1. Introduction

Light actuating materials have potential to lead to light fueled robots and machines, whereby materials deform and move in intensity gradients by photo-active molecules; e.g. molecular machines [1]. Azobenzene derivatives are such a molecules, and they undergo a high rate of light-induced cyclic molecular shape change from an elongated, trans form, to a more globular, cis form (Fig. 1

 

Fig. 1 (Left) Trans-cis photoisomerization of an azo dye; e.g. DR1. (Right) Schematic of the photoisomerization force on a colloidal particle near the focus of a Gaussian beam.

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Azo-polymers produce observable flow of matter because photoisomerization strongly enhances the mobility of the molecules. Photoisomerization reduces by many orders of magnitude the viscosity of an organic glassy host at temperatures 100 K below its glass transition temperature $\left(T_g\right)$ with a negligible increase in temperature [9,10]; a phenomenon which is referred to as athermal fluidization, or photo-fluidization. Such an observations, and others [1,7,8], imply that the photoactive molecules acquire the ability to move when photoisomerized. To put the work in a broader context, it is worth mentioning that there are four major types of mobility induced by photoisomerization of azobenzene derivatives in polymer hosts reported in the literature so far. These types of induced mobility are as follows. First, sub-Tg reorientation of the molecules, leading to birefringence and data storage and holography [11–13]; second, sub-$T_g$ poling in dc-electric field assisted by photoisomerization [14,15]; or all-optically [16], even in high $T_g$ polymers leading to second order nonlinear optical effects [17]; third, mass flow in gradients of light intensity leading to morphological changes of the films [4,5]; and fourth, bending of films and beams leading to actuation [1]. Poling requires the concomitant action of a polar component of the electric field with photoisomerization, and mass flow is observable only in the presence of intensity gradients of photoactive light. That is the second and third types of mobility enhancement are clear evidence of strongly enhanced molecular mobility by photoisomerization tens to hundreds of ${}^{\circ }C$ below the Tg of the polymer.

Recently, we have introduced the concept of the photochemical force that leads to molecular motion of a particle in a gradient of light intensity [18]. If a particle containing light addressable molecules is located in an inhomogeneous distribution of light intensity, a photochemical potential will move the particles away from bright area into dark area. In this paper, vectorial mobility is described theoretically, and simple analytical expressions are derived to describe molecular motion allowing for a rigorous description of a complex problem of physics that occurs at the interface of optical physics and photochemistry. In particular, it is shown that while the presence of an intensity gradient is essential for driving photo-mobile molecules, vectorial mobility in symmetric gradients is peaked into the direction of light polarization. The theory is applied to the case of molecular machines consisting of the well-known azobenzene derivatives, which transform absorbed light into mechanical work; and a comparison is made between the photochemical motion and that resulting from optical forces due to radiation pressure [19–29]. In particular, I show that the calculated magnitude of the photochemical force is orders of magnitudes larger than radiation pressure forces allowing for the transport of particles with gravities larger than the $\mu N$ range.

Vectorial motion of particles in an arbitrary gradient of light intensity can be described by mapping photo-selection on the gradient of the irradiating light at the particles. Azobenzene containing polymers are good examples of materials showing vectorial mobility since they are photo-softened and move preferentially in the direction of light polarization [1,6,9,10]. Both in-plane and of-plane motion have been observed in these materials when exposed to intensity gradients [30]. It is out of the scope of this paper to discuss a detailed microscopic model of the change of the mechanical properties of such materials by photoisomerization; rather, an in-depth study shows that the force due to photoisomerization creates molecular motion even in films of solid polymers.

2. Photochemical potential energy

The photochemical potential energy U is the internal energy of the system, and it is given by the energy of the total number of absorbed photons which is transformed into mechanical work of the molecular machines; e.g. for example, by the photoisomeric cycling of azobenzene derivatives. Inhomogeneous light absorption confers to the material an energy, which is position dependent, and it acts as a potential energy to move the particles; much like molecules or particles move from higher to lower chemical potential, light-fueled molecular machines move from bright to dark area. This is a generalization of the physics concept of, for example, gravitational potential energy. If a particle containing light fueled molecular machines is present in an intensity distribution it will be subjected to a potential

Equation (1) shows that isomerization cyclic motion alone does not produce net motion; e.g. via $k$ alone, rather isomerization must take place in an inhomogeneous light irradiation; e.g. in a position dependent actinic light intensity; e.g. via $I(r)$. Both $k$ and $I(r)$ need not to be null in order for the potential $U(r)$ to create motion; and the microscopic mechanism for the mobility enhancement in a viscous material system that contains, besides the photoisomerization potential, competing potentials of, for example, optical trapping and material’s elasticity, is seen in the competition of the photoisomerization force and the viscous force as well as elastic and optical trapping forces (vide infra). In fact, $k$ mirrors the efficiency of trans-cis isomerization cycling, and a push-pull azobenzene derivative can be considered as a spring vibrating at high rate when undergoing cyclic photoisomerization. The molecule undergoes a large number of cycles per second upon light absorption owing to a fast trans-cis isomerization, typically in the picosecond time scale [32].

During the continuous shape change of the molecule, e.g. during light action, atoms rearrangement in the azo-dye (Fig. 1) pushes indefinitely those of the surrounding medium and creates additional free volume thereby decreasing friction and enhancing mobility; a concept supported by detailed molecular simulations performed by Tiberio et al. [33] and Teboul et al. [34,35] who showed, respectively, that changes of molecular shape of azobenzenes in solution are opposed by intermolecular viscous forces, and lead to enhancement of mean square molecular displacement in clusters of the nano-cogent environment of the isomerizing molecules. That is azo dye derivatives can exert mechanical forces on their environment. When an azobenzene molecule isomerizes, the long molecular axis shortens by $3.5\AA$; e.g. from $9\AA$ in the trans form to $5.5\AA$ in the cis form, leading to a mechanical force; e.g. $F~0.11 nN$. This rough estimate of the force is obtained by dividing the energy of the absorbed photon at 532nm by the change of the length of the molecule; e.g. $2.331eV/3.5\AA$ and considering 0.11 for the quantum yield of trans-cis photoisomerization.

Force measurements on single azobenzene and azo-polymer molecules are of the same order of magnitude; e.g. within the $1 nN$ $~ 100 pN$ range [36,37]. Considering trans-azobenzene (radius $a ~ 9 \AA$) in water at room temperature (density ${\rho }_f~1000 kg m^{-3}$ and viscosity ${\mu }_f~{10}^{-3} Pa s$), the friction coefficient of the molecule is $\xi =6\pi {\mu }_{f}a$ $~ 16.96\times {10}^{-12} N s m^{-1}$, and the mechanical force leads to a characteristic velocity; e.g. $v=F/\xi$ $~ 6.5 m s^{-1}$. Such values correspond to a hydrodynamic Reynolds number $Re={\rho }_{f}va/{\mu }_f~ 0.58\times {10}^{-2}$, and a characteristic frequency $f=v/a~ 0.72\times {10}^{10} Hz$; a frequency which confirms that azobenzene type molecules undergo cyclic photoisomerization at a very high rate; e.g. billions of kicks per second, owing to the picosecond time scale of the photoreaction. The low-Re estimate indicates that photoisomerization must decrease the viscosity of water by two orders of magnitude to get to the point where inertial effects compare to viscous dissipation. The theory presented in this paper assumes that the particles become mobile when photoisomerized.

3. Photoisomerization force and pressure

A uniaxial chromophore is excited by polarized light preferentially if the transition dipole moment μ is parallel to the electric field E of the light, and the excitation rate is proportional to ${\left|\mu .E\right|}^2$, and for a macroscopic system, the orientational distribution of the molecules $N(\theta ,\phi )$, has to be considered; with $\theta$ and $\phi$ the polar coordinates in an orthonormal laboratory frame of Cartesian coordinates (x, y, z) (Fig. 2

 

Fig. 2 Representation of the orientation of the transition dipole moment of the molecule, μ, and the vector gradient of the light intensity k_G, and the electric field of the actinic light, E, in an orthonormal Cartesian coordinate system (x, y, z). E is chosen parallel to z to simplify the calculations. The red thick curved arrow between k_G and the potential energy $U=k{x}^2/2$, which is approximated by a harmonic-type potential for week forces, indicates that a molecule is dragged by the potential in the direction of k_G after acquiring mobility by photo-selective isomerization. The spring schematically depicts the vibrational motion of the molecule upon cyclic trans-cis isomerization and k is the spring constant.

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The potential energy that causes the molecule to move is obtained by projecting E on μ, so the molecule becomes mobile, and re-projecting; e.g. the projected dipole ($\mu Ecos\theta$), on $k_G$. So, the energy rate to move a molecule is then proportional to ${\left|E\right|}^2{\left|\mu \right|}^{2}co{s}^2\chi co{s}^2\theta$, and the energy that moves the macroscopic system, e.g. the ensemble of $N(\theta ,\phi )$ molecules, is then obtained by averaging over all possible orientations. $N/4\pi$ is the isotropic orientational distribution of molecules in the ground state. Here reorientation effects are neglected; a feature which facilitates the understanding of the process without loss of generality. When the photoactive molecule is assumed to be rod-like; only ${\sigma }_{\parallel }$ is considered, and if the excitation light is linearly polarized, say along the z-axis, the energy reads

If the field; e.g. the irradiation is inhomogeneous, then the potential will lead to a position dependent gradient force $F_{ph}^p$ per unit area; e.g. pressure, expressed in $N m^{-2}$, which will move the particles in the direction of the intensity gradient vector. Note that both the potential and the intensity are defined per unit area.

$F_{Ph}^P$ is the photoisomerization pressure (PP), and it moves the particles from bright to dark area in the light gradient $\nabla I\left(r\right)$, expressed in $m^{-1}$, and $K$ is the force constant, expressed in units of $N m^{-1}$. $K$ depends on the respective orientation of the vector field of light and the intensity gradient vector via the term $co{s}^2{\theta }_G$; and it mirrors the properties of the molecule and the photoreaction. The PP force, denoted by $F_{ph}$, on particles of different shapes and surface area $A$, has a magnitude that can be calculated by $F_{ph}=F_{ph}^{p}A$, where $v=\delta A$ is the particle’s volume.

A direct consequence of Eqs. (4) and (5) is that movement occurs in the direction of the gradient vector, independent of the state of the incident light polarization, and with an efficiency that depends on $co{s}^2{\theta }_G$. For example, if a line-shaped parallel beam of an actinic linearly polarized light is propagating perpendicular to the surface of a photo-reactive sample, $k_G$ is perpendicular to the line-shape, and the efficiency of the displacement of matter should be in the ratios 1, 2, 3 for E perpendicular $({\theta }_G=\pi /2)$, at 45${}^{\circ }$ $({\theta }_G=45°)$ and parallel $({\theta }_G=0)$ situations. If a disk-shaped parallel beam is propagating perpendicular to the film sample, then $k_G$ has equal components in the (x, y) plane and no or negligible z-component, then movement will occur in-plane along the polarization direction of the actinic light. If the light beam is focused with $k_G$ having components in all three directions; e.g. x, and y, and z, then a longitudinal field; e.g. z-polarized, will produce z-movement; and in-plane polarized field; e.g. polarized in the (x, y) plane linearly or circularly, will produce in-plane, linear or circular motion, respectively; and vortex-type beams –which contain radial and azimuthal components of the E-field- will produce vortex-type motion. All of these situations are supported by experimental evidence [6,30,39–41]. When a parallel is made with optical trapping and manipulation of micro/nano-particles, photochemical tweezing by longitudinal fields must lead to levitation and propulsion of particles; and the photochemical transport of matter by beams with shaped intensity gradients is conceptually possible [42]. Next, the dynamics of the photo-induced motion matter are discussed, and a dimensionless parameter is introduced to account for the competition of inertial and viscous forces.

4. Motion dynamics

In this section, I consider a visco-elastic material; e.g. a polymer film, where both viscous and elastic forces are included to account for a material moving like a fluid and deforming like a solid; e.g. a viscoelastic material. The application of this theory to a solid azo-polymer film is discussed in the following sections. If the particle, which is acted upon by $F_{ph}$, is present in a viscous environment, it will be subjected to the frictional force $F_f=-\xi v$; e.g. drag due to the viscosity of the medium, with $v$ is the velocity of the particle, and $\xi$ the friction constant, and its reciprocal, $1/\xi$, is called the mobility. For a spherical particle of radius $a$ in a fluid of viscosity ${\mu }_f$, hydrodynamic calculation indicate $\xi =6\pi {\mu }_{f}a$. The particle is also subjected to a random force $F_r$, which is stochastic in nature, due to thermal fluctuations. The dynamics of the motion produced by $F_{ph}$ can be studied by the Smoluchowski equation [18,43]. Besides $F_{ph}$, if other forces act on the particle, such as the optical gradient force $F_{grad}$, due to a contrast of refractive index between the particle and the surrounding medium, when irradiation is performed by a highly focused laser beam, and the force $F_{el}$, due to the elasticity of the polymer, when a polymer film is considered, then Smoluchowski equation describes Brownian motion in a potential, $U_{Total}$, due to all these forces. $U_{Total}$ is assumed harmonic in the small displacement regime; e.g.

$k_{grad}$ is the stiffness of the trap in optical tweezers, and $k_{el}$ is the elastic constant of the polymer material and $x$ is the displacement. $k_T=K$ when $k_{grad}$ and $k_{el}$ are small compared to $K$; e.g. when $F_{grad}$, and $F_{el}$ are small compared to $F_{ph}$. When we consider $\text{Ψ}\left(x, t\right)$ the probability distribution function, for weakly interacting particles, that a particular particle is found at point $x$ at time t, the dynamics of the motion of the particles is described by

$G(x, x^{\text{'}};t)$ is the Green function and it is the probability that the system which was in the state $x^{\text{'}}$ at time $t=0$ is in the state $x$ at time $t$. With $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature, and $\tau =\xi /k_T$. Note that $t/\tau$ is a dimensionless quantity that embodies the competition of the medium’s inertia and viscosity, and it is the equivalent of Re-number in hydrodynamics [44,45]. Two limiting cases can be distinguished at small and large values of $t/\tau$; denoted by h; equivalent to low- and high-Re number, respectively. For low-$h$, viscosity is dominant and the motion of the particles is slow, and the Green function becomes the distribution function of free diffusion, which is the usual situation of Brownian motion. For high-$h$, inertial forces dominate and the motion of the particles is fast and the Green function is the distribution of Boltzmann in accordance with the second law of thermodynamics; e.g. the distribution function of the particles at equilibrium, ${\text{Ψ}}_{eq}\left(x\right)$.

${\text{Ψ}}_{eq}\left(x\right)$ is also the distribution function, at thermodynamic equilibrium, for the displacement of a trapped particle in an optical tweezer. Angle change and interaction between particles could be included in Smoluchowski equation, and a sophisticated model could be considered, however this will add complexity without improving the understanding and the description of a complex physics phenomenon. I will go on to discuss the motion of a particle in a Gaussian beam and to estimate the magnitudes of $F_{ph}$ and $h$ in a viscous fluid.

5. Brownian Walker model under the photoisomerization and optical forces

In this section, I neglect elasticity to consider viscous flow and thermal fluctuations as the driving mechanisms of the motion of the particle together with applied external forces; a situation which corresponds to a Brownian walker model when a nanoparticle is moving slowly in a viscous fluid under week external forces; e.g. $F_{ph}$ and $F_{grad}$, and undergoing small displacements. At any given time, the motion of the particle is described by the Gaussian Green function; e.g. Eq. (7), with a variance

In the small time limit, free diffusion prevails and the standard deviation $\sqrt{B\left(t\right)}$ is given by $\sqrt{2Dt}$, where $D=k_{B}T/\xi$ is the diffusion coefficient of the particle; and at large times; e.g. at equilibrium, $\sqrt{B\left(t\right)}$ is given by $\sqrt{k_{B}T/k_T}$. That is, at small times, the particle is moving randomly, and as time passes, directional motion of the particle builds up in the direction of the intensity gradient owing to the external forces. $B\left(t\right)$ is a motion parameter that includes contribution from random forces; via $k_{B}T$, and viscous forces, via $\xi$, and optical trapping and photoisomerization forces, via the stiffness of the trap, $k_{grad}$, and the strength $K$ of the photoisomerization force; respectively. The average displacement, e.g. ${\lambda }_x$, that a particle undergoes in the direction of the light intensity gradient, in a given period of time, is ${\lambda }_x=\sqrt{B\left(t\right)}$. Note that ${\lambda }_x$, shows a square-root dependence on an exponential function of time, and it is governed by the competition of viscous forces and external forces; e.g. governed by $\tau =\xi /k_T$. Consequently, a clear cut experiment to perform for the study of the effects of external forces; e.g. $F_{ph}$ and $F_{grad}$, on the displacement and the velocity of the particle, is the control of the energy dose absorbed by the particle under actinic light gradients in mediums of varying viscosity.

Such an experiment have been performed by Abid et al [2] who studied the displacement, in actinic intensity gradients, of 16 nm polystyrene nanoparticles decorated with azo-dye molecules (azo-NPs) in aqueous solutions of poly-vinyl-alcohol with controlled viscosity at room temperature; e.g. $0.68\times {10}^{-3}$ and $3.85\times {10}^{-3} Pa s$. They found that the azo-NPs diffuse randomly in area of homogeneous illumination, and that they are propelled into the dark when they reach the gradient region of the illumination in a directional manner. Motion was observed to occur in the direction of the intensity gradient with a power-tunable velocity. The azo-NPs could be transported over tens of microns into the dark regions of the optical gradient with a velocity up to $15 \mu m s^{-1}$. They also found that the force that drove the motion of the particle; e.g. the photoisomerization force, is power dependent, and within the power range and azo-dyes concentration they used, this force; e.g. ~0.012 - 0.105 pN, was 3 to 4 orders of magnitude larger than optical gradient and gravity forces. In addition, they measured the mean square displacement; e.g. $B\left(t\right)$, of the azo-NPs in small time intervals; e.g. 5 - 10 s, and found a nonlinear dependence on time. These experimental findings are in good agreement with the random walker model developed in this paper, and additional experiments are needed to use the full potential of the theory; e.g. for example, control of azo-dye concentration and light polarization and intensity gradients, and use of $B\left(t\right)$ to study displacements and velocities of NPs versus light power in larger time intervals. Next, we discuss the motion of a NP in a strongly focused beam, and we show, indeed, that the photoisomerization force acting on the NP can be orders of magnitude larger than the optical gradient and gravitational forces.

6. Motion of a nanoparticle in a Gaussian beam

Consider, for example, a polystyrene microsphere (radius $a=1\mu m$; density, $\rho =1.05 g.c{m}^{-3}$) in a focused laser beam. The gravitational force, $F_g=4\pi \rho g{a}^3/3$, is $F_g~43.15 fN$, where $g=9.81 m s^{-2}$ is the standard gravity; and the trapping forces in optical tweezers, e.g. which are proportional to the absorbed laser power, $P_{abs}$, divided by the speed of light in vacuum $c$, on such a particle are in the $pN$ range. The radiation pressure (RP) force is $F_{RP}= Q{P}_{in}/c$, where $P_{abs}=Q{P}_{in}$ with $Q$ a factor that represents the efficiency of light absorption, and generally $Q~0.03$. A Gaussian laser beam (20 mW; 1.064 μm) focused to a beam waist of ~0.5 μm exerts a RP force $F_{RP}~15 pN$ on a 1 μm-diameter polystyrene sphere suspended in water [46]. The expression of the mean of PP is $〈F_{ph}^p〉= 4K_n{P}_{in}/\pi {\omega }_0^3$, where $K_n=K/I_0$; and $〈F_{ph}〉= 16K_n{P}_{in}{a}^2/{\omega }_0^3$ is the mean PP force exerted on a nano-sphere of radius $a$ by a Gaussian beam with an intensity profile $I(r)= I_{0}exp\left(-2r^2/{\omega }_0^2\right)$, where $r$ is the radial distance from the center axis and ${\omega }_0$ is the beam waist and $I_0=2P_{in}/\pi {\omega }_0^2$. Equation (5) yields $F_{ph}^p(r)= \left(4r/{\omega }_0^2\right)K_n{I}_{0}exp\left(-2r^2/{\omega }_0^2\right)$, and $〈F_{ph}^p〉$ is obtained by averaging $F_{ph}^p$ over the beam radius; e.g. over $2{\omega }_0$. $F_{ph}^p$ is positive for positive $r$ and negative for negative $r$, indicating that the PP pushes the particle into intensity minima [Fig. 3(a)

 

Fig. 3 Calculated intensity profile $I\left(x\right)$ and photoisomerization pressure $F_{ph}^p$ produced by a Gaussian beam (a) and an interference pattern for an s-s configuration (b) versus the direction of matter motion; e.g. along the direction of the gradient vector of the actinic light intensity. $I$ and $F_{ph}^p$ are normalized in (a) by $I_0$ and $K_n{I}_0/{\omega }_0^2$; respectively, and in (b) by $2I_0$ and $4\pi K_n{I}_0/\text{Λ}$; respectively. The motion of a colloidal particle is schematically depicted to move from intensity maxima to intensity minima according to the sign of the force.

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Since PP, e.g. $F_{ph}^p(r)$, exerted on the particle by a Gaussian beam depends on the position $r$ of the particle in the intensity gradient of the beam, its magnitude is calculated by averaging over all positions in the section for a Gaussian beam with no, or negligible, intensity gradient in the longitudinal direction

This force; e.g. $F_{ph}$, is orders of magnitude larger than the range of optical trapping forces, and it is estimated with theoretical experiments conditions looser than those used for $F_{RP}$. A rough estimate of the ratio of the PP force and the RP force for the same incident laser power, shows that $F_{ph}$ dominates $F_{RP}$ by several orders of magnitude $F_{Ph}/F_{RP}=16c{K}_n{a}^2/Q{\omega }_0^3$ $~4.84\times {10}^6$. In fact, experimental observations indicate that $F_{ph}$ easily overcomes $F_{RP}$ and $F_g$, and it is readily observable for intensity gradients of few $mW/c{m}^2/\mu m$ [2–6]; a range which is used for the above estimation. This estimate of $F_{ph}$ is reasonably close to the force; e.g. in the $1-3 mN$ range, produced by an azopolymer film sample during bending by photoisomerization [3]. The magnitude of $F_{ph}$ in azo-polymers is remarkable, knowing that cardiac myocytes produce forces on the scale of $0.1 mN$ [47], and smooth muscle cells only apply forces on the scale of 10-100 $nN$ to their surroundings [48]. Materials parameters, especially the molecules density, which is typically of the order of $1 molecule n{m}^{-3}$ for azobenzene containing polymers, leave room for the increase of $F_{ph}$ by few orders of magnitude.

$K_n=1.21\times {10}^{-5}$ is estimated by using the parameters corresponding to disperse red one (DR1); an azo dye molecule which is well known for its ability to undergo efficient cyclic trans-cis photoisomerization at 460nm irradiation: ${\sigma }_T^{460nm}~11.6\times {10}^{-17}c{m}^{2}molecul{e}^{-1}$, and ${\varphi }_{TC}~0.11$; with $N=N_v$, and $\delta =a/3$, and by assuming the incident light beam linearly polarized in the (x,y) plane, say x-polarized; e.g. the major component of the intensity distribution is x-polarized $({\theta }_G=0)$. With these assumptions, the $100 nm$-diameter particle is driven out of the laser spot in the direction of the polarization of the laser, with a PP force $~0.2 mN$. When only photoisomerization and viscous forces are considered, $h$ is given by $h=\left(2co{s}^2{\theta }_G+1\right) \eta I_0/6\pi {\mu }_{f}a$. With the above conditions of irradiation, $h$ corresponding to the nanosphere in, for example glycerol $\left({\mu }_f~1.2 Pa .s\right)$ near ${20}^{\circ }C$, is high: $h~1.36\times {10}^9$ confirming that $F_{ph}$ dominate other forces acting on the nanoparticle. $h$ should decrease with decreasing average intensity and increasing fluid viscosity and particle size. For example, if the particle diameter increases from 100 nm to 100 μm, with every other thing being equal, the value of $h$ decreases by 3 orders of magnitude, but it remains high; e.g. $h~1.36\times {10}^6$.

7. Motion of a nanoparticle and a polymer film in vectorial fields of interference patterns

Next, I discuss the motion of a nanoparticle and a polymer film in the vectorial fields of an interference pattern; an irradiation scheme which may function as an amplifier of the photoisomerization force. If the intensity pattern is generated by the interference of two parallel beams, say of the same wavelength $\lambda$ and equal intensities and incident on the sample at an angle $\beta$; e.g. with a spatial period of $\text{Λ}=\lambda /\left(2\text{sin}(\beta /2)\right)$ and amplitude$k_x=$ $\pi /\text{Λ}$ for the grating vector $k_x$, the electric field strength E is given by superposition of the two plane waves, and the resulting intensity distribution is

This light intensity is modulated in the x-direction, e.g. parallel to $k_x$, and it is not modulated in the y, in-plane, and z, out-of-plane, perpendicular directions. $\Delta I$ is the significant parameter for this analysis and it is obtained by using the electric field amplitudes of the interfering beams [49]. $\Delta I$ has the following expressions: e.g. $I_0$ for interfering beams of parallel polarizations TE-TE and ${}_{+}{}^{+}45^{°}$; and $I_0\left|cos\beta \right|$ for TM-TM polarizations; and $I_{0}si{n}^2(\beta /2)$ for orthogonal polarizations, both linear (${}_{-}{}^{+}45^{°}$) and circular (right and left, ${}_L{}^{R}CP$) [50]. TE and TM refer to transverse electric (or s) and transverse magnetic (or p) waves, respectively, and the parallel bars refer to the absolute value. Note that the modulation of intensity vanishes for the p-p configuration when the two beams are propagating orthogonally to each other $(\beta =\pi /2)$, and for the ${}_{-}{}^{+}45^{°}$ and ${}_L{}^{R}CP$ cases when the two beams are collinear $(\beta =0)$. Beams of parallel polarizations, linear at ${}_{-}{}^{-}45^{°}$ and circular left ${}_L{}^{L}CP$ and right ${}_R{}^{R}CP$ lead to a configuration which is equivalent to the ${}_{+}{}^{+}45^{°}$ one; and $\Delta I=0$ for the s-p configuration.

It follows from Eqs. (5) and (11), that a PP force, $F_{ph}$, parallel to the grating vector $k_x$ will act on a particle of area $A$, with dimensions smaller than $\text{Λ}$, located in the illuminated area of the interference pattern. In this case, PP is

Interestingly, the PP force $F_{ph}$, is negative for $-\text{Λ}/2\le x\le 0$, and positive for $0\le x\le \text{Λ}/2$; and it is maximum, in absolute value, for $x=-\text{Λ}/4$ and $\text{Λ}/4$, and minimum near $x=0$ and $-\text{Λ}/2$ and $\text{Λ}/2$, resulting in a movement in the direction of decreasing intensity, as expected (Fig. 3(b)). This behavior is typical of the so-called surface relief grating (SRG) experiments where azo-polymers move from bright to dark area when exposed to an interference pattern [4,5]. The mean PP force over a spatial period dictates the efficiency of the movement. The mean PP $〈F_{ph}^p〉$, denoted by $〈F^J〉$ in the remaining of this section with $J$ referring to the configuration of the interfering beams, calculated using Eq. (10) over a half-period; e.g. for $0\le x\le \text{Λ}/2$, for different arrangements of interfering beams polarizations is $〈F^{ss}〉=8K/\text{Λ}$; and $〈F^{{}_{+}{}^{+}45^{°}}〉=〈F^{{}_{-}{}^{-}45^{°}}〉=〈F^{{}_R{}^{R}CP}〉=〈F^{{}_L{}^{L}CP}〉=2〈F^{ss}〉$. For the p-p and ${}_L{}^{R}CP$ configurations, ${\theta }_G$ changes with the phase difference $\varphi =2\pi x/\text{Λ}$. The interference field polarization in the p-p configuration points in the x-direction when $\varphi =0$ and becomes circular in the (x, z) plane for $\varphi =\pi /2$, and points in the z-direction; i.e. a longitudinal field, when $\varphi =\pi$; and in the ${}_L{}^{R}CP$ configuration, it becomes linear and it rotates in the (x, y) plane, across the interference pattern, with the spatial period $\text{Λ}$. The ${}_{-}{}^{+}45^{°}$ interference field polarization behaves similarly to the p-p one. It points in the y-direction when $\varphi =0$ and becomes circular in the (x, y) plane for $\varphi =\pi /2$, and points in the x-direction when $\varphi =\pi$. Note that for the ${}_{-}{}^{+}45^{°}$ and ${}_L{}^{R}CP$ configurations, $\varphi$ is $\pi$-shifted compared to the p-p one; i.e. the field polarization is parallel to x-direction for p-p and perpendicular to it for ${}_{-}{}^{+}45^{°}$ and ${}_L{}^{R}CP$ when $\varphi =0$, and the opposite happens when $\varphi =\pi$. So, using Eqs. (4) and (10), and noting that $cos{\theta }_G=(1+e^{i\varphi }) \text{sin}(\beta /2)$ for p-p, and $cos{\theta }_G=(1-e^{i\varphi }) \text{cos}(\beta /2)$ for ${}_{-}{}^{+}45^{°}$, and $cos{\theta }_G=(1+e^{i\varphi }) \text{cos}(\beta /2)$ for ${}_L{}^{R}CP$, where, $i^2=-1$, $i$ is a complex number, $〈F^{pp}〉=(1+4si{n}^2(\beta /2))〈F^{ss}〉$, and $〈F^{{}_{-}{}^{+}45^{°}}〉=〈F^{{}_L{}^{R}CP}〉=(1+4co{s}^2(\beta /2))〈F^{ss}〉$.

This finding supports SRG experiments on azo-polymers which demonstrate grating heights observed at small interference angles that increase with the increasing $\beta$, and which are nearly 5 times higher for the ${}_{-}{}^{+}45^{°}$ and ${}_L{}^{R}CP$ configurations compared to those observed with the p-p configuration for small $\beta$ [51]. More importantly, the analysis shows that light confinement schemes, such as interference patterns, may function as amplifiers for vectorial motion. For the $ss$ configuration, using the characteristics of the nanosphere,$\eta =4\times {10}^{-6}$, with $\beta =26°$; $\text{Λ}=1.02 \mu m$; $I_0=100 mW c{m}^{-2}$; intensity gradient $~100 mW/c{m}^2/\mu m$; $〈F^{ss}〉=32 fW\mu m^{-3}$ and, for 1 s of irradiation, the mean PP force is $~1.02 nN$.

If a 50 nm thin DR1-polymer film ($N_v= 1 molecule n{m}^{-3}$; $\eta =6.38\times {10}^{-2}$) is subjected to the same intensity gradient as the nano-sphere, $〈F^{ss}〉=510 pW\mu m^{-3}$, and the mean PP force on a $1 \mu m^2$ area is ~$5.1 \mu N$. Physically, the PP force will move both the sphere and the polymer into the direction of grating vector, and since the sphere is much smaller than the spatial period of interference, it will move either side of the intensity maximum; and for the polymer, half of the irradiated volume will move each side of intensity maximum.

In solid polymer films, $F_{ph}$ is experimentally observed to overcome the elastic force, $F_{el}$, of the polymer. Poly-methyl-methacrylate (PMMA) in the form of sheets exhibit an elastic constant of $k_{el}~2\mu N/nm$ measured in the linear part of the load versus displacement, within 500 nm displacement range, with a load maintained for 10 s during indentation experiments [52]; and a 380 nm DR1-PMMA type film exhibits experimental values of $k_{el}~50nN/nm$ in the dark, and $k_{el}~20nN/nm$ under uniform actinic light irradiation [10]. For a DR1-PMMA film, with the characteristics above, under irradiation ($I_0=10 mW.c{m}^{-2};10 s$), the theoretical estimate of $k~6.38\times {10}^{2}pN/nm$ for small displacements, yields $F_{el}/F_{ph}$ in the range of $20~3100$.

Clearly, photosoftening of films of polymer by photoisomerization alone cannot explain observed mass motion in azo-polymers, and a gradient of light intensity must exist in order to put the polymer into motion as predicted by Eqs. (1) and (5). Furthermore, using $h=k.t/\xi$ and $a~37 \mu m$ as the cubic dimension of the irradiated volume of the polymer, yields $h~0.76\times {10}^{-6}$ in PMMA ($h~0.76\times {10}^4$ in glycerol for comparison). For the estimate of $h$ in PMMA, ${\mu }_f~1.0\times {10}^{10} Pa .s$ was used near ${25}^{\circ }C$ for a $~50 nm$ thin-PMMA film $\left(T_g~{82}^{\circ }C\right)$ [53]. These findings suggest that the frictional force should dominate the photoisomerization force in the thin film configuration; however experiments demonstrate the opposite. Indeed the friction coefficient $\xi$ must be decreased by several orders of magnitude in the expression of $h$ to reach high-$h$ number just as experimentally observed in [9] where the materials viscosity was reported to decrease by several orders of magnitude by photoisomerization. This confirms the assumption that the particles become mobile under the action of the PP force.

Temperature increase by local heating due to light absorption is negligible in isomerizable materials. Unlike the photophoretic force caused by the transformation of light absorption by the particles into heat [28], the PP force is caused by the dissipation of the energy of the absorbed photons into mechanical energy of isomeric shape change of the molecules [2]. The PP force is due to mechanical work done by molecular machines. Different particles containing the same photoactive molecule and under the same intensity gradient, are acted upon by the same PP force per molecule per unit time of irradiation. This analysis yields a PP force ~$10pN/molecule/s$ for both the sphere and the polymer, and it importantly shows that it takes only one molecule per second of irradiation to reach the range of the RP force.

8. Conclusion

The theory presented in this paper opens a new direction of investigations in the field of optical manipulation and transport of matter by using the mechanical work of light-fueled molecules. To name a few examples, the formalism presented here could be used for light potentials of bottle beams and speckle and vortex type beams; and inasmuch as such a research will benefit from the already established and very well developed area of optical trapping and manipulation of micro- nano- particles by laser beams, including beam shaping, and vectorial optical beams; the large magnitude of the photoisomerization force allows for the manipulation; for example levitation and propulsion; of particles with gravitational forces larger than the $\mu N$ range, and its vectorial nature allows for particles directional sorting by light polarization. I envision that this theory will expand horizons in the transport of matter by light.