\textrm{/s}\textrm{.}$ The comparison of Figs. 6(a) and 6(b), as well as Figs. 7(a) and 7(b) shows that the diffusion of precursor molecules is not a limiting factor for the shell growth process. It can be seen from these figures that for realistic values of parameters ${K_A} \approx {10^{12}} \div {10^{13}}{\textrm{s}^{ - 1}}$ and ${T_A} \approx 16000 \div 17000\textrm{K}$ in the strong matrix it can be possible to grow for a reliable time the shell of thickness close to the radius of the nanoparticle. In the weak matrix, this thickness would be an order of magnitude smaller.

In some papers (see, e.g. [24,42]) a temperature drop on the surface between the heated nanoparticle and the surrounding media is considered. This interfacial boundary resistance and inverse quantity interfacial thermal conductance usually denoted as G was estimated mainly from the analysis of experimental data on cooling of heated by femtosecond pulses metal nanoparticles embedded in different dielectric materials mainly in water and alcohol. The estimated value of G is about 100MW·m⁻²·K⁻¹ at temperatures close or below 100°C. There is no data on G value at temperatures near 1000K studied in the present paper. It is only known that G is a strongly increasing function of temperature, which decreases the relative temperature drop. At values of G about 10³- 10⁴MW·m⁻²·K⁻¹ the effect should be either small or negligible. In our paper we do not consider the effect of finite conductance G and use the ‘classical’ boundary condition at the interface between the metal particle and polymer $T({r_\textrm{p}} - 0) = T({r_\textrm{p}} + 0)$ (see e.g. [44,45]).

In the above consideration we did not take into account the reciprocal effect of shell growth on the flux J_B . The growth of the shell will red shift the absorption peak of the particle. This can decrease the absorption cross-section of the particle at the wavelength of irradiation. In praxis, it can be compensated by moderate increase in pulse energy. Besides, the increase in particle radius will accelerate shell growth increasing the flux J_B (see Eq. (17)). All these factors can be taken into account by numerical calculations. The present paper aims at analytical estimations to provide the realistic physical ranges for the parameters involved.

The values of maximum temperature of the nanoparticle in the above consideration are often higher than the typical temperature at the laser ablation threshold for polymers [46,47]. However, the specific feature of the above-considered process is a very short time of effective cooling. Indeed, the expression for the fraction of broken bonds per pulse just near the surface of the sphere, can be represented as ${G_C}_\textrm{p} = {\tau _C}{K_C}\textrm{exp} ( - \frac{{{T_C}}}{{{T_{\textrm{therm}}} + \Delta {T_\textrm{p}}}}),$ where ${\tau _C} = \frac{{r_\textrm{p}^2}}{{{\beta ^2}{D_T}}}\frac{{\mathrm{\pi }{{({{T_{\textrm{therm}}} + \Delta {T_\textrm{p}}} )}^4}}}{{{{({\Delta {T_\textrm{p}} \cdot {T_C}} )}^2}}}$ is the characteristic cooling time for the bond cleavage process. Estimations for $\Delta {T_\textrm{p}} = 1000\textrm{K}$ yield ${\tau _C} \approx 10\textrm{ps,}$ which is much shorter than the cooling time near the threshold of laser ablation of PMMA by ultrashort laser pulses. Thus, at such irradiation regimes when the heat comes to the matrix from the metallic sphere, the matrix can be stable at higher temperatures than are needed for laser ablation due to very fast cooling.

4. Conclusions

We consider the growth process of a shell structure around the metallic nanoparticle within the polymer matrix. The polymer sample, containing the nanoparticles and a molecular precursor of the shell material dissolved within the matrix, is irradiated by picosecond pulses at the wavelength close to the plasmon resonance. Absorption of the laser pulse energy results in immediate heating of metallic sphere followed by the heat diffusion process within the surrounding matrix. Heating of the matrix results in thermally activated decomposition of the dissolved precursor molecules, leading to the extraction of elementary species that form the shell structure around the metallic sphere. This process occurs just after the pulse during the cooling of the heated nanoparticle. Between the laser pulses two diffusion processes take place - the diffusion of the elementary species towards the sphere, resulting in the shell growth, and the diffusion of the precursor molecules replacing the decomposed ones. In order to eliminate diffusion limitations the sample should be kept at a temperature near the temperature of softening of the matrix. This ‘thermostat’ temperature can be realized by putting the sample between massive quartz plates kept at the fixed temperature and by the averaged heating of the sample by laser pulses. The reliability of the laser-induced shell growth requires that the precursor molecules should be stable during the irradiation process at the thermostat temperature and the pulse heating of the metallic sphere should not lead to the destruction of the matrix. These requirements impose restrictions on the maximal thickness of the shell which can be grown on the surface of the metallic nanoparticles. We present a mathematical approach allowing to obtain the maximal thickness of the grown shell as a function of kinetic parameters of the precursor destruction. We also consider the achievable thickness of the shell if additional restrictions, namely, limited irradiation time and limited maximal metal nanoparticle temperature, are imposed. The consideration of two matrices, strong and weak, shows that the stability of the matrix significantly influences the efficiency of the growth process. Another limiting factor is the diffusion of the precursor molecules replenishing their concentration just near the metal nanoparticles. It is shown that the diffusion limitation of the growth process is not significant at typical values of precursor diffusion coefficients of the order of 10⁻¹²cm²/s.