${\eta}_{\alpha}=\frac{{\sigma}_{\alpha}}{\text{\pi}{R}_{p}^{2}}\approx 1.9\times {10}^{-2}$ (28)

Now substituting the above values in Equations (27) and (28), yields
${\sigma}_{\alpha}\approx 2.18\times {10}^{-18}\text{\hspace{0.17em}}{\text{m}}^{2}$ and^{
${\eta}_{\alpha}\approx 1.9\times {10}^{-2}$ }. Since, x ≈ 0.08 < 1, it is assumed that each MNP is quasi-transparent to the incident light. By taking the experimental values of laser power P ≈ 150 mW, spot area
$A\approx 7.83\times {10}^{-3}\text{\hspace{0.17em}}{\text{cm}}^{\text{2}}$ and
$\alpha \approx 2.6\times {10}^{-5}\text{\hspace{0.17em}}{\text{cm}}^{-1}$ then the heat produced per unit volume
$Q={I}_{0}\alpha \approx $
$0.2\text{\hspace{0.17em}}\mu \text{W}\cdot {\text{cm}}^{-3}$ . Similarly, the heat power generated is
$P={V}_{p}Q={I}_{0}{\sigma}_{\alpha}\approx $
$0.4\times {10}^{-12}\text{\hspace{0.17em}}\text{W}$ where V_{p} is the NP volume. To calculate the heat generated inside a NP, it is assumed that the size of a MNP is smaller than the laser wavelength so that electrons inside the MNPs respond collectively to the applied electric field of the laser radiation
${E}_{0}\left[\frac{3{\epsilon}_{w}}{{\epsilon}_{p}+2{\epsilon}_{w}}\right]$ . The heat source is derived from the heat power density
${h}_{\rho}\left(r\right)={\displaystyle {\int}_{v}{h}_{\rho}\left(r\right){\text{d}}^{3}r}$ , where the integral is over V_{p}. (i.e. total heat generated
${Q}_{T}={V}_{p}Q$ ).

2) Temperature distribution

When a laser beam with a Gaussian profile, intensity ${I}_{0}\left(r,t\right)$ and beam diameter 2a interacts with the NFF in water, the radiation is absorbed by the sample (i.e., $\alpha \gg \beta $ ) and subsequent nonradiative decay of excited MNPs electrons results in local heating of the medium. Secondly, ${I}_{0}\left(r,t\right)$ is exponentially attenuated at a radial distance r within the medium and in the propagation direction (depth) z. described by Equation (27)

$I=1/2c{\epsilon}_{w}{\left|{E}_{0}\right|}^{2}{\left|\frac{3{\epsilon}_{w}}{{\epsilon}_{p}+2{\epsilon}_{w}}\right|}^{2}\cdot \mathrm{Im}\left(\frac{{\epsilon}_{p}-{\epsilon}_{w}}{{\epsilon}_{p}+2{\epsilon}_{w}}\right){\text{e}}^{-2{r}^{2}/{a}^{2}}\cdot {\text{e}}^{-\alpha z}$ (29)

The temperature distribution in the medium resembles the profile of the excitation beam and hence a refractive index gradient is created. The temperature distribution around the MNPs (i.e., heat source) placed in a surrounding medium (i.e., water) is described by the parabolic Fourier’s heat conduction equation [43]

${\rho}_{p}\left(r\right){c}_{p}\left(r\right)\frac{\partial T\left(r,t\right)}{\partial t}={K}_{w}{\nabla}^{2}T\left(r,t\right)+{Q}_{s}\left(r,t\right)$ (30)

where $T\left(r,t\right)\text{\hspace{0.17em}}\left(\text{K}\right)$ is local temperature,

$Qs=P/{V}_{p}=\frac{\omega}{\text{8\pi}}{\left|{E}_{0}\right|}^{2}{\left|\frac{3{\epsilon}_{w}}{{\epsilon}_{p}+2{\epsilon}_{w}}\right|}^{2}\cdot \mathrm{Im}\left(\frac{{\epsilon}_{p}-{\epsilon}_{w}}{{\epsilon}_{p}+2{\epsilon}_{w}}\right)\text{\hspace{0.17em}}\left(\text{W}/{\text{m}}^{\text{3}}\right)$ is the heating source, P

is the power of heat generation (W),
${\rho}_{p}\approx 5240\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{3}}$ , and
${c}_{p}\approx 640\text{\hspace{0.17em}}\text{J}/\text{kg}\cdot \text{K}$ are density, and specific heat of Fe_{3}O_{4}, respectively,
${K}_{w}\approx 0.6\text{\hspace{0.17em}}\text{W}\cdot {\text{m}}^{-1}\cdot {\text{K}}^{-1}$ is the thermal conductivity of the water and r is the radial distance from the heated nanoparticles. A characteristic time t_{c}, to establish the temperature profile around a single NP is
${\delta}_{0}^{2}/4{D}_{t}$ _{ }where
${\delta}_{0}\approx {\alpha}^{-1}\approx 38\text{\hspace{0.17em}}\text{nm}$ and
${D}_{t}\approx 1.4\times {10}^{-7}\text{\hspace{0.17em}}{\text{m}}^{2}\cdot {\text{s}}^{-1}$ are optical penetration depth and thermal diffusivity of water respectively, so
${t}_{c}\approx 2.6\text{\hspace{0.17em}}\text{ns}$ , which clearly is a very fast time. Thus, one can determine the thermal diffusion depth into MNP by substituting the value of
${\left({D}_{t}\right)}_{p}={K}_{p}/{\rho}_{p}{c}_{p}\approx 1.8\times {10}^{-6}\text{\hspace{0.17em}}{\text{m}}^{-2}\cdot {\text{s}}^{-1}$ ^{}in
${X}_{T}={\left[4{\left({D}_{t}\right)}_{p}\tau \right]}^{1/2}\approx 2.6$ mm for an exposure time τ = 1 s. Therefore, the condition
${R}_{p}\ll {\delta}_{0}\ll {X}_{T}$ or
$t\gg {t}_{c}$ applies in our case i.e., a non-adiabatic case. In the steady-state regime, the local temperature around a NP (i.e.,
$r\ge {R}_{p}$ ) is described by [44]

$\Delta T\left(r\right)=\frac{P}{4\text{\pi}{K}_{w}}=\frac{{V}_{p}Q}{4\text{\pi}{K}_{w}r}=\frac{{\displaystyle \int I\text{d}{A}_{p}}}{4\text{\pi}{K}_{w}r}=\frac{\left|I\right|{A}_{p}}{4\text{\pi}{K}_{w}r}$ (31)

where A_{p} is the area of NP and the according to Equation (27) the intensity decreases exponentially both in r and z directions. The temperature increases at the surface of NP (i.e., at r = R_{p}) is

$\Delta T\left(r\right)=\frac{{R}_{p}^{2}\text{}{K}_{p}}{{K}_{w}}\cdot \frac{1}{2}c{\epsilon}_{w}{\left|{E}_{0}\right|}^{2}{\left|\frac{3{\epsilon}_{w}}{{\epsilon}_{p}\text{}+\text{}2{\epsilon}_{w}}\right|}^{2}\cdot \mathrm{Im}\left(\frac{{\epsilon}_{p}-{\epsilon}_{w}}{{\epsilon}_{p}+2{\epsilon}_{w}}\right)$ (32)

$\Delta T\left(r\right)=\frac{I{R}_{p}^{2}{K}_{p}}{{K}_{w}}\cdot \mathrm{Im}\left(\frac{{\epsilon}_{p}-{\epsilon}_{w}}{{\epsilon}_{p}+2{\epsilon}_{w}}\right)$ (33)

where K_{p} is the thermal conductivity of MNP So,
$\Delta T\left(r\right)\propto {R}_{p}^{2}$ and the total heat current from the surface of NP is given by
${K}_{w}{A}_{p}\partial \Delta T/\partial r$ . It is interesting to note that the size dependence of the temperature increase is governed by the total rate of heat produced and by the heat transfer through the NP. Based on this fact, the temperature increases at later times observed in Figure 6 can be explained caused by for example the agglomeration effect.

3) Thermal conductivity

Since Maxwell’s equation of thermal conductivity is only for first-order approximation, it applies only for mixtures with low particle volume fraction V_{f }and small values of
${K}_{p}/{K}_{w}<10$ , which in this case is ≈ 0.1, so we can write [45]

${K}_{m}={K}_{w}\left[1+{V}_{f}\left(\frac{{K}_{p}}{{K}_{w}}\right)-1\right]$ (34)

Though the K value of NFF depends on factors such as volume fraction, NP size, morphology, additives, pH, temperature, base fluid and NP material [46] [47] . Here,
${K}_{p}=0.\text{6}\text{\hspace{0.17em}}\text{W}\cdot {\text{m}}^{-\text{1}}\cdot {\text{K}}^{-\text{1}}$ ,
${K}_{w}=\text{6}\text{\hspace{0.17em}}\text{W}\cdot {\text{m}}^{-\text{1}}\cdot {\text{K}}^{-\text{1}}$ and the volume fraction V_{f} = 0.012 (for 100 μL MNP solution). Using the above values in Equation (34) it gives
${K}_{m}=1.18\text{\hspace{0.17em}}\text{W}\cdot {\text{m}}^{-\text{1}}\cdot {\text{K}}^{-\text{1}}$ at T = 300 K. Therefore, on would expect a higher thermal conductivity by using smaller MNPs.

4) Change of refractive index and beam trajectory path

The heating can produce thermal gradient within the medium due to absorption of light energy and redistribute the concentration of MNPs. These factors can change the refractive index of NFF.

$\frac{\partial \left[\Delta T\left(r,z\right)\right]}{\partial r}=\frac{B{a}^{2}}{4{K}_{p}}\frac{1}{r}\left({\text{e}}^{-2{r}^{2}/{a}^{2}}-1\right){\text{e}}^{-\alpha z}$ (35)

where $B=\alpha P/\text{\pi}{a}^{2}$ and P is the laser power, a is the laser beam radius. Therefore, thermal and concentration diffusion of MNPs occur due to local heating by the laser beam inside the NFF [15] [21] [48] . It is noteworthy that ∆n can be caused by both thermal and nonthermal effects where in the first case, the change of refractive index caused by thermal heating and concentration redistribution is given by

$\frac{\text{d}n\left(r,z\right)}{\text{d}T}={\left(\frac{\partial n}{\partial T}\right)}_{c}+\frac{\partial n}{\partial c}\frac{\partial c}{\partial T}$ (36)

and in the latter case it is due to transitions of Fe_{3}O_{4} NP electrons to higher energy states by the action of photons with energies higher than the bandgap energy of Fe_{3}O_{4} NP 0.2 eV, which are considered as intraband transitions causing ∆n [49] . It can be seen from Equation (33) that the effect of
$\Delta T\left(r\right)\propto {R}_{p}^{2}$ can consequently cause the change of refractive index hence the beam divergence angle, θ_{d} i.e., the angle between centered axis of the laser and the diverged beam rays [50] [51] .

${\theta}_{d}=1-{\left(1+2I\right)}^{1/2}$ (37)

which in this case yields a value of θ_{d} = 5.24˚ ≡ 91 mrad. Applying the values of θ_{d} and K_{m} in Equation (38)

$\frac{\text{d}n}{\text{d}T}=\frac{{\theta}_{d}\lambda {K}_{m}}{P}=1\times {10}^{-5}\text{\hspace{0.17em}}{\text{K}}^{-1}\ll 9\times {10}^{-5}\text{\hspace{0.17em}}{\text{K}}^{-1}$ for water. (38)

However, because the NFF concentration used in the experiment is very small one may assume that the initial diffusion coefficient ${D}_{0}=D\left(c\right)$ at a given concentration i.e., it is a concentration independent case. Thus, a step-like variation of concentration in a plane within the medium can be written as

$\left(\frac{\partial C}{\partial y}\right)=-\frac{{C}_{0}}{2\sqrt{\text{\pi}Dt}}{\text{e}}^{\left(-{y}^{2}/4Dt\right)}$ (39)

This is a Gaussian function and has the same shape as the deflected beam trajectory inside the base fluid [52] i.e.,

$Z\left(y\right)=\Pi {\text{e}}^{\left(-{y}^{2}/4Dt\right)}$ (40)

where Π is a constant. It can be seen from Equation (39) as time elapses, the boundary smears out until the concentration gradient vanishes consequently, the broadening of the Gaussian function occurs. Self-assembly of NPS under influence of electromagnetic field with the frequencies in the optical range has been studied by Park et al. [53] and as suggested by Slabko et al. [54] , when NPs are irradiated by the laser radiation, dipole moment is induced which enhances the formation of structural geometry hence forming an agglomeration. In our case, the downward motion of the agglomerates is demonstrated by FITC fluorescence. However, in the case of Brownian dynamics (i.e., no laser), trajectories of an ensemble of NPs in base medium are described by well-known Langevin equation described Equation (3) where the interaction between NPs with environment with fluctuating density results in random change of trajectory movement.

6. Conclusion

Dynamics of laser-transport nanoferrofluid was studied by using FITC-conjugated MNPs as marker based on LIF. Based on the Brownian diffusion and DLVO theory, the NPs are more dispersed and free to move within the medium at earlier times. At later stages they become less mobile due to agglomeration. Also, the results showed a laser-induced enhanced velocity of NPs almost twice as much without laser. An initial rapid forward movement was observed when the laser was switched on. The measured diffusion coefficients showed a higher value for the case with laser action. The mechanisms for the enhanced mobility and laser transport of NPs are thought to be due to e.m.w induced force (i.e. an oscillatory motion) and laser absorptive force (i.e., photothermophoresis). Also, the laser beam showed a trajectory path due to thermal heating causing the change of refractive index of medium and redistribution of NPs concentration.

Cite this paper

Khosroshahi, M. , Asemani, M. (2017) Dynamics Study and Analysis of Laser-Induced Transport of Nanoferrofluid in Water Using Fluorescein Isothiocyanate (FITC) as Fluorescence Marker.*Journal of Modern Physics*, **8**, 2219-2244. doi: 10.4236/jmp.2017.814137.

Khosroshahi, M. , Asemani, M. (2017) Dynamics Study and Analysis of Laser-Induced Transport of Nanoferrofluid in Water Using Fluorescein Isothiocyanate (FITC) as Fluorescence Marker.

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