In the present study, combining the conventional photothermal analysis and the concept of interaction of solvent molecules in interfacial layer used for thermophoresis in liquid, a theory for photophoresis of a hydrophobic particle suspended in liquids is developed. To characterize hydrophobicity of the micro-particle, slip length in Navier’s formula is used as an index. Analytical expressions are derived and a parametric analysis for photophoretic velocity is performed with emphasis on the influences of particle characteristics such as size, optical properties, hydrophobicity, and thermal conductivity. Heat source function and the corresponding asymmetry factor at various conditions are evaluated to interpret the mechanisms of negative and positive photophoresis and the conditions for transition between them. The present theory discloses that the particle surface hydrophobicity or fluid slippage at particle-liquid interface may lead to a remarkable enhancement in the particle photophoretic velocity in liquids. Higher particle thermal conductivity and larger size of liquid molecules both result in weaker photophoretic motion.
©2010 Optical Society of America
As a particle suspended in a background fluid subjected an intensive light beam, electromagnetic energy from the incident light can be absorbed by the particle and turned into heat distribution in the particle volume. The uneven heat distributions within the particle may cause a force to drive the particle moving either toward (negative photophoresis) or away from (positive photophoresis) the light source . This phenomenon is the so-called particle photophoresis. Over the last few decades, a number of studies on the photophoresis in gaseous media have been conducted, e.g. theoretical studies for free molecule flow regime [2–7] and slip flow regime [8–12], and experimental works [13–19].
On the other hand, due to vigorous development of microdevices such as bio-chips, manipulation of micron-sized particles in liquids has become one of the focuses in the emerging technology. The migration of particles in liquids could be induced with the presence of temperature gradient, electrical, magnetic, and optical fields. Among these phoretic motions, photophoresis is a special one having an advantage of less damage to the particle and the buffer fluid. Although a few works on the fabrication of the optofluidic devices and/or experimental measurements on the photophoresis in liquids have been published [20–26], only one theoretical study  on the photophoresis in liquids can be found in literature. Difficulty in theoretical modeling of photophoretic force on the particle surrounded by the dense liquid molecules is probably the major reason. The phoretic motion of particles suspended in liquids is more complicated than in gases due to closer proximity and more frequent collisions of molecules. In liquids, a molecule would be densely surrounded by others including solid atoms and under the influence of their intermolecular potential all the time . As noted by Anderson , the theory of phoretic motion in gaseous media, which is well understood from the kinetic theory of gases and relies on the collisions between gas molecules and particle surface, is not appropriate for describing phoretic motion in liquids, since the liquid state is dominated by intermolecular potential energies rather than by kinetic exchange of momentum . For the case of photophoresis in liquid, the intermolecular interaction between liquids and the micro particle plays a critical role.
What we have learned about the photophoresis in liquids is relatively less than that in gas media. In the only theoretical analysis , the photophoretic motion of hot solid spherical hydrosol particles in a viscous fluid was analyzed with thermal slip theory of photophoresis in gaseous media. Experimental evidences of positive photophoresis of particles suspended in liquids were provided by the pioneering work of Monjushiro’s group using organic droplets [20,23,25] and solid particles [21,24] with laser irradiation. Negative photophoresis in liquids was first observed with smaller droplets in water and was attributed to the photo-thermal effect in the presence of laser irradiation .
Up to date, due to the complexities involved in the mechanism, a generally accepted theory for micro-particle photophoresis in liquids has not been established. Although the photophoresis is different from the thermophoresis in the energy source, but both of them have the driving force generated through the thermal/mechanical interactions at the particle-fluid interface. With somewhat similarity, some ideas in development of theory for thermophoresis in liquid might be useful in theoretical modeling of the photophoresis in liquid. For example, it was proposed in some works that the van der Waals forces between solute and solvent may be the main reason for particle thermophoresis in liquids [28,31,32]. Semenov  presented a theoretical model of particle thermophoresis in pure liquids based on the surface interaction potential and temperature distribution of the liquid molecules in a thin interfacial layer around the particle. This concept of accounting for phoretic force through intermolecular interaction between the background liquid and the particle seems also applicable to the photophoresis in liquid.
In the present work we try to build up a theoretical model for the photophoresis of a hydrophobic micro-particle in liquids. To interpret the driving mechanism of photophoresis, heat source function (HSF) indicating the energy distribution in the particle and an index named asymmetry factor pertaining to the strength and direction of the photophoretic force are evaluated at various conditions of particle size and optical properties. The transition between negative and positive photophoresis is also explored. Then, the concepts of molecular interaction and interfacial layer borrowed from previous analyses of thermophoresis are employed to develop theoretical expressions of photophoretic mobility. Influences of hydrophobicity indexed by slip length and relative thermal conductivity of the particle on photophoretic velocity are analyzed.
2. Theoretical background
As shown in Fig. 1 , an isotropic homogenous particle is suspended in a viscous incompressible and quiescent liquid with irradiation of a monochromatic, parallel, and linearly polarized light beam along a fixed direction. In the analysis, we adopt the Anderson’s model of interfacial layer , which is of thickness δ attached at the particle-liquid interface and the thickness is assumed negligibly small with respect to the radius of particle R, i.e. δ << R. In the Navier boundary condition, the slip velocity V S is assumed proportional to the tangential viscous stress τ 0 and the degree of slip is measured by a slip coefficient or named slip length L S based on its dimension of length, i.e.,
It has been experimentally verified at various conditions [33–36] that the slip length usually ranges from L S = 1nm to 100nm and even up to microns in some situations [37–39]. Hydrophobicity of a solid-liquid interface can be quantified with contact angle or fluid slippage as an index. Since the slippage is easier to be introduced into the boundary condition in equations of motion for velocity solution than the contact angle is, therefore, the slip length is employed herein as a quantitative index in the analysis to characterize the hydrophobicity of the solid particle under consideration.
2.1 Temperature distribution
Since the fluid motion is very slow and the forced convection is neglected for low Reynolds number, the energy transfer between the fluid and particle is assumed to be dominated simply by heat conduction. In addition, to facilitate the analysis, the light absorption of the fluid is neglected. Then the fluid temperature distribution Tf with constant thermal conductivity is governed by the Laplace equation,
The temperature distribution Tp inside the radiation-absorbing particle is described by solution to the following Poisson equation,40],
In the above equation, is the electric conductivity of the particle, where n p and κ p are the refractivity and absorptivity expressed as real and imaginary parts of the complex refractive index, m p = n p + κ pi, of the particle, is the wavelength in vacuum, the magnetic permeability, and c the speed of light in vacuum. Based on the above expressions, the following form of heat source function (HSF) can be derived,
The thermal boundary conditions on the particle surface (r = R) and in far-field (), respectively, areEq. (5) as in terms of , can be expressed as
The function in Eq. (5) stands for the distribution of energy absorbed by the particle from the incident light. Since, geometrically, the particle has an illuminated as well as a shaded side; optically, the energy focusing strongly depends on the magnitudes of particle absorptivity and refractivity. Therefore, the energy distribution is generally not symmetric and its asymmetry can be quantified by the factor , which is an important parameter for development of photophoretic mobility.43]
In Eq. (10), is the Hankel function, and and denote the Ricatti-Bessel functions of the first kind and second kind, respectively.
2.2 Photophoretic velocity of the particle
Since the driving force of the photophoretic motion in liquid is related to the interfacial phenomena in a thin region close to the particle surface, in which the velocity field is determined by solving Navier–Stokes equations of fluid motion with slip boundary condition.
At low Reynolds numbers, ignoring Brownian motion, the Stokes flow around a sphere can be described by the incompressible creeping flow equations with μ standing for the fluid viscosity, v for the velocity vector and p for pressure, viz.,
It was assumed in previous works [28,44–47] that the temperature dependence of the intermolecular interaction potential is the main source of either a pressure gradient or a volume force . The thickness of the surface layer is assumed to be much smaller than the radius of the particle and, approximately, the particle surface can be assumed flat in order to facilitate the solution of the hydrodynamic problem . For a spherical solvent molecule near the flat particle surface, following the approach used in previous thermophoresis analysis, the London-van der Waals interaction energy is given by ,
The force of the particle acting on the surrounding fluid is expressed by an effective force density fZ [51,52], which is equivalent to temperature-induced pressure gradient, , and is of the following form ,Eq. (7).
The boundary velocity VB is the sum of two contributions, i.e. the intrinsic slip velocity and the velocity change through the interfacial layer Δ V, and is expressed as a function of slip length ,
In the above equation, the changes in velocity and shear stress can be developed as follows,
Then, the boundary velocity in Eq. (14) has the form of
Considering transformation of reference frame, the photophoretic velocity has opposite sign of the fluid velocity, and following the previous derivation , the photophoretic velocity is related to an average velocity along the particle surface, i.e. , where . Thus, the photophoretic velocity of the particle is then given by
Defining a reference velocity and then the particle photophoretic velocity can be normalized in the form of and expressed as
We define the influence of the particle hydrophobicity on the photophoretic mobility by the ratio of photophoretic velocity with fluid slippage at the solid-liquid interface to that without slip effect, i.e.,
Comparing with the only existing theory of photophoresis in liquids , the present theory contains more realistic effects such as particle-liquid interactions and hydrophobicity of the particle. Influences of these parameters could be significant in some situations and should be included in the development of a theoretical model.
3. Results and discussion
3.1 Asymmetry factor with various particle size parameters
As shown in Eq. (18), the photophoretic velocity is proportional to the asymmetry factor and the positive (negative) stands for negative (positive) photophoresis. The factor is a strong function of particle size parameter α. Since there is no previous data of available for photophoresis in liquids, photophoresis of a micro particle with m p = 1.57 + 0.048i in air is considered for verification of the present approach. For the data shown in Fig. 2 the air is assumed non-absorbing and of the refractivity n a = 1. The present calculations of are plotted versus α and compared with the results from previous studies [55,56]. It is observed that our results agree very well with the calculations of Pluchino  as well as the experimental data from Pluchino and Arnold .
The -α curve shown in Fig. 2 is a typical one and the characteristic features can be stated as follows. In general, the particle of small size (small α) has microlens effect and tends to generate hot spot in the trailing part of the particle volume. This pattern of HSF distribution usually leads to a positive , which corresponds to a negative photophoresis. On contrary, in the case of a large particle, negative asymmetry factor or positive photophoretic motion is most likely to appear. In between, therefore, there must be a certain particle size, at which the asymmetry factor changes its sign and transition of positive-negative photophoresis occurs. As the value of α reduces continuously, first reaches a maximum and then down to zero as the size parameter further lowers towards the range of α < 1.
The curves in Fig. 3 show variations of asymmetry factor of the particle suspended in water versus the particle size parameter α at three different patterns depending on the ranges of particle absorption, κ p. Here we consider water is of refractivity 1.33 but ignore its light absorption for simplicity. Asymmetry factors for particles of refraction index m p = 1.57 + κ pi and = 2.0 + κ pi are presented in Figs. 3(a) and 3(b), respectively. At very low absorptivity of the particle, κ p = 0.001, negative photophoresis with > 0 occurs over a large extend of particle size. It is the pattern of negative photophoresis dominant. As the absorptivity becomes high, e.g. κ p = 0.5, < 0 over all values of size parameter and a positive photophoresis is prevailing at various particle size. In between of these two extreme cases, for κ p = 0.05 has a local maximum for negative photophoresis and changes sign at a certain intermediate value of size parameter. It is the pattern containing a normal transition of negative-positive photophoresis at a certain medium value of α.
3.2 HSF distributions at various conditions
The normalized HSF contours in Fig. 4 reveal the variation of energy distribution with particle size corresponding to the negative photophoresis dominant pattern with m p = 1.57 + 0.001i in Fig. 3. In the cases of low absorptivity of the particle, the refraction is dominant. It is obvious that more energy is contained in the volume of the trailing part. The liquid near the trailing (shaded) surface of the particle is then heated more intensively than that near the leading (illuminated) surface. Thereby, the particle migrates towards the light source. For the particle of m p = 1.57 + 0.5i, the HSF contours in Fig. 5 show that the energy is concentrated on the leading side for the strong light absorption of the particle, which tends to result in a positive photophoretic motion.
For the intermediate absorption, m p = 1.57 + 0.05i, the HSF distribution is almost uniform within the very small particle of α = 0.5 as that shown in Fig. 6(a) . With the particle size increased a little, as the case of α = 5 in Fig. 6(b), the particle acts as a microlens and the light is refracted and focused onto the area close to the shaded (trailing) surface with a noticeable HSF peak. This HSF distribution is obviously corresponding to a negative photophoresis. As the particle size parameter is raised up near a critical value, i.e. α = 8≈αcr in Fig. 6(c), the integral of the absorbed energy over the particle volume is nearly zero, ≈ 0. From Eq. (18), the photophoretic velocity is proportional to the asymmetry factor and the particle has no photophoretic motion at the critical condition of = 0. Beyond this critical condition, e.g. the larger size of α = 30 in Fig. 6(d), the energy absorbed by the leading surface becomes important and the HSF peak due to refraction reduces continuously. In this situation, the HSF distribution leads to a negative and thus positive photophoresis with light absorption on the illuminating side of the particle dominant.
3.3 Comparison of the present theory with existing measurements
It is difficult to make comparison of the present theoretical predictions with the measured data of particle photophoresis in liquids. Firstly, the existing experimental data are limited. Secondly, the values of some parameters such as absorptivity κ p of the test particle were not available in literature. To facilitate the analysis, uniform light is assumed in the theory, which is different from the laser irradiation used in previous experiments. Even though, in Fig. 7 comparisons of the theory and measurements of Monjushiro et al.  are presented. Since the particle absorptivity and slip length are not provided in the paper, we take the cases of = 0.1~0.4 with = 0 in the comparison. As to the light intensity, by using the formula with the power Po = 0.12W and semi-width = 21μm of the laser beam used in the experiment of Monjushiro et al. , we obtain I = 173W/mm2. Also, one more condition of I = 240W/mm2 is considered for comparison. Consider water as the fluid around the particle, with thermal conductivity = 0.58, Hamaker constant J, molecular radius Å (Å = Angstrom), specific volume Å3 , cubic thermal expansion , viscosity at 25°C, and particle thermal conductivity = 0.08, one then has from Eq. (18). The data show that, for I = 173W/mm2, the theoretical predictions of photophoretic velocity are close to the measured data of κ p = 0.3 except the case with larger particle (R = 1.5μm), in which the uncertainty in measurement is more pronounced. At a higher intensity, I = 240W/mm2, the present predictions with a lower absorptivity, κ p = 0.25, fit the measurements better. It is also of a qualitatively reasonable tendency.
So far, only few measurements demonstrated occurrence of negative photophoresis in liquid. Probably the work of Monjushiro et al.  with smaller organic droplets in water was the first one and the same results were also presented in their later work . In Fig. 7, it is revealed that the present theory predicts occurrence of the negative photophoresis for the case of low absorptivity (κ p = 0.1) and small size of particles (R < 0.5μm). Therefore, the negative photophoresis of small particles in liquid disclosed in the work of Monjushiro et al.  and Watarai et al.  can be regarded as an indirect evidence for the adequacy of the present theory.
3.4 Parametric analysis of photophoretic velocity
Figure 8 presents the influences of the thermal conductivity ratio k* on the dimensionless photophoretic velocity with various slip length, = 0 (no-slip), 1, 5 and 10. It can be observed that is a monotonically decreasing function of k* for all cases. A particle of high thermal conductivity is able to spread heat over the particle and renders the non-uniform heat energy distribution smooth inside the particle volume. Therefore, the absolute value of asymmetry factor and in turn the photophoretic velocity can be reduced. The data show that drops a little fast as k* is of order unity or higher. In general, as k* is of order O(10), the normalized photophoretic velocity becomes very small and has a tendency of approaching zero asymptotically with further increasing k*.
At various values of k*, the effects of boundary slip on the particle photophoretic mobility are examined with the plots of the ratio versus the slip length in Fig. 9 . To highlight the influences of slip length, the range of is set from to , which corresponds to the dimensional slip length below 10nm for a typical particle size of R = 1μm. The present theory predicts an increasing trend of the photophoretic velocity with enhancement of the slip characteristics at the particle-liquid interface. It implies that hydrophobizing the particle surface has a favorable consequence of increasing the photophoretic mobility. The results also disclose that a fast increase in can be obtained as the value of reaches the order of or higher.
Since the present theory involves the interactions between the liquid molecules and the particle in microscopic view, the size of the liquid molecule becomes one of parameters influential to the photophoretic mobility. In Fig. 9, the effects of the molecular size of liquid are explored by employing Eq. (22) for , in which only the dimensionless molecular size and slip length appear explicitly but the other parameters about the liquid are hidden. In the study of Semenov , four liquids with different molecular radius, i.e. water of = 1.54Å, methanol of = 2.02Å, acetonitrile of = 2.18Å, and cyclohexane of = 2.56Å, are considered. For the particle of R = 1, the corresponding values of are 1.54, 2.02, 2.18, and 2.56, respectively. In Fig. 10 , therefore, we plot the variations of photophoretic velocity with respect to the parameter in the range of (1-10). The results reveal that, at conditions of all other parameters fixed, the normalized photophoretic velocity is higher in a solvent of smaller molecules.
4. Concluding remarks
- 1. In the present study, a theory for photophoresis of a hydrophobic particle suspended in liquids has been developed by combining the photothermal analysis for the conventional photophoresis and the concept of interaction of solvent molecules in interfacial layer. Different from the existing theory, the present one contains more realistic effects such as particle-liquid interactions and hydrophobicity of the particle.
- 2. Heat source function and the corresponding asymmetry factor at various conditions are evaluated to interpret the mechanisms of negative and positive photophoresis. The present theory predicts occurrence of the negative photophoresis with weak absorption and small size of particles.
- 3. Analytical expressions of photophoretic velocity are derived and thereby the influences of particle characteristics such as size, optical properties, hydrophobicity, and thermal conductivity on the photophoretic velocity were explored. The present results disclose that the particle surface hydrophobicity or fluid slippage at particle-liquid interface may lead to a remarkable enhancement in the particle photophoretic velocity in liquids. Higher particle thermal conductivity and larger size of liquid molecules both result in weaker photophoretic motion.
- 4. Comparing with the limited measurements on photophoretic velocity in existing literature, the present theory has been demonstrated reasonable qualitatively as well as quantitatively. For further verification of this novel model, more in-depth experiments are worthwhile in the future.
This study was supported by National Science Council of the Republic of China (Taiwan) through the grant NSC-98-2221-E-035-068-MY3. The second author (WKL) would like to gratefully acknowledge Prof. A. Würger and Prof. S. N. Semenov for their useful discussion and suggestions.
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