Patent ID: 12252598

DETAILED DESCRIPTION

The disclosed subject matter provides levitation and flight of structures using a photophoretic effect.

FIGS.1A-1Billustrate an exemplary structure100comprising a sheet side110, having a structured side120. InFIG.1A, a plurality of gas particles130,140can be incident on either surface. Each of the plurality of gas particles130,140collides with one of the surfaces. Gas particles incident on the structured side140can exit the structure with a velocity greater than the gas particles incident on the sheet side110, resulting in a net recoil force150. This net recoil force150can cause levitation in the exemplary structure100. In certain embodiments the sheet side110can include a BoPET film160and the structured side120can include carbon nanotubes1B).

In certain embodiments the structured side120can provide rigidity to the structure100, includes several advantages that would be recognized by persons or ordinary skill in the art. For the purpose of example, and not limitation, the rigidity can reduce or eliminate deformation of the structure100which can result from application of a force or pressure. This force or pressure can result from uneven application of the net recoil force150across the structure, or, alternatively, from a plurality of external forces, including, for example, environmental forces or differential thermal expansion.

In certain embodiments, the net recoil force is produced in an upward direction by a new momentum transfer between the sample and the incident gas molecules. In such embodiments, the levitation is not caused by a temperature difference between the top and bottom surfaces of the structure, but rather due to the difference in thermal accommodation coefficients between the top and bottom surfaces. The force generated by the difference in thermal accommodation coefficients (Act-force) can be in excess of the force needed to lift the structure, and thus additional components, for example cameras or other sensors, can be attached to the structure to provide greater functionality. In the free-molecular regime, Act-force can increase proportionally with pressure, and can reach a maximum when the Knudsen number (that is, the ratio of the mean free path to the size of the structure) is approximately 1. Since Act-force is generated primarily through the collision of molecules, an increase in pressure can reduce the Act-force. Further, various simulations can be used to predict the different force components of the resultant Act-force. In certain embodiments, Monte Carlo simulations or numerical solutions of the Boltzmann equation can provide further insight into the force generated from Act-force.

In certain embodiments, either the sheet side110or the structured side120or both can comprise one or more ultrathin materials. For example, and not limitation, ultrathin materials can include materials with a thickness of between 1 and 100 nanometers.

FIG.2illustrates an exemplary embodiment of a structured material200according to the present embodiments. A smooth surface210is provided which can be a BoPET film or other similar material. Onto this smooth surface210can be deposited a structured surface220. In certain embodiments, this structured surface220can include a carbon nanotube solution which is dropcast onto the smooth surface210. The structured surface220can be configured to trap incoming gas molecules or otherwise inhibit their movement, resulting in multiple gas-CNT collisions and better thermal accommodation of the incident gas molecules than for corresponding incoming gas molecules on the smooth side. In certain embodiments, the carbon nanotube solution can include a 0.2 weight % water-based single-wall carbon nanotube (with 1-2 nanometer diameter and 5-30 micrometer length) diluted with deionized water by a volumetric ratio of 3:1. The BoPET sheet can then be placed on a silicon wafer, and heated to a temperature of 50 degrees Celsius. The carbon nanotube solution is then placed on the heated BoPET sheet, allowing the water to evaporate, and resulting in the structured material200.

In certain free-molecular regimes, gas molecules colliding with a heated structure absorb energy from the structure and can leave with a higher average temperature. The measure of this energy transfer is known as the thermal accommodation coefficient, and can be represented by the equation:

α=Tr-TiTs-Ti(2)
Wherein, Trcan represent the temperature of departing gas molecules, Tscan represent the temperature of the structure, and Tican represent the temperature of the incident molecule.

This carbon nanotube (“CNT”) layer220can act as a lightweight light absorber (with, for example, absorptivity of ˜90%) while also increasing the structural stability of the resultant structure, for example, by providing rigidity to the resultant structure. In such an embodiment, the CNT layer220can tend to trap incoming gas molecules130, which can allow the gas molecules130to collide with the CNT layer220several times before exiting and to absorb more heat from the structure before exiting. As a result, the thermal accommodation coefficient for the CNT layer220of the resultant structure can be higher than the BoPET layer210of the resultant structure, resulting in a lift force being generated. Light absorber materials with less absorptivity (for example 80%) or greater absorptivity (for example 99%) are also contemplated.

In certain embodiments, the smooth surface210comprises a surface which is configured to reflect all or substantially all incident particles130without trapping or otherwise impairing reflection of the incident particles130. In certain embodiments, the smooth surface210can reflect about 100% of the incident particles without trapping or otherwise impairing reflection of the incident particles130, resulting in the lowest potential accommodation coefficient. In certain embodiments, the smooth surface210can reflect less than 100% of the incident particles without trapping or otherwise impairing the incident particles130. In certain exemplary embodiments, the smooth surface210can reflect a first portion of incident particles130without trapping or otherwise impairing reflection of the incident particles130, wherein the structured surface220can reflect a second portion of incident particle130, with trapping or otherwise impairing at least a portion of the incident particles130.

FIG.3illustrates a sequence of photos showing a sample according to the present embodiments that is levitated using a photophoretic effect. As shown in310, at zero seconds, the samples340, which have not been irradiated and have no achieved thermal equilibrium, do not levitate and rest at the bottom of the test chamber. As shown in320, at seven seconds, the samples340have achieve lift due to the net recoil force150produced by the photophoretic effect. Finally, as shown in330, the samples340, after light is no longer applies, no longer levitate, and so fall to the bottom of the test chamber again.

FIGS.4A-4Bshow the results of a series of exemplary tests of the system according to certain disclosed embodiments. As shown in410(FIG.4A), using a Δα-force with Δα=0.15±0.05 and a flux of ˜0.5 W/cm2, an object of various sample areal densities can be levitated. Additionally, as shown in420(FIG.4B), the pressure-dependent lift force, as compared to the weight of the sample, can provide lift up to a limited pressure, as pressures beyond that limit can result in distortion of the sample and reduction in the lift force.

In certain embodiments, various other components can be attached to the structured material, without inhibiting the ability of the material to achieve lift. These components can include, for example and not limitation, electronic components, mechanical components, computer components, and sensor components. As further examples, the structured material can also contain one or more cameras, one or more sensors, one or more thermometers, one or more wings, and/or one or more payloads. The inclusion of different components can allow the structured materials to achieve a variety of different functions, including, but not limited to, surveillance, weather monitoring, video recording, navigation, and/or measuring.

FIGS.5A-5Bshow an exemplary light trap system500according to certain embodiments. According to certain embodiments, the light trap500can include a plurality of LEDs510located beneath a test chamber550. Within the test chamber (FIG.5A), a partially transparent mesh540can be placed above the bottom surface of the test chamber. On top of this transparent mesh540a test sample560can be placed. This test sample560can then be illuminated by the LEDs510, resulting in levitation of the test sample560. As shown in520and530(FIG.5B), the intensity of light from the LEDs510can be significantly greater at 7 centimeters above the LEDs510(as shown in520) than at 10 centimeters above the LEDs (shown in530).

FIGS.6A-6Dshows a plurality of contour plots which can represent the potential altitude reached by a sample which is levitated according to various variables. For example, and not limitation,610(FIG.6A) shows a contour plot having an x-axis measuring radius of a sample disk, a y-axis measuring altitude which can be achieved, and contour shades showing log10of the areal density of the disk.620(FIG.6B) shows a contour plot having an x-axis measuring radius of a sample disk, a y-axis measuring altitude which can be achieved, and contour shades showing log10of an exemplary payload mass (in milligrams). For example, and not limitation,630(FIG.6C) shows a contour plot having an x-axis measuring radius of a sample disk, a y-axis measuring altitude which can be achieved, and contour shades showing sample disk temperature (in Kelvin).640(FIG.6D) shows a contour plot having an x-axis measuring radius of a sample disk, a y-axis measuring altitude which can be achieved, and contour shades showing the difference between disk temperature and ambient temperature (in Kelvin).

FIGS.7A-7Cshow an example chamber.FIG.7Ashows a side view of chamber700and samples701.FIG.7Bshows a top view of the setup with the8arrays of the LED light ring and 74% transparent mesh.FIG.7Cshows 74% (left,702) and 85% (right,703) transparent meshes.

The disclosed subject matter provides various models for developing air levitation of a target structure. For example, the disclosed subject matter provides a theoretical model that can be used for the development of mid-air levitation of a target structure based on the heat transfer between the structure and environment. In this model, the heat transfer analysis can be performed for the entire range of pressure. The temperature distribution of the surface of the target structure can be identified to find the temperature of the gas molecules impinging on and reflecting from the surface. Then, the total amount of force experienced by the target structure (e.g., a disk) with two different surface properties on either side can be identified.

In certain embodiments, the disclosed subject matter provides a heat transfer model. Force generation in free molecular and continuum regimes obey distinct physics. Hence, the disclosed heat transfer model can be configured to properly describe the physical phenomena in both regimes. The heat transfer model can assess the energy balance for a target structure (e.g., a disk) and derive the equations for the surface temperature of the disk. In the disclosed model, the structure (e.g., disk) is absorbing radiation on one side and dissipating heat on both sides via radiation, convection, and/or conduction. In non-limiting embodiments, air can be considered an ideal gas with the properties listed below:

Heat capacity at constant pressure:

Cp[kJkg⁢K]=28.11+(0.1956×10-2)⁢T[K]+(0.4802×10-5)⁢T[K]2-(1.966×10-9))⁢T[K]3,(3)
thermal conductivity:

kair[WmK]=(0.238×10-3)⁢T[K]0.8218,(4)
thermal diffusivity (D), thermal expansion coefficient (β), dynamic viscosity (μ), and density is given, respectively, by:

D=kairρ⁢cp,β=1T0,μ=μo(TT0)2/3,and⁢ρ=PRair⁢T,(5)
where T is the temperature, T0=273 K is the reference temperature, μ0=1.716×10−5Pa·s is dynamic viscosity at the reference temperature, P is the pressure, and

Rair=RuMair=287.1Jkg⁢K
is the ideal gas constant for air, obtained from the universal gas constant

Ru=8.3⁢1⁢4⁢Jmol⁢K
and the molar mass of air

Mair=0.0⁢2⁢8⁢9⁢6⁢kgmol.

The incident energy can be absorbed on one side of the disk and can be balanced by the total heat transfer from the disk, which includes radiation and conduction/convection terms:
Qinc=Qrad+Qcc.  (6)
Here,

Qi⁢n⁢c=Iinc⁢S2,
where lincis the incident flux shone on one side of the disk and S=2πa2is the total surface area of the disk. In non-limiting embodiments, the disk can be assumed to have a uniform temperature, Ts, which can be a reasonable approximation when it is compared to the results of finite-element simulations in COMSOL under a variety of conditions. The radiative heat transfer from both sides of the disk can be then given by:

Qrad=S⁢σ⁢ε(TS4-T∞4),where⁢σ=5.6⁢7×1⁢0-8⁢Wm2⁢K4,(7)
ε is the emissivity of the surface and is assumed to be 0.95 for the disclosed models (consistent with the temperature measurements of the disks using a thermal infrared camera), and T∞is the ambient temperature.

In the free molecular regime, the conduction heat transfer for a disk with a total area S can be given by:
Qfm=hmolS(Ts−T∞),  (8)
In equation (8), the molecular heat transfer coefficient

hm⁢o⁢l=α8¯⁢γ+1γ-1⁢P⁢v¯T
was used with the average thermal accommodation coefficient of the top and bottom sides of the disk

α¯=αtop+αbottom2,
the adiabatic constant

γ=CPCV=CPCP-R,
and the average speed of gas molecules

v¯=8⁢kB⁢Tπ⁢m=8⁢Rair⁢Tπ,
where kBand m are the Boltzmann constant and molecular mass of the gas molecules. In calculating CP, CV, kairandv, the temperature can be the average temperature between ambient temperature T∞and the surface temperature Ts.

In the continuum regime, the convection/conduction heat transfer can be written as
Qcont=Nu aπkair(Ts−T∞),  (9)
with Nu=0.417Ra0.25+8/π. The Rayleigh Number can be given by

R⁢a=g⁢β⁡(Ts-T∞)⁢ρ⁢d3μ⁢D,
where g is gravitational acceleration, d is the diameter of the disk, ρ is density, and α, β, and μ are defined in equation 5. Equation (5) includes a constant term 8/π representing conduction and a pressure-dependent term representing advection 0.417Ra0.25, which scales with pressure as Qadv∝Ra0.25∝(P2)0.25∝P0.5and vanishes in the free molecular regime (P→0).

Interpolating between the free-molecular and continuum regimes, the following expression can be obtained for the convection/conduction term valid for all pressures:

Qc⁢c=11/Qf⁢m+1/Qc⁢o⁢n⁢t.(10)
By inserting the three heat transfer mode equations (7) and (10) into equation (6), the temperature of the surface of a disk, Ts, numerically as a function of radius, pressure, and incident intensity can be identified.

Force Formulation model: The photophoretic force acting on a disk with a temperature difference between the top and bottom sides can be defined asT-force. Modifying the surface to achieve different accommodation coefficients on the top and bottom can result in a force on the same order of magnitude (i.e., Δα-force). Surface modification for a thin lightweight disk can be simpler than fabricating thicker ultralight structures with decreased thermal conductivity, such as nano cardboard.

In both the free molecular regime and the continuum regime, due to the similar physical origin of the photophoretic force, the net force on the structure can be expressed as
F=Δθ(P)ψ(P),  (11)
where Δθ is the temperature variation of gas molecules next to the surface, and

ψ[NK]
represents the force per unit change in temperature of the colliding molecules. Δθ(P) and ψ(P) are both functions of pressure. Equation (11) is based on the interaction between the disk surface and the gas molecules next to the surface.

Free Molecular Regime: In the free molecular limit with Kn→∞, the average temperature of the gas molecules next to the surface is approximated as

θ=12⁢(Ti+Tr),
in which Tiand Trare temperatures of the gas molecules before and after the collision, respectively. θ can be defined using the definition of thermal accommodation coefficient between gas molecules and surface,

α=Tr-TiTs-Ti,
which can result in

θ=Ti+12⁢α⁡(Ts-Ti).
Thus, the temperature variation between the two sides of the disk can be:

Δ⁢θ⁡(P)=12⁢Δ⁢α⁡(Ts-Ti),(12)
for a disk with an accommodation coefficient difference of Δα=αbottom−αtop. In this condition, the collision of gas molecules with a surface can be far more probable than the collision of gas molecules with each other hence the temperature of gas molecules before colliding with a surface can be equal to far-field temperature, or Ti=T∞. The force can be found in the free molecular regime with Kn→∞. The derivation can start by finding the force due to the momentum transfer between the gas molecules and the surface. Assuming a uniform temperature across the thickness of the disk and an accommodation coefficient difference of Δα, the Maxwell distribution, ƒ(v), can be integrated over the entire range of velocity and assume an area of πa2and volume of V of the air with N number of gas molecules. The net force on one side can be:

<F>=π⁢a2⁢N⁢∫0∞{(2⁢m⁢v)⁢f⁡(v)⁢vV}⁢d⁢v(13)
and molecule flux of

<J>=NV⁢∫0∞f⁡(v)⁢v⁢dv(14)
representing the flux of air molecules hitting and reflecting from the surface. In these relations, ƒ(v) is Maxwell distribution and is defined by:

f⁡(v)=(m2⁢π⁢k⁢T)1/2⁢e-m⁢ν22⁢k⁢T.(15)
This approach results in the following net Act-force on a thin plate with uniform temperature and different accommodation coefficients on two sides:

Ffm=π⁢a24⁢T∞⁢P⁢Δ⁢α⁡(Ts-T∞).(16)
Equation (16) represents a linear increase with pressure, which is valid when the air molecules do not collide with each other as frequently as they do with the surface (Kn>>1).

Continuum Regime: The derivation of the photophoretic force acting on a sphere can be extended to the case of an oblate spheroid and then take the limit to approach a flat disk with negligible thickness. In order to find the force for the entire range of pressure, equation (11) can be used. Knowing the temperature solution from the heat transfer model, Δθ(P) can be identified and then an expression for the force can be derived.

Continuum regime, part A: constructing Δθ(P): For an oblate spheroid with semi-axes a and b (a>b), similar to the free molecular regime, the average temperature of the gas molecules next to the surface in the continuum regime can be approximated as

θ=12⁢(Ti+Tr)=Ti+12⁢α⁡(Ts-Ti).
In the case of the spheroid, mathematical modeling can disallow for a discontinuity in the value of the accommodation coefficient. Thus, in order to achieve a smooth transition from one value of accommodation coefficient to the other instead of the two constant accommodation coefficients on the two sides, the variation of the accommodation coefficient over the surface of the spheroid can be approximated by the Legendre expansion:
α=ΣanPn(cos η)=a0+a1cos η+ . . .  (17)

Finding the coefficient of the Legendre expansion can give

a0=αtop+αb⁢o⁢t⁢t⁢o⁢m2⁢and⁢a1=34⁢Δα.
The amplitude of temperature variation along the surface, Δθ, of the spheroid can be expressed as:

Δ⁢θ=38⁢Δ⁢α⁡(Ts-Ti).(18)
In order to express (Ts−Ti) in terms of (Ts−T∞), a form for the conductive heat transfer can be constructed from the disk as a function of (Ts−T∞), then the amount of heat being removed from the surface by the interaction of the surface with the gas molecules with Tican be equated as their initial temperature right before colliding with the surface. This heat transfer can be expressed by[50,51].

d⁢QdS=hm⁢o⁢l(Ts-Ti),(19)
with dS being the surface area element, and hmolmolecular heat transfer coefficient defined above. Note that equation (19) holds for the entire range of pressure because the conduction from the surface to the adjacent gas molecules directly on the surface happens via molecular interaction. Once Q is found as a function of (Ts−T∞), equation (18) can be used to find (Ts−Ti) which gives Δθ according to equation (18).

In the continuum regime, Qcond,codoes not have a trivial solution for a spheroid. The problem of steady heat conduction can set up around a spheroid of surface temperature Ts+Δθ cos (η), which can be a superposition of a uniform value and a surface varying component with η as the polar angle of the spheroid, in a medium of ambient temperature T∞and without volumetric heat generation within the spheroid. The governing equation can be:

1cos⁢h⁢ξ⁢∂∂ξ[cos⁢h⁢ξ⁢∂T∂ξ]+1sin⁢η⁢∂∂η[sin⁢η⁢∂T∂η]=0,(20)
where ξ and η are radial and angular parts of spheroidal coordinates (seeFIG.9). The boundary conditions for this problem can be:

T=Ts+Δ⁢θ⁢cos⁡(η)⁢at⁢ξ=ξ0(surface⁢of⁢the⁢spheroid,ξ0=tan⁢h-1⁢ba),(21)T=T∞⁢as⁢ξ→∞.
The temperature solution for equation (16) with boundary conditions shown in (21) becomes:

T⁡(ξ,η)-T∞=(Ts-T∞)⁢tan-1(sin⁢h⁢ξ)-π2tan-1(sin⁢h⁢ξ0)-π2+Δ⁢θ⁢cos⁡(η)⁢sin⁢h⁢(ξ)⁢(tan-1(sin⁢h⁢ξ)-π2)+1sin⁢h(ξ0)⁢(tan-1(sin⁢h⁢ξ0)-π2)+1.(22)
The local amount of heat transfer can be found using solution (22) and its proper boundary conditions:

[∇T]ξ=ξ0=1l[∂T∂ξ⁢e^ξ+∂T∂η⁢e^η]ξ=ξ0=1l[(Ts-T∞)⁢1cos⁢h⁢ξ0⁢{tan-1(sin⁢h⁢ξ0)-π2}⁢e^ξ-Δ⁢θ⁢sin⁡(η)⁢e^η](23)
with l=√{square root over ((a2−b2)(sin h2ξ0+cos2η))}. The normal component of the temperature gradient due to a surface-varying component of the temperature is not included in (23) since its integration over the surface of the spheroid is zero. Thus, the total heat flow from the surface of the oblate spheroid can be:

Qcond,co=∫(-ka⁢i⁢r⁢∇T·eˆξ)⁢d⁢S=8⁢π⁢a⁢kair(TS-T∞)s⁢h,(24)
where, sh is a shape factor that depends solely on the geometry of the sample and defined as:

s⁢h=(π-2⁢tan-1(sin⁢h⁢ζ0))cos⁢h⁢ξ0.(25)

In the two limiting cases of disk and sphere, the shape factor reduces to:

limba→0(sh)=πfor⁢a⁢disk.(26)limba→1(sh)=2for⁢a⁢sphere.(27)
In the case of a disk, equation (24) reduces to: Qcond,co=8akair(Ts−T∞) which can be the conductive term in equation (9). By relating this heat conduction from the disk to the total heat transfer from (19), the following question can be identified:
hmol(Ts−Ti)2πa2=8akair(Ts−T∞),  (28)
thus, the temperature difference in equation (18) reduces to:

Δθdisk=32⁢π⁢kaira⁢hm⁢o⁢l⁢Δ⁢α⁡(Ts-T∞).(29)

Continuum regime, part B: developing the force formula, F=Δθ(P)ψ(P): The photophoretic force on a particle in continuum regime can be caused by thermal creep. When the gas over a surface has a tangential temperature gradient, it flows over the surface from the cooler side to the hotter side with slip velocity, vs, defined by:

vs=ks⁢μairρ⁢T∞⁢∇sT,(30)
where κs=1.14 is thermal slip coefficient, μ is viscosity, ρ is density, and ∇sT is the tangential temperature gradient in the gas layer. Using (23), the following equation can be derived:

∇sT=-Δ⁢θ(a2-b2)⁢(sin⁢h2⁢ξ0+cos2⁢η)⁢sin⁡(η)⁢eˆη.(31)
In order to calculate the force, the Lorentz reciprocal theorem can be for the Stokes flow. The migration velocity of the spheroidal particle along its symmetry axis can be identified by the following equation:

U=-14⁢π⁢b⁢a2⁢∫(n·r)⁢(vs·eˆz)⁢d⁢S=κs⁢μρ⁢T∞⁢Δ⁢θa⁢sin⁢h⁢ξ0⁢cos⁢h⁢ξ0(cos⁢h2⁢ξ0sin⁢h⁢ξ0⁢tan-1(1sin⁢h⁢ξ0)-1)(32)
By taking the limit of

ba→0,
the value or migration velocity for a disk can be:

Udisk=π⁢κs⁢μair2⁢a⁢ρ⁢T∞⁢Δ⁢θ=π⁢κs⁢μair2⁢a⁢ρ⁢T∞⁢32⁢π⁢kaira⁢α_⁢γ+1⁢P⁢v_8⁢γ-1⁢T⁢Δ⁢α⁡(Ts-T∞)=6⁢κs⁢μ⁢Ts⁢kaira2⁢ρ⁢P⁢v_⁢γ-1γ+1⁢Δ⁢αα_⁢(Ts-T∞)T∞.(33)
Using kinetic theory of gases,

ka⁢i⁢r=f⁢Cv⁢μa⁢i⁢r⁢and⁢μa⁢i⁢r=12⁢v¯⁢ρ⁢λ
can be substituted, where the standard definition can be used for the mean free path

λ=μairP⁢π⁢Rair⁢T2,
the f-factor can be given by

f=1+9⁢Rair4⁢Cv=1+94⁢(γ-1),
and substituted

RCν=Cp-CνCν=CpCν-1=γ-1.
The resulting expression can be:

Udisk=v_⁢κs⁢f⁢ρ⁢Ts⁢CvP⁢(λa)2⁢γ-1γ+1⁢Δ⁢αα_⁢Ts-T∞T∞=v_⁢κs⁢f⁢CvRair⁢(λa)2⁢γ-1γ+1⁢Δ⁢αα_⁢Ts-T∞T∞=v_⁢κs⁢f⁡(λa)2⁢1γ+1⁢Δ⁢αα_⁢Ts-T∞T∞=v_⁢κs(λa)2⁢1+94⁢(γ-1)γ+1⁢Δ⁢αα_⁢Ts-T∞T∞,(34)
where the ideal gas law, P=ρTsRaircan be used. The last expression in equation (34) suggests that in the transition regime

(K⁢n=λa∼1),
the air can flow around the disk at a significant fraction of the average speed of the air molecules (i.e., tens of meters per second).

Equation (33) can represent the velocity of a disk that is free to move in a gaseous medium without any forces acting on it. For a fixed (immobile) disk, the corresponding force acting on the disk can be obtained using the Stokes drag formula with an effective Stokes radius for a disk[54],

r=83⁢π⁢a:

Fc⁢o=6⁢π⁢μair(83⁢π⁢a)⁢Udisk=1⁢6⁢μair⁢a⁢Udisk=1⁢6⁢μ⁢6⁢κs⁢μair⁢Ts⁢kaira⁢ρ⁢P⁢v¯⁢γ-1γ+1⁢Δ⁢αα¯⁢(Ts-T∞)T∞.(35)
As a check to see whether the Stokes flow assumption is correct, the Reynolds Number can be evaluated using the migration velocity in equation (33),

Re=ρ⁢Ud⁢i⁢s⁢k⁢2⁢aμ=π⁢κs⁢ΔθT∞≪1,
which can justify the Stokes Flow assumption.

Force formula for the entire range of pressure: the photophoretic force can be combined in the free molecular (16) and the continuum (35) regimes to generate an interpolation valid for the entire range of pressure,

F=11Ffm+1Fco=π4⁢Δ⁢α⁡(Ts-T∞)T∞⁢a2⁢P*P22⁢P*2+P*P(36)
where P* is the pressure at which force can be maximized:

P*=1a⁢(1⁢9⁢2π⁢μair2⁢kB(γ-1)⁢kair⁢Tavg2m⁡(α¯)⁢(γ+1)⁢v¯)13=v¯⁢ρ⁢λ2⁢a⁢(4⁢8⁢(γ-1)⁢(9⁢γ-5)π⁢m⁢α¯⁢v¯⁢Cv⁢kB⁢Ta⁢v⁢g2)13.(37)
All the parameters can be the same as defined as disclosed.

P*∝1a
and, therefore, the maximum force:

FΔα_⁢max=π6⁢Δ⁢α⁡(Ts-T∞)T∞⁢a2⁢P*(38)
Can scale linearly with the radius of the disk if all other parameters are held constant

(P*∝1a→FΔ⁢α_⁢max∝a).

Force Generation from Temperature Gradient Through a Thin Disk, ΔT-Force:

In the case of a disk with

Thicknessradius≪1,
where the accommodation coefficient is equal on both sides, one can write the ΔT-force in the free molecular regime as

Ffm⁢_⁢Δ⁢T=π⁢a24⁢T∞⁢αa⁢v⁢g⁢P⁡(Ts⁢_⁢H-Ts_C)
where Ts_Hand Ts_Care the surface temperature on the hot and cold side, respectively, and αavgis an average value of the thermal accommodation coefficient of both sides of the sample. The continuum regime force can also be deduced from equations (33) and (35) by replacing Δα(Ts−T∞) with α(Ts_H−Ts_C). This can result in:

Udisk⁢_Δ⁢T=π⁢κs⁢μa⁢i⁢r2⁢a⁢ρ⁢T∞⁢Δ⁢θ=π⁢κs⁢μa⁢i⁢r2⁢a⁢ρ⁢T∞⁢α⁡(Ts⁢_⁢H-Ts_C)(39)Fco⁢_Δ⁢T=8⁢π⁢κs⁢μa⁢i⁢r2ρ⁢T∞⁢α⁡(Ts⁢_⁢H-Ts_C)(40)FΔ⁢T=11Ffm⁢_⁢Δ⁢T+1Fco⁢_Δ⁢T=π⁢a24⁢α⁡(Ts⁢_⁢H-Ts_C)T∞⁢32⁢κs⁢μa⁢i⁢r2⁢R⁢T∞a2⁢P+32⁢κs⁢μair2⁢RT∞P=π4⁢α⁡(Ts⁢_⁢H-Ts_C)T∞⁢32⁢κs⁢μa⁢i⁢r2⁢R⁢T∞⁢PP2+32⁢κs⁢μair2⁢RT∞a2.(41)
The force can be defined as:

FΔ⁢T=Fˆ⁢P⁢PˆP2+Pˆ2(42)Pˆ=4⁢μa⁢2⁢κ⁢R⁢T∞;(43)Fˆ=π⁢a⁢μ⁢α⁡(Ts⁢_⁢H-Ts⁢_⁢C)⁢2⁢κ⁢R⁢T∞T∞(44)
where {circumflex over (P)} is the pressure at which FΔTcan be maximized, and the maximum can be written as:

FΔ⁢T⁢_⁢max=Fˆ2=π⁢a⁢μα⁡(Ts⁢_⁢H-Ts⁢_⁢C)⁢2⁢κ⁢R⁢T∞2⁢T∞(45)
This formula can show that for a (Ts_H−Ts_C)˜0.1 K, even with full thermal accommodation, using values shown inFIGS.8A-8B, and

Fmax⁢_Δ⁢TFmax⁢_Δα∼(Ts⁢_⁢H-Ts⁢_⁢C)Δ⁢α⁡(Ts-T∞)<0.0⁢1,
which demonstrates that FΔTcan be negligible compared to FΔαand insufficient to explain the observed levitation. FIGS.8A-8B show example temperature estimation and thermally deformed samples.FIG.8Ashows a graph showing a calculated temperature of mylar disk under

0.5⁢wcm2
incident light ∈=0.95. This plot can be used to predict at what pressures, and with what radii, samples exceed a temperature threshold and deform.FIG.8Bshows i) side and ii) angled view of a 6 mm disk under

0.6⁢wcm2
in 50 Pa environments.FIG.8Balso shows iii) undeformed (left) and deformed (right) 6 mm disk. The highly curled sample, which has rolled up into a cylinder with submillimeter diameter, can be under

0.8⁢wcm2
in 50 Pa environment. All scale bars are 3 mm.

Altitude dependency of the properties for the Earth's atmosphere: In order to model changes in temperature and pressure of ambient air as a function of altitude, altitude dependency can be incorporated in all parts of the model that are functions of temperature and pressure.FIGS.10A-10Bshow how the temperature (10B) and pressure (10A) depend on the altitude. The graphs represent annual and spatial averages, and the values can vary depending on the exact location and time of year.

Predicted payload for various combinations of emissivity and Δα: In addition to the predicted payload shown inFIG.6, the maximum payload can be identified for other combinations of emissivity and the difference in the thermal accommodation coefficient.FIGS.11A-11Dprovide graphs for determining of Δα for CNT covered mylar. Areal density that can be levitated under

0.5⁢wcm2
is shown with ∈=0.95 and11A) Δα=0.0511B) Δα=0.111C) Δα=0.211D) Δα=0.25. The range of 0.1<Δα<0.2 results in an acceptable match between the observations and theoretical predictions.

FIGS.12A-12Dshow the predicted payload for the parameters that provided the best fit to our actual experiments (with Δα=0.15, ∈=0.95). The maximum payload can be comparable to the weight of the disk itself (˜0.1 mg) and can be achieved for disk radius of ˜1 cm at altitudes of ˜80 km.FIGS.13A-13Dshow that reducing the thermal emissivity using a selective solar absorber with ∈=0.5 increases the maximum payload to ˜0.5 mg, still achieved for a radius of ˜1 cm at altitudes of ˜80 km.FIGS.14A-14Dshow that using an even lower emissivity of 0.1 makes the temperatures exceed 500 K, which can require the use of materials other than Mylar. However, such low emissivity can also allow levitation and significant payloads with much lower light intensities than full natural sunlight (FIGS.15A-15D). Increasing the Δα to 0.3 improves the maximum payload to a few mg, achieved for radii of a few cm at altitudes of ˜85 km (FIGS.16A-16D). Using Δα=0.5 can result in maximum payloads of up to 10 mg for radii of ˜3 cm at altitudes of ˜90 km (as shown inFIG.6).

All patents, patent applications, publications, product descriptions, and protocols, cited in this specification are hereby incorporated by reference in their entireties. In case of a conflict in terminology, the present disclosure controls.

While it will become apparent that the subject matter herein described is well calculated to achieve the benefits and advantages set forth above, the presently disclosed subject matter is not to be limited in scope by the specific embodiments described herein. It will be appreciated that the disclosed subject matter is susceptible to modification, variation, and change without departing from the spirit thereof. Those skilled in the art will recognize or be able to ascertain using no more than routine experimentation, many equivalents to the specific embodiments described herein. Such equivalents are intended to be encompassed by the following claims.