Supplementary information: Efficient mass transport by optical advection

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1 Supplementary nformaton: Effcent ma tranport by optcal advecton Veerachart Kaorndenukul, Sergey Sukhov, and Artde Dogaru CREOL, The College of Optc and Photonc Unverty of Central lorda, 4 Central lorda Boulevard Orlando, lorda 3816, USA 1. Velocty of partcle n a collod llumnated by Gauan beam Here we analytcally etmate the velocty of optcally nduced advectve flow n collodal ytem dependng on parameter of llumnaton. Let u conder nfnte lqud medum wth em-nfnte pace < contanng collodal partcle. We aume that Gauan beam ncdent normally to the lqud-collod nterface. Each partcle nde the collod experence optcal force nduced by the beam. We aume that the beam not focued o that gradent force would be neglgble n comparon to catterng force. We alo aume that the collod dlute o that there are no gradent force becaue of fat decay of ntenty nto collodal ytem 1. In th approxmaton the force act along the propagaton drecton of the beam, drecton. The catterng force determned by the erent part of ntenty I ee man t for explanaton) and defned by the followng expreon : r) = σ I r) c, σ = σ σ coθ, S1) / where σ, σ, σ ca are the radaton preure, ncton, and catterng cro-ecton, repectvely, co θ the catterng aymmetry parameter, c the peed of lght n the urroundng medum. Intenty of the erent part of propagatng beam I r) defned by the followng formula 3 : I ca, ρ ) = I exp ρ / w ) exp / l ), S) where ρ the radal coordnate, w the wat e of the Gauan beam, 1 l = nσ ) the catterng 3 length and n the number denty of collodal partcle, 1/ n = 4π a /3 f ), a and f are the radu and volume fracton of collodal partcle. or a dlute collod, the velocty of collodal partcle n the locaton defned by radu-vector r can be found n an approxmaton of hydrodynamc par nteracton of pont-lke partcle 4 : v r ) = μ r, r ) r ). S3) μ r, r ) the Oeen tenor 4 μ r, r ) = I + Rˆ Rˆ ) /8πηR ) and μ r, r ) = I /6πηa). Here R = r r the dtance between center of the -th and the -th partcle, R ˆ = r r ) / R a unt vector n a drecton from partcle to partcle, and I untary matrx, ymbol denote outer product, η the dynamc vcoty of urroundng lqud. To obtan analytcal expreon, we conder partcle located at the ax of Gauan beam. rom ymmetry conderaton, only -component of the velocty for thee partcle. Takng nto account that

2 force alo ha only -component, only µ component of the matrx μ r, r ) urvve n expreon S3). In a typcal tuaton, the dtance between collodal partcle much maller than charactertc dmenon of the beam w, l ) and ummaton n Eq.S3) can be replaced by ntegraton. In an ntal ntant of tme when the lght ut turned on and plan lqud-collod nterface tll unperturbed, the velocty of collodal partcle near the beam center can be evaluated from the followng expreon: ) V σ ) 1 v, ρ = ) = + µ r r r dr, ) 6πηa 8πησ. S4) The ntegraton n Eq.S4) performed over the whole volume of collod excludng volume σ = 1/ n around the partcle at obervaton pont. or obervaton pont cloe to the urface ), the ntegral over em-nfnte volume V n Eq.S4) can be calculated analytcally evaluaton of ntegral wa performed n Wolfram Mathematca 9.): where 1 w 8πησ V P σ w µ = Φ r, r ) r ) dr, S5) 8π cησ l 1 P = π I the total power of Gauan beam, Φ x ) = π x 1) exp x )1 x )Ch x ) π erf x) + Sh x )), Ch ) the hyperbolc cone ntegral functon Ch ) = γ + ln + t 1 dt t wth γ = beng Euler contant, Sh ) = nh t) / t dt the hyperbolc ne ntegral functon, erf ) = erf ) / the magnary error functon. The ntegral over excluon volume σ can be etmated under aumpton that excluon volume much maller than charactertc cale of change of optcal force a f << w, l ): / 1 f µ, ) ) d ) 8πησ r r r r = r. S6) σ η a After ubttuton of Eq.S5), S6) nto S4) one get the fnal expreon for the velocty of collodal partcle near the nterface: ) 1 P + Φ w v = r σ ) f ηa 6π 8π cησ. S7) l uncton Φ x) monotonouly decreae wth x. Thu, v decreae wth ncreae of beam radu w f one keep power P contant. It alo of nteret to nvetgate the dependence of v on concentraton of collodal partcle. or wde Gauan beam w > l ) the aymptotc expreon for the functon Φ x) Φ x >> 1) = π /x). Takng nto account that the econd term n Eq.S7) w / a) tme larger than the frt one, the expreon for the velocty of collodal partcle take the form 1 σ P v r > l. S8) ), w 16 π σ cηw

3 One can ee from th expreon that for wde beam the velocty of collodal partcle doe not depend on ther concentraton. Th tatement may be not necearly true for focued beam, but the prevouly made aumpton that make t mpoble to acertan th on the bae of Eq.S7).. Addtonal experment on dfferent collodal ytem To how the unveralty of advectve tranport, we performed experment wth dfferent collodal upenon. The upenon of nm-dameter-lca partcle Geltech, Inc.) wa made to reach the concentraton of 1.5% v/v that correpond to the average center to center nteartcle eparaton of 4.4 partcle dameter. 4.5-μm-dameter polytyrene partcle placed on the urface of the upenon were ued for tetng the tranport ablte. The mlar procedure a decrbed n the man t wa followed to depot mcrophere on the lqud nterface. The prepared collodal ytem wa llumnated by -polared monochromatc wavelength 53 nm, optcal power.3w) unfocued beam at the ncdent angle of 7. Upon the wtch on of llumnaton, the target PS partcle on the urface tarted ntantaneouly travel along the beam propagaton drecton ee Supplementary Vdeo ). The patal dtrbuton of the average velocty along the urface hown n g. S1. A explaned n the man t, th moton motly caued by the advectve flow of collodal partcle. The maxmum velocty of lca partcle along the beam propagaton drecton meaured n experment 16μm/. The oberved velocty n the cae of lca collod lghtly le than that of polytyrene PS) one.3µm/). The lower velocty of lca partcle 91µm/ for lca v. 19µm/ for PS partcle) alo found n numercal calculaton. gure S1 Advectve tranport of target partcle. a) Traectore of target partcle durng the 1 of laer llumnaton. The end of traectore are ndcated by mall crcle. b) Spatal dtrbuton of tme-averaged velocte of target PS partcle on the urface of collod. The length of the arrow proportonal to the peed magntude whle ther orentaton ndcate the local drecton of partcle moton. 3. Maxmum achevable velocty To etmate the maxmum achevable velocty of advectve flow, we performed ended numercal calculaton wth the ame parameter a n experment of the man t. A the volume fracton of partcle ued n the experment very low, the three- and hgher multbody correcton are ngnfcant and we can ue tokelet approxmaton 4. To take nto account ar-water nterface, the moblty tenor n Eq. S3) hould be modfed to take nto account the preence of the urface. or a free nterface, th can be done by ntroducng mage partcle 5, 6 :. S9)

$S1), the collod regarded a an effectve medum wth a complex refractve ndex n c where Re n c ) the refractve ndex of water and the magnary part of refractve ndex Im n c ) = 1/kl ) depend on the$ $catterng length and decrbe the attenuaton of erent wave nto collod. Thu, the refracton of a Gauan beam nto the collodal ytem can be treated lke n the cae of an aborbng medum 7.$ Ung the Wolfram Mathematca oftware package, we etmated th ntegral numercally and typcal reult are llutrated n gure S where the dtrbuton of advectve flow hown both at the urface and n the bulk of the

Ung the Wolfram Mathematca oftware package, we etmated th ntegral numercally and typcal reult are llutrated n gure S where the dtrbuton of advectve flow hown both at the urface and n the bulk of the

Dtrbuton of ntenty color) and advectve flow arrow) along the urface a) and peendcular to the urface n a cro-ecton through the center of the beam b).

The nert how poton of cro-ecton plane. A can be een, the flow pattern qute mlar to the flow of target partcle oberved n the experment.

4 Here μ r, r ) the new moblty tenor, R = 1 ˆ ˆ the reflecton operator wth repect to water-ar nterface =, r = R r 5. To fnd the dtrbuton of erent part of ntenty I n Eq.S1), the collod regarded a an effectve medum wth a complex refractve ndex n c where Re n c ) the refractve ndex of water and the magnary part of refractve ndex Im n c ) = 1/kl ) depend on the catterng length and decrbe the attenuaton of erent wave nto collod. Thu, the refracton of a Gauan beam nto the collodal ytem can be treated lke n the cae of an aborbng medum 7. A the average nteartcle eparaton of 7nm much maller than the charactertc dmenon of the Gauan beam ued n experment w = 1. 1 mm, l =. 4 mm), the ummaton n Eq.S3) can be replaced by ntegraton. Ung the Wolfram Mathematca oftware package, we etmated th ntegral numercally and typcal reult are llutrated n gure S where the dtrbuton of advectve flow hown both at the urface and n the bulk of the collodal ytem. gure S Numercal calculaton of advectve flow. Dtrbuton of ntenty color) and advectve flow arrow) along the urface a) and peendcular to the urface n a cro-ecton through the center of the beam b). The length of the arrow proportonal to the local peed of the flow. The dahed rectangle n a) how the correpondng feld of vew n the experment hown n g.1 of the man t. The nert how poton of cro-ecton plane. A can be een, the flow pattern qute mlar to the flow of target partcle oberved n the experment. Moreover, the advectve flow end far beyond the range of acton of optcal force demontratng the effect of long-range hydrodynamc nteracton. Calculaton how that, becaue of thee hydrodynamc nteracton, the velocty of collodal partcle can acheve a maxmum peed of the urface flow 19μm/ at the center of the beam. 4. Decrpton of Supplementary Vdeo 1 The vdeo how the moton of targeted 4.5µm PS partcle on a urface of collodal upenon contanng.3%v/v collodal concentraton of -nm-dameter polytyrene partcle. The.3W -polared Gauan laer beam wth a dameter.mm ncdent at 7. The moton of target partcle oberved over an area of approxmately 3 µm µm n the vcnty of the beam center located at the center of the frame.

5 5. Decrpton of Supplementary Vdeo The vdeo how the moton of targeted 4.5µm PS partcle on a urface of collodal upenon contanng 1.5%v/v collodal concentraton of -nm-dameter lca partcle. The.3W -polared Gauan laer beam wth a dameter.mm ncdent at 7. The moton of target partcle oberved over an area of approxmately 3 µm µm n the vcnty of the beam center located at the center of the frame. Reference 1. Greenfeld E, Nemrovky J, El-Ganany R, Chrtodoulde DN, Segev M. Shockwave baed nonlnear optcal manpulaton n denely catterng opaque upenon. Optc Expre 13, 1): Bohren C, Huffman DR. Aboton and catterng of lght by mall partcle. Wlley-Intercence publcaton: NY, Varadan V, Brng V, Varadan V, Ihmaru A. Multple catterng theory for wave n dcrete random meda and comparon wth experment. Rado Scence 1983, 183): Happel J, Brenner H. Low Reynold number hydrodynamc: wth pecal applcaton to partculate meda. Martnu Nhoff: The Hague, Cchock B, Ekel-Jeżewka ML, Wanryb E. Hydrodynamc nteracton between phere n a vcou flud wth a flat free urface or hard wall. The Journal of Chemcal Phyc 7, 1618): Perkn G, Jone R. Hydrodynamc nteracton of a phercal partcle wth a planar boundary: II. Hard wall. Phyca A: Stattcal Mechanc and t Applcaton 199, 1893): Serdyuk VM, Ttovtky JA. A Smple Analytc Approxmaton for the Refracted eld at Gauan Beam Incdence upon a Boundary of Aborbng Medum. Journal of Electromagnetc Analy and Applcaton 1, 11):

Electromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology

Electromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology Electromagnetc catterng Graduate Coure Electrcal Engneerng (Communcaton) 1 t Semeter, 1390-1391 Sharf Unverty of Technology Content of lecture Lecture : Bac catterng parameter Formulaton of the problem