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ISSN 1063 7710, Acoustical Physics, 2010, Vol. 56, No. 6, pp. 861н870. й Pleiades Publishing, Ltd., 2010.

The Field of Radiative Forces and the Acoustic Streaming in a Liquid Layer on a Solid Half Space
V. A. Gusev and O. V. Rudenko
Faculty of Physics, Moscow State University, Moscow, 119991 Russia e mail: vgusev@bk.ru
Received May 21, 2010

Abstract--The acoustic field and the field of radiative forces that are formed in a liquid layer on a solid sub strate are calculated for the case of wave propagation along the interface. The calculations take into account the effects produced by surface tension, viscous stresses at the boundary, and attenuation in the liquid volume on the field characteristics. The dispersion equations and the velocities of wave propagation are determined. The radiative forces acting on a liquid volume element in a standing wave are calculated. The structure of streaming is studied. The effect of streaming on small size particles is considered, and the possibilities of ordered structure formation from them are discussed. DOI: 10.1134/S1063771010060102

INTRODUCTION Today, an increase in the number of publications devoted to the acoustic effect on microdrops and to the streaming induced in them is being observed. Devices (microboards, microchips) implementing the motion of drops under the effect of surface acoustic waves (SAWs) are under development [1н10]. As a rule, SAWs are excited in a plate made from piezoelec tric material. After the first experiments [11], many studies concerned with this subject had been carried out, which resulted in the formation of an important field of applied research: acoustoelectronics [12]. In this field, the studies were stimulated by the needs of biology, cytology, chemistry, and, in the last few years, nanotechnologies. One of the topical problems is the effect of SAWs on the content of a microdrop and, in particular, its mixing by SAW induced streaming. The effect of SAWs on a drop is also used for breaking the drop into smaller droplets, i.e., for "microatomiza tion" due to the capillary wave generation. An important application of this is the formation of structures with a preset morphology on a substrate by drying a liquid with suspended particles in a wave field. Today, methods of structure formation due to the self assembly of nanoparticles in the course of evaporation of a solution are widely used. The acoustic effect not only introduces a new ordering mechanism, but also allows controlling the parameters of the structure to be formed. Most of the studies concerned with the aforemen tioned problems are experimental ones. This is related to the technological needs. An appropriate theoretical description may be useful for understanding the phys ics of the processes and also for optimizing the param eters of both the system and the acoustic action. To

develop the theory, it is necessary to calculate the acoustic fields, the radiative forces, the acoustic streaming, and, finally, their effects on the particles. Below, we consider the following system. A liquid layer with a thickness h overlies a solid substrate. A wave travels along the interface and penetrates into both the solid half space and the liquid layer. The field formed in the layer acts upon the particles suspended in it so that it causes the formation of a periodic struc ture. ACOUSTIC FIELDS IN THE LIQUID LAYER AND IN THE SUBSTRATE The Basic Problem First, we consider the simplest statement of the problem [13]. The (x, y) plane of the Cartesian coor dinate system coincides with the boundary z = 0 between the solid substrate and the liquid layer. The z axis is directed vertically downwards. The upper boundary of the liquid layer is the plane z = нh. The acoustic fields in such a system were considered earlier and described by I.A. Viktorov [13]. However, for new applications, a development of the theory is necessary. Below, we briefly review the known results in order to generalize them. The displacement U of an element medium is expressed through the scalar potentials. For the two dimensional which the displacement along the y axis the potentials are independent of the y of the solid and vector problem (in is absent and coordinate),

861

862

GUSEV, RUDENKO

the displacements along the x and z axes have the form U = ( U, V = 0 , W ) = + rot , U = н , x z W = + . z x

= D ( z ) exp ( н i t + ikx ) ,
2 2 dD + ( k0 н k ) D = 0 . 2 dz 2

Here, k0 = /c0 is the wave number in the liquid. Determining the function D(z), we write the solution = ( C1 e + C2 e r= (1)
irz н irz

The potentials satisfy the wave equations and, for monochromatic waves, the Helmholtz equations: + x2 z
2 2

) exp ( н i t + ikx ) ,
2 2

(5)

2 + kl = 0 . 2 k2 t

k0 н k .

Equations (1) involve the wave numbers kl = /cl for longitudinal waves, and kt = /ct for transverse waves. The velocities of the waves are expressed through the Lame parameters and : cl = ( + 2)/, ct = /.

Solutions to Eqs. (1) are sought in the form of plane waves traveling along the horizontal axis, i.e., the x axis, and decreasing with the z coordinate: = A exp ( н q z ) exp ( н i t + ikx ) . B exp ( н s z )
2 2 l 2 2 t

Thus, the acoustic field in the substrate is described by potential (2), while the acoustic field in the liquid layer is described by potential (5). The relations between wave amplitudes, as well as the wave number k, are determined from the characteristic system and the dispersion equation that correspond to the bound ary conditions of the problem. The following conditions should be satisfied. (i) At the interface z = 0, the vertical displacements should be identical, i.e., W
z=0

= + z x

z=0

(2) = н uz i

= ( н qA + ikB ) exp ( н i t + ikx ) r = н 1 = н ( C 1 н C 2 ) exp ( н i t + ikx ) . i z

z=0

Here, q = k н k and s = k н k are the scales characterizing the decrease in the longitudinal and transverse field components with depth in the sub strate. According to Eq. (2), we seek a wave with a fre quency and a horizontal wave number k determined from the dispersion equation. Let us consider the field in the liquid layer нh < z < 0. First, we assume that the liquid is an ideal one and apply the linearized Euler and continuity equations:
0

This relation should be satisfied for any x and t. Hence, we obtain the relation between the constants A, B, C1, and C2: C 1 н C 2 = н ( н qA + ikB ) . r (6)

u + p' = 0, t

2 p' + c 0 0 div u = 0 . t

(3)

In what follows, the exponential factor corresponding to the wave propagation along the interface will be omitted. (ii) At the boundary z = 0, the normal stresses should be identical: zz = нp. The normal stress in the solid is
zz

Here, u is the oscillation velocity vector of the liquid, 0 and ' are the equilibrium density and its acoustic increment, p' is the acoustic pressure, and c0 is the velocity of sound. For the liquid, it is convenient to introduce the sca lar potential . The corresponding expressions for acoustic variables and the wave equation derived from Eq. (3) have the form u = , p' = н
0

= 2 + 2 + 2 2 + , x z x z z = (н k + q )A + 2(q A н iksB). = i 0 = i 0 ( C 1 + C 2 ) . t
2 2 2

2

2

2

2

zz z = 0

The pressure in the liquid at the boundary is p ' = н
0

, t

2 н c 0 = 0 . 2 t

2

(4)

Equating the last two expressions, we arrive at the sec ond relation between the constants: ( н k + q + 2 q ) A н 2 i ksB = н i 0 ( C 1 + C 2 ) .
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010
2 2 2

We seek the solution to Eqs. (4) in the form of a wave traveling along the boundary with an unknown depen dence D(z) of its amplitude on the vertical coordinate:

(7)

THE FIELD OF RADIATIVE FORCES AND THE ACOUSTIC STREAMING

863
2

(iii) At the boundary z = 0, the tangential stresses in the solid are zero, because the liquid is assumed to be ideal. The tangential stress is
xz z = 0

1нc c = т c2 2 н 1 c0
2 c 1 н 2 c 0

2 2 l

2 2 1 н c 2 н 1 н c 2 ct 2 ct 4 2 2 l

= 2 + н x z x2 z2
2 2

2

2

2

z=0

= н 2 ikqA н k B н s B = 0 . This yields the third relation, B = н 2 i 2kq 2 A . k +s (8)

1 0 c 4 ct
н 1/2

1н

c c

(11)

2 c tan H 0 c н 1 , cc 2 c tanh H 0 1 н c 2 , c c 0

c > c0 ,

н 1/2

(iv) At the free surface of the liquid, i.e., at the upper boundary of the liquid layer z = нh, the acoustic pressure p' is zero: p'
z = нh

c < c0 .

= i 0

z = нh

= C1 e

н irh

+ C2 e

ir h

= 0.

This yields the fourth (and last) relation, C 2 = н C 1 exp ( н 2 irh ) . (9)

Thus, we have four homogeneous equations (6)н (9) for determining the unknown amplitudes A, B, C1, and C2 and the wave number k. The dispersion equa tion is derived from the condition that the determinant of the system is zero: 4 k qs н ( k + s ) =
2 2 22

0 4 r tan ( rh ) qk t r н1 tanh ( r h ) . * *
н1

(10)

We note that Eq. (10) differs from the dispersion equa tion obtained in [13] (see Eq. (1.58) in [13]). If, in Eq. (10), we set the liquid density 0 or the layer thick ness h to zero, we obtain a simple dispersion equation for Rayleigh waves traveling along the solidнvacuum boundary. The right hand side of Eq. (10) takes into account the effect of the liquid layer. The upper row on the right hand side of Eq. (10) corresponds to such a solution to this equation that the wave velocity in the system is greater than the sound velocity in the liquid but smaller than the longitudi nal and transverse wave velocities in the solid: c > c0, c < ct < cl. The lower row on the right hand side of Eq. (10) corresponds to a wave velocity smaller than the sound velocity in the liquid: c < c0. In this case, r = ir, r = k нk . The Dispersion Curves Substituting the expressions k = /c, kl = /cl, kt = /ct, q = c н c l , s = c н c t , and r = c 0 н c in Eq. (10), we represent dispersion equa tion (10) in the form
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010
н2 н2 н2 н2 н2 н2 2 2 0

Here, H = h/c0 is the wave thickness of the layer. Since the variable H contains the frequency, for this type of wave a dispersion takes place. To analyze Eqs. (11), it is convenient to normalize all the velocities in Eqs. (11) by the transverse wave velocity ct. In order to avoid the inaccurate statements encountered in [13], we begin with considering the 2 2 simple specific case where c l / c t = 2. In this case, the root of the dispersion equation that corresponds to the Rayleigh wave (traveling along the boundary between the solid half space and the vacuum) is determined 2 2 analytically: c R / c t XR = 3 н 5 0.764. For this case, dispersion equation (11) is simplified. The explicit expression for the wave thickness of the layer has the form H= 2X 1 н 2X

4 1 н 2X X 3/2 т arctanh 1 н X н 1 н , 0 X2 2 X S = 0.442 < X < X 0 = H= 1 , 2 (12)

2X 4 2X н 1 n + arctanh 2X н 1 0 X2 X0 < X < 1 .

3/2 т 1 н X н 1 н X , 2 2

Here, X = c2/ c t ; for definiteness, we assumed that c 0 / c t X0 = 0.5. The quantity XS 0.442 is identical to the value of X at which the argument of the hyper bolic arctangent in Eqs. (12) is unity. The correspond ing velocity of the wave proves to be identical to the velocity of the Stoneley wave cS at the boundary between the solid and liquid half spaces. This velocity
2 2

864 H(n = 0) 8

GUSEV, RUDENKO H(n = 1, 2) 20

6

15

n=2 4 n=1 2 n=0 5 10

X 0 0.4 XS 0.5
Fig. axis h/

0

0.6

0.7 XR 0.8 X

0.9

1.0

1. Illustration of dispersion relation (12). The vertical represents the wave thickness of the liquid layer H = c0, and the horizontal axis, the normalized square of
2 2

2 the wave propagation velocity in the structure X = c2/ c t .

The following parameters are preset: 4/0 = 10, c l / c t = 2, and X0 = c 0 / c t = 0.5. In this case, for the Rayleigh wave, XR = c R / c t 0.764; for the Stoneley wave, XS = c S / c t 0.442.
2 2 2 2 2 2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 H 0 н0.1 (c н cR)/c
R

н0.2 н0.3 н0.4 н0.5

is known [13] to be somewhat smaller than the wave velocity in the unbounded liquid, cS < c0, for which X = X0 = 0.5 in Eqs. (12). The point X0 = 0.5 separates the regions of applicability of the first and second formu las (12). The value n = 0 corresponds to the zero order mode, and the values n = 1, 2, ... correspond to modes of higher orders. In Fig. 1, dispersion curves (12) are plotted for the modes with the numbers 0, 1, and 2. It should be noted that, as the layer thickness H increases, the propaga tion velocities of higher order modes tend to the velocity of sound in the liquid, whereas the velocity of the zero order mode tends to the velocity of the Stoneley wave. Figure 2 shows the dispersion dependence for the system that was used in the experiments described in [14]: a film of an aqueous solution on a lithium nio bate substrate. The following parameters are preset: the liquid is represented by water with the density 0 = 1 g/cm3 and the velocity of sound c0 = 1500 m/s; the lithium niobate substrate has the density = 4.7 g/cm3, the longitudinal wave velocity cl = 7250 m/s, the transverse wave velocity ct = 3750 m/s, and the Rayleigh wave velocity cR = 3480 m/s; the frequency of waves is 15 MHz. Figure 2 shows the dependence of the relative deviation of the wave velocity c from the Rayleigh wave velocity cR on the liquid layer thick ness H. For the principal mode, when the thickness H is small, these velocities differ only slightly. As H increases, the velocity c decreases; when the thickness is large, this velocity tends to the Stoneley wave veloc ity cS, which, at the given parameters, is very close to the velocity of sound in the liquid c0. At the same time, as H increases, the next (first order) mode arises, its velocity tending to the sound velocity in the liquid when H . At the point of the first mode genera tion, the velocity of this mode is identical to the trans verse wave velocity in the substrate ct. As a rule, in the experiments, the layer thickness is much smaller than the wavelength (H 1) and the velocity c little differs from the Rayleigh wave velocity cR. Thus, one can expect that the period of the structure formed from nanoparticles will be approximately identical to half the Rayleigh wavelength. The Acoustic Field in the Liquid Layer In view of Eq. (5), after determining the con stants C1 and C2, we write k sin r ( z + h ) e = нi A q 2 t 2 cos rh rk + s
2 нi ( t н k x )

.

(13)

Fig. 2. Dependence of the velocity of wave propagation along the boundary between water and lithium niobate on the thickness of the liquid layer. The abscissa axis repre sents the layer thickness normalized to the wavelength in the liquid, and the ordinate axis, the relative deviation of the velocities of the surface and Rayleigh waves.

From Eq. (13), we determine the components of the oscillation velocity u = (D(z)e )e
ikx нi t

= ikD ; D e z
Vol. 56

н i t + ikx

.

ACOUSTICAL PHYSICS

No. 6

2010

THE FIELD OF RADIATIVE FORCES AND THE ACOUSTIC STREAMING

865

Thus, the amplitudes of the horizontal and vertical velocity components are ux = k q k 2 sin r ( z + h ) A , t 2 cos rh k +s r
2 2 cos r ( z + h ) i qk t A. 2 cos rh k +s 2

(~10н11 m). However, in some specific cases, for example, when particles suspended in a solution form a polymer film as a result of solvent evaporation, the effect of surface tension may be of interest. The Inclusion of Sound Attenuation in the Liquid One more factor that may affect the field structure is the attenuation of sound in the liquid. In the pres ence of attenuation, equation of motion (3) acquires an additional term that contains the effective viscosity coefficient b. Then, wave equation (4) takes the form b 2 н c 0 н = 0. 2 0 t t with the parameter r being replaced by r1 = where k 1 = k 0 (1 н i)
2 2 2 2

uz = н

The Inclusion of Surface Tension In calculating the acoustic field and the radiation forces, it is necessary to estimate the effect of addi tional factors. One of them is the surface tension. With this factor taken into account, the basic problem acquires another condition instead of condition (iv) (Eq. (9)) at the free surface of the liquid. All the other boundary conditions (Eqs. (6)н(8)) remain the same. The new condition involves the pressure at the free boundary z = нh in the presence of surface tension, which is determined by the Laplace formula p н p 0 = 2 + 2 . x y
2 2

(18)

The solution to Eq. (18) has the form of expression (5) k1 н k ,
2

н1

is the complex wave number
2

(14)

Here, is the surface tension coefficient and is the surface displacement. Let us differentiate Eq. (14) with respect to time. Then, we express the pressure p given by Eq. (14) through the velocity potential and use the evident relation /t = uz = /z for the ver tical velocity component. As a result, we obtain + 2 0 t2 x z
2 2

= 0.
z = нh

(15)

in the liquid. The parameter = b/0 c 0 is small if the absorption in the liquid at distances on the order of wavelength can be considered to be weak. The same replacement r r1 should be made in dispersion equation (10). Since the object of most interest is a thin layer, namely, a drying liquid film, we restrict our consider ation to the presence of zero order mode alone in dis persion equations (10) and (11). Expanding tan(r1h) in a series for a small layer thickness and assuming that the values of parameters are the same as those in Eq. (12), we obtain a simplified dispersion relation: 1 н X н 1 н X 2
3/2

Substituting potential (5) in boundary condition (15), we arrive at the equation 02D + k2D' = 0. This leads to the following generalization of relation (9) between the constants: C 2 = н C 1 1 + i exp ( н 2 irh ) = н C 1 exp ( н 2 irh eff ) , 1 н i (16) = k r / 0 . In Eq. (16), we introduced the effective thickness of the layer: h
eff 2 2

2 2 H 0 3/2 H 2X н 1 H = X 1+ н i . 3 2X 3 4 2

(19)

= h н 1 arctan . r

(17)

The dispersion equation is similar to that obtained in the problem without surface tension, namely, in the previous equation (10), it is necessary to make the sub stitution h heff. Since the combination of parameters is small in all the cases, effective layer thickness (17) differs from the true thickness h by approximately /r = /(0c2). The latter quantity is independent of h and . For a thin layer of pure water, this quantity is very small
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010

One can see that the only imaginary term on the right hand side of Eq. (19) is proportional not only to the small parameter , but also to the cube of the small wave thickness of the layer H. A similar dependence on the parameters and H will occur for the imaginary additions to X, c, and the wave number. This means that, as the liquid film dries, attenuation rapidly decreases and becomes negligibly small. Evidently, in all the cases, attenuation in the system is smaller than in the unbounded liquid, because the major part of wave energy is concentrated in the ideal medium, i.e., in the solid half space. The Inclusion of Viscous Stresses at the Boundary between the Liquid Layer and the Substrate Shear viscosity creates an additional mechanism of wave interaction at the boundary between the liquid layer and the substrate. If shear viscous stresses are

866

GUSEV, RUDENKO

taken into account, only the third boundary condition given by Eq. (8) changes. The new condition is the equality of tangential stresses at z = 0:
xz

tion (нx) can be obtained by the formal replacement k нk in Eq. (13). The total potential is as follows: = 1 + 2 = 2 D ( z ) cos kxe
2 нi t

= 2 + н x z x2

2

2

2

= 2 . 2 xz z

2

Substituting expressions (2) and (5) for the potentials in this formula and applying the first boundary condi tion given by Eq. (6), we obtain the relation B = н 2 ikq 1 + i A, 2 2 k + s + 2ik
2

k sin r ( z + h ) = н iA 2q 2 t 2 cos kxe cos rh r k +s

нi t

(22) .

= .

Potential (22) corresponds to a standing wave with an amplitude periodically varying with the horizontal coordinate x. From Eq. (22), we determine the real components of oscillation velocity and the acoustic pressure: ux = k U 0 sin r ( z + h ) sin kx sin t , r

Using the remaining boundary conditions, we deter mine the constants C
1, 2

= ▒ q A exp ( ▒ irh ) 2 s 2 н k 2r cos rh k + s + 2 i k
2

2

2

2

u z = н U 0 cos r ( z + h ) cos kx sin t , p' = н 0 U 0 sin r ( z + h ) cos kx cos t . r

and, finally, the dispersion equation
2 2 22 2 4 k qs ( 1 + i ) н ( k + s ) 1 + i 2 k 2 k +s

4 tan rh = 0 qk t . r

(20)

Here, we introduce the notation for the amplitude factor k U0 2 q 2 t 2 A . cos rh ( k + s ) Evidently, attenuation should cause variations in the wave numbers k and r, each of which will acquire an imaginary addition. In an ideal medium, potential (22) has only the cosine component in the dependence on the x coordinate, whereas, in the presence of attenua tion, a sine component appears. THE RADIATION PRESSURE OF AN ACOUSTIC WAVE AND THE STREAMING IN THE LIQUID LAYER Above, we calculated the characteristics of the acoustic field in the liquid layer. The characteristics vary according to the harmonic law, and their average values are zero. Therefore, the period average force acting on a liquid volume element and on the particles suspended in the liquid should also be zero. In this approximation, the expected formation of structures should not occur. Nonzero average values appear when the quadratically nonlinear terms are taken into account in the initial hydrodynamic equations. Since the average values of quadratic combinations of oscil lating variables are not identically zero, we obtain a nonzero average force that leads to structuring of the ensemble of particles. The Behavior of Small Size Particles in the Liquid The radiation pressure of sound was described in many reviews (see, e.g., [15]). The pressure acting
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010
2

The contribution of tangential stresses appears on the left hand side of Eq. (20) and is determined by the parameter , which is usually small. With allowance for the smallness of and the wave thickness H of the liquid layer, we obtain a dispersion relation similar to Eq. (19): X 1 н X н 1 н 2
3/2

X + i 1 н X н 1 н 2

3/2 = H 0X . 4 2

(21)

In Eq. (21), we replace X by X(1 + i), where is a small imaginary addition. After calculating , we cal culate the attenuation coefficient, i.e., the imaginary part of the wave number: k X2 3 X k '' = н = 1 н 1 н 1 н 2 2 2 2 2 2 cc t
2 2 н1

.

One can see that, when the thickness of the liquid layer is small, the wave attenuation occurs approximately in the same way as it would occur for a shear wave in a solid with a viscosity identical to the viscosity of the liquid. A Standing Wave in the Liquid Layer Now, let two waves propagate in the substrate in opposite directions. The potential of the wave traveling in the positive direction of the x axis is determined by Eq. (13). The wave propagating in the negative direc

THE FIELD OF RADIATIVE FORCES AND THE ACOUSTIC STREAMING

867

from the side of an acoustic field of arbitrary config uration on a suspended particle was calculated in [16]. The origin of the radiative force is the response of a particle to the scattering of the incident wave. However, for small size particles, such a mechanism is presumably not the governing one. As is known, the fraction of the wave energy scattered by a particle is proportional to (kR)4, where R is the particle radius (see, e.g., [17]). For example, for polystyrene particles with a radius of 100 nm under an incident wave with a frequency of 15 MHz [14], this parame ter is on the order of 10н9. Therefore, the predomi nant mechanism is as follows: because of their small size, the particles can be carried away by the acoustic streaming in the liquid and can move together with the latter. Let us consider a small particle in an oscillating liq uid. The equation of motion has the form (see, e.g., [18]) m d 2 (X н ) = н m' d + F 2 dt dt
2 2 diss

for the radiation pressure on a particle in a standing wave field, we reduce Eq. (24) to the form R(kR) S + ( 2/3 ) ( S н 0 ) 1 c 0 0 н2 u 3 cS S 2 S + 0 = 61 + R 0 (v н u). 2 Here, u0 is the amplitude of the oscillation velocity of the liquid. Estimates by this formula show that, for the particles used in the experiments [14], the difference between the velocities did not exceed 10н7н10н8 m/s. Thus, the hypothesis that small size particles are almost completely carried away by the acoustic streaming while the radiative force acting on the parti cles can be neglected seems plausible. The Radiation Pressure on the Liquid
2 2 0

+ F.

(23)

Here, X is the displacement of a particle, is the acoustic displacement of the liquid at the particle site, Fdiss is the dissipative force due to the flow around the particle, and F is the radiative force. If the particle has a spherical shape, we have 1 m = mS + m0 , 2 m ' = mS н m0 = 43 R ( S н 0 ) , 3

The radiative force Fi is expressed through the radi ation stress tensor ik [19]: Fi = н ik , xk
ik

=

2 p ' ik + 0 u i u k . (25) 2 0 c0

Here, the angular brackets denote averaging over the period of the acoustic wave and is the nonlinearity parameter of the liquid. For the nonzero components of the tensor, from Eq. (25) we obtain the expressions
xx

where mS and R are the mass of the particle and its radius, m0 is the mass of the displaced liquid, and S and 0 are the respective densities. If the densities are different, the particle placed in the acoustic field expe riences the action of forces both depending on the vis cosity of the medium and independent of it. We represent Eq. (23) in terms of the velocities of the particle v = dX/dt and the liquid u = d/dt, and, specifying the expression for the dissipative force [18], we obtain the equation
3 4 R 3 0 d du + ( v н u ) = н 4 R ( S н 0 ) S 2 dt 3 dt 3

=

2 2 p ' + 0 u x , 2 0 c0 zx

xz

=

= 0 u x u z ,

zz

=

p ' 2 + u 2 . 0 z 2 0 c0

Thus, the contribution to the force is made by both diagonal and nondiagonal elements of the radiation stress tensor. Calculating the average values of acoustic quantities, we obtain
2 2 2 2 u x = k 2 U 0 sin r ( z + h ) sin kx , 2r 2

н 6 R 1 + R 0 ( v н u ) 2 н 3R
2

(24)

2 0 1 + 2R 0 d (v н u) + F. 9 2 dt

uz =

2

U0 2 2 cos r ( z + h ) cos kx , 2

2

In a steady state flow, where the velocities v and u are time independent, their difference is determined by the effects of the radiative force F and the Stokes force. The radiative force tends to increase the velocity of the particles with respect to the flow, whereas the Stokes force tends to decrease it. Using the Gor'kov formula
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010

2 u x u z = н k U 0 sin 2 r ( z + h ) sin 2 kx , 8r

p' =

2

0 2r
2

2

2

U 0 sin r ( z + h ) cos kx .

2

2

2

868 (a) V 4 3 2 1

GUSEV, RUDENKO (b) V 3

2

1 0 н1 0 н0.5 2kz н1.0 н6 н4 н2 02 2kx 4 6 н6 н4 н2 0 2 4 6 2kx

Fig. 3. Radiation pressure potential: (a) the three dimensional representation and (b) the sections of the potential profile at the interface z = 0 (the solid line) and at the free surface z = нh (the dotted line).

Correspondingly, the components of the radiation stress tensor are
xx

=

0 U 2r
2

2 0

sin r ( z + h ) ( k 0 + ( k н k 0 ) sin kx ) , =н k 0 2 U 0 sin 2 r ( z + h ) sin 2 kx , 8r

2

2

2

2

2

xz

=

zx

zz

=

2 2 0 U0 2 k0 2 cos kx 2 н 1 sin r ( z + h ) + 1 . 2 r

Now, we calculate the radiative forces: нFx = xx xz 0 kU + = x z 4
2 0

(26) 2 2 k0 k0 т sin 2 kx ( н 1 ) 2 cos 2 r ( z + h ) н ( н 1 ) 2 + 1 , r r нFz = zz zx 0 rU + = z x 4
2 0

of Fx is only determined by the horizontal coordinate x. Since the dependence on x is periodic, the force exhibits maxima and minima, the latter corresponding to zero force value. Presumably, the particles sus pended in the liquid are mainly grouped in the regions of these minima. For the vertical force, the factor in parentheses is positive for any value of x, so that the direction of the vertical force component does not depend on the horizontal coordinate x and is deter mined by the z coordinate alone. An analysis of the dispersion curves shows that the parameter rh can vary from 0 to /2 with a subsequent periodic shift by n, where n is an integer. This means that the factor sin2r(z + h) is positive and the vertical component of the radiative force tends to gather the suspended parti cles at the free surface of the liquid. In other words, as the thickness of the layer decreases in the course of evaporation, the grouped particles are deposited on the substrate surface. Since the radiative forces are quadratic in the acoustic field, the period of the spatial structure formed under their action in the horizontal direction is identical not to the acoustic wavelength, but to half the acoustic wavelength, because 2k = 2 т 2/ = 2/(/2). The Radiation Pressure Potential In the general case, the radiation pressure is a tensor quantity, but, in the problem under study, rot F = 0; i.e., the radiation pressure force can be represented as a gradient of a certain potential:
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010

(27) 2 2 2 ( н 1 ) k0 + k k0 т sin 2 r ( z + h ) ( н 1 ) 2 cos 2 kx + . 2 r r Expressions (26) and (27) are convenient for analyzing the radiative forces in a liquid layer with a small thick ness, when the surface wave velocity is greater than the velocity of sound in the liquid, because, in this case, all the coefficients are positive and the signs of the terms appearing in the formulas can be easily determined. For the horizontal force Fx, the factor in parentheses is always negative for any values of z, so that the direction

THE FIELD OF RADIATIVE FORCES AND THE ACOUSTIC STREAMING

869 4 x

F = нgrad V. Let us determine this potential by using Eqs. (26) and (27): V= 0 U0 k0 ( н 1 ) 2 cos 2 kx cos 2 r ( z + h ) 8 r ( н 1 ) k0 + k r
2 2 2 2 2

н6

н4

н2

0

2

н0.2 н0.4 н0.6 н0.8 н1.0 z
Fig. 4. Streamlines. The abscissa axis represents the quan tity x1 = 2kx, and the ordinate axis, z1 = 2kz.

+

( cos 2 r ( z + h ) н 1 )

2 k0 н ( н 1 ) 2 + 1 cos 2 kx . r

Here, the integration constant is introduced in such a way that, at the layer surface z = нh, the hydrody namic pressure related to acoustic streaming (see the next section) is zero. Figure 3a shows the characteris tic form of the potential for the parameters of the media considered above. Figure 3b shows the profiles of the potential for two sections: near the boundary between the solid substrate and the liquid layer and near the free surface of the liquid layer. The solid line corresponds to the potential near the interface, and the dotted line, near the surface. The vertical lines indicate the positions of the extrema of the potential. One can see that the horizontal structure of the poten tial does not depend on the vertical coordinate and contains minima, which points to the possibility of particle concentration in these regions. The absolute minima of the potential are close to the free surface of the liquid layer; at the same time, near the interface, the potential well is narrower. Hence, as the layer thickness decreases, the particles are additionally con centrated in the regions corresponding to the minima of the potential. The Acoustic Streaming Caused by the Radiation Pressure The radiation pressure sets the liquid in motion and causes it to flow. The structure of the steady state streaming at small hydrodynamic Reynolds numbers is calculated using the system of equations [20] н U = н P + F , d iv U = 0 . (28) Here, U is the velocity of acoustic streaming and P is the flow pressure. Applying the rot operation to the first of Eqs. (28) and taking into account the potential nature of the radiation force F, we obtain a biharmonic equation for the stream function: = 0 , U x = / z , U z = н / x . (29)

Taking into account the structure of the calcu lated field of radiation forces, we seek the solution with the following dependence on the horizontal coordinate: = 0 ( z ) sin ( 2 kx ) , U x = A ( z ) sin ( 2 kx ) , U z = B ( z ) cos ( 2 kx ) . Formulas (29), which determine the stream function, suggest the relations A = '0 , B = н2k0, and B ' = н2kA. Integrating the equation for the streamlines dx/Ux = dz/Uz and taking into account the aforementioned relations, we obtain the equation 0sin(2kx) = const. One can see that the flow velocity is zero on the lines x = n/2k lying in the (x, z) plane. These lines sep arate the regions with oppositely directed velocities (see Fig. 4). This qualitative result is confirmed by the result of streamline calculation. From biharmonic equation (29), we obtain an ordinary differential equation for the function 0(z), which can be solved: d н 4 k 2 ( z ) = 0 , dz 2 0 0 = [ sinh ( 2 kz ) + cosh ( 2 kz ) + ( 2 kz ) sinh ( 2 kz ) + ( 2 kz ) cosh ( 2 kz ) ] . Here, the constants , , , and are determined by the boundary conditions. From the conditions at the interface z = 0, we find = 0 and = н1. Then, for the stream function and the velocity components of streaming, we obtain = [ sinh ( 2 kz ) + ( 2 kz ) sinh ( 2 kz ) н ( 2 kz ) cosh ( 2 kz ) ] sin ( 2 kx ) ,
2 2

These equations should be complemented with boundary conditions. At the boundary z = 0 of the liq uid with the solid half space, the flow velocity is zero (Ux = Uz = 0); at the free surface of the liquid z = нh, the vertical flow is absent (Uz = 0) and the pressure on the surface is also absent (P = 0).
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010

870

GUSEV, RUDENKO

U x = 2 k [ sinh ( 2 kz ) + ( 2 kz ) cosh ( 2 kz ) н ( 2 kz ) sinh ( 2 kz ) ] sin ( 2 kx ) , U z = н 2 k [ sinh ( 2 kz ) + ( 2 kz ) sinh ( 2 kz ) н ( 2 kz ) cosh ( 2 kz ) ] cos ( 2 kx ) . The condition Uz(z = нh) = 0 determines the form of , and the constant is expressed through the radia tion pressure: sinh ( 2 kh ) н ( 2 kh ) cosh ( 2 kh ) = , ( 2 kh ) sinh ( 2 kh ) =н 0 U0 h 32 k sinh ( 2 kh )
2 2

tion of particles drawn together under the effect of streaming. ACKNOWLEDGMENTS This work was supported by the Presidential Pro gram in Support of Leading Scientific Schools (grant no. NSh 4590.2010.2) and the Russian Foundation for Basic Research (projects nos. 09 02 00925 a and 09 02 00967 a). REFERENCES
1. S. Shiokawa, Y. Matsui, and T. Moriizumi, Jpn. J. Appl. Phys. 28 (Suppl. 1), 126 (1989). 2. S. Shiokawa, Y. Matsui, and T. Ueda, Jpn. J. Appl. Phys. 29 (Suppl. 2 1), 137 (1990). 3. T. Uchida, T. Suzuki, and S. Shiokawa, Proc. IEEE Ultrason. Symp. 2, 1081 (1995). 4. C. J. Strobl et al., Proc. IEEE Ultrason. Symp. 1, 255 (2002). 5. A. Wixforth, J. Scriba, and C. Gauer, MST News. 5, 42 (2002). 6. A. Wixforth, Superlatt. Microstruct. 33, 389 (2003). 7. C. J. Strobl, Z. V. Guttenberg, and A. Wixforth, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 51, 1432 (2004). 8. A. Renaudin, P. Tabourier, V. Zhang, J. C. Camart, and C. Druon, Sens. Actuat. B 113, 389 (2006). 9. M. K. Tan, J. R. Friend, and L. Y. Yeo, in Proc. of the 16th Austral. Fluid Mechanics Conf. (Australia, 2007), pp. 790н793. 10. M. K. Tan, J. R. Friend, and L. Y. Yeo, in Proc. of the 5th Austral. Congress on Appl. Mechanics ACAM 2007, pp. 348н353. 11. R. M. White and F. W. Voltmer, Appl. Phys. Lett. 7, 314 (1965). 12. Yu. V. Gulyaev and F. S. Hickernell, Acoust. Phys. 51, 81 (2005). 13. I. A. Viktorov, Sonic Surface Waves in Solids (Nauka, Moscow, 1981) [in Russian]. 14. O. V. Rudenko, P. V. Lebedev Stepanov, V. A. Gusev, A. I. Korobov, B. A. Korshak, N. I. Odina, M. Yu. Izo simova, S. P. Molchanov, and M. V. Alfimov, Akust. Zh. 56 (6) (2010, in press). 15. Z. A. Gol'dberg, in High Intensity Ultrasonic Fields, Ed. by L. D. Rozenberg (Nauka, Moscow, 1968) [in Russian]. 16. L. P. Gor'kov, Dokl. Akad. Nauk SSSR 140, 88 (1961) [Sov. Phys. Dokl. 6, 773 (1961)]. 17. Acoustics in Problems, Ed. by S. N. Gurbatov and O. V. Rudenko (Fizmatlit, Moscow, 2009) [in Russian]. 18. P. V. Lebedev Stepanov and O. V. Rudenko, Acoust. Phys 55, 729 (2009). 19. A. P. Sarvazyan, O. V. Rudenko, and W. L. Nyborg, Ultrasound Med. Biol. (2010, in press). 20. O. V. Rudenko and S. I. Soluyan, Theoretical Founda tions of Nonlinear Acoustics (Nauka, Moscow, 1975; Consultants Bureau, New York, 1977).

.

The final solution for the stream function has the form =н + 0 U0 h 32 k sinh ( 2 kh ) sinh ( 2 kz )

sinh ( 2 kh ) н ( 2 kh ) cosh ( 2 kh ) ( 2 kz ) sinh ( 2 kz ) ( 2 kh ) sinh ( 2 kh ) н ( 2 kz ) cosh ( 2 kz ) sin ( 2 kx ) .

This expression was used to plot the streamlines of the acoustic streaming. As one can see from Fig. 4, the streamlines are denser near the lines 2kx = n. This means that, in the presence of particle interaction forces, which may be either of hydrodynamic or some other origin (e.g., of electric or chemical nature), sus pended particles should mainly concentrate in these regions. Thus, a standing surface wave can serve as the controlling factor for the formation of ordered struc tures from nanoparticles suspended in the liquid. CONCLUSIONS We developed a theory that qualitatively explains the experiments on the formation of ordered struc tures of particles in the course of drying of a colloidal solution on a solid substrate. We demonstrated the possibility to control the process by exciting a wave that propagates along the liquidнsolid interface. We calculated the wave field, the radiation forces, and the acoustic streaming in the liquid layer. We considered the main factors that affect the formation of the acous tic field, the vortex streaming, and the nanoparticle structures. The radiative forces arising in the liquid layer serve as the main factor of ordering. The period of the structure formed on the substrate is identical to half the wave length. The radiation pressure that acts on the particles carried by the liquid and is due to the acoustic wave scattering by the particles can presum ably be neglected. An important role can be played by forces of nonacoustic origin, which lead to aggrega

Translated by E. Golyamina
ACOUSTICAL PHYSICS Vol. 56 No. 6 2010