Acoustic Second Harmonic Generation from Rough Surfaces under Shear Excitation in Liquids

Transcription

1 10346 Langmuir 2004, 20, Acoutic Second Harmonic Generation from Rough Surface under Shear Excitation in Liquid Katrin Wondracek, Andrea Bund, and Diethelm Johannmann*, Intitute of Phyical Chemitry, Clauthal Univerity of Technology, Arnold-Sommerfeld-Str. 4, D Clauthal-Zellerfeld, Germany, and Intitute of Phyical Chemitry and Electrochemitry, Dreden Univerity of Technology, Mommentrae 13, D Dreden, Germany Received June 23, In Final Form: July 26, 2004 Emiion of compreional acoutic wave at the econd harmonic frequency (econd harmonic generation, SHG) i poible from rough urface undergoing ocillatory hear in liquid. Thi nonlinear repone i a conequence of the inertial term in the Navier-Stoke equation. On a corrugated urface, the treamline of the heared liquid are not trictly parallel to the urface, leading to variation of preure along the treamline and a concomitant Bernoulli preure. Being quadratic in peed, the Bernoulli preure contain a tatic term and a term at the econd harmonic frequency, 2ω. Preure fluctuation at 2ω generate compreional wave. Introduction Shear ocillating reonator have found widepread application in enor technology in the form of the quart crytal microbalance (QCM). 1 The ma of a film depoited on the crytal urface can be inferred from the induced decreae in the reonance frequency. Film thickne monitor baed on the QCM are routinely ued in evaporation machine. Operating the QCM in liquid i more difficult, on one hand, but on the other hand, thi mode i very attractive for electrochemitry and bioening. Technical problem related to the large damping have today been overcome by the ue of advanced driving circuit, 2 impedance analyi, 3 or the ring-down technique. 4 A particularly well-etablihed application i the electrochemical quart crytal microbalance (EQCM; for review, ee ref 5 and 6). Surface roughne ha turned out to be one of the mot eriou problem of the QCM operated in liquid. From intuition, one realie eaily that roughne will trap ome of the liquid. 7 Material contained in the valley i locked to the movement of the crytal and therefore doe not diipate energy. However, the ituation i more complicated. A Urbakh and Daikhin have pointed out, roughne affect not only the frequency hift but alo the amount of diipation. 8 In cae the rough layer i vicoelatic, the ituation turn even more complicated. 9 * Correponding author. Phone: +49 (0) Fax: +49 (0) johannmann@pc.tu-clauthal.de. Clauthal Univerity of Technology. Dreden Univerity of Technology. (1) Lu, C., Canderna, A. W., Ed. Application of Pieoelectric Quart Crytal Microbalance; Elevier Publiher: Amterdam, The Netherland, Sauerbrey, G. Z. Phy. 1959, 155, (2) Arnau, A.; Ferrari, V.; Soare, D.; Perrot, H. In Pieoelectric Tranducer and Application; Arnau, A., Ed.; Springer: Berlin, (3) Johannmann, D. Macromol. Chem. Phy. 1999, 200, 501. (4) Rodahl, M.; Höök, F.; Fredrikon, C.; Keller, C. A.; Kroer, A.; Breinki, P.; Voinova, M.; Kaemo, B. Faraday Dicu. 1997, 107, 229. (5) Buttry, D. A. In Electroanalytical Chemitrya Serie of Advance; Bard, A. J., Ed.; Marcel Dekker: New York, 1991; Vol. 17, p 1. (6) Hillmann, R. In Encyclopedia of Electrochemitry, Intrumentation and Electroanalytical Chemitry; Bard, A. J., Stratmann, M., Unwin, P. R., Ed. Wiley-VCH: Weinheim, Germany, 2003; Vol. 3, p 230. (7) Martin, S. J.; Frye, G. C.; Ricco, A. J.; Senturia, S. D. Anal. Chem. 1993, 65, In thi paper, we prove that rough urface under hear can generate compreional wave at the econd harmonic frequency (econd harmonic generation, SHG) if the amplitude of ocillation i large enough. Such a phenomenon would allow for an independent meaurement of the urface roughne. It alo i of fundamental interet, incea will be hown belowacoutic SHG hould be connected to a tatic Bernoulli preure. Nonlinear interaction of hear ocillating quart crytal with their environment have been invetigated by variou group. For intance, Klenerman and co-worker have ued 3rd harmonic generation to probe the detachment of mall phere from the reonator urface. 10,11 Nonlinearitie are particularly trong when a particle or a tip touche the reonator urface. In uch cae, there i a tre concentration at the point of contact. 12 Generally peaking, nonlinearitie are le common in liquid than in the dry tate becaue the large damping reduce the amplitude of motion. Neverthele, we could recently detect acoutic SHG while electrochemically growing a rough metal urface. 13 Here, we preent the theoretical calculation. Flow Field above a Rough Surface The calculation i baed on the Navier-Stoke equation. The main aumption i a mall (but nonero) Reynold number. A a conequence, nonlinearitie can be treated a a mall perturbation to the law of linear hydrodynamic. For mall Reynold number, the equation of hydrodynamic are linear in peed and the uperpoition principle hold. In the firt tep, we decompoe the flow field ν(x, y, ) and the urface topography h(x, y) into their Fourier component. The linearied hydrodynamic equation a well a the boundary condition are olved for each Fourier (8) Daikhin, L.; Gileadi, E.; Kat, G.; Tionky, V.; Urbakh, M.; Zagidulin, D. Anal. Chem. 2002, 74, (9) Bund, A.; Schneider, M. J. Electrochem. Soc. 2002, 149, E331- E339. (10) Dultev, F. N.; Otanin, V. P.; Klenerman, D. Langmuir 2000,, (11) Dultev, F. N.; Speight, R. E.; Fiorini, M. T.; Blackburn, J. M.; Abell, C.; Otanin, V. P.; Klenerman, D. Anal. Chem. 2001, 73, (12) Berg, S.; Johannmann, D. Phy. Rev. Lett. 2003, 91, (13) Wehner, S.; Wondracek, K.; Johannmann, D.; Bund, A. Langmuir 2004, 20, /la CCC: $ American Chemical Society Publihed on Web 10/09/2004

2 Acoutic SHG from Rough Surface Langmuir, Vol. 20, No. 23, component, eparately. The urface corrugation h(x, y) i aumed to be hallow with the gradient dh/dx and dh/ dy much maller than unity. In the econd tep, we turn on a mall nonlinearity but aume that the flow field remain eentially unchanged. In that cae, the nonlinear term F(v )v in the Navier- Stoke equation give rie to an ocillating preure at the econd harmonic frequency, 2ω, which radiate compreional wave. Linear Hydrodynamic. The linearied Navier- Stoke equation i given by F v t - η 2 v + p ) 0 (1) where v i the peed, F i the denity, η i the vicoity, and p i the preure. The lateral motion occur along the x-direction at an angular frequency of ω. The urface normal i. The fluid i aumed to be incompreible, leading to volume conervation a expreed by v x x + v ) 0 (2) A mooth, laterally ocillating urface with a no-lip boundary condition induce a pure hear flow (index: p), whoe x-component varie in the -direction according to v p (x,, t) ) v exp(i(ωt - k)) (3) where v repreent the lateral peed of the ubtrate and k ) (-iωf/η) 1/2 ) k - ik ) (1 - i) -1 repreent the wave vector of hear ound. ) (2η/(ωF)) 1/2 i the penetration depth of the hear wave in the liquid. In principle, v i a vector with only one nonero component, which i the x-component. In the following, we term the value of the x-component v a well for brevity and mean (v,0,0) when peaking of v in the ene of a vector. The hear gradient, v p /, i given by v p )-ikv )-(1 + i)v For a corrugated urface, the field v p doe not fulfill the boundary condition. Before treating the boundary condition in detail, we introduce a econdary field, v(x,, t), and write v tot ) v p + v. The total field, v tot, will atify the boundary condition. Even without pecifying the boundary condition, certain tatement on the field ν(x, y,, t) can be made: ince we aume linear hydrodynamic, the field v(x,, t) can again be decompoed into the Fourier erie v(x,, t) ) q ν q () exp(iqx) exp(iωt). Each component [v q () exp(iqx) exp(iωt)] mut atify linear hydrodynamic. We omit the index q in the following. For the flow field with a lateral wave vector of q (along the direction of motion), we ue the anat v x (x, ) ) x co(qx) exp(-r) v y (x, ) ) 0 v (x, ) ) in(qx) exp(-r) p(x, ) ) p 0 in(qx) exp(-r) where p i the preure and R i a complex decay contant. Inerting thi anat into eq 1 and 2 yield (4) (5a) (5b) (5c) (5d) iωfx co(qx) - η(-q 2 +R 2 )x co(qx) + qp 0 co(qx) ) 0 (6a) iωf in(qx) - η(-q 2 +R 2 ) in(qx) - Rp 0 in(qx) ) 0 (6b) or equivalently Uing iωf/η )-k 2, thi read Equation 9 ha the following two olution: Poitive quare root have been ued in order to enure that the wave decay at infinity. R I and R II correpond to two eparate mode. The two mode have the amplitude xi and xii, repectively. The parameter i fixed by eq 6c. For mode I, we find For mode II, one ha -qx in(qx) -R in(qx) ) 0 Boundary Condition. We now ue the boundary condition of the -component at the urface (v )0 0) to fix the relative amplitude of mode I and of mode II. The condition of vanihing vertical flow at the urface implie that (ee eq 5c) Uing eq 11 and 12, thi lead to (6c) [iωf +(q 2 -R 2 )η 0 q 0 iωf +(q 2 -R 2 )η -R ][v0x ]) 0 (7) -q -R 0 p 0 Equation 7 i a linear homogeneou equation ytem which only ha a olution if the determinant vanihe: -(iωf +(q 2 -R 2 )η)r 2 + (iωf +(q 2 -R 2 )η)q 2 ) 0 (8) (-k 2 + q 2 -R 2 )(q 2 -R 2 ) ) 0 (9) R I ) q 2 - k 2 ) q 2 + 2i/ 2 R II ) q (10) I )- q R I xi (11) II )-xii (12) (I + II ) in(qx) ) 0 (13) xii )- q R I xi )-(1 + 2i(q) -2 ) -1/2 xi (14) xi i the only free amplitude. For implicity, we term it in the following. We again write down the full flow field: v x ) [ exp(-r I ) - q R I exp(-q)] co(qx) v ) -q R I [exp(-r I ) - exp(-q)] in(qx) (15) The amplitude of the econdary flow field,, i fixed by a no-lip condition at the quart-liquid interface. The urface i periodically corrugated (q parallel to x). The

3 10348 Langmuir, Vol. 20, No. 23, 2004 Wondracek et al. urface topography h(x) i expanded into a Fourier erie: v p B )-F(v )v )-F(v x x + v v h q (x) ) h 0 ) co(qx) () ) F 2 R exp(-2(r I + q))(exp(r I) - exp(q)) I (R I exp(r I ) - q exp(q))q 2 v 2 0 co 2 (ωt) (22) where h 0 i the vertical cale of roughne. The econdary flow field mut compenate for the mimatch between the peed of the ubtrate, ν, and the peed of the pure hear wave, ν p, at the height h q (x). We aume that the aperitie are rigid. On top of an aperity, we have a peed of v tot (x, h q (x)) that i equal to v, even though * 0. For the difference v ) v tot - v p, we therefore have v x (x, h q (x)) ) v - v p (x, h q (x)) ) v (1 - exp[-ikh 0 co(qx)]) (17) Inerting the flow field from eq 15, we find [ exp(-r I h 0 co(qx)) - q R I exp(-qh 0 co(qx))] Taylor expanding in h 0 and keeping only leading order reult in or co(qx) ) v (1 - exp(-ikh 0 co(qx))) (18) [ 1 - q R I ] co(qx) ) v ikh 0 ) iv R I kh 0 R I - q co(qx) (19) (20) The flow field from eq 15 ha alo been ued by Urbakh and Daikhin in ref 14. Thee author olved the boundary condition to econd order in the cale of the roughne, h 0. They were intereted in the effect of roughne on frequency and bandwidth. Since the effect mut depend at leat quadratically on roughne for reaon of ymmetry, the perturbation mut be carried to econd order in h 0. In the context of acoutic SHG, it i ufficient to treat the boundary condition in firt order, which i much impler. A quadratic dependence on the vertical cale of roughne will eventually be found becaue the SHG depend on the quare of the amplitude. Nonlinearitie and Bernoulli Preure We now allow for a mall nonlinear term of the form F(v )v in the Navier-Stoke equation. The new term mut be compenated by a new, mall preure gradient in order to atify the force balance. We term thi preure gradient p B, where the B tand for Bernoulli. For the x-component of thi nonlinear preure gradient, we have v x x p B )-F(v )v x )-F(v x x + v v x ) (21) ) F 2R exp(-(r 2 I + q))(r I - q)2 qv 2 0 I co 2 (ωt) in(2qx) Note that the preure gradient contain the term in- (2qx). It average out laterally. When calculating the emitted ound wave at the econd harmonic frequency (which ha q x ) 0), we may diregard thi term. For the -component of the preure gradient, we find Importantly, the velocity, v, enter the new term a a quare. Becaue of the quadratic dependence on peed, the nonlinear term generate new frequencie and new patial Fourier component. In the cae of the -component, it turn out that there i only one new patial Fourier component, which i q x ) 0. p B i independent of x and will therefore not average out laterally. A for the frequency, the preure contain a tatic part a well a a part ocillating at 2ω becaue of the trigonometric relation co 2 (ωt) ) 1 / 2 [1 + co(2ωt)]. The 2ω-component emit compreional wave; that i, it generate econd harmonic wave. We indicate all field ocillating at 2ω with a tilde ( ) in the following. Note that acoutic SHG (which i of interet here) i alway coupled to a tatic preure ( acoutic rectification ), which i a tatic Bernoulli preure. In low frequency hydrodynamic, a Bernoulli preure only arie if there are velocity gradient along the treamline, that i, if the flowing material experience acceleration. The ame i true here: Acoutic econd harmonic generation only occur in the preence of roughne. Volume conervation require that the liquid above the peak flow fater than the liquid in the valley of a corrugated urface (Figure 1). Pure hear flow, on the other hand, doe not generate a Bernoulli preure. For the 2ω-component at q x ) 0, we write p B,q)0 ) 1 Fq 2 2 exp(-2(r 2 2 I + q))(exp(r I ) - R I exp(q))(r I exp(r I ) - q exp(q)) co(2ωt) (23) The factor of 1 / 2 arie becaue we only conider the 2ωcomponent. The q-vector on the right-hand ide i related to the patial tructure of the flow field at the fundamental frequency. It i not ero. Compreional Wave To decribe the generation of compreional wave at 2ω, we have to relax the condition of incompreibility. There i a mall preure fluctuation, p, and a mall denity fluctuation, F. Both ocillate at a frequency of 2ω. They are related to each other by the bulk modulu of compreion, K: p ) K F F (24) where F i the time-averaged denity. The denity fluctuation i related to the divergence of the flow field by df ) 2iωF )-F( ṽ) (25) dt The time derivative introduce a factor of 2iω becaue we are concerned with the econd harmonic frequency. The preure gradient can be written a p ) K K F )- ( ṽ) (26) F 2iω (14) Urbakh, M.; Daikhin, L. Phy. Rev. B 1994, 49, Inerting thi term into the Navier-Stoke equation and

4 Acoutic SHG from Rough Surface Langmuir, Vol. 20, No. 23, Figure 1. Schematic repreentation of acoutic econd harmonic generation. The fact that the liquid flow fater above the protruding ection of the urface give rie to a Bernoulli preure. Since the preure i quadratic in peed, it emit compreional wave at 2ω. only conidering the component at q x ) 0 and 2ω, one find 2iωFṽ +F(ṽ )ṽ - K 2iω ( ṽ) - η 2 ṽ )- p B,q)0 (27) Since x p B,q)0 ) 0, there i no ource term for the x-component and we can confine ourelve to the -component. Alo, ince q x ) 0, all derivative with repect to x vanih. Further, we neglect the term F(ṽ )ṽ becaue the Reynold number i mall and the flow field ṽ i much maller than the flow field v. Finally, the bulk modulu of compreion, K, i much larger than the hear modulu G ) iωη, and we can neglect the vicou term. With thee implification, we arrive at 2iωFṽ - K 2 ṽ 2iω )- 2 p B,q)0 (28) Introducing the peed of compreional ound, c ) (K/F) 1/2, we can write 2iω 2] ṽ ) [( 2ω c ) Fc 2 p B,q)0 (29) Thi inhomogeneou wave equation i olved by a Green function ṽ() ) 0 d { G(, )2iω Fc 2 p B,q)0 } ( ) (30) where the Green function obey 2] G(, ) ) ( - ) (31) [( 2ω c ) Here, ( - ) i the Dirac delta function. Equation 31 i olved by G( - ) ) 1 [θ( - ) exp(ik 4ik c ( - )) + c θ( - ) exp(ik c ( - ))] (32) where k c ) 2ω/c i the wave vector of compreional ound of the acoutic econd harmonic, and the tep function θ() i defined a θ() ){ 0, < 0 1, > 0 (33) The Green function contain a wave traveling upward and a wave traveling downward. Preumably, the wave traveling downward i reflected at the olid urface to ome extent. For implicity, we ignore thi effect for now. The reflection would introduce a factor of (1 + r) (with r being the reflectivity) into all equation below. Preure at the Microphone At ome location of the microphone, m, outide the penetration depth of the hear wave, the flow field i given by ṽ( m ) ) 0 md { 1 4ik c 2iω Fc 2 exp(ik c ( - m )) p B,q)0 ( ) } ) 1 4Fc exp(-ik c m ) md {exp(ikc 0 ) p B,q)0 ( )} (34) In the econd line of eq 34 above, the relation 2ω/k c ) c wa ued. The integrand vanihe at m ( m. ), and the upper limit of integration can be replaced by ). Since the penetration depth,, i much le than the wavelength of compreional ound, the exponential term can be Taylor-expanded a exp(ik c ) 1 + ik c. Uing eq 23, eq 34 evaluate to ṽ( m ) ) 1 4Fc exp(-ik c m ) 0 d {exp(ikc ) p B,q)0 } 1 4Fc exp(-ik c m )(Fq2 2 /2R I 2 ) 0 d {(1 + ikc ) exp(-2(r I + q))(exp(r I ) - exp(q))(r I exp(q) - q exp(r I ))} ) ik c qv 2 0 (R I - q) 2 exp(-ik 32R 3 c m ) (35) I (R I + q) The modulu of the preure fluctuation at the microphone, p, i given by p ( m ) ) -K v ( m ) 2iω m ) -K(-ik c )qv 2 0 2iω ( ifk c (R I - q)2 (36) 32R 3 I (R I + q)) ) Fv k c q(r I - q)2 R 3 I (R I + q) To obtain the lat line of eq 36, the relation k c ) 2ω/c and K )Fc 2 have been ued. Replacing by the expreion from eq 20 and uing k 2 )-2i/ 2, we obtain p ( m ) ) Fv 2 ) 2 k c q (37) R I (R I + q) Uing eq 14, R I can be replaced by R I ) q(1 + 2i/(q) 2 ) 1/2 to give p ( m ) ) Fv 2 ) Fv 2 ) 2 k c ) 2 k c q q 1 q 2 + 2i/ 2 ( q 2 + 2i/ 2 + q) 1 + 2i(q) -2 ( 1 + 2i(q) ) (38)

5 10350 Langmuir, Vol. 20, No. 23, 2004 Wondracek et al. Again, the term Fv 2 ha the appearance of a dynamic Bernoulli preure. Acoutic SHG cale a the quare of the urface roughne, (h 0 /) 2. Thi can be expected becaue it hould not depend on the ign of the roughne. Here, ν can be inferred from the peed at the electrode which i related to the drive level, d, in unit of dbm by 15, v ) 8 πn f 0 Qd 26 U el ) 2f 0 Qd 26 (0.22 V)10d/20 (39) where f 0 i the frequency of the fundamental, Q i the quality factor, d 26 ) pm/v i the pieoelectric coefficient, and U i the overtone order. Limit of Large- and Small-Scale Roughne It i intructive to conider the high-q and the low-q limit. For roughne with large lateral cale (q, 1), eq 37 implifie to p 1 32 k c qh Fv The amplitude of ound cale a the q-vector and frequency (k c ω). For mall-cale roughne (q. 1), the repective relation i p 1 2 k c h 0 32 q Fv 2 2 (40) (41) For large q, the amplitude of ound cale quadratically with overtone order (k c ω, ω -1/2 ) and inverely with q. Note, however, that the calculation aume hallow grating, that i, qh 0, 1. A mall lateral cale thu implie (15) Borovky, B. L.; Krim, J. J. Appl. Phy. 2000, 88, () Thurton, R. N. In Mechanic of Solid; Truedell, C., Ed.; Springer-Verlag: Heidelberg, 1984; Vol. 4, Chapter 36, p 257. a mall vertical cale of roughne a well. Conequently, the high-q limit probably i of a le practical relevance than the low-q limit. In the following, we inert ome typical number for a quart crytal immered in water into eq 38 and 39. Uing F)1 g/cm 3, a drive level of 20 dbm, a Q-factor of , ω ) 2π(5) MH, η ) 10-3 Pa, k c ) 2ω/c, with c ) 1480 m/, h 0 ) 0.2 µm, q ) 2π/(1 µm), and n ) 1, we find a preure of the order of 1 Pa. Thi preure can be eaily picked up with a hydrophone. The tentative interpretation of the SHG ignal reported in ref 13 i confirmed by thi calculation. Finally, we again emphaie that acoutic econd harmonic generation i the companion of a tatic Bernoulli preure. The latter phenomenon may be of importance when the QCM i in contact with a colloidal diperion of olid particle. The treamline around thee particle will be qualitatively the ame a the treamline above a rough urface. Since there i a velocity gradient along the treamline, there will be a Bernoulli preure, which will pull the particle toward the urface. One would therefore be able to apply a net vertical force onto the particle by mean of a large-amplitude hear movement of the ubtrate. Thi effect may be of interet for microfluidic application. Concluion A calculation making ue of perturbation theory and the Navier-Stoke equation how that acoutic econd harmonic generation i poible on rough urface undergoing hear ocillation. Such ocillatory hear typically occur on the front of quart crytal reonator. The econd harmonic field conit of compreional wave which are emitted into the bulk medium. They are meaurable with conventional hydrophone. Acoutic econd harmonic generation can be ued to quantify roughne. LA