# Broadband study of the scattering of ultrasound by polystyrene-latex-in-water suspensions.

код для вставкиСкачатьAnn. Physik 5 (1996) 13-33 Annalen der Physik 0 Johann Ambrosius Barth 1996 Broadband study of the scattering of ultrasound by polystyrene-latex-in-watersuspensions U. Kaatze, C. Tiachimow, R. Pottel, and M. Brai Drittes Physikalisches Institut, Universitat GBttingen, Burgerstraae 4244, D-37073 Gottingen, Germany Received 19 January 1995, revised version 26 October 1995, accepted 29 October 1995 Abstract Ultrasonic absorption spectra have been measured between 0.1 and 2000 MHz of aqueous suspensions of plystyrene latex globules. Samples of different diameter (60 to 100 nm) and different concentration of the polystyrene latex particles (volume fraction between 0.05 and 0.38) have been studied. The broadband spectra exhibit a Debye-type relaxation term in the frequency range around some MHz and a contribution with special characteristics at higher frequencies. The former tern is assumed to be due to counterion effects in the surface layer of the globules. No special attention is payed to this effect. The latter contribution is attributed to scattering and mode conversion mechanisms. An approximate explicit and an exact formulation of the theoretical description of the sound wave attenuation by the scattering process are discussed. The explicit formulation enables valuable insights into the mode conversion mechanisms but does not allow for a complete representation of the measured spectra. Application of the exact version leads to the conclusion that in the frequency range under consideration monopole and quadrupole oscillations of the polystyrene latex globules the dominant cause of ultrasonic wave attenuation. It is found that even at the elevated concentrations used effects of multiple scattering are of minor importance here. Keywords: Uluasonic attenuation; Sound scattering; Acoustic mode conversion; Suspensions. 1 Introduction Since the early days of the pioneer work by Lord Rayleigh [l] scattering of sound waves fascinates and likewise intrigues physicists, particularly because of the complexity of the phenomenon and of its substantial differences with electromagnetic wave scattering. In a heterogeneous system acoustic energy is not only dissipated by fourth-power-law “Rayleigh” scattering but also by a variety of additional processes. As will be discussed more detailed below, when an acoustic wave propagating in a suspension of spherical particles is incident on an obstacle it does not simply give rise to a reflected compressional wave and to a compressional wave within the particle itself but also to a number of further modes with special dependence upon sound wavelength. Among these modes are predominantly so-called viscous and thermal boundary layer waves at the continuous phase/particle interface. It is only briefly mentioned that Rayleigh scattering in a narrow sense is found if the wavelength 1 of the incident sound wave distinctly exceeds the dimensions of the obstacles. 14 Ann. Physik 5 (1996) The rather complicated features of sound wave propagation in heterogeneous material stimulated a multitude of theoretical treatments of which only some early bench-. mark studies [2] - [5] and a few more recent investigations [6] - [15] are quoted here. The theoretical formulations differ from one another mainly by the assumptions taken as a basis and by the simplifications made in the mathematical approaches. In contrast to the intense theoretical effort that has been undertaken in the past only a few experimental studies exist so far [8], [16] - [20], mainly due to difficulties in the broadband ultrasonic spectrometry of liquids. Besides such difficulties problems result from the fact that additional mechanisms like fluctuations in concentration [21] [26] and chemical reactions [27] - [31] may also add frequency-dependent contributions to the acoustic spectrum of heterogeneous liquid samples [32] - [341. During the past years, much interest has been directed toward resonant scattering of acoustic waves, and the localization phenomena resulting thereby, in media containing random distributions of identical finite-size obstacles [35] - [MI. These interesting effects are predicted to occur on special conditions. The wavelength 1 of the incident sonic wave, for instance, should commensurate with the size of the inhomogeneities [36] and the attenuation coefficient should be small, in order to enable noticeable effects of coherent interference of scattered waves, a precondition for localization. Here we are not dealing with such special conditions, but we consider more conventional scattering phenomena, with particular emphasis on strongly absorbing suspending media. The insufficient experimental verification of conventional theoretical scattering models is the more an unfortunate deficiency as the correct knowledge of the details in the sound propagation mechanism is not only of great significance for fundamental aspects. There exists a current wide interest in the scattering of sound in a variety of situations existing in nature and in industrial, biological and medical applications. Rapid characterization of particles in fog, smoke and clay suspensions, for instance, attracted considerable attention and also multiphase creaming and sedimentation monitoring. Flow rate measurements in industrial plants and by Doppler sonography in medicine, as well as non-invasive sample analysis including biological cells are other examples. The situation has prompted us, before considering in future investigations the interaction of sound with more complicated systems of special relevance, to study the scattering of ultrasonic waves by comparatively well-defined aqueous suspensions of spherical polystyrene-latex particles. We measured the acoustic attenuation coefficient in the frequency range from 100 kHz to 2 GHz for this purpose and compared the absorption spectra obtained thereby with theoretical models, including predictions based on evident simplifications. 2 Experimental 2.I Sample suspensions Three especially prepared polystyrene-in-water suspensions with different particle diameter have been donated by the BASF, Ludwigshafen. The volume fraction (D, of polystyrene amounts to about 0.37 in the original suspensions which, in addition, contain ionic surfactants to prevent the particles from sticking together and forming clusters. To look for the effect of multiple scattering in the ultrasonic studies the highly concentrated original suspensions have been diluted using bidistilled water to 15 U. Kaatze et al.. Broadband study of the scattering of ultrasound by polystyrene-latex Table 1 Shorthand notation of original samples, volume fraction pVofpolystyrene-latex in the suspensions, particle radii (R) with standard deviation (AR) according to the mass distributions ( m ) and number distribution (n) found in ultracentrifuging experiments as well as determined from electron micrographs (e) and by quasielastic light scattering (0. i? denotes a mean of the radius values by the different methods. ~ Sample; pv Rm, mrn Rn, A& A; 0.2, 0.371 B; 0.1,0.2, 0.365 31. 8 44.9 46.5, 10 21. 7 40, 8 40.5, 9 nm C; 0.05, 0.175, 0.376 nm Re, me nm RI, mi Kf6nm 27.4, 5 35, 7.5 35.9, 5.4 45, 8.2 52.4, 8 30.5 41 46.5 nm nm obtain samples with smaller volume fraction rpv. A survey of the samples used in the measurements is given in Table 1. Also presented in that table are data for the radius R of the polystyrene-latex globules and for its standard deviation AR, as measured by altogether four methods. For all original suspensions the manufacturer determined by ultracentrifuging the radius R m following from the mass distribution and that (RJ from the number distribution. For samples A and B, in addition, the radius R e derived from electron micrographs is given in Table 1. The electron micrographs, also prepared by the BASF, have been taken after desiccation of the sample and successive evaporation of a thin layer of suitable metal. Finally, the particle radius Rl resulting from quasielastic light scattering measurements of this laboratory is listed. There seem to exist some systematic differences in the radius data that can be explained. RI, for instance, is somewhat larger than the others since it is a hydrodynamic parameter that may include parts of the particle surroundings, like the surfactant layer at the polystyrol-latex globule surface. Re appears to be rather small, probably as a result of shrinking during the desiccation procedure. Nevertheless, for all samples the radius values found by the different methods are close together so that a mean R can be given to within f 6 nm (Table 1). 2.2 Ultrasonic measurements The sound attenuation coefficient a of liquids, if contributions ad from concentration fluctuations are not considered, according to the simple additivity rule a= -k &hem -k asc (1) is composed of three different parts. Of them the so-called classical contribution acl due to internal viscous friction and thermal conductivity is always present. Part @-hem Considers energy dissipation by possibly proceeding chemical reactions, while a,,represents the scattering processes at which we aim here. The classical part of the attenuation coefficient increases with frequency v as [471 where c(v) denotes the velocity of sound and B a parameter which is independent of Y. Measurements over the frequency band from 100 kHz to 2 GHz have thus not only to cover a broad range of wavelengths within the sample but have to also deal 16 Ann. Physik 5 (1996) ~~~ ~~ with the considerable dynamic range exp(a(2 GHz)Z)/ exp(a(100 kHz)l) that is of the order of exp(4 108a(100 kHz)Z). The remarkable requirements in broadband ultrasonic spectrometry resulting thereby can be granted by applying different methods and by using specimen cells that are matched to part of the frequency band [48]. At low frequencies (0.1 MHz 5 v 5 5 MHz), where according to Eq. (2) the sound attenuation coefficient is comparatively small, a resonator method is appropriate in which the pathlength of interaction between the ultrasonic signal and the sample is effectively increased by multiple reflections. We used a biplanar circular cylindrically shaped resonator cell [49]the faces of which are established by piezoelectric X-cut quartz discs of 60 mm diameter and with a fundamental frequency of 1 MHz for its thickness vibrations. Unavoidable instrumental loss has been considered by reference measurements using liquids of known sound attenuation coefficient and with sound velocity and density appropriately adjusted to as close as possible simulate the resonator field configuration of the actual sample measurement. Undesired higher-order radial modes within the cell [50] have been carefully considered by always measuring, with the aid of a computer-controlled network analyzer, the complete resonator transfer function over the relevant frequency range. The optimum mode of operation with respect to both, tolerable measuring time and high degree of accuracy, consisted in carrying out two spans. One with increased sampling frequency distance has been extended over a broad frequency band to record the amount and phase of the transfer function around the principal resonance under consideration. The data obtained thereby have been used to correct the second span for the effect of higher-order modes. In order to fully utilize the high resolution and accuracy of the analyzer with respect to measurements of the signal level in this span only the amount of the transfer function has been recorded at small sample distances around the principal resonance frequency [51]. At high frequencies absolute measurements of the liquid attenuation coefficient a have been performed by a variable-sample-length method. Applying this method the amplitude of an ultrasonic wave transmitted through a suitable liquid-filled cell is automatically recorded at variable transmitter-receiver spacing. Pulsed signals are used so that electrical cross-talk and undesired waveforms resulting from multiple reflections within the cell can be faded out by delay effects. Stability of the electronic apparatus is controlled by a direct comparator technique in which the sample cell is substituted for a variable high-precision reference attenuator. We used three different cells to cover the frequency range. Two cells were operated at overtones of the fundamental frequency VT of the transducer thickness vibrations. Of these cells one was appropriate for the frequency range from 3 to 45 MHz here (quartz transducer discs of 40 mm diameter, VT = 1 MHz [52]), the other one for the range from 30 to 480 MHz (lithium niobate transducer discs with diameter of 12 mm and VT = 10 MHz [53]). At very high frequencies (480MHz 5 v 5 2 GHz 1421) we utilized broadband surface excitation of small piezoelectric lithium niobate rods according to the method by Bommel and Dransfeld [S]. No deviations have been observed from the exponential decay of the sound amplitude as a function of sample length. In the lower part of the frequency range of measurements the sound velocity c of the samples has been determined from the distance between successive principal resonance frequencies of the cavity cells applying suitable theoretical expressions [56]. Above 3 MHz c has been derived from standing wave patterns resulting at small transducer spacing from multiple reflections within the transmission cells. U. Kaatze et al., Broadband study of the scattering of ultrasound by polystyrene-latex 17 2.3 Accuracy of attenuation coeficient and sound velocity data Due to multiple data recording followed by averaging, digital filtering, and suitable regression analysis statistical errors in the measured a- and c-values of the samples are negligibly small. Also negligibly small are fluctuations in the measuring frequency v. The temperature T of the specimen cells was controlled to within 0.01 K and was measured with the aid of a Pt-100 thermometer that had been also calibrated to within 0.01 K. Temperature of different cells did thus not deviate by more than 0.05 K, corresponding to an error of 0.15% in the a-value of the liquids. Possible changes in the sample concentration that could have been resulted from preferential evaporation during the measuring procedure have been carefully considered by repeated data recording. It was found that experimental errors in the absorption coefficient a by alterations of the sample composition are also smaller than 0.1%. Together with possible systematic errors that are specific for the particular apparatus [49], [52] - [54] the uncertainties listed in Table 2 result. Over a significant part of the frequency range of measurements the a-and c-values are accurate to within globally 1%. The error in the absorption coefficient below 3 MHz is indeed distinctly larger. As will be shown below, however, those data are in any case of minor interest here. Table 2 Experimental emr of absorption coemcient o and sound velocity c of samples. V A 0f a w c 0.1 0.07 0.02 0.005 0.01 0.02 0.005 0.005 MHz 0.1 -0.5 0.5-3 3-30 30-400 400-1200 1200-2000 0.005 0.005 0.01 0.01 2.4 Obvious characteristics of measured spectra of the attenuation coefficient With the present suspensions the dispersion in the sound velocity is small. As common practise in ultrasonic spectrometry of liquids, we thus restrict the discussion to the attenuation spectra, an example of which is shown in Fig. 1. Only the excess attenuation per wavelength a,L, defined by a,(v) * I, = a(v). I , - CYcl(V) ' 1, (3) is shown in that diagram since the classical contribution ctc,(v) I, = Bv is of minor interest here. The measured a-spectrum exhibits a relative maximum at around some MHz. We suggest this maximum to reflect a chemical contribution 18 Ann. Physik 5 (1996) Fig. 1 Lidog-plot of the excess attenuation per wavelength, cr,l= c r l - Bv, versus frequency v for a polystyrene latex suspension with R = 40 nm,p, = 0.1 and T = 25OC ( B = 28.1 . lO-'*s). The dashed curve indicates a contribution which is suggested to be due to disturbation of a stoichiometrically defined chemical equilibrium. that probably results from a catiodanion association mechanism or from a sound wave induced shift of the counterion cloud relative to the electrically charged polystyrene globule [61]. In Eq. (4) is o = 27rv the angular frequency, r&em denotes a relaxation time and Achem an amplitude. This explanation of the chemical relaxation has to be verified by another series of measurements in which attention is particularly focussed on the low-frequency part of the spectrum. Since we aim at a study of possibly present scattering phenomena the chemical contribution will be also subtracted from the total absorption per wavelength and only the remaining part h ( v ) * 1= a,(v) . 1 - a c h e m ( v ) * 1 (5) will be discussed in the following. This pait attracts attention for two reasons. There is another relative maximum at frequencies around 100 MHz that cannot be easily attributed to a chemical relaxation and there is, in addition, at frequencies above about 1 GHz a strikingly strong increase in 8a(v), respectively a,(v). The a / $ data even increase at high frequencies though the classical part in a predicts a constant value ( C Y , ~v)/$ ~ ~ ( = B / c ) . This unusual frequency dependence of attenuation coefficient data has been found with all spectra of polystyrene-latex suspensions investigated. Most interesting it cannot be explained by a forth-power law. The attempt is nevertheless made in the following to discuss the &(v) . A-spectra resulting from the measurements in terms of scattering models. In doing so we always consider the originally measured complete attenuation spectra. Hence in the nonlinear least-squares regression analysis of the experimental data we used a function of the type a(v)* 1= a,(v) . 1 + + Bv Achem~~chem 1 + (archem) U. Kaatze et al., Broadband study of the scattering of ultrasound by polystyrene-latex 19 with a,(v) chosen according to the model under consideration. Merely for reasons of clearness the results of the fitting procedures will be presented as plots of 6a(v).1as a function of frequency. Since, in the frequency range of measurements there are no indications of concentration fluctuation, an a c f ( v )- I-term (Eq. 1) has not been included in the model spectral functions. 3 Theoretical model of ultrasonic scattering 3.1 Single scattering approach The model that will be used to analytically represent the measured spectra is based on the formulations by Epstein and Carhart [57] and by Allegra and Hawley [16]. When a compressional wave interacts with a particle within the suspension, it gives rise to a reflected compressional wave, a compressional wave within the particle, and viscous as well as thermal waves in both the fluid and the particle. In the model utilized in this treatment of the problem, the wave equations describing the propagation of these waves in the isotropic media are obtained from six fundamental relations, namely the laws of conservations, of mass, and of energy, a stress-strain relation, and two thermodynamic equations of state [161. Since the particles are presumed to be spherically shaped, the wave equations are solved in terms of a series expansion of spherical Bessel functions with a priory unknown coefficients. These coefficients are determined in the usual manner by utilizing the boundary conditions at the particle/ Suspending liquid interface. The coefficient's derived for a single particle are then related to the attenuation coefficient for the suspension by linear superposition. This single scattering approximation will be shown to allow for an appropriate representation of the measured spectra within the limits of experimental error. Only the fust two coefficients of the series expansion are normally considered in the literature, and these are used in explicit analytical terms that are derived as a long-wavelength approximation. The explicit formulation enables the physical processes reflected by the analytical terms to be highlighted. These terms are, therefore, discussed below. We first briefly present, however, the general formulation of the attenuation coefficient, which appears to be the adequate description of the broadband spectra, particularly in the small-wavelength regime above about 100 MHz. 3.2 Scattering coeflcients and their relationship to sound attenuation In the frequency range of interest, the particle size is much smaller than the wavelength of the incident compressional wave. For reasons of simplification, let us first presume the distance d between particles to be substantially larger than wavelengths & and AT of a viscosity wave and a thermal wave, respectively, defined below. This means that the heoretical approach is restricted to suspensions for which the preconditions 20 Ann. Physik 5 (1996) hold, where q, denotes the shear viscosity, p the density, A the thermal conductivity and C, the specific heat capacity at constant pressure of the continuous phase. At the highest frequency considered in this paper (2 GHz) A, = 80 nm and AT = 30 nm, while at 9 = 0.1 the inter-particle distance d = 150 nm. Hence the above preconditions are fairly well fulfilled in the high-frequency part of the spectrum but not at lower frequencies. It will be shown below that the theoretical description nevertheless applies to the measured spectra, as a consequence of the high attenuation coefficient of the visco-inertial and thermal waves. Along a path of one wavelength the amplitude of these waves decreases by about 55 dB! One indication for the applicability of the theoretical treatment even at As > d and AT > d is the finding that there does not exist a dependence on the volume fraction qV (and thus inter-particle distance d) in the analytical description of the measured spectra. In this calculation of scattering coefficients, let us further assume the amplitude of the incident compressional wave to be so small, that it is sufficient to consider a linear relationship t =t o +At (8) (t = p , p , T , u ) for the density p , pressure p, temperature T and specific internal energy u of the system at applied sound field. In Eq. (8) lodenotes the equilibrium value and A t the displacement from equilibrium. From the three laws of conservation of mass, momentum and energy, that are used in linearized form for small displacement At, from a stress-strain relation and two thermodynamic equations of state mentioned above, the following wave equations can be separated (v2-k k?)$Jc * 0, (v2-k k ~ ~ ) &=r 0, (0'+ k:)A, = 0, (9) where qbC, C$T, and A, denote the compressional, thermal and (vector) shear wave potential. At frequencies below about 10" Hz the wave numbers in Eq. (9) are given by the relations with i = (-1)i and with CL and CYL denoting the sound velocity and absorption coefficient for the pure submedium under consideration. The wave equations can be solved in spherical coordinates r,O. If primed quantities refer to the particles and unprimed quantities to the suspending liquid (except the Bessel functions for which primes indicate differentiation) the following Eqs. (11) to (17) with unknown coefficients are solutions of the wave equations in spherical coordinates r, 0, U. Kaatze et al., Broadband study of the scattering of ultrasound by polystyrene-latex 21 where (1 1) represents the incident sound wave and (12) to (14)the reflected compressional, thermal, and viscous wave, respectively. In the disperse phase, the compressional, thermal and viscous wave, respectively, are described by 00 4; = F(2n + 1)A2n(kir)Pn(cos0) In Eqs. ( 1 1) to (17) are P n ( C 0 S 0) and P;(COS8)Legendre functions and associated Legendre functions of the first kind andj, and h, are the spherical Bessel functions of the first and third kind. Assuming the velocity and stress components, the temperature and the heat flow to be continuous at the particlehspending fluid interface a set of six complex equations is obtained through which the unknown complex coefficients An,Bn,Cn,A;, BL and CL in the above series expansions are given as indicated below, 22 Ann. Physik 5 (1996) Herein R is the particle radius, n = 0, 1,2,3, . . ., and the following abbreviations have been used for primed and unprimed quantities, r where p and are Lam6 constants, p is the coefficient of cubic expansion and K is the ratio of specific heat at constant pressure and constant volume. Thermal and viscous waves are strongly damped and thus restricted to a close range near the particle/suspending fluid interface as mentioned above. For this reason, a macroscopic detector monitoring the sound field within the suspension will receive contributions from the incident (40) and reflected (4,) compressional wave only. From these contributions the scattering part of the sound attenuation coefficient of a suspension follows as the series expansion where qV again denotes the volume fraction of the suspended particles and where the scattering coefficients A,, n = 0, 1,2, . . ., are given by Eqs. (18) to (23). 3.3 Explicit formulation of visco-inertial and thermal boundary effects The theory of scattering summarized above leads to a rather universally valid description of contributions as, a,/ to the ultrasonic absorption coefficient of micro-heterogeneous liquids. The complexity of the analytical formulation, however, tends to mask the physical significance of the dominating mechanisms that result in non-vanishing scattering parts in the attenuation of sound waves. In order to illuminate the process of sound attenuation by scattering and to show up significant differences to scattering and attenuation of electromagnetic waves we shall briefly discuss explicit solutions of the equations for the usually most important first coefficients A0 and A1 in the above series expansions. Again we shall follow the treatments by Epstein and Carhart [45] as well as by Allegra and Hawley [ 161. With the above supposition that the wavelength A, of the incident and scattered sound waves is much larger than the particle radius R and with various further reasonable assumptions [ 16, 571, the following relations are found, + Ao= where [ .";( a,-6a, -1- 2 !2)] + [.ra,-; ;(1-6' b') HI] : = A o l + A O 2 b, H2 U. Kaatze et al., Broadband study of the scattering of ultrasound by polystyrene-latex 23 and (31) Q(u:) = aS.1 (a:) - 2( 1 - ~)j z (a:). The first terms in the series expansion of a,, (Eq.26 without A,Ai terms) are thus given by the explicit expressions 9 suspending phase and the particles, reIf the absorption coefficients QL and a; of spectively, are small so that terms ( a L c L / o ) and (a;c;/w)* can be neglected follows. Disregarding a possibly present small dispersion in CL and c;, a01 shows the same frequency dependence as a~ and a;. Term a01 constitutes a mixture relation that shows how the absorption coefficient a ~of . the suspending liquid is changed if particles with absorption coefficient a;, sound velocity ci, density pl, and volume fraction qv are added. Hence the parameter in the classical part of the absorption coefficient of the mixture is given by The physical mechanisms that are represented by the first terms (Eq. 32) in the series expansion of asc(Eq. 26) will be discussed in the next section. It is only briefly mentioned here that the dependence upon frequency of the so-called thermal absorption per wavelength, ~ T I Z ( = a d ,Eq. 32), and of the so-called visco-inertial absorption per wavelength, a,l = ail, Eq. 32), as following from this long-wavelength approximation theory can be well represented [32, 341 by a restricted version Of the semi-empirical Hill [58, 591 relaxation function Herein AH denotes an amplitude, ZH a characteristic relaxation time and < SH 5 1) a relaxation time distribution parameter that determines the shape of the relaxation spectrum in the frequency range around the relaxation frequency ( 2 n z ~ ) - ' Towards . low frequencies, the restricted Hill spectral function behaves like a Debye-type relaxation spectral function with discrete relaxation time, SH(O 24 Ann. Physik 5 (1996) lim RH(v) = A ~ 2 nv.r ~ v-0 At high frequencies the restricted Hill term decreases as Hence its negative slope is distinctly smaller than in the case of a discrete relaxation time. Rayleigh scattering of the incident sound wave into different directions is described by the contribution 00 aRay 3 4% x(2n = zRe(k:)R3 n=O + l)A,A: to asc(Eq. 26). 3.4 Specific attenuation mechanisms Let us commend on relevant terms in the above multipole development of the sound fields, in order to give an idea of the underlying mechanisms of attenuation. We shall restrict ourselves to the discussion of the Re(Ao), Re(AI) and the A,Ai terms in the series expansion of a,y,(Eq. 26) which, normally, constitute the main contributions to the total absorption coefficient. First of all, as mentioned above, if a scattering particle is added to the suspending medium the classical part of the sound absorption coefficient will be altered. Term A01 describes the resulting classical absorption (Eqs. 33, 34) of the binary mixture of constituents of different sonic attenuation coefficient, sound velocity, and density. , 32) that represents a monopole field Term A02 (corresponding to a02 = a ~Eq. distribution describes a thermal part in the sound attenuation coefficient of the microheterogeneous mixture. Its origin may be illustrated as follows. For particles with diameter distinctly smaller than the wavelength 1, of the incident sound wave, a homogeneous distribution of temperature around a particle results. In conformity with the frequency of the sound wave, however, the temperature tends to periodically change with time. C p # CL and A # A' leads to a difference in the temperature of the particles and of the suspending phase within the sound field. Accompanied by pulsation of the particles, an irreversible heat transport between particles and the suspending liquid results so that sonic energy is dissipated as heat. The thermal wave resulting from the particle is strongly damped and its attenuation coefficient (YT depends in a characteristic manner on the frequency of the incident sound wave. This effect may be illustrated by the following consideration. At low frequencies, temperature is almost completely equilibrated so that no appreciable effect of irreversible heat transport exists. Due to the finite heat conductivity temperature variations occur, on the other hand, only in a thin layer around the particle at high frequencies. Hence at high frequencies the sound wave does not exchange a noticeable amount of energy within the layer. In the intermediate frequency range, where the thermal wavelength AT (Eq. 7) almost agrees with the particle size, the complete particle will participate in the process of heat exchange and QT. therefore, adopts significant values. U. Kaatze et al., Broadband study of the scattering of ultrasound by polystyrene-latex 25 Term A1 (corresponding to a1 = av,Eq. 32), that is related to a dipole-type oscillation, represents the part of the total attenuation coefficient that results from socalled transversal viscosity waves which may be generated if density of the particles differs from that of the suspending fluid. In the velocity field of a sound wave, a net inertial force acts on the particles which are caused to oscillate thereby. Due to the nonvanishing viscosity of the suspending fluid the particle oscillations are damped and again the contribution to the total attenuation coefficient depends in a characteristic manner on the frequency of the incident wave. The reasons for the frequency dependence are as follows. At low v a uniform motion of particles and of the liquid surrounding them is set up by the sound field. Thus there is hardly acoustic energy dissipated by viscous friction. At high frequencies, the amplitudes of the particle oscillations, relative to the surrounding fluid, are small. Hence again only a small amount of sound energy is dissipated by the stimulated motions of the particles. In correspondence with the thermal conductivity effect, however, losses due to the visco-inertial effect reach noticeable values in the intermediate frequency range where the wavelength A, (Eq.7) of the viscosity wave nearly coincides with the particle size. Term A,A; in EQ. (26) represents the Rayleigh scattering in a strict sense, namely scattering of sound waves into different directions. As will be detailed discussed below, attenuation by scattering into different direction, due to the strong sound absorption, plays a minor role here in the frequency range under consideration, 4 Results and discussion 4. I Attempt to represent spectra by explicit expressions for mode conversion efsects In Fig. 2, a further plot is presented of the spectrum that has been already displayed in another form in Fig. 1. For reasons of clearness, a contribution due to a chemical relaxation has been subtracted from the measured absorption per wavelength exceeding the classical contribution. According to Eqs. (5, 6) the remaining part S a ( u ) .A is assumed to represent the contribution resulting from scattering. The dashed curve indicates the combined spectrum of the thermal and visco-inertial boundary effect. This spectrum has been calculated with parameter values as found by a fitting procedure in which the measured total a(v)* A spectrum was treated. It is clearly illustrated by Fig. 2 that at high frequencies ( u 2 1 GHz) the measured data are inadequately represented by the explicit expressions for a~(= a 0 2 , Eq. 32) and aq(= a l ,Eq. 32). This discrepancy between the measured spectrum and the explicit description of the thermal and visco-inertial boundary effects may reflect imperfections of the long-wavelength approximation of the theoretical model. To begin with, the strong increase with frequency in the Sa(v) values above 1 GHz could reflect Rayleigh scattering into different directions. However, the measured spectra Seem not to follow the A-4 law as predicted by Eq. (38). Secondly, as a result of the comparatively high polystyrene latex particle concentration, contributions due to multiple scattering could be important that have not been considered in the theoretical treatment of the attenuation coefficient. However, no dependence of the measured spectra on the volume fraction (pv has been found as characteristic for effects from multiple scattering 17, 10, 11, 45, 461. Finally, approximations made in the explicit formulation in the model are not generally valid at high frequencies. Among these approximations it is assumed that 26 Ann. Phvsik 5 (1996) -3 10 30 100 300MHz1000 3000 V Fig. 2 Contribution Sal = a,,l - a,~cml (Eq. 5 ) to the ultrasonic attenuation per wavelength displayed as a function of frequency v for the example shown in Fig. 1 (R = 40nm, (pv = 0.1, T = 25OC). The dashed curve is the graph of the sum of the explicit expressions for the thermal part q A = (Eq. 32) and the visco-inertial part a,l = a12(Eq.32) of the total attenuation per wavelength, with parameters found in a nonlinear least-squares regression analysis. At frequencies around 1 GHz and above, however, many of these assumptions are no longer valid. For the mixture for which the spectrum is shown in Figs. 1 and 2, for instance, Re(aa) = 0.26 at 2 GHz (instead of 1.2. lo-' at 100 kHz) and Zrn(k,) = 8.5 lo4 at 2 GHz (whereas Zm(k,) = 2.1 at 100 kHz). It is thus of considerable interest to compare the explicit expressions for the effects of mode conversion with the exact formulation of the scattering theory. - 4.2 The explicit relations of the t h e m 1 and visco-inertial boundary effects in the light of exact formulation of scattering theory In our evaluation of the exact formulation of the scattering theory, Eqs. (18) to (23) are treated numerically. For this purpose, the coefficients in the system of equations are determined for each frequency of interest and for each n up to n = 2 and are arranged in a matrix. Numerical matrix inversion allows for a determination of the A,values which, with the aid of Eq. (26), yield the attenuation coefficient asc(v). The numerical treatment of the system of Eqs. (18) to (23) allows also for consideration of a single term of interest in the series expansion of a,?,( v) (Eq. 26). An example is shown in Fig. 3 where an ao2-spectrum resulting from the explicit formulation of the attenuation due to thermal boundary effect (mode conversion accompanied by particle oscillations, Eq. 32) is compared to the exact data as obtained by the matrix inversion procedure. It has been presumed that 27 U. Kaatze et d.,Broadband study of the scattering of ultrasound by polystyrene-latex Fig. 3 Term a02 in the series expansion of the ultrasonic attenuation coefficient displayed as a function of frequency. Triangles indicate data calculated according to the explicit formulation (Eq. 32), the full curve represents the spectrum following from the matrix inversion treatment of the exact system of h s . (1 8) to (23). 45 lo-& 35 30 < 25 e7 20 15 10 5 - 0 1 3 10 30 100 300 MHz 3000 V In addition, to obtain correspondence in the results from the exact and the approximate explicit version of the theory, respectively, approximations had to be made. Further assumptions are necessary for parameters p' and c' that are not contained in the explicit theory. The values of these parameters have been fixed so that the deviations between the explicit and the exact formulation are minimized (p' << lo9 Pa. s, c' >> c). The curves displayed in Fig. 3 nicely agree at frequencies below 500 MHz. A similar result is found for the al-spectrum. As to be expected, deviations between the numerical treatment of the exact systems of equations and the explicit formulation increase with v at high frequencies, where presumptions of the latter are invalid (v > 500 MHz). This result, on the one hand, may be taken to indicate the validity of the matrix inversion procedure. On the other hand, it is clearly illustrated that the idea of the existence of thermal and visco-inertial boundary effects only, as connected with the explicit formulation of the scattering phenomenon, is invalid at high frequencies. Particularly, the presumptions given by Eq. (39) are not fulfilled in that regime. 4.3 Representation of measured spectra by exact expressions for the first terms in the series expansion of the attenuation coefficient In order to avoid the restrictions in the explicit formulation of a!,(v) we fitted the measured data to a more generally valid spectral function which includes the scattering phenomena according to the matrix inversion procedure described above. In , bk (Eq. 25) have been doing so, the non-approximated relations for b , , b ; , b ~and used throughout. Only the first terms ( n = 0,1,2) of the series expansion of a,, (Eq. 26) have been considered in order to reduce the problem to a sufficiently small number of matrix inversions perSrequency. It turned out, however, that the model spec- 28 Ann.Physik 5 (1996) tral function covers a too small frequency band to appropriately account for the measured spectra. Following the variance in the radii of the polystyrene latex particles we therefore took into account a distribution in the size of the globules. The existing continuous distribution function has been simply approximated by assuming three discrete particle radii Ri with suitable weight factors w ,(i = 1 , 2 , 3 ; R1 < R2 < R3, R2 = R , R I = R-9 nm, R3 = R 6.5 nm, w1 = 17/75, w2 = 35/75, w3 = 23/75). Hence a spectral function + i= 1 i= 1 has been finally used. Nine inversions of 12 x 12 matrices had to be performed for each frequency of interest. Most relevant parameters of the polystyrene-latex globules are known from the literature (Table 3). Parameters of the suspending liquid have been largely fixed at the values for water. Density p , shear viscosity qs, and sound velocity c, however, may sensitively depend on small amounts of additives, The values for these quantities are therefore unknown. Since, in addition, the B-parameter is only insufficiently known for the polystyrene-latex particles (and also for the suspending liquid) we considered the classical absorption per wavelength by treating r x ~and a; as unknown parameters. The mean radius R of the globules is indeed well known (Table 1). It has nevertheless been considered an unknown quantity in order to look for the agreement in the particle size as resulting from the ultrasonic spectra and from other methods. There were nine free parameters in the fitting procedure by which the variance was minimized where P denotes the number of adjustable parameters. In this equation, N denotes the number of frequencies of measurement and the total model spectral function. Quantities A c h e m , t c h e m r and B are of low interest here. Values for the parameters p , v , ~c,, Q ' ( V + o o ) / v 2 , p 2 , and R, however, which are related to the scattering mechanism are collected in Table 4.These parameter values allow the measured spectra to be represented almost within the limits of experimental Table 3 Values at 25 "C of known parameters in the exact description of the measured spectra. p', kg .m-3 A', Wm-' K-' Ck, Jkg'K-' K-' p', Pa. s2 dL,ms-' $1 1053 0.113 1202 2.64. I O - ~ 1.27.109 2380 4 ,ms-' hJ A, Wm-'K-' Cp, Jkg-' K-' P I K K-' 1100 1.069 0.59 4180 2.38.10-4 1.0058 29 U. Kaatze et al., Broadband study of the scattering of ultrasound by polystyrene-latex Table 4 Parameter values found by the regression analysis of the measured spectra using the exact formulation of a in the scattering model. All data refer to 25OC. *, Value fixed in the fitting procedure. Sample Iv R o'(a3) A A 0.2 0.37 1 0.1 0.2 0.05 0.175 0.376 27.7 27.7* 39.3 39.3* 46.2 46.21; 46.2 4.4 4.4 4.4 * 4.4 * 4.4 * 4.4 * 4.4 * B B C C C nm 3 10-dm-l cc' l09Pu. s P kgm-3 VS c 3.56 3.59 3.56 3.57 3.56 3.56 3.57 970 980 970 970 991 950 950 9.7 10.2 1347 1522 I332 1345 I O - ~ .Ps ~ ms-' 10.0 . 5.5 9.6 8.3 10.6 1500 1323 143 1 error. Examples to illustrate the correspondence of the model spectral function and the measured a(v)data are shown in Figs. 4 and 5 where the contributions from the different terms in the series expansion of a,.(v) (Eq.26) are also indicated. Due to the frequency range of interest the a,,(v) part of the spectrum is obviously dominated by the coefficients A02 and A2. Depending on the particle size the A02 contribution exhibits a relative maximum at around 100 MHz. As already discussed in the preceding section it strongly increases at high frequencies. The A02 contribution can, of course, not infinitely increase as suggested by the theoretical predictions. It should be recalled in this connection that the theoretical approach is based on the assumption of a particle size distinctly smaller than the wavelength of the incident compressional wave (Sect. 3.1). At 3 GHz, however, A = c/v x 500 nm which is only five to ten times the diameter of the polystyrene-latex globules. Hence, for these suspensions, even the exact formulation of a in the theoretical model may thus be inappropriate beyond our frequency range of measurements. V Fig. 4 Bilogarithmic plot of the 8aL spectrum for a suspension of sample B (pV= 0.2) at 25OC. Figure symbols indicate the measured data, the full curve represents the sum of the different contributions to the scattering function that are indicated by dashed or dotted curves. 30 Ann. Phvsik 5 (1996) 1000 1 I I I I lo-& 300 100 . '"10 30 ,'A2 100 300 MHz 1000 3000 V Fig. 5 Part 6aA of the ultrasonic attenuation spectrum bilogarithmically plotted for a suspension of sample C (pv = 0.376) at 25OC. Figure symbols indicate the measured data. Dashed and dotted curves show the contributions due to the series expansion (Eq. 26) of the attenuation coefficient and the full curve represents the sum of these contributions. The A2-term represents losses accompanied by quadrupole oscillations of the polystyrene latex particles. In the frequency range around 1 GHz these oscillations appear to be a much more effective attenuation mechanism than the dipole oscillations of the globules. The situation is different with liquid emulsions of similar particle size. If, in a computer simulation of the theoretical model, the Lam6 constant p' is changed to refer to liquid particles instead of solid, that is if p' is assumed to be an imaginary quantity instead of a real, the A2 contribution shifts toward higher frequencies. It is interesting to notice that the strong increase in our attenuation data at high frequencies does in fact hardly contain contributions from Rayleigh scattering into different directions. The agreement between the measured spectra and the theoretical predictions is still not perfect. The remaining small deviations, at least in parts, may result from the neglection at low frequencies of multiple scattering of the thermal and viscosity waves in the theoretical approach. Clearly not fulfilled with the present samples is the presumption that the wavelengths 2, and AT of the viscosity wave and the thermal wave should be much smaller than the distance d between the scattering particles (Eq. 7). Another reason for the remaining small deviations may be some clustering of the polystyrene-latex globules. We nevertheless take the reasonable fit of the measured spectra to the model function to verify the general trends in the scattering and mode conversion (accompanied by particle oscillation and vibration) mechanism. This result is supported by the finding of suggestive parameter values in the fitting procedures. The data for the particle radius fit to those derived from ultracentrifugation and elastic light scattering experiments and also from electron micrographs (Table 1). U. Kaatze et al., Broadband study of the scattering of ultrasound by polystyrene-latex 31 The values for the B coefficient of the suspensions, as well as for the density p and sound velocity c of the suspending phase, are somewhat smaller than the corresponding water data (B = 31.7 ps, p = 0.99705 kgmP3 [60], c = 1496.6ms-’; 25°C). This result may be due to the additives within the liquid. With the exception of suspension B of volume fraction (pv = 0.2 the shear viscosity v], of the suspending liquid nearly agrees with the value for water (ys = 8.903Pa. s; 25°C [60]). The Lam6 constant distinctly exceeds the literature value p’ = 1.27 . lo9 Pa - s [16] for polystyrene latex. 5 Conclusions Broadband ultrasonic spectra of polystyrene-latex-in-water suspensions with well-defined particle size show that attenuation at low frequencies, in addition to a chemical contribution, is dominated by the thermal conductivity effect at internal interfaces. Up to a certain particle diameterlwavelength ratio the spectrum can be adequately represented by an explicit formulation of the attenuation due to the scattering and mode conversion mechanism that is related to particle oscillations and vibrations. With the samples studied in this investigation (particle diameter between about 60 and 100 nm) this ratio corresponds to a frequency of about 100 MHz. Above this frequency a more general, but more complicated, treatment of the theoretical model enables an adequate description of the measured spectra in an ‘intermediate’ frequency range. Within this range, the wavelength of the incident compressional wave is indeed small, but it is too large for preponderant attenuation by Rayleigh scattering (Figs. 4,5). In this intermediate range of wavelength, it is already sufficient to additionally consider the quadrupole contribution in the multipole development of the Sound fields in order to adequately represent the measured spectra. It is not necessary to take into account more complicated additional effects. If contributions from Rayleigh scattering are small at all, then it is justified to also neglect effects of multiple scattering. Notice, however, that the scenario of different attenuation mechanisms reflected by Figs. 4 and 5 result from fitting the single scattering model to the measured specm. we can, therefore, not definitely exclude, that a somewhat different contribution from Rayleigh scattering might be found on the basis of multiple scattering model. However, as mentioned above, effects from multiple Rayleigh scattering are anyhow small, as following from the missing characteristic dependence of the measured data on the volume fraction of polystyrene-latex scatterers. Hence the use of a still more complicated theoretical model does not seem to be justified in the case of these suspensions. 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